MATHCOUNTS Support search Stanford college Harvard Yale

Hello, my 1337 classmates of Mastering MATHCOUNTS/AMC 8
and other students who frequent the Mathcounts forum at AoPS:

I’m qvc122, a student in Mastering MATHCOUNTS/AMC 8. I am a female in 7th grade participating in Mathcounts for the second time this year (2008-2009). This is my first time taking a Mathcounts class from AoPS.

I need your help with my science project. In my science project, I will conduct the following survey, hoping to generate some meaningful results. I have the permission of Mr. Ruscyzk, who expressed his interest in the result of this survey. Thank you for your support, O Mighty Ruscyzk.

Qualification for this survey:
1.	Anyone in the Fall 2008 Mastering MATHCOUNTS/AMC 8 class (I assume that you are all 6th - 8th graders).
2.	Anyone in 6th to 8th grade, who frequently visit the Mathcounts forum, intends to participate, and thinks he or she has a reasonable chance to make the school Mathcounts team.

I understand that there is a fallacy since making the school team can be as hard as making a state team if you are in a school with Darryl Wu or Bobby Shen. I have sympathy for you, but they can each only take one slot. However, I know you are all smart/ 1337. I wish you all good luck with 2009 Mathcounts.

[b]This survey is anonymous. Participation is voluntary, but I invite all of you who meet the qualification above to please join my survey with the questions below. Thank you in advance for your time and effort. Please post your answer in this Mathcounts forum. Please PM me if you don't feel comfortable to post your answers.[/b]

[u][b]Survey Questions[/b][/u]

1.	You are (please state whatever your birth certificate says)
a.	Male
b.	Female

2.	You are in
a.	6th grade
b.	7th grade
c.	8th grade.

3.	You participated/ will participate in Mathcounts for the
a.	1st time (2009)
b.	2nd time
c.	3rd time

4.	If you have participated in Mathcounts before, how far did you get (please specify the grade you were in when you participated)
a.	Did not make school team (strong school or average good school).
b.	Made the school team (strong school or average good school).
c.	Top 10 in chapter (which place/ written and final if countdown is official)
d.	Made state (by being in the top two teams or by being top 4 individual in the chapter).
e.	Made national, and your place (for the anonymous purposes, please do not post your national place on this board, you can PM me only if you feel comfortable, otherwise just say you made national).

5.  Your Math grade in middle school (A=4, B=3, C=2, D=1, F=0)
 GPA/math= (please give a number answer: such as 3.86).

6.	Language Arts/English grade (A=4, B=3, C=2, D=1, F=0)
GPA/English = (please give a number answer).

7.	Your overall grade in middle school (A=4, B=3, C=2, D=1, F=0)
GPA/Overall = (please give a number answer).

8.	Your motivation in math
a.	Highly motivated (disciplined on your own), study math more than two hours on average (if math is your life/ your joy, and you study more than 2 hours, please indicate).
b.	Somewhat motivated (study somewhat but less consistently and you need your parents to push/ encourage you from time to time; if you have such parents, consider yourself lucky)
c.	You like math/ math competition because you are naturally good at them but you do not have the passion to put in real effort (Those who come to the forum at AoPS for games/ chat but do very little math belong to this group).

9.	Your overall motivation in academics
a.	Highly motivated (disciplined on your own), study more than two hours on average in non-math subjects (if academics is your life/ your joy, and you do much more than 2 hours, please indicate the amount of time).
b.	Somewhat motivated (study somewhat but less consistently and you need your parents to push/ encourage you from time to time)
c.	You like academics/ academic competition because you are naturally good at them but you do not have the passion to put in real effort.

10.	 How much time do you spend playing computer games/other games (DDR, Game consoles. etc) and watching anime/ other cartoons/TV
a.	A lot, more than 4 hours on average/ you are addicted (include weekends in calculation)
b.	2-4 hour on average
c.	Less than 2 hours
d.	Very little because you don’t like games (What do you do? Sports? Excessive homework? Please specify).

11.	How many books (non-mathematical) do you read:
a.	more than 20 books per year
b.	more than 15 books per year
c.	more than 10 books per year
d.	between 5-10 books per year
e.	less than 5 books per year

12.	Hobbies (Please be specific: Sports (list the type and awards you achieved, if any), Technology/Computer Gaming, Musical Instruments/Music Lessons (list type and awards you achieved, if any), Charity Work (if any), Jobs (such as babysitting), Shopping (for software, clothing, bags, shoes, etc., specify))

13.	Do you consider yourself sociable:
a.	I made friends at AoPS and in summer “nerd” camp but have some difficulty in mixing with my regular school classmates.
b.	I am in a magnet (or similar school with many motivated students) and do have some friends.
c.	I am easy going and easily make friends/ act friendly with classmates of all academic levels.

14.	Did you participate in SAT/ ACT (such as through talent search programs)? If so, what are your scores
a.	SAT scores: Verbal/ Critical reading/ Math /Writing  (in one sitting)
b.	ACT scores: English/ Math/ Reading/ Science/ Composite
c.	N/A

15.	What do you want to do when you grow up:
a.	Math research.
b.	Things in math-related profitable fields (such as Wall Street)
c.	Engineer: (in traditional hardcore engineering or biomedical engineering).
d.	Scientist doing research and discovering work.
e.	Entrepreneur /Business.
f.	Things related to medicine
g.	Lawyer/Law-related occupation
h.	Other (Specify whether it’s sales person for computer / technology company/ drug company; used car dealer, etc.)

16.	What are your dream colleges (please list your top three choices)

17.	Do you have any other interesting “quirks” (please specify).

18.	 Please list your 3 favorite books (if any math book makes that list, it is fine)/ movies/songs.

I took the class over the summer, and am an active user of this forum.

1. A
2. B
3. A
4. N/A, but this year so far, I'm projected to get first at our school competition
5. A, 4/4=1
6. A, 4/4=1
7. A, 4/4=1
8. I would say A, but I don't have time for two hours a day, maybe 1.5?
9. C (besides math)
10. C
11. D, maybe C
12. Please don't read if you don't want to.
[hide]
SPORTS:
Run cross country / track.  Qualified for national cross country meet in 5th (placed 133rd in nation), 6th (didn't go) and 7th (leaving tomorrow morning!).  Qualified for national track meet in summer between 6th and 7th (9th place in nation for 4x400 relay)

MUSIC:
Play piano, won a few local competitions.
[/hide]
13. Between A and C
14. C (taken practice tests but not at one sitting)
15. Seriously, I don't know right now.
16. In no particular order: Stanford, Harvard, ??? for third
17. Erm, not that I can think of now.
18. Harry Potter series, AoPS series, and any other fiction book.

[hide]1.a
2.a
3.a
4.
5.
6.
7.a
8.b
9.a
10.c
11.a
12.chess, piano, violin
13.a
14.c
15.a,b,c
16.Princeton, Harvard, Yale
17.
18.Percy and the Olympians Series, Harry Potter Series, The Alchemist[/hide]

[hide]

1.a
2.c
3.c
4.e
5.4
6.4
7.4
8.a
9.a
10.b
11.a
12.Soccer, online flash games, rpgs, violin, guitar hero, math, playing with friends, chatting/email
13.c
14.a.640/800/650(no verbal)
15.a,b,d,e
16.harvard, princeton, mit
18.inheritance(eragon, eldest, brisingr, unnamed 4th book), percy jackson books, harry potter i guess, all AoPS books, system of a down, rock and classical music, quantum of solace, batman dark knight, transformers, simpsons movie, day after tommorrow,  pirates of the carribean, i-robot, titanic, alot more that i can't think of
[/hide]

[hide]
1. A
2. C
3. C
4. D
5. A/4.0
6. A/4.0
7. A/4.0
8. A
9. A
10. between C and D
11. B
12.
Sports: basketball (school team second string), tennis (playing for school next year), track (for school, didnt make anything big)
Music: saxophone (playing since fifth grade, playing for district honors band)
Volunteer at temple regularly, take saxophone classes, play for multiple bball team, other similar stuff.
13. C
14. SAT:
7th grade: 600 CR, 770 Math (fail), 590 WRI [b]COMPOSITE[/b]: 1960 (really bad, i  know)
ACT; took in sixth grade, pretty bad and dont remember anyways.
15. not quite decided; considering engineering and medical.
16. harvard, MIT, duke
17. none
18. harry potter, the giver, house of the scorpion, the westing game. thats more than three and thats not all of them so...yeah.

[/hide]

[hide]1. You are (please state whatever your birth certificate says)
Male
2. You are in
8th grade.

3. You participated/ will participate in Mathcounts for the
3rd time

4. If you have participated in Mathcounts before, how far did you get (please specify the grade you were in when you participated)
d. Made state (by being in the top two teams or by being top 4 individual in the chapter).

5. Your Math grade in middle school (A=4, B=3, C=2, D=1, F=0)
3.54

6. Language Arts/English grade (A=4, B=3, C=2, D=1, F=0)
3.79

7. Your overall grade in middle school (A=4, B=3, C=2, D=1, F=0)
3.67

8. Your motivation in math
a. Highly motivated (disciplined on your own), study math more than two hours on average (if math is your life/ your joy, and you study more than 2 hours, please indicate).

9. Your overall motivation in academics
a. Highly motivated (disciplined on your own), study more than two hours on average in non-math subjects (if academics is your life/ your joy, and you do much more than 2 hours, please indicate the amount of time).

10. How much time do you spend playing computer games/other games (DDR, Game consoles. etc) and watching anime/ other cartoons/TV
d. Very little because you don’t like games (What do you do? Sports? Excessive homework? Please specify).
I wrestle and play soccer with practice every single night so between math and that I have like no time at all for electronics except for music
11. How many books (non-mathematical) do you read:
a. more than 20 books per year
I read 20 per 9 weeks +
12. Hobbies (Please be specific: Sports (list the type and awards you achieved, if any), Technology/Computer Gaming, Musical Instruments/Music Lessons (list type and awards you achieved, if any), Charity Work (if any), Jobs (such as babysitting), Shopping (for software, clothing, bags, shoes, etc., specify))
Wrestling, Soccer
Piano Saxephone
Student Council President
Shoping for technology stuff

13. Do you consider yourself sociable:
c. I am easy going and easily make friends/ act friendly with classmates of all academic levels.

14. Did you participate in SAT/ ACT (such as through talent search programs)? If so, what are your scores
a. SAT scores: Verbal/ Critical reading/ Math /Writing (in one sitting)
[hide]Math- 700
Reading- 630
Writing- 550[/hide]

15. What do you want to do when you grow up:
I want to be an Opthalmologist
16. What are your dream colleges (please list your top three choices)
Davidson
Duke
Wake Forest
17. Do you have any other interesting “quirks” (please specify).
Can't go anywhere without my iPod or cell phone
18. Please list your 3 favorite books (if any math book makes that list, it is fine)/ movies/songs.
 The Hunger Games
Harry Potter 4
Wanted The Movie[/hide]
_________________

Hope ur science project goes well

1.  A
2.  C
3.  C
4.  D
5.  Math Grade- 99
6.  English Grade- 95
7.  Average- 95.8
8.  between A and B
9.  between A and B (most probably A)
10.  C
11.  A
12.  Sports: tennis, swimming; Hobbies: ping-pong, pool, air-hockey :P
13.  C
14.  C
15.  F (doctor)
16.  I'm not really thinking, but in no particular order: Yale, Harvard, Princeton.  This is because they are close to my family.
17.  Can't think of any now.
18.  Books: I like Percy Jackson, Inheritance, etc.

[hide]
1-a
2-c
3-c
4-e
5-A, 4/4=1
6-A, 4/4=1
7-A, 4/4=1
8-b...I'm a lazy person.
9-b
10-b
11-a
12-Reading and wasting time on gmail
13-a
14-6th grade scores were 700 Math (very bad), 540 critical reading (gah didn't know any vocab), and 530 writing (...). Add them up yourself.
15-I...don't know? Definitely either a doctor, or a quant. Perhaps something else related to math, not likely though.
16- Harvard, Princeton, something else goes here
17-I lost
18- Oh snap...anything related to Ender's Game, Eragon, Alex Rider, etc...[/hide]

[hide][quote="qvc122"]Hello, my 1337 classmates of Mastering MATHCOUNTS/AMC 8
and other students who frequent the Mathcounts forum at AoPS:

I’m qvc122, a student in Mastering MATHCOUNTS/AMC 8. I am a female in 7th grade participating in Mathcounts for the second time this year (2008-2009). This is my first time taking a Mathcounts class from AoPS.

I need your help with my science project. In my science project, I will conduct the following survey, hoping to generate some meaningful results. I have the permission of Mr. Ruscyzk, who expressed his interest in the result of this survey. Thank you for your support, O Mighty Ruscyzk.

Qualification for this survey:
1.	Anyone in the Fall 2008 Mastering MATHCOUNTS/AMC 8 class (I assume that you are all 6th - 8th graders).
2.	Anyone in 6th to 8th grade, who frequently visit the Mathcounts forum, intends to participate, and thinks he or she has a reasonable chance to make the school Mathcounts team.

I understand that there is a fallacy since making the school team can be as hard as making a state team if you are in a school with Darryl Wu or Bobby Shen. I have sympathy for you, but they can each only take one slot. However, I know you are all smart/ 1337. I wish you all good luck with 2009 Mathcounts.

[b]This survey is anonymous. Participation is voluntary, but I invite all of you who meet the qualification above to please join my survey with the questions below. Thank you in advance for your time and effort. Please post your answer in this Mathcounts forum. Please PM me if you don't feel comfortable to post your answers.[/b]

[u][b]Survey Questions[/b][/u]

1.	You are (please state whatever your birth certificate says)
[color=red]a.	Male[/color]
b.	Female

2.	You are in
a.	6th grade
b.	7th grade
[color=red]c.	8th grade.[/color]

3.	You participated/ will participate in Mathcounts for the
a.	1st time (2009)
b.	2nd time
[color=red]c.	3rd time[/color]

4.	If you have participated in Mathcounts before, how far did you get (please specify the grade you were in when you participated)
a.	Did not make school team (strong school or average good school).
b.	Made the school team (strong school or average good school).
c.	Top 10 in chapter (which place/ written and final if countdown is official)
d.	Made state (by being in the top two teams or by being top 4 individual in the chapter).
e.	Made national, and your place (for the anonymous purposes, please do not post your national place on this board, you can PM me only if you feel comfortable, otherwise just say you made national).
[color=red]actually i made state as an alternate so.....
[/color]
5.  Your Math grade in middle school (A=4, B=3, C=2, D=1, F=0)
 GPA/math= (please give a number answer: such as 3.86).
[color=red]4[/color]
6.	Language Arts/English grade (A=4, B=3, C=2, D=1, F=0)
GPA/English = (please give a number answer).
[color=red]4[/color]
7.	Your overall grade in middle school (A=4, B=3, C=2, D=1, F=0)
GPA/Overall = (please give a number answer).
[color=red]4[/color]
8.	Your motivation in math
a.	Highly motivated (disciplined on your own), study math more than two hours on average (if math is your life/ your joy, and you study more than 2 hours, please indicate).
b.	[color=red]Somewhat motivated (study somewhat but less consistently and you need your parents to push/ encourage you from time to time; if you have such parents, consider yourself lucky)[/color]
c.	You like math/ math competition because you are naturally good at them but you do not have the passion to put in real effort (Those who come to the forum at AoPS for games/ chat but do very little math belong to this group).

9.	Your overall motivation in academics
a.	Highly motivated (disciplined on your own), study more than two hours on average in non-math subjects (if academics is your life/ your joy, and you do much more than 2 hours, please indicate the amount of time).
b.	[color=red]Somewhat motivated (study somewhat but less consistently and you need your parents to push/ encourage you from time to time)[/color]
c.	You like academics/ academic competition because you are naturally good at them but you do not have the passion to put in real effort.

10.	 How much time do you spend playing computer games/other games (DDR, Game consoles. etc) and watching anime/ other cartoons/TV
a.	A lot, more than 4 hours on average/ you are addicted (include weekends in calculation)
b.	2-4 hour on average
c.	Less than 2 hours
d.	[color=red]Very little because you don’t like games (What do you do? Sports? Excessive homework? Please specify). sports, playing with friends and other stuff
[/color]
11.	How many books (non-mathematical) do you read:
[color=red]a.	more than 20 books per year[/color]
b.	more than 15 books per year
c.	more than 10 books per year
d.	between 5-10 books per year
e.	less than 5 books per year

12.	Hobbies (Please be specific: Sports (list the type and awards you achieved, if any), Technology/Computer Gaming, Musical Instruments/Music Lessons (list type and awards you achieved, if any), Charity Work (if any), Jobs (such as babysitting), Shopping (for software, clothing, bags, shoes, etc., specify))
[color=red]Uh, sports not on team or not in competition but play a lot-ping pong, tennis, and a lot of others(I like sports) on teams- swimming-made cut for zones, but found out that our team didn't participate....., track and field our school only does like dual meets with other schools so there is no awards...
I play violin but I never competed
I have done some charity stuff like raking, donating
I am volunteering as a tennis coach
I hate shopping unless it involves food[/color]
13.	Do you consider yourself sociable:
a.	I made friends at AoPS and in summer “nerd” camp but have some difficulty in mixing with my regular school classmates.
[color=red]b.	I am in a magnet (or similar school with many motivated students) and do have some friends.[/color]
c.	I am easy going and easily make friends/ act friendly with classmates of all academic levels.

14.	Did you participate in SAT/ ACT (such as through talent search programs)? If so, what are your scores
[color=red]no[/color]
a.	SAT scores: Verbal/ Critical reading/ Math /Writing  (in one sitting)
b.	ACT scores: English/ Math/ Reading/ Science/ Composite
c.	N/A

15.	What do you want to do when you grow up:
a.	Math research.
b.	Things in math-related profitable fields (such as Wall Street)
c.	Engineer: (in traditional hardcore engineering or biomedical engineering).
[color=red]d.	Scientist doing research and discovering work.[/color]
e.	Entrepreneur /Business.
f.	Things related to medicine
g.	Lawyer/Law-related occupation
h.	Other (Specify whether it’s sales person for computer / technology company/ drug company; used car dealer, etc.)

16.	What are your dream colleges (please list your top three choices)
[color=red]mit, havard, princeton[/color]
17.	Do you have any other interesting “quirks” (please specify).
[color=red]at this time no besides, eating a lot-though I am really skinny I find the scary, and my energy and personality varies a lot from time to time though mostly from bored to hyper-that's in class[/color]
18.	 Please list your 3 favorite books (if any math book makes that list, it is fine)/ movies/songs.[color=red]Intermediate C&P, National Book of Secrets(I think that's the title...), Pendragon series eh, I lot are tied.[/color][/quote][/hide]

[hide][quote="qvc122"]Hello, my 1337 classmates of Mastering MATHCOUNTS/AMC 8
and other students who frequent the Mathcounts forum at AoPS:

I’m qvc122, a student in Mastering MATHCOUNTS/AMC 8. I am a female in 7th grade participating in Mathcounts for the second time this year (2008-2009). This is my first time taking a Mathcounts class from AoPS.

I need your help with my science project. In my science project, I will conduct the following survey, hoping to generate some meaningful results. I have the permission of Mr. Ruscyzk, who expressed his interest in the result of this survey. Thank you for your support, O Mighty Ruscyzk.

Qualification for this survey:
1.	Anyone in the Fall 2008 Mastering MATHCOUNTS/AMC 8 class (I assume that you are all 6th - 8th graders).
2.	Anyone in 6th to 8th grade, who frequently visit the Mathcounts forum, intends to participate, and thinks he or she has a reasonable chance to make the school Mathcounts team.

I understand that there is a fallacy since making the school team can be as hard as making a state team if you are in a school with Darryl Wu or Bobby Shen. I have sympathy for you, but they can each only take one slot. However, I know you are all smart/ 1337. I wish you all good luck with 2009 Mathcounts.

[b]This survey is anonymous. Participation is voluntary, but I invite all of you who meet the qualification above to please join my survey with the questions below. Thank you in advance for your time and effort. Please post your answer in this Mathcounts forum. Please PM me if you don't feel comfortable to post your answers.[/b]

[u][b]Survey Questions[/b][/u]

1.	You are (please state whatever your birth certificate says)
a.	Male
[color=red]b.	Female[/color]

2.	You are in
a.	6th grade
b.	7th grade
[color=red]c.	8th grade.[/color]

3.	You participated/ will participate in Mathcounts for the
a.	1st time (2009)
b.	2nd time
[color=red]c.	3rd time[/color]

4.	If you have participated in Mathcounts before, how far did you get (please specify the grade you were in when you participated)
a.	Did not make school team (strong school or average good school).
b.	Made the school team (strong school or average good school).
c.	Top 10 in chapter (which place/ written and final if countdown is official)
[color=red]d.	Made state (by being in the top two teams or by being top 4 individual in the chapter).[/color]
e.	Made national, and your place (for the anonymous purposes, please do not post your national place on this board, you can PM me only if you feel comfortable, otherwise just say you made national).

5.  Your Math grade in middle school (A=4, B=3, C=2, D=1, F=0)
 GPA/math= (please give a number answer: such as 3.86).
[color=red]4[/color]
6.	Language Arts/English grade (A=4, B=3, C=2, D=1, F=0)
GPA/English = (please give a number answer).
[color=red]4[/color]
7.	Your overall grade in middle school (A=4, B=3, C=2, D=1, F=0)
GPA/Overall = (please give a number answer).
[color=red]4[/color]
8.	Your motivation in math
a.	Highly motivated (disciplined on your own), study math more than two hours on average (if math is your life/ your joy, and you study more than 2 hours, please indicate).
b.	[color=red]Somewhat motivated (study somewhat but less consistently and you need your parents to push/ encourage you from time to time; if you have such parents, consider yourself lucky)[/color]
c.	You like math/ math competition because you are naturally good at them but you do not have the passion to put in real effort (Those who come to the forum at AoPS for games/ chat but do very little math belong to this group).

9.	Your overall motivation in academics
a.	[color=red]Highly motivated (disciplined on your own), study more than two hours on average in non-math subjects (if academics is your life/ your joy, and you do much more than 2 hours, please indicate the amount of time). (Well,... I don't really put in much effort, but I do care that I get A's rather than my parents telling me to do homework/study.)[/color]
b.	Somewhat motivated (study somewhat but less consistently and you need your parents to push/ encourage you from time to time)
c.	You like academics/ academic competition because you are naturally good at them but you do not have the passion to put in real effort.

10.	 How much time do you spend playing computer games/other games (DDR, Game consoles. etc) and watching anime/ other cartoons/TV
a.	A lot, more than 4 hours on average/ you are addicted (include weekends in calculation)
b.	2-4 hour on average
[color=red]c.	Less than 2 hours[/color]
d.	Very little because you don’t like games (What do you do? Sports? Excessive homework? Please specify).

11.	How many books (non-mathematical) do you read:
[color=red]a.	more than 20 books per year[/color]
b.	more than 15 books per year
c.	more than 10 books per year
d.	between 5-10 books per year
e.	less than 5 books per year

12.	Hobbies (Please be specific: Sports (list the type and awards you achieved, if any), Technology/Computer Gaming, Musical Instruments/Music Lessons (list type and awards you achieved, if any), Charity Work (if any), Jobs (such as babysitting), Shopping (for software, clothing, bags, shoes, etc., specify))
[color=red]
Swimming, tennis, no awards for that lol
Piano- won WMTA twice (quite a few years ago though)
NR any charity work yet
Babysitting my little sis?
Shopping... occasionally with friends
 [/color]
13.	Do you consider yourself sociable:
a.	I made friends at AoPS and in summer “nerd” camp but have some difficulty in mixing with my regular school classmates.
b.	I am in a magnet (or similar school with many motivated students) and do have some friends.
[color=red]c.	I am easy going and easily make friends/ act friendly with classmates of all academic levels..[/color]

14.	Did you participate in SAT/ ACT (such as through talent search programs)? If so, what are your scores
a.	SAT scores: Verbal/ Critical reading/ Math /Writing  (in one sitting)
b.	ACT scores: English/Math/Reading/Science/ Composite
c.	N/A [color=red](yet)[/color]

15.	What do you want to do when you grow up:
a.	Math research.
b.	Things in math-related profitable fields (such as Wall Street)
c.	Engineer: (in traditional hardcore engineering or biomedical engineering).
d.	Scientist doing research and discovering work.
e.	Entrepreneur /Business.
[color=red]f.	Things related to medicine[/color]
g.	Lawyer/Law-related occupation
h.	Other (Specify whether it’s sales person for computer / technology company/ drug company; used car dealer, etc.)

16.	What are your dream colleges (please list your top three choices)
[color=red]Harvard, Johns Hopkins, University of Pennsylvania[/color]
17.	Do you have any other interesting “quirks” (please specify).
[color=red]Um... IDK really... [/color]
18.	 Please list your 3 favorite books (if any math book makes that list, it is fine)/ movies/songs.[color=red]the Twilight series, Harry Potter, and... something else[/color][/quote][/hide]

[hide]1.a
2.b
3.a
4.n/a
5.99% in Geometry 10 (3 grades higher)
6.90%
7.96%
8.a (I frequently go onto G&FF, but do much math)
9.a
10.c "averaged out" is the operative :)
11.a
12. [Programming (games, applications), Swimming (placed 8th), Running for fun]
13.c
14.a (710/780/600)
15.c/d/e
16.MIT/Stanford/No third choice  :lol:
17. Memorization: 60 digits of pi, 15 digits of e, 72 elements in periodic table, countless license plates!
18. Engel's Problem-Solving Strategies, Terence Tao's Solving Mathemetical Problems, The IMO Compendium[/hide]

AIME15
andrewjjiang
AIME_is_hard
nikeballa96
matt919
ernie
xpmath
shentang
Stargirl240
james4l

Thanks to all who participated so far! :D  Hopefully, I will get more responses.

Here are my answers to this survey.
[hide]1.b
2.b
3.b
4.d (purely by luck;my written score was low)
5.4
6.4
7.4
8.b
9.b
10.a
11.a
12.piano and violin lessons (no competitions so far), browser-based games, anime, Nintendo DS, DDR
13.b
14.c (I don't have my scores yet.)
15.f
16.Stanford (the [i]good[/i] Harvard, no offense), Harvard, Yale
17.?
18.HP series, Dr. Jekyll and Mr. Hyde, some Art of Problem Solving books[/hide]

Hey! Here are my answers:[hide]1. b.
2. b
3. b.
4. d.
Sorry I can't really answer questions 5,6, and 7. I'm homeschooled.
8. b. I guess
9. b. again?
10. d. I only go on AoPS and I watch movies every once in a while. Music is probably one of the things that takes up the most time... I also like to do sports.
11. a.
12. Most any sport... skiing is probably the sport that I am the most into. Frisbee, hiking, swimming, anything outdoors I like to do for the most part. For music this is my sixth year in piano and my first in violin. I have been in two piano competitions in the past year. (I have been in about 12-15 recitals other than that.) In a statewide one this spring I didn't play very well so I got nothing, but in the MTNA competition about a month ago I played pretty well and got an Honorable Mention.
13. c.
14. I haven't participated in either... but I will probably be taking the SAT sometime next school year.
15. I don't know at the moment.
16. I don't have any
17. I'm a Christian. =)
18. The Bible... and umm I don't have any others.  [/hide]

I wanted to update the list of people who responded.

AIME15
andrewjjiang
AIME_is_hard
nikeballa96
matt919
ernie
xpmath
shentang
Stargirl240
james4l
kookamunga

I also got some posts through PM.

Thanks to all who participated!
[b]I'm waiting for more responses over the weekend. I know that all of you are busy, but please take a few minutes to participate. [/b]

1. You are (please state whatever your birth certificate says)
b. Female

2. You are in
c. 8th Grade

3. You participated/ will participate in Mathcounts for the
c. 3rd time

4. If you have participated in Mathcounts before, how far did you get
d. Made state (took 6th place, got there via 4th place in chapter)

5. Your Math grade in middle school (A=4, B=3, C=2, D=1, F=0)
4

6. Language Arts/English grade (A=4, B=3, C=2, D=1, F=0)
4

7. Your overall grade in middle school (A=4, B=3, C=2, D=1, F=0)
4

8. Your motivation in math
a. Highly motivated. I constantly study it in my free time.

9. Your overall motivation in academics
c. I naturally do well in non-math academics, but I'm not as passionate in them.

10. How much time do you spend playing computer games/other games (DDR, Game consoles. etc) and watching anime/ other cartoons/TV
a. I play computer games more than 10 hours a week. Yeah, I'm an addict.

11. How many books (non-mathematical) do you read:
a. More than 20 books per year.
12. Hobbies (Please be specific: Sports (list the type and awards you achieved, if any), Technology/Computer Gaming, Musical Instruments/Music Lessons (list type and awards you achieved, if any), Charity Work (if any), Jobs (such as babysitting), Shopping (for software, clothing, bags, shoes, etc., specify))
Other than math: Computer gaming, tech, violin, spelling, chess, Rubik's cubing

13. Do you consider yourself sociable:
a. Lol, yes. I hate the stupid people at my school. Especially those tall skinny girls who just shake their @sses off at everyone.

14. Did you participate in SAT/ ACT (such as through talent search programs)? If so, what are your scores
c. N/A. BTW I'm taking the SAT in January.

15. What do you want to do when you grow up:
b. (Though I'm not exactly sure)

16. What are your dream colleges (please list your top three choices)
MIT, Harvard, Stanford

17. Do you have any other interesting “quirks” (please specify).
Not really.

18. Please list your 3 favorite books (if any math book makes that list, it is fine)/ movies/songs.
Movie: Proof
Math book: AoPS Volume 2
Other books: Born on a Blue Day, Flatland

1. a
2. b
3. a
4. N/A
5. 4
6. 4
7. 5
8. b
9. b
10. b
11. a
12. Tennis, piano, clarinet
13. b
14.
a)Haven't got my scores
b)N/A
15. I dunno
16. Don't have any
17. Not sure what you mean
18. Harry Potter, Series of Unfortunate Events, Inkheart.

1.	You are (please state whatever your birth certificate says)
a.	Male

2.	You are in
b.	7th grade

3.	You participated/ will participate in Mathcounts for the
b.	2nd time

4.	If you have participated in Mathcounts before, how far did you get (please specify the grade you were in when you participated)
a.	Did not make school team (strong school or average good school).

5. Your Math grade in middle school (A=4, B=3, C=2, D=1, F=0)
4

6.	Language Arts/English grade (A=4, B=3, C=2, D=1, F=0)
4

7.	Your overall grade in middle school (A=4, B=3, C=2, D=1, F=0)
3.83

8.	Your motivation in math
b.	Somewhat motivated (study somewhat but less consistently and you need your parents to push/ encourage you from time to time; if you have such parents, consider yourself lucky)

9.	Your overall motivation in academics \
b.	Somewhat motivated (study somewhat but less consistently and you need your parents to push/ encourage you from time to time)

10.	 How much time do you spend playing computer games/other games (DDR, Game consoles. etc) and watching anime/ other cartoons/TV
c.	Less than 2 hours

11.	How many books (non-mathematical) do you read:
a.	more than 20 books per year

12.	Hobbies (Please be specific):
Swimming (state champion qualified)
Golf (3rd in junior world for age group)
Chess (Multiple state champ, 1-time national champ)

13.	Do you consider yourself sociable:
c.	I am easy going and easily make friends/ act friendly with classmates of all academic levels.

14.	Did you participate in SAT/ ACT (such as through talent search programs)? If so, what are your scores
a.	SAT scores: Verbal(750)/ Critical reading(570)/ Math(800) /Writing (3)(in one sitting)

15.	What do you want to do when you grow up:
West Point, serve my country for a few years, then med school and doctor.

16.	What are your dream colleges (please list your top three choices)
West Point, Stanford, HArvard

17.	Do you have any other interesting “quirks” (please specify).
Sports geek, can rattle off stats just like that.

18.	 Please list your 3 favorite books (if any math book makes that list, it is fine)/ movies/songs.
Any emo punk

Thanks to the recent posters!

gauss1181
froggy
sabretooth

[b]My project is due on [i]Friday[/i]. Any responses posted as late as [i]Thursday[/i] will be incorporated in my project.[/b]

I forgot to add in my PM:

One of my hobbies is Chess.

I am currently rated 1912. I've won Nationals once and State 4 (?) times.

I've won Foxwoods ($ \$1000$) before.

I got an extension on my project. :D  I will be accepting replies [b]until the next school semester.[/b]

I would like to have more responses for this project. I hope to get [b]fifty[/b] replies. It's winter break, so I know you have a few minutes to spare. I hope you (whomever is reading this) will participate!

[b]Please help!![/b]

Happy holidays to everyone!

1. male
2. 8th grade
3. 1st time
4. nope
5. 4
6. 4
7. 4
8. B
9. B
10. B
11. B
12. I won district in chess, did travel-team soccer for one year, started up tennis, i enjoy football, played rec basketball for one year, trying to get some kids to tutor in math, do charity work occasionally, pretty good at piano, second year alto sax, won a state medal thing for piano, and i play games but i'm only allowed over weekends.
13. C
14. A, but i dont know where my sheet is...
15. H (financial economist, or anything to do with finances. this might fit in E...  oops)
16. haha... I'd like to get into princeton, MIT, or Chicago.
17. i like to cook, but sometimes it just doesn't turn out the way i want it to. i'm usually not that determined doing things, but sometimes i spend so much time doing the most useless stuff like lacing my shoes for 30 minutes or carving something outta wood or going on some week-long hiking trip/campout on some godforsaken mountain range in new mexico.
18. I like [u]The Three Musketeers[/u] (book), [u]The Dark Knight[/u] (movie), and [i]Must Have Done Something Right[/i] (song)

good luck

Great news! My project has been selected to go on to the county competition!

Thanks again to all participants! Here is a summary of the results. Sorry it's long.

[hide]The hypothesis was supported by the findings. Two people who made national Mathcounts were both 8th grade males.

However, the pool of the sample is small (with 20 responses received) because the sampling is through self-selection. Therefore, the results may not represent a random pool of students in 6th –8th grades with interests in mathematics and mathematic competitions. Nevertheless, the data shows that this small pool of students has certain features.

Among the 19 students responded to the survey, approximately 75% of the participants were male students (male: female ratio 3:1). Each year, about two hundred twenty-eight students qualify for the national Mathcounts (four people from each state times fifty states and regions). If society doesn’t want to waste 50% of its potential, it should encourage more females to do math and science.

All 19 students seem to excel in schoolwork with average GPAs close to 4.0 in both math and language arts.

All students have relatively strong motivation to math and all academic areas with near 50% classify themselves as strongly motivated and disciplined and another 50% motivated but needed parents encouragement.

Despite the common stereotypical image of people excelling at math being obsessive computer nerds and friendless geeks, most of students in this small pool play computer games, though not excessively (on average of 0-4 hours per day). Most of them are avid readers; fifteen out of nineteen people said that they read more than twenty non-math books a years. The majority of them do have friends.

Among the 19 students, ten of them have taken the SATs (a test for high school juniors and seniors) in 7th and 8th grade through talent search programs. Eight students provided their scores (two could not remember). The averages of this small group were 634/738/605 (in the format Critical Reading/Math/Writing). Even though it had reading and writing scores that were somewhat lower than their math scores, this small group achieved significantly higher scores than those of average high school graduates.

In terms of future career choices, surprisingly, about over 50% of students wanted to be in medicine /medicine related fields and less than 50% wanted to have careers in math research, profitable math-related fields, engineering, science, business or others. No one wanted to have a law-related occupation.

Among the choices for colleges, Harvard came first, MIT second and Stanford and Princeton third.

While math “geeks” have similarities, they certainly have differences. When asked their favorite books and music, other than Harry Potter (which made the list of the favorites of most of the students) other favorite books and music were all over the board.

Contrary to the stereotypical math “geek”, most of the students in this survey are not “book worms”. Many of them are well-rounded and participate in a variety of extracurricular activities, including musical instruments (piano /the most common one, violin, saxophone, clarinet, and cello); sports (tennis, swimming, and track and field. Other mentioned sports include soccer, basketball, wrestling, ping-pong, pool, air-hockey, skiing, Frisbee, hiking, golf, and football), computer-related activities (like programming or role-playing games) or video games (like Guitar Hero or Dance-Dance Revolution (both are music-related games)), volunteering by (donating or tutoring others), and occasional ‘nerdy’ hobbies like chess or solving Rubik’s cubes. A few of them have awards in activities other than math, including sports and music competitions.
[/hide]

1. b. female
2. b. 7th grade
3. b. 2nd year
4. d. top 12 at state
5-7 N/A (I'm homeschooled. sorry!)
8. a. (OK, maybe i don't study for exactly 2 hours every day, but i'm COMPLETELY self-motivated when it comes to MathCounts.)
9. b. or c., probably b.
10. probably d. Well, i study math, (obviously) play the violin, solve rubik's cubes (i'm kinda slow lol), knit, hang out w/my friends sometimes, play with my cats.... (that's just the beginning!!!)
11. c., i guess. I don't love to read, and am not very good at it, either.
12. I play the violin a lot, and am pretty good at it, (no competitions so far, but i've been First chair in my youth orchestra several times) same stuff that I said in #10.
13. c. I'm pretty easy going, and have a lot of friends, but find it hard to socialize with cliquish people. (who, incidentally, are usually the not-so-smart people)
14. Yeah I took the SAT's a month ago for CTY and flunked :-P I got 490-770-510 (CR-Math-Writing)
15. h. i really have no idea what i want to be when i grow up.
16. um, i don't know yet, but i want to go somewhere that has a good math program, and a good music program. Maybe Johns Hopkins/Peabody.
17. I'm a vegan, but i don't really consider that a quirk. other people do, to some extent.
18. Harry potter, Ender's Game, i guess the first Warriors series.

I'm not sure that survey results posted after preliminary results are posted are really as valid; they're likely self-selected to not fit with your previous data.

Survey Questions

1. You are (please state whatever your birth certificate says)
a. Male

2. You are in
c. 8th grade.

3. You participated/ will participate in Mathcounts for the
c. 3rd time

4. If you have participated in Mathcounts before, how far did you get (please specify the grade you were in when you participated)

c. Top 10 in chapter (which place/ written and final if countdown is official) - 6th grade
d. Made state (by being in the top two teams or by being top 4 individual in the chapter). - 7th
e. Made national, and your place (for the anonymous purposes, please do not post your national place on this board, you can PM me only if you feel comfortable, otherwise just say you made national).

5. Homeschooled.  If I had to calculate, it would be an A
6. Same
7. Same

8. Your motivation in math
b. Somewhat motivated (study somewhat but less consistently and you need your parents to push/ encourage you from time to time; if you have such parents, consider yourself lucky)

9. Your overall motivation in academics
b. Somewhat motivated (study somewhat but less consistently and you need your parents to push/ encourage you from time to time)

10. How much time do you spend playing computer games/other games (DDR, Game consoles. etc) and watching anime/ other cartoons/TV
c. Less than 2 hours

11. How many books (non-mathematical) do you read:
a. more than 20 books per year

12. Hobbies :  Theater, guitar, soccer, volunteer at church

13. Do you consider yourself sociable:
c. I am easy going and easily make friends/ act friendly with classmates of all academic levels.

14. Did you participate in SAT/ ACT (such as through talent search programs)? If so, what are your scores

b. ACT scores: English/ Math/ Reading/ Science/ Composite
24/27/23/23/24

15. What do you want to do when you grow up:
Undecided

16. What are your dream colleges (please list your top three choices)
Who knows?

17. Do you have any other interesting “quirks” (please specify).
I like to make comics with my brother.

18. Please list your 3 favorite books (if any math book makes that list, it is fine)/ movies/songs.
Dilbert, Dave Barry and They Might Be Giants

_________________
"There are 10 types people in the world: people who understand binary, and people who have friends."


[quote="ashrafmod"]$sin 10*sin 20*sin 30*......*sin 170$[/quote]
[hide="a start"]I have a feeling that the solution involves complex numbers so I can't finish this, but
$\sin^{2}10 \cdot \sin^{2}20 \cdot ... \cdot \sin^{2}90$ (this is because $\sin x = \sin (180-x)$
$\sin^{2}(10) \cos^{2}(10) ... \sin^{2}(80) \cos^{2}(80)$ (note that we can completely ignore $\sin^{2}90$ since it's one)
$\frac{\sin^{2}(20) \cdot \sin^{2}(40) \cdot ... \cdot \sin^{2}160}{65536}$
$\frac{\sin^{4}20 \cdot \sin^{4}40 \cdot \sin^{4}60 \cdot \sin^{4}80}{65536}$
Now I'm stuck...[/hide]
Any suggestions? Like I said, this looks like it involves complex numbers.

It is possible to brute-force this equation by bashing trig but using  this nice theorem,

$\prod_{k=1}^{n-1}\sin\left(\frac{k\pi}{n}\right)=\frac{n}{2^{n-1}}$

we can just plug the special case in and get $\frac{9}{65536}$.


linear algebra
matrix
algebra
function
domain
Ring Theory

Let $B$ be an $n$x$n$ invertible matrix. Define $T: M_{nxn}(F)\to M_{nxn}(F)$ by $T(A)=B^{-1}AB$. Prove that $T$ is an isomorphism.

what I am thinking is that since the domain and range have the same dimension i could show that $T$ is either 1-1 or onto and have it follow that $T$ is isomorphic. but am having trouble showing either condition.
does anyone have any pointers?
thanks.

If $T(X)=T(Y)$ then $B^{-1}XB=B^{-1}YB\Longrightarrow B(B^{-1}XB)B^{-1}=B(B^{-1}YB)B^{-1}\Longrightarrow X=Y.$ And $f(kX+mY)=kf(X)+mf(Y)$ . Am I wrong?  :maybe:

that does show it is 1-1 but im am wondering if you are allowed to just multiply by $B^{-1}$ on the right and $B$ on the left

Just write down the inverse of $T$: $T^{-1}(A)=BAB^{-1}$.

$T$ is actually a ring isomorphism- it preserves addition and multiplication.

Say you have two finite dimensional linear spaces V and W and an isomorphism U:V->W. Let L(V,V) and L(W,W) denote the spaces of linear operators on V and W respectively. Then M:L(V,V)->L(W,W), defined $M(T) = UTU^{-1}$is an isomorphism between the two spaces.

For example, if C is an element of some basis B for L(W,W), then $M(U^{-1}CU) = C$ shows that M is onto.


Let a triangle with integer sides; if the inradius is 1, prove that the triangle is a right triangle.

EDIT: Could you give me the link?  Thanks.

[quote="mathwizarddude"]Let a triangle with integer sides; if the inradius is 1, prove that the triangle is a right triangle.[/quote]
Famous and easy one,from some China Regional Olympiad.
I think it is more N.T problem than Geometry. :roll:


algebra
polynomial
inequalities
algebra unsolved

Let $ a,b,c\in R$ be such that the polynomial $ x^{3}+ax^{2}+bx+c$ has three real roots. Show that $ 12ab+27c\leq 6a^{3}+10(a^{2}-2b)^\frac{3}{2}$.

When does equality occur?

This is wrong. Try $ a=3,b=4,c=1.$ Then the given condition is fulfilled, yet the polynomial $ x^{3}+3x^{2}+4x+1$ has only one real root.

Does that actually make it wrong? It looks like the logic is

$ x^{3}+ax^{2}+bx+c$ has three real roots
$ \Rightarrow$
$ 12ab+27c \leq 6a^{3}+10(a^{2}-2b)^{\frac32}$

Not the other way around.


trigonometry
geometry
cyclic quadrilateral
angle bisector
geometry solved

$\triangle {ABC},\ \frac A5=\frac B2=\frac C1\ \ \Longrightarrow\ \ 2b^2=a^2+c^2,\  \ \frac ac -\frac ca=2$

[b]Generalization.[/b]

$\triangle {ABC},\ p\in N,\ p\ge 3,\ \frac Ap=\frac B2=\frac C1\Longrightarrow b^2=c(a+c),\ \cos {\frac {\pi}{p+3}}=\frac {b}{2c}.$

Very nice problem!

[color=blue][b]Problem.[/b] Let the angles A, B, C of a triangle ABC satisfy the equation $\frac{A}{5}=\frac{B}{2}=\frac{C}{1}$. Prove that $b^2=\left(a+c\right)c$, $2b^2=a^2+c^2$ and $\frac{a}{c}-\frac{c}{a}=2$.[/color]

[i]Solution.[/i] Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a regular 8-gon. Denote its sidelength by

$z=A_1A_2=A_2A_3=A_3A_4=A_4A_5=A_5A_6=A_6A_7=A_7A_8=A_8A_1$,

and its diagonals by

$y=A_1A_3=A_2A_4=A_3A_5=A_4A_6=A_5A_7=A_6A_8=A_7A_1=A_8A_2$;
$x=A_1A_4=A_2A_5=A_3A_6=A_4A_7=A_5A_8=A_6A_1=A_7A_2=A_8A_3$;
$w=A_1A_5=A_2A_6=A_3A_7=A_4A_8$.

If O is the center of the regular 8-gon $A_1A_2A_3A_4A_5A_6A_7A_8$, then this point O is the center of the circle through the points $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$, $A_7$, $A_8$; thus, by the central angle theorem, $\measuredangle A_5A_1A_7=\frac12\cdot\measuredangle A_5OA_7$, but, on the other hand, $\measuredangle A_5OA_7=2\frac{2\pi}{8}$, so that $\measuredangle A_5A_1A_7=\frac12\cdot 2\frac{2\pi}{8}=2\frac{\pi}{8}=\frac{\pi}{4}$. Similarly, $\measuredangle A_7A_5A_1=\frac{\pi}{4}$. Thus, the triangle $A_1A_7A_5$ is an isosceles right triangle, and therefore, $A_1A_5=\sqrt2\cdot A_7A_1$. In other words, $w=\sqrt2\cdot y$. Hence, $w^2=2y^2$.

Similarly to how we found $\measuredangle A_5A_1A_7=\frac{\pi}{4}$, we can see that $\measuredangle A_1A_8A_5=\frac{\pi}{2}$. Thus, the triangle $A_1A_8A_5$ is right-angled at $A_8$; consequently, by the Pythagoras theorem, $A_1A_5\ ^2=A_5A_8\ ^2+A_8A_1\ ^2$. In other words, $w^2=x^2+z^2$. Since $w^2=2y^2$, this becomes $2y^2=x^2+z^2$, so that $x^2-z^2=\left(x^2+z^2\right)-2z^2=2y^2-2z^2$.

Applying the Ptolemy theorem to the cyclic quadrilateral $A_6A_7A_8A_1$ (which is cyclic since the 8-gon $A_1A_2A_3A_4A_5A_6A_7A_8$ is regular), we get $A_6A_8\cdot A_7A_1=A_7A_8\cdot A_6A_1+A_6A_7\cdot A_8A_1$. In other words, $y\cdot y=z\cdot x+z\cdot z$. Equivalently, $y^2=\left(x+z\right)z$. Hence, $y^2=xz+z^2$, so that $xz=y^2-z^2$.

Thus,

$\frac{x}{z}-\frac{z}{x}=\frac{x^2-z^2}{xz}=\frac{2y^2-2z^2}{y^2-z^2}=2$.

Note that the triangle $A_8A_1A_6$ has the sidelengths $A_6A_1=x$, $A_6A_8=y$ and $A_8A_1=z$ and the angles $\measuredangle A_6A_8A_1=\frac{5\pi}{8}$, $\measuredangle A_8A_1A_6=\frac{\pi}{4}$ and $\measuredangle A_1A_6A_8=\frac{\pi}{8}$ (these angles are obtained in the same way as we showed $\measuredangle A_5A_1A_7=\frac{\pi}{4}$).

Now, consider our triangle ABC with its angles A, B, C. These angles must satisfy the proportion $\frac{A}{5}=\frac{B}{2}=\frac{C}{1}$ on the one hand, and the equation $A+B+C=\pi$ (sum of angles in a triangle) on the other hand. This together yields $A=\frac{5\pi}{8}$, $B=\frac{\pi}{4}$ and $C=\frac{\pi}{8}$. Thus, the triangle ABC has the same angles as the triangle $A_8A_1A_6$. Hence, the triangles ABC and $A_8A_1A_6$ are similar, so that their corresponding sides are proportional: $a:b:c=x:y:z$. Using this proportion, we see that the equations $2y^2=x^2+z^2$, $y^2=\left(x+z\right)z$ and $\frac{x}{z}-\frac{z}{x}=2$ imply $2b^2=a^2+c^2$, $b^2=\left(a+c\right)c$ and $\frac{a}{c}-\frac{c}{a}=2$. And the problem is solved.

  Darij

[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=45481&view=next]www.mathlinks.ro/Forum/viewtopic.php?t=45481&view=next[/url]
The first one Darij posted. This is a very useful theorem, especially when we solve triangles.

[quote="levi"]
[b]Generalization.[/b]

$\triangle {ABC},\ p\in N,\ p\ge 3,\ \frac Ap=\frac B2=\frac C1\Longrightarrow b^2=c(a+c),\ \cos {\frac {\pi}{p+3}}=\frac {b}{2c}.$[/quote]
Let $D\in AC$ such that $AD$ bisects $\angle {ABC}$. A short computation using the angle bisector theorem yields $CD=\frac {ab}{a+c},AD=\frac{bc}{a+c}$. On the other hand, triangles $ABC$ and $ADB$ are similar, and this implies $\frac cb=\frac{b}{a+c}$, that is, $b^2=c(a+c)$; while triangle $BCD$ is isosceles, which imply $cos\frac{\pi}{p+3}=\cos \angle{C}=\frac{\frac{BC}{2}}{CD}=\frac{a+c}{2b}$. Using the previous equality, the latter equals $\frac{b}{2c}$.


algebra
polynomial
number theory
modular arithmetic
combinatorics proposed
combinatorics

A regular polygon has $ 6n \plus{} 1$ vertices, and these vertices are coloured in two colours (say red and blue). Fix the number of blue vertices. Prove that the number of isosceles monochromatic triangles which were formed does not depend on how the vertices were coloured.

Some nice, interesting things, probably clues to a solution.
[hide][b]Lemma[/b]: For every two points $X$, $Y$, there are exactly $3$ other distinct points $A_1$, $A_2$ and $A_3$ for which $XYA_i$ is isosceles. Even better: the triangle $A_1A_2A_3$ is also isosceles!
[b]Proof[/b]: Given $X$ and $Y$ you can construct an isosceles triangle in two ways. The third point can be in the middle of the other points (seen along the polygon). Since $6n+1$ is odd, there is exactly one such point, $A_1$. The other way gives us $A_2$ and $A_3$: there are equally many points between $A_2$ and $X$ as between $X$ and $Y$ as between $Y$ and $A_3$. Since $6n+1$ is no multiple of $3$, $A_2$ and $A_3$ can't be the same point, and none of both can be the same point as $A_1$. Since this constructions is symmetric with axis = the line through $A_1$ and the middle of $XY$, the triangle $A_1A_2A_3$ is also isosceles with top $A_1$.
Probably there is a more elegant proof using modular arithmetic, but I don't feel like doing that now :-).

[b]Lemma[/b]: Every point is contained in exactly $9n$ isosceles triangles.
[b]Proof[/b]: A point $X$ and every of the other $6n$ points are contained in $3$ triangles (above lemma), but by this method we count each traingle twice. So we have $\frac{6n\cdot 3}{2} = 9n$ isosceles triangles that contain $X$.[/hide]

This problem was earlier given at the final round of Russian Mathematical Olympiad in 1994, as the fourth problem for 10th grade. The proposer is [i]D. Tamarkin.[/i] Here's my solution:

Let $ B$ denote the number of blue vertices. Let $ c_i\in\{0,1\},i = \overline{0,6n}$ be the color of vertex $ i$ and consider the polynomial $ P(X) = \displaystyle\sum_{i = 0}^{6n}\sum_{k = 1}^{3n}(X - c_i)(X - c_{i + k})(X - c_{i - k})$, where the indices are taken modulo $ 6n + 1$. It is easy to see that $ P(1)$ counts the number of isosceles with vertices of color $ 0$ and $ - P(0)$ counts the number of isosceles triangles of color $ 1$. Indeed, since there are no equilateral triangles ($ 3\not|6n + 1$), each equilateral triangle has the form $ A_{i - k}A_iA_{i + k}$.
Hence the number $ E$ of monochromatic isosceles triangles equals $ E = P(1) - P(0)$. Note that $ P(X) = \displaystyle\sum_{i = 0}^{6n}\sum_{k = 1}^{3n}(X - c_i)(X - c_{i + k})(X - c_{i - k})$ $ \displaystyle = \sum_{i = 0}^{6n}\sum_{k = 1}^{3n}[X^3 - X^2(c_{i - k} + c_i + c_{i + k}) + X(c_{i - k}c_i + c_ic_{i + k} + c_{i - k}c_{i + k}) - c_{i - k}c_ic_{i + k}]$, hence
$ \displaystyle E = P(1) - P(0) = \sum_{i = 0}^{6n}\sum_{k = 1}^{3n}[1 - (c_{i - k} + c_i + c_{i + k}) + (c_{i - k}c_i + c_ic_{i + k} + c_{i - k}c_{i + k})]$.
Let $ S = \displaystyle\sum_{i = 0}^{6n}\sum_{k = 1}^{3n}(c_{i - k} + c_i + c_{i + k})$ and $ T = \displaystyle\sum_{i = 0}^{6n}\sum_{k = 1}^{3n}(c_{i - k}c_i + c_ic_{i + k} + c_{i - k}c_{i + k})$.
Then $ E = 18n^2 + 3n - S + T$. Let's computer $ S$ and $ T$. Remember that all we know about $ c_i\in\{0,1\}$ is that $ \displaystyle\sum_{i = 0}^{6n}c_i = \sum_{i = 0}^{6n}c_i^2 = B$. Since $ \sum_{k = 1}^{3n}(c_{i - k} + c_i + c_{i + k}) = (3n - 1)c_i + (c_0 + c_2 + \ldots + c_{6n}) = (3n - 1)c_i + B$, we get $ S = (3n - 1)\sum c_i + (6n + 1)B = 9nB$.
Also $ \sum_{k = 1}^{3n}(c_{i - k}c_i + c_ic_{i + k} + c_{i - k}c_{i + k}) = c_i( - c_i + \sum c_j) + \sum_{k = 1}^{3n}c_{i - k}c_{i + k}$. Note that for each $ a < b$ there is exactly one pair $ i\in\{0,\ldots,6n\}$ and $ k\in\{1,\ldots,3n\}$ so that $ \{c_a,c_b\} = \{c_{i - k},c_{i + k}\}$, hence $ T = \displaystyle  \sum_{i = 0}^{6n}[-c_i^2 + c_iB] + \sum_{a < b}c_ac_b = B^2 - B + \frac {B^2 - B}2 = \frac32(B^2 - B)$.
So $ \boxed{E = 18n^2 + 3n - S + T = 18n^2 + 3n - 9nB + \frac32(B^2 - B)}$, a constant value.
[i]Navras[/i]

The special case of this problem was at our country's TST.Instead of $ 6n \plus{} 1$ there was $ 2003$ and the fixed number of blue vertexes was $ 500$,though its number isn't important,as the number of vertexes in regular polygon,we just need that this number is not divisible by $ 3$.
I will give a short,not detailed proof for the problem,as it was stated at Kazakhstan TST.
Proof:
We will prove that the number of bichromatic iscosceles triangles doesn't depend on how the vertexes are colored.
Indeed,the number of dioganals with differently colored ends(vertexes) is a fixed number.Denote their set as $ L$.
Then for every dioganal $ l\in L$ there exist exactly $ 3$ iscosceles bichromatic triangles.Because the number of vertexes is not divisible by $ 3$,these triangles are distinct.Hence the number of bichromatic iscosceles triangles doesn't depend on how the vertexes were colored,but as we know the number of iscosceles triangles in a regular $ 2003$-polygon is a fixed number too,hence the result follows.


algebra
polynomial
probability
quadratics
binomial theorem
Rational Root Theorem

here's a problem I came up with.  Might be a little challenging:

Let f(x)= :pm: 1 :pm: x :pm: x^2... :pm: x^(2n-1) where the sign in front of each term is assigned randomly.  What is the probability that f(x) is irreducible over Z (all integers)?

enjoy!

nice problem!

hint: [hide]rational root theorem[/hide]

Rational Root Theorem says that the only possible roots of f are -1,1.

Clearly, there is a root iff the number of (+)'s equals the number of (-)'s (as can be seen by evaluating f(-1), f(1))

By the binomial theorem, we are trying to calculate the probability of an equal number of (+)s and (-)s, and this is given by:

C(2n,n)*(.5)^n*(.5)^n = C(2n,n)*(.25)^n

for n > 1.  (If n=1, there is always a root).

Looks like I still have some work to do.

I'm a bit confused.

If -1 or 1 is a root then the polynomial is reductible. But the reverse is not true.
(X+1)^2 is reductible, yet it has no real roots.

Isn't it a little more delicate (or I don't quite understand blah:^3: proof) ?

[quote="blahblahblah"]Rational Root Theorem says that the only possible roots of f are -1,1.
[/quote]

That's right.

A polynomial may not have rational roots if it's reducible. For example x^6+x^4+x^2+1 = (x^4+1)(x^2+1) is reducible over Z, but it has no rational roots.

Oops, somebody beats me by seconds :-)

Hey, we have exactly the same time stamp 10:19pm!

[quote="amd2357"]fuzzylogic, you make a good point but it has no bearing on this problem.  In the example you gave the expanded product does not include every term.  x^5, x^3, and x terms are all missing.  In the problem, all powers of x from 0 to 2n-1 are included.  blahblahblah is on the right track.[/quote]

That doesn't mean anything.  For example, (x^2 - x - 1)(x^2 + 2x - 1) has no rational roots, contains all terms from x^0 to x^4, and is reducible.  He hasn't necessarily shown that, say, 1 + x + x^2 + x^3 - x^4 + x^5 can't be factored as a cubic times a quadratic, only that it can't be factored as a linear times a quartic factor.

you're right JBL.  I forgot to include that for the sake of the problem I was only taking about powers of x whose coefficient is either 1 or -1, sorry.

Right.  Either way, though, he has to show that such a polynomial doesn't factor without linear factors.


What will be the domain of the following real function?

$ f(x)\equal{} \frac{2x\plus{}1}{x^2\minus{}1},x$ belongs to the set of real numbers.

I got the answer $ R\minus{}\{\minus{}1,1\}$ (where $ R$ is the set of real numbers) but the answer given in the book is $ R\minus{}\{\minus{}3,3\}$. I think it is wrong. Am I right? Please help, tell me what is right and if the second one is right, then tell me why? Thank you.

[quote="modularmarc101"]Yes you are correct. The denominator cannot equal 0 so,

 $ x^2 \minus{} 1 \neq 0$

$ x^2 \neq 1$

$ x \neq \pm 1$[/quote]

Thank you. :-)


Spring will come and Iranian new year will begin in about about 2 hours.

[size=150]Happy new year to all of Iranians[/size] :welcome:  :gathering:

Look hera at [url=http://www.google.com/intl/fa/]Google[/url] Persian

You see the beautiful Iranian "Sofre 7-Seen"


In a farm there is a 100 kL tank and a 25 kL tank. As it rains, the 100 kL tank collects water from a 200 sq-m. roof and the 25 kL tank collects water from an 80 sq-m. roof. Currently, the 100 kL-tank is filled with 35 kL and the 50 kL-tank is filled with 13 kL.

Rain is forecast, and he farmer wants to collect as much waer as possible? He should:

(a) pour all the contents of the 100-kL tank to the 25-kL tank
(b) pour all the contents of the 25-kL tank to the 100-kL tank
(c) transfer 10 kL from the 100 kL-tank to the 25 kL-tank
(d) transfer 10 kL from the 25 kL-tank to the 100 kL-tank
(e) do nothing

Can you clear up how water collects from a roof?

Doesnt it depend how long its going to rain since if it rains for a short amount of time and it wont fill up both tanks, you want both to tanks to be collecting water the whole time if possible.  But if it rains long enough for both tanks to fill up either way, then... it doesnt matter....

but it does matter if it rains long enough for one tank to fill up and not the other.  The optimal situation is that they both finish filling up at the same time.  Let $ x$ be the amount of water we leave in the 100kl tank and $ y$ be the amount we leave in the other tank.  So we have $ x\plus{}y\equal{}35\plus{}13\equal{}48$.

Now we assume that each square meter of roof surface area gets rained on equal, so the rate at which each tank fills is proportional to the amount of roof space that it collects from.  So we want the ratios of roof surface to the amount of additional water that the tanks can hold to be constant, so we have $ \frac{100\minus{}x}{200} \equal{} \frac{25\minus{}y}{80}$.

Solving:
$ y\equal{}48\minus{}x$
$ \frac{100\minus{}x}{5} \equal{} \frac{\minus{}23\plus{}x}{2}$
$ 200\minus{}2x \equal{} \minus{}115\plus{}5x \implies x\equal{}45$

So the farmer should move 10kL into the larger tank.

(I'm assuming that the smaller tank is 25kL and that the 50kL you mentioned at the end is a typo)


email
number theory unsolved
number theory

Hi
My teacher has given me an exercise:

Check if there is a permutation $(x_{1},x_{2},x_{3},...,x_{p-1})$ of a set $\{1,2,3,...,p-1\}$ that all expressions: $x_{1}, x_{1}x_{2}, x_{1}x_{2}x_{3}, ... , x_{1}x_{2}x_{3}...x_{p-1}$ give different residues when divided by $p$, where $p$ is a prime

After a day of failed tries i asked my teacher for an answer which is YES but now I need to prove it...

This is a problem from current Polish MO. The contestants have to send their [b]own[/b] solutions of 12 problems ([url]http://www.om.edu.pl/zadania/om/om58_1.pdf[/url]). [i]II seria (10 pazdziernika 2006 r. – 6 listopada 2006 r.)[/i] means that that current (i.e. II) series of the competition starts at 10.10 and ends at 6.11, so please delete your solution until 6.11.

@tondor: it seems that you are a cheater! Your two posts up to now are this post and [url]http://www.mathlinks.ro/Forum/viewtopic.php?p=662486#662486[/url], where you asked for the solution of another problem from this competition. The location [i]Spain[/i] is false, because you email [email]tondor@poczta.fm[/email] is registered on a Polish server.

The deadline passed, so we can discuss the problem. My solution:

Let's consider $1,\ldots,p-1$ as elements of the field $\mathbb{Z}_{p}$. Then we simply put $x_{1}=1,\; x_{2}=\frac21,\; x_{3}=\frac32,\; \ldots,\; x_{p-1}=\frac{p-1}{p-2}$. It's easy to see that $x_{i}=x_{j}$ for $i\neq j$ gives an immediate contradiction, so $(x_{1},\ldots,x_{p-1})$ is a permutation of the set $\{1,\ldots,p-1\}$. For any $i$ we have $x_{1}\cdot\ldots\cdot x_{i}= 1\cdot\frac21\cdot\ldots\cdot \frac{i}{i-1}= \frac{i!}{(i-1)!}= i$, so $x_{1},\; x_{1}x_{2},\; x_{1}x_{2}x_{3},\; \ldots,\; x_{1}\ldots x_{p-1}$ have different residues modulo $p$. Therefore there exist a permutation of the set $\{1,\ldots,p-1\}$ satisfying given conditions which ends the proof.

@[b]ZetaX[/b]: Thank you for unlocking the thread.

My solution:

Let $ \zeta$ be a primitive root $ \mod p$. If we set $ x_{k} \equal{} \zeta^{a_{k \plus{} 1}}$, we just have two things to fulfill:
- all $ a_{k}$ (with $ k \in \{0,1,...,p \minus{} 2\}$ have to be different $ \mod (p \minus{} 1)$.
- all $ b_{n} \equal{} \sum_{k \equal{} 0}^{n}a_{k}$ have to be different $ \mod (p \minus{} 1)$.

Now we have an additive problem, and we will replace $ \mod (p \minus{} 1)$ by $ \mod 2m$ ($ p$ is clearly odd for all interesting cases) and now look for numbers $ a_{0},a_{1},...,a_{2m \minus{} 1}$ such that:
- all $ a_{k}$ are different $ \mod 2m$.
- all $ b_{n} \equal{} \sum_{k \equal{} 0}^{n}a_{k}$ are different $ \mod 2m$.

Set $ a_{k} \equal{} ( \minus{} 1)^{k}\cdot k$:
- a simple check shows: they are different $ \mod 2m$ (here we need that $ 2m$ is even).
- we have $ b_{2s} \equal{} \sum_{k \equal{} 0}^{2s}a_{k} \equal{} \sum_{k \equal{} 0}^{2s}( \minus{} 1)^{k}\cdot k \equal{} \sum_{k \equal{} 1}^{s}2k \minus{} (2k \minus{} 1) \equal{} s$ and $ b_{2s \minus{} 1} \equal{} \sum_{k \equal{} 0}^{2s \minus{} 1}a_{k} \equal{} \sum_{k \equal{} 0}^{2s}( \minus{} 1)^{k}\cdot k \equal{} \sum_{k \equal{} 1}^{s} \minus{} (2k \minus{} 1) \plus{} (2k \minus{} 2) \equal{} \minus{} s$, thus we get all residues $ \mod 2m$ (again using that $ 2m$ is even).

@[b]Rzeszut[/b]: thanks for notifying.


If $ x,y$ are positive integers find the solutions of

$ (x\minus{}2)(x\plus{}3)\equal{}8(y\minus{}1)(y^2\plus{}y\plus{}1)$


[quote="iamagenius"]i'm just wondering what you guys usually associate with smart people. what sorts of things to smart people (not necessarily nerds) do?

for example,

-chess
-rubik's cube
-advanced math (duh)

...

just wondering.  :)[/quote]

hmm wrong forum?

my friends call me a nerd cuz im doing like limits (we're in alg 2 now) and i can't go their house

[quote="usaha"]you do limits in algebra 2? I dont remember doing them..[/quote]

no i just go out of the curriculm and so some calculus in class cuz our curriculum is sooo boring

[quote="max_tm"]The Demon, I couldn't help but notice that the limit in your sig should be $\Delta x$ approaches $0$ not $x$. But yes, nerdy stuff... I donno, a couple of my friends and I play D&D every once and while (mind you, we're horrible at it and we usually play it the morning after having gone to a huge party or something... it's nerdy all the same  :P.)[/quote]

thanks im always making mistakes in my sigs :lol:

They see me mowin'... my front lawn
I know they're all thinkin' I'm so [url=http://en.wikipedia.org/wiki/White_%26_Nerdy]white and nerdy[/url]...

But yeah, a moderator should move this to G&FF.

it has to be advanced math.
Any noob can play chess, some just suck
any noob can learn rubik's cube, some are just slow
not everyone can do advanced math

[quote="DiscreetFourierTransform"]Siggie'd!
[quote]Any noob can play chess, some just suck
any noob can learn rubik's cube, some are just slow
not everyone can do advanced math[/quote][/quote]
Hahahaha the second isn't true.

someone really needs to move this to GFF :D

[quote="iamagenius"]i'm just wondering what you guys usually associate with smart people. what sorts of things to smart people (not necessarily nerds) do?

for example,

-chess
-rubik's cube
-advanced math (duh)

...

just wondering.  :)[/quote]

I think that "nerds" have braces, and they have spiky hair, AND they have really neat handwriting.....okay, before all of you get on me because you do or have one of these things, I based this on this one kid that I know...Seriously, he's the ultimate geek....He's always trying to beat me in Algebra ( although it never works b/c I study Algebra 2 in my spare time.), and ends up frustrated that his plans aren't working....that's just my opinion!

++i am planning to buy the rubik's cube..ok that may make me a nerd, but I also do Sudoku++

sudoku is fun

spiky hair? not so much..

braces? yes


trigonometry
number theory proposed
number theory

[quote="JRav"]If you prove that $ \sin{1}\cdot\cos{1}$ is irrational, then you're done (why?).[/quote]
Because $ \sin 2 \equal{} 2 \cdot \sin 1 \cdot \cos 1 \equal{} (\cos 1 \plus{} \sin 1)^2 \minus{} 1$ ! :P
So either both $ \sin 1 \cdot \cos 1$ and $ (\cos 1 \plus{} \sin 1)^2$ are rational, or $ \sin 2$ are irrationel :)

To Green_Eyes: Did you want these lousy hints or did you want an actual solution ? :)

You have to be willing to put some effort in.  You can't just say "I want a solution!" and expect people to bend over backwards for you.  Show us what you've done, or suggest something, and then you'll get help.  And we've already given you some hints.

A word on that "solution": It is impossible to prove the result using only the information presented here. "Irrational" is far too weak.

[quote="green_eyes"]We know that $ \sin 1$ and $ \cos 1$ are irrational. Prove that $ \sin 2$ is irrational.

We talk about radians. :roll:[/quote]
we can prove that: $ \sin 1$ and $ \cos 1$ are irrational :)
Assume $ \sin 1 \in\mathbb{Q}$,then $ \sin3 = 3\sin1 - 4\sin^3 1 \in\mathbb{Q}$
Similar, we have:$ \sin9 \in\mathbb{Q}$
On the other hand :$ \sin9 = \sqrt {\frac {1 - \cos18}{2}}$
And $ \cos18 = \sqrt {1 - \sin^2 18} = \sqrt {1 - (\frac {\sqrt {5} - 1}{4})^2} = \sqrt {\frac {5 + \sqrt {5}}{8}}$
So, $ \sin 9 = \sqrt {1 - \sqrt {\frac {5 + \sqrt {5}}{8}}} = x$
There fore:
$ \sin9 \in\mathbb{Q}$ ,$ < = > x \in\mathbb{Q}$
$ = > x^2 \in\mathbb{Q} = > \frac {5 + \sqrt {5}}{8} \in\mathbb{Q}$
$ = > \sqrt {5} \in\mathbb{Q} (FALSE)$
=> $ \sin1$ is irrational
Similar,we have:$ \cos1$ is irrational (Q.E.D)
Furthermore $ \tan1$ ,$ \cot1$ is irrational  :)  :lol:

Just a note on your syntax:
1) If you need to put a fraction in parentheses, \left( and \right) look a lot nicer than ( and ).
2) To produce an implication arrow, use \Rightarrow ($ \Rightarrow$).  To produce a double implication/equivalence arrow, use \Leftrightarrow ($ \Leftrightarrow$).

Just a note on your semantics:
1) Radians are not degrees. A full circle in radians is $ 2 \pi$, one in degrees is $ 360^\circ$.
2) You could have shown that $ \sin(2^\circ)$ is irrational that way.

Well, for the original problem, Lindemann-Weierstrass gives that all those number are transcendental, giving a much stronger (and more usefull) result.


Hi all!

I've searched the forum a bit and was unable to find the solution to my problem. I would like to have some commands I defined to be bold the first time I use them in my paper. I use \boldsymbol command, but it doesn't work.
I have such commands:

\usepackage{amsmath}
\usepackage{amsfonts}

\newcommand{\B}[1]{\boldsymbol{#1}}

\newcommand{\ord}{\ensuremath{\textrm{ord}}}

Then, when I use \B{\ord_P(f)} (I use dolar signs in my paper, i left them out here becouse div was not seen in my post if I used them), only P and f are bold, ord is not.

Can anyone please help?
Thank you!

I'm not sure what you want to do, but I'll try to help.  If you want to display code in your document, you need to use the verbatim environnement.  I don't think you can actually use and display the command at the same time so you would need to do that in two steps with the command \backslash.

I don't know if it helped.  if not, post a more complete example of what you want.

I wasn't clear enough about what my problem is. Maybe because I used some code in my post. I don't want to display code in my document. My problem is that boldsymbol command is not working as I want it to. If I use boldsymbol command on a command I defined in my document with newcommand command, then it is not displayed as bold but as normal font.

example:
\boldsymbol{\ord_P(f)} doesn't output $\boldsymbol{ord_P(F)}$ but only P and f are bold.

ord command is defined like \newcommand{\ord}{\ensuremath{\textrm{ord}}} so that in math mode 'ord' is in roman font.

English is not my native language soI have some problems expressing myself  :oops:
Hope you can help and thanks.

ok, I get it now.  Try this:

\mathbf{ord}\boldsymbol{_P(f)}

if you only use it once, it's not that long to use


function
complex analysis
complex analysis unsolved

Let $f ,g$ be entire functions. Then there exists $k>0$ such that for all $r>0$ one has $M_{fg}(r)\geq M_f( k\cdot M_g(\frac{r}{2}))$ where $fg$ is the composition between $f$ and $g$ and $M_h(r)=max_{|z|=r}{|h(z)|}$.

The statement should be corrected somehow. Otherwise $f(z)=z-1$ and $g(z)=1$ is a trivial counterexample.


found all (x,y) positive integer so that :
   x^3+y^3-(xy)² /(x+y)² be a positive integer

I'd assume the latter, because then the first part of the former would be irrelevant.

of course the latter one Farenhajt  :lol:

[hide="Solution"]Rewrite the expression thus:

$x+y-{3xy\over x+y}-{x^{2}y^{2}\over(x+y)^{2}}$

If $N : ={xy\over x+y}\in\mathbb{Q}$, then $3N+N^{2}$ is integer iff $N\in\mathbb{Z}$. From $xy=Nx+Ny$ we get $(x-N)(y-N)=N^{2}\iff x=N+k\land y=N+{N^{2}\over k}$ where $k|N^{2}$

Assume $k>0$. Now the given expression becomes

$2N+k+{N^{2}\over k}-3N-N^{2}=\frac{k^{2}-(N^{2}+N)k+N^{2}}{k}$

For it to be positive, $k$ must lie outside the interval between its zeros, which means that either the lower zero must be greater than one or the upper zero less than $N^{2}$:

\begin{eqnarray*}\frac{N^{2}+N-\sqrt{N^{2}(N+1)^{2}-4N^{2}}}{2}>1\implies (N^{2}+N-2)^{2}>N^{2}(N^{2}+2N-3)\implies\dots\implies N<1\end{eqnarray*}

\begin{eqnarray*}\frac{N^{2}+N+\sqrt{N^{2}(N+1)^{2}-4N^{2}}}{2}<N^{2}\implies N\sqrt{N^{2}+2N-3}<N^{2}-N\implies\dots\implies N<1\end{eqnarray*}

If $k<0$, put $l=-k>0$ to get $x=N-l,y=N-{N^{2}\over l}$, the given expression becoming $-l-N-N^{2}-{N^{2}\over l}$, which is obviously negative.

Therefore, for no natural $N$ we can find $k$ such that the given expression is positive, hence no desired $x,y$ exist.[/hide]

[quote="yassinus"]that's good , but i think u have made a mistakes in the last part of your proof , because with my proof i found the only couple (2,2)  :P[/quote]

${2^{3}+2^{3}-2^{2}2^{2}\over (2+2)^{2}}=0$, which is not positive.

If we're looking for solutions which give INTEGER (be it positive or negative or zero) value, then there are infinitely many of those, namely $(x,y)\in\{(N+k,N+{N^{2}\over k})|N\in\mathbb{N},k\in\mathbb{N},k|N^{2}\}$

positive integer dosen't mean >=0  ??
in fact, the last infinite couple gives the couple (2,2) because...we must have N=1....
congratulation  :D

As far as I know, "positive" means $>0$, and $\geqslant 0$ is called "non-negative".

That's what i have lern from that problem   :blush:


geometry
rectangle
trapezoid
geometry solved

Let $M,N$ be respectively midpoints of $AD$ and $BC$ in a rectangle $ABCD$. Point $P$ lies on the half-line $CD^{\rightarrow}$, not on the segment $CD$. Lines $PM$ and $AC$ intersect in $Q$. Prove $\angle QNM =\angle MNP$

See http://www.mathlinks.ro/Forum/viewtopic.php?t=14246 problem 2 and http://www.mathlinks.ro/Forum/viewtopic.php?t=55017 .

  Darij

[hide="Here is the my solution and a easy generalization."]
Denote the intersection $R\in PM\cap BC$. Thus, $\frac{PM}{PR}=\frac{CN}{CR}=\frac{MA}{CR}=\frac{QM}{QR}\Longrightarrow \frac{PM}{PR}=\frac{QM}{QR}$.
Therefore, the division $(P,M,Q,R)$ is harmonically and in this case $NM\perp NR\Longrightarrow$ the ray $[NM$ is the bisector of the angle $\widehat {PNQ}\ .$

[b]Generalization[/b] (the proof is identically with the above solution).
Given is a right trapezoid $(AB\parallel CD,\ BA\perp BC)$. Denote:
$\blacktriangleright$ the midpoints $M$, $N$ of the sides $[AD]$, $[BC]$ respectively;
$\blacktriangleright$ the projection $T$ of the point $M$ on the line $AB$;
$\blacktriangleright$ for any point $P\in CD$, the intersection $Q\in PM\cap TC$.
Prove that $\widehat {PNM}\equiv \widehat {QNM}$.[/hide]


We fix a cyllinder of radius $R$ on the floor so that it cant move. We put a homogenous metal sheet of thickness $a$ and length $b$ on it.
Find the period of small oscillations of the sheet. (The friction is big enough).

I know this one :) This is one great mess to do, I don't feel like typing it now  :roll: Maybe someone who isn't familiar with the problem would like to give it a shot?

$x=R\theta$

$mgxcos\theta+I\frac{d^{2}\theta}{dt^{2}}=0$

$mgR\theta+I\frac{d^{2}\theta}{dt^{2}}=0$

$I=\frac{mb^{2}}{12}$

$T=\frac{\pi b}{\sqrt(3gR)}$

If it was only that easy :) What you wrote is the case of a uniform stick, not a rectangle.

Do you mean that the moment of inertia is different?

The complete equation of motion is different (it has the same form, $kx+\omega^{2}\ddot x = 0$, but $k$ and $\omega$ are not obvious, you need a careful analysis).

Can you post the answer? (Not the solution,only answer :roll: )

Here is the answer:
$T=2\pi \sqrt{\frac{b^{2}+4a^{2}}{12g(R-\frac{a}{2})}}$

$I_{0}=\frac{m}{12}(b^{2}+a^{2})$

This is the moment of inertia for the center of mass.We will write the moment of inertia of the touching point ti the sphere.

$I=I_{0}+\frac{ma^{2}}{4}=\frac{m}{12}(b^{2}+4a^{2})$

Now we will take the moment of forces of touching point.

$mgR\theta-mg\frac{a}{2}\theta+I\frac{d^{2}\theta}{dt^{2}}=0$

$mgR\theta$ and $mg\frac{a}{2}\theta$  are moments of projection of $mg$ due to that point.

Then the answer is ,

$T=2\pi(\frac{b^{2}+4a^{2}}{12g(R-\frac{a}{2})})^\frac{1}{2}$

Sorry, I don't see how you got the equation of motion... Could you please explain the first two terms?


This is the CTC sign ups. Moderator positions, Aops Mods (that are in charge) and regulars will be posted here. I'm making this different forum topic because a want to save the other one for rules, announcements, and complaints.

Aops mods:

Isabella2296
Vallon22

Tourney mods:

TBA (to be announced)

Regulars:

Mathblitz

See [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?t=214516]here[/url]

[url=http://www.mathblitz.justgotowned.com]Click to reveal hidden content[/url]

I could add those people but the tourneys i hope to start are hopefully going to be started by izzy and vallon this winter break and some of those people hardly ever post anywhere besides G or FF. I hope to make the sign ups for each individual tourney like mafia but thats a long way off so I'm starting simple. Oh and most of CTC will be given to the Aops mods and maybe regular representatives hopefully in the middle of winter break.

Also I agree with mathblitz. We should start soemthing new, not CTC. BEcause lots of people quit because we were not satisfied with mewto for coughreasons. I guess we could start CTC again exactly the same BUt with new elections.

jjx1 that will have to w8 until summer because I dont have enough time to make a FAIR election for everyone and tourneys wont be entirely active until then. but if you want you can help me set up the goverment. This was by mathblitz. I let him use my account.

sry drake didnt mean to. but anyway jjx1 mathmaster and I are working on the goverment. you can help if you want


real analysis
function
calculus
integration
limit
real analysis unsolved

If the function is non-negative. Does Riemann integrable imply Lebesgue integrable? And their values agree?

Here Riemann integration includes improper integration as well.

I know this is not true for function which is not non-negative. for example, sinx/x.

Personally, I would prefer to restrict the use of the words "Riemann integrable" to bounded functions on a bounded interval. An improper integral is not itself a Riemann integral; it is a limit of integrals.

Now, if we can rephrase your question to make it the question you want to ask:

Suppose $f\ge0$ on an interval $I.$ Suppose there is an increasing sequence of intervals $I_{n}$ such that $\cup I_{n}=I,$ $f$ is (Riemann) integrable on each $I_{n},$ and $\lim\int_{I_{n}}f=L$ exists. Then $f$ is Lebesgue integrable on $I$ and $\int_{I}f=L.$

This is a simple consequence of the Monotone Convergence Theorem - just consider the sequence of functions $f\chi_{I_{n}}$ on $I.$

In the end, this isn't about Riemann integration at all - it's about improper integration, which is a different thing. I stuck the hypothesis of Riemann integrability on $I_{n}$ in parentheses because the question of what kind of integral that is is really somewhat irrelevant.


geometry
3D geometry
complementary counting

We count the possibilities:

Painted on opposite sides:

The 9 middle cubes on 4 faces=36

The middle edge cubes (does that make sense) 4 groups of 3=12

so 48.

Wouldn't the 9 middle cubes be on 5 faces, for a total of 57?

No, there is a completely hollow region in the cube; it goes through it completely.

It says 3x3x5 so u know it goes all of the way through. Is my answer correct?

Yes, I believe so. I used complementary counting to get the same answer...(noting that the borders surrounding the openings are the only blocks not with 2).

the correct answer is 48.
i would draw a pretty diagram with colors, but unfortunately that's beyond my skill level.

Imagine the cube as 5 layers of 5x5 squares with 3x3 squares cut out.
the middle 3 layers are made of cubes which all have exactly 2 faces painted (it's not hard to see why)
each layer has $5^2-3^2=16$ cubes. 3 layers of 16 cubes makes a total of $3\cdot16=48$ cubes.


calculus
limit
function
ratio
calculus computations

Hello once again,

I am self-studying calculus so I have two more questions on which I am having trouble.

Thank you in advance!

---

1. Evaluate the following limit:

$ \lim_{k \to 0}\frac {(x \plus{} h)^3 \minus{} x^3}{h}$

2. Show that $ \lim_{x \to \plus{} \infty} x \minus{} \sqrt {x^2 \minus{} 1} \equal{} 0$
(Hint: Multiply and divide by $ x \plus{} \sqrt {x^2 \minus{} 1}$)

---

1. I cannot immediately substitute h = 0 into the function because then, the denominator would become 0.

So expanding the fraction:

$ \frac {(x \plus{} h)^3 \minus{} x^3}{h}$

= $ \frac {(x \plus{} h)^3 \minus{} x^3}{h} \times \frac {(x \plus{} h)^3 \plus{} x^3}{(x \plus{} h)^3 \plus{} x^3}$

= $ \frac {(x \plus{} h)^6 \minus{} x^6}{h[(x \plus{} h)^3 \plus{} x^3]}$

This does not seem to work because substituting h = 0 would still cause the denominator to be 0.

2. $ x \minus{} \sqrt {x^2 \minus{} 1} \equal{} 0$

Multiplying by conjugate:

$ (x \minus{} \sqrt {x^2 \minus{} 1}) \times (x \plus{} \sqrt {x^2 \minus{} 1})$

$ \equal{} x^2 \minus{} (x^2 \minus{} 1)$

$ \equal{} 1$

Dividing by conjugate:

$ \frac {1}{x \plus{} \sqrt {x^2 \minus{} 1}}$

Here, the square root seems to be the trouble. I cannot square the fraction as this is an expression. What should I do?

Ok, lets do this fast:

$ 1. \lim_{h \to 0}\frac {(x \plus{} h)^3 \minus{} x^3}{h}$ $ \equal{} \lim_{h \to 0}\frac {x^3 \plus{} 3x^2 h \plus{} 3xh^2 \plus{} h^3 \minus{} x^3}{h}$ $ \equal{} \lim_{h \to 0}\frac {3x^2 h \plus{} 3xh^2 \plus{} h^3}{h}$ $ \equal{} \lim_{h \to 0}3x^2 \plus{} 3xh$ $ \equal{} 3x^2$.

(you could have done it also by using the expression for $ a^3\minus{}b^3$, instead of the $ (a\plus{}b)^3$ I used)).

2. You did it well: $ \frac {1}{x \plus{} \sqrt {x^2 \minus{} 1}}$. What happens for large $ x$ ($ x\to \infty$)? $ x \plus{} \sqrt {x^2 \minus{} 1}\to \infty$ so your limit is $ 0$.

$ \lim_{x\to\infty} x\minus{}\sqrt{x^2\minus{}1}\equal{}0$
$ \lim_{x\to\infty}x\equal{}\lim_{x\to\infty}\sqrt{x^2\minus{}1}$
$ \lim_{x\to\infty} \frac{x}{\sqrt{x^2\minus{}1}}\equal{}1$
$ \lim_{x\to_infty} \frac{x^2}{x^2\minus{}1}\equal{}1$

Thank you for your replies, hsiljak and igiul!

@hsiljak: I especially like your solution for the second problem, where you used logic to solve it.

FYI, I had trouble with the first question because I expanded $ (x + h)^3$ incorrectly.

[quote="igiul"]$ \lim_{x\to\infty} x - \sqrt {x^2 - 1} = 0$
$ \lim_{x\to\infty}x = \lim_{x\to\infty}\sqrt {x^2 - 1}$
$ \lim_{x\to\infty} \frac {x}{\sqrt {x^2 - 1}} = 1$
$ \lim_{x\to_infty} \frac {x^2}{x^2 - 1} = 1$[/quote]

@igiul: For your algebraic solution, I never knew you could treat $ \lim_{x\to\infty}\sqrt {x^2 - 1}$ as a variable and thus, add it to both sides.

So, for example, if I have:

$ \lim_{x\to\infty} (x + y) = 3$

Can I can rewrite it as such:

$ \lim_{x\to\infty} x + \lim_{x\to\infty} y = 3$

OR

${ \lim_{x\to\infty} y = 3 - \lim_{x\to\infty}x}$

[quote="igiul"]$ \lim_{x\to\infty} x \minus{} \sqrt {x^2 \minus{} 1} \equal{} 0$
$ \lim_{x\to\infty}x \equal{} \lim_{x\to\infty}\sqrt {x^2 \minus{} 1}$
$ \lim_{x\to\infty} \frac {x}{\sqrt {x^2 \minus{} 1}} \equal{} 1$
$ \lim_{x\to_infty} \frac {x^2}{x^2 \minus{} 1} \equal{} 1$[/quote]

$ \lim_{x\to\infty}f(x) \equal{} \lim_{x\to\infty}g(x)\equal{}\infty$
doesn't imply that
$ \lim_{x\to\infty}f(x)\minus{}g(x)\equal{}0$.
For example, take $ f(x)\equal{}x^2$, $ g(x)\equal{}x$

I think you misunderstand my approach...
\[ \lim_{x\to c}\frac {f(x)}{g(x)} \equal{} 1\Leftrightarrow \lim_{x\to c} f(x) \minus{} g(x) \equal{} 0
\]
Your contradiction does not satisfy the left, and so cannot satisfy the right. I demonstrated that because the given cases of $ f$ and $ g$ satisfied the left, that they satisfied the right as well. Additionally,
\[ \lim_{x\to c} \frac {f(x)}{g(x)} \minus{} 1 \equal{} 0 \equal{} \lim_{x\to c} \frac {f(x) \minus{} g(x)}{g(x)} \equal{} 0
\]
\[ \Leftrightarrow \lim_{x\to c} f(x) \minus{} g(x) \equal{} 0
\]
And furthermore \[ \lim_{x\to c} f(x)\equal{}\lim_{x\to c} g(x)\] The occurence of $ f(x)\equal{}g(x)$ when both tend towards infinity at c cannot be determined by a simple limit saying that $ f(x)\equal{}\infty\equal{}g(x)$. You must show that the limit of the ratio of f over g as x tends to c is equivalent to one.

[quote="igiul"]I think you misunderstand my approach...
\[ \lim_{x\to c}\frac {f(x)}{g(x)} \equal{} 1\Leftrightarrow \lim_{x\to c} f(x) \minus{} g(x) \equal{} 0
\]
Your contradiction does not satisfy the left, and so cannot satisfy the right. I demonstrated that because the given cases of $ f$ and $ g$ satisfied the left, that they satisfied the right as well. Additionally,
\[ \lim_{x\to c} \frac {f(x) \minus{} g(x)}{g(x)} \equal{} 0
\]

\[ \Leftrightarrow \lim_{x\to c} f(x) \minus{} g(x) \equal{} 0
\]
And furthermore
\[ \lim_{x\to c} f(x) \equal{} \lim_{x\to c} g(x)
\]
The occurence of $ f(x) \equal{} g(x)$ when both tend towards infinity at c cannot be determined by a simple limit saying that $ f(x) \equal{} \infty \equal{} g(x)$. You must show that the limit of the ratio of f over g as x tends to c is equivalent to one.[/quote]

\[ \lim_{x\to c} \frac {f(x)}{g(x)} \minus{} 1 \equal{} 0 \equal{} \lim_{x\to c} \frac {f(x) \minus{} g(x)}{g(x)} \equal{} 0
\]

would imply that

\[ \lim_{x\to c} f(x) \minus{} g(x) \equal{} 0
\]

except that $ g(x)$ goes to infinity here, so $ f(x)\minus{}g(x)$ doesn't need to go to 0 for the first limit to be 0. (It could go to infinity)

And here's an example that shows your statement is wrong: $ f(x)\equal{}x\plus{}1$, $ g(x)\equal{}x$. As you tend to infinity the ratio tends to 1 but the difference is 1.

But again, if both tend to infinity which they do in this problem, then $ \lim f(x)\equal{}\lim g(x)\equal{}\infty$ does not mean that $ \lim f(x)\minus{}g(x)\equal{}0$, so your second line does not imply your first line in your original proof and neither does your third line.

$ \lim_{h \to 0}\frac{(x\plus{}h)^3 \minus{} x^3}{h} \equal{} \frac{d}{dx}(x^3) \equal{} 3x^2.$

$ \lim_{x \to \infty}(x \minus{} \sqrt{x^2\minus{}1}) \equal{} \lim_{x \to \infty} x\left(1 \minus{} \sqrt{1 \minus{} \frac{1}{x^2}}\right)$

$ \equal{} \lim_{x \to \infty} x\left[1 \minus{} \left(1 \minus{} \frac{1}{2x^2} \plus{} O(x^{\minus{}4})\right)\right]$

$ \equal{} \lim_{x \to \infty} x\left(1 \minus{} 1 \plus{} \frac{1}{2x^2} \minus{} O(x^{\minus{}4})\right)$

$ \equal{} \lim_{x \to \infty} \left(\frac{1}{2x} \minus{} O(x^{\minus{}3})\right) \equal{} 0 \plus{} 0 \equal{} 0.$


inequalities
graph theory
combinatorics proposed
combinatorics

In a country there are 870123 cities, some of them joined by roads. each city has at least one road, and no city is conected to all others. Let a dominating set $D$ be a set of cities, such that if a city is not in $D$ it is joined to a city in $D$. If all dominating sets in this country have size at least $k$, show that we can divide the country into $870123-k$ disjoint republics, such that  2 cities in the same republic are never joined.

We assume that there are $n \geq 5$ cities instead of 870123.

Recall Brooks' theorem :
If $G$ is a connected graph other than a complete graph or an odd cycle, we have $\chi (G) \leq \Delta (G)$,
where $\chi (G)$ and $\Delta (G) $ are the chromatic number and the maximal degree in $G$, respectively.

Now, let the graph $G$ whose vertices are the $n$ cities and the egdes are the roads. Wlog, we may assume that $G$ is connected, otherwise we prove the result on each connected component.
Let $\delta (G)$ be the minimal size of a dominating set. Then $\delta (G) \geq k$.

We want to prove that $\chi (G) \leq n - k$. For that it suffices to prove that $\chi (G) \leq n - \delta (G)$. (1)

Since no vertex is connected with all the others, $G$ is not a complete graph.
If $G$ is the odd cycle $C_n$ then $\delta (G) = \frac {n-1} 2$ and $\chi (G) = 3$. Since $n \geq 5$ the inequality (1) follows easily.
From now, we assume that $G$ is not an odd cycle.

Using Brooks' theorem, it follows that we only have to prove the following result :

\[\Delta (G) + \delta (G) \leq n\]

Proof : Let $A$ be a vertex with maximal degree, and $N(A)$ be the set of the $\Delta (G)$ vertices adjacent to $A$.
Then, the set of all vertices which do not belong to $N(A)$ is a dominating set, because $A$ dominates itself and all the vertices in $N(A)$, and each of the other vertices which are not in $N(A) \cup \{A \}$ dominates itself.
It follows that $\delta (G) \leq n - \Delta (G)$, as desired, and we are done.

Pierre.

Does anybody has an idea about the reason the proposer has chosen 870123?
Does it make some special sense in Russia???

Pierre.

for an odd cycle the minimum dominating set would be of size [n/3]+1 if n is not divisible by 3 and [n/3] if it is.

87/01/23=1987 january 23th. My birthday :D  :D  :D  :D  :D . I wanted to put a number you had never seen on problems!

By the way Pierre, I am reading your graph.pdf in french (pretending to at least), it is really nice!

Oops...you are right Seshadri....I confused between 2 and 3...But, I think the rest of the proof can be achieved in the above way.

Pascual : I see... :D Nice choice.

Pierre.

I am sure that original proof doesn't appeal to any exterior theorems (it is feature of all ARO problems).

Like my solution, it only uses that if the maximal degree is t then the graph is t+1 coloreable.


Three points are randomly located on a circle. What is the probability that the shortest distance between each point is less than or equal to the radius of the circle?

[quote="r15s11z55y89w21"]Unless I misunderstood you question, the answer is $ \fbox{0}$ simply because there is no way you can make a total of $ 360$ degrees for the circumference of a circle without having at least one chord be longer than the radius, since the length of the radius is equivalent to a chord cutting off a $ 60$-degree segment.[/quote]

I think the answer would be $ 1/18$ becuase for the first point, you can choose any part of the circle.  for the second point, you can choose $ 60$ degrees to the left or right of the first point, which is $ 1/3$ of the entire circle.  For the last point, you only have $ 60$ from the farthest away point, so that this point has an area of probability $ 1/6$

So, $ (1)(1/3)(1/6)\equal{}1/18$

I'm probably wrong though...

[hide] $ \frac{2}{9}?$ It can't be zero because you could have that small probability that all three points are in a line or really close or something. A little offset of what rd5493 said, because you could have any point for the first, within 60 degrees for the second, but the third has to be within 60 degrees of the second as well, so that has a probability of 2/3. So the probability would be $ (1)(\frac{1}{3})(\frac{2}{3}) \equal{} \frac{2}{9}$[/hide]

my turn to try  :D

first one can be anywhere: $ 1$
second has to be within 60 degrees of the last one: $ \frac{1}{3}$
the last points possibilities depend on the second point. if the second point is 60 degrees away from the first, there is a range of 60 degrees for the third, making it $ \frac{1}{6}$. if the second point were "on" the first point, there would be a probability of $ \frac{1}{3}$, the same as the second point. since any solution in between is equally likely, I think the answer should be:

$ 1*\frac{1}{3}*\frac{\frac{1}{6}\plus{}\frac{1}{3}}{2}\equal{}\frac{1}{12}$

i agree with rd5493. if you have the answer or have the means of finding them, please do. It  would definetely be interesting  to know the solution  :)

I got $ \frac12$ by means of a geometric probability argument. :maybe:

[hide="General Formula"]
In general, the probability that when $ 3$ randomly chosen points on the circumference of a circle are such that no two points are farther than $ a^\circ$ apart, or equivalently, lie on an arc with measure less than or equal to $ a^\circ$, where $ 0\le a\le180$, is $ \frac {a^2}{43200}$. For $ 180\le a\le240$, it is $ 1\minus{}\left(\frac{240\minus{}a}{120}\right)^2\equal{}\frac{(a\minus{}120)(360\minus{}a)}{120^2}$, and for $ 240\le a\le360$, it is $ 1$.

[hide="Proof"]
Let the three arc measures formed be $ x^\circ$, $ y^\circ$, and $ (360 \minus{} x \minus{} y)^\circ$ where $ x,y\ge0$ and $ x \plus{} y\le360$. In order to have them all lie on an arc of $ a^\circ$, we must have the largest of $ x^\circ$, $ y^\circ$, and $ (360 \minus{} x \minus{} y)^\circ$ be greater than or equal to $ (360 \minus{} a)^\circ$, so the three relevant inequalities are $ x\ge360 \minus{} a$, $ y\ge360 \minus{} a$, or $ 360 \minus{} x \minus{} y\ge360 \minus{} a\implies x \plus{} y\le a$ (any one of them being true forces the rest to work out, even though the other two inequalities will be false). Graphing this, we see that the "successful" region consists of $ 3$ congruent isosceles right triangle each with area $ \frac12a^2$ and the "total" region is an isosceles right triangle with area $ \frac12\cdot360^2$, so the probability is $ \frac {\frac32a^2}{\frac12\cdot360^2} \equal{} \boxed{\frac {a^2}{43200}}$. For $ 180\le a\le240$, we can use a similar method; the "bad" region turns out to be an isosceles triangle with side length $ 720\minus{}3a$, and the probability is $ 1\minus{}\left(\frac{720\minus{}3a}{360}\right)^2\equal{}\boxed{1\minus{}\left(\frac{240\minus{}a}{120}\right)^2\equal{}\frac{(a\minus{}120)(360\minus{}a)}{120^2}}$. For the final case $ 240\le a\le360$, note that since the sum of the three arcs formed is $ 360^\circ$, the minimum possible value of the maximum of the three is $ 120^\circ$, and thus it will always be true that the three points are on an arc of measure at least $ 240^\circ$.
[/hide]
[/hide]

In this problem, note that all three points must lie within an arc of length $ 60^\circ$ for the given condition to be satisfied. The probability is thus $ \frac {60^2}{43200} \equal{} \boxed{\frac1{12}}$.

[quote="rd5493"]FantasyLover, do you actually know the answer to this problem.  We have 4 people that answered, and also 4 different answers, though I don't think I am right because of the point that ajai made.[/quote]

No, I had not responded because I do not know the answer.

Thanks everyone who answered. :)

i got the same thing as rd by same idea. not sayin much tho-im not that great at these things. but im pretty sure it is

[b]e=mc2.kroll[/b] is correct.

[b]rd5493[/b] and [b]bobma99[/b]: Consider the case where the first and second points are the same point: clearly there is more than $ 60^\circ$ (in fact, there is $ 120^\circ$) for the third point to be.

Seriously, shouldn't it be $ \fbox{0}$ since $ 360>180$?

What? If all the points are within $ 60^\circ$ of each other, it works...I don't get what you're saying. :huh:

[quote="math154"]If all the points are within $ 60^\circ$ of each other, it works...[/quote]

That's exactly what I'm trying to get at. There are only $ 3$ points, and $ 3\cdot60 \equal{} 180$, but that is only half of a circle. For arc amounts larger than $ 180$ degrees, including the full circle of $ 360$ degrees, there is no way you can make all gaps smaller than the radius.

EDIT: Oh, I see! So that's what it was asking for...

Consider this:
 [geogebra]39df01bc9fb701ec2d99a69bf99c3f46abbbaac0[/geogebra]
Clearly this works; I'm also not sure what you're saying. :maybe:


Does anyone know any of any packages or tricks that will give me all the capital greek letters in math mode? Currently only 11 of the 30 greek letters have an upper case  (found in the amssymb package, i think).

Hi Gary,

because the other capital Greek letters have the same appearance like some Latin letters the don't need special symbols. Just think of A for Alpha, B for Beta, E for Epsilon, ... , X for Chi.

Stefan


linear algebra
matrix
linear algebra unsolved

[u]problem 1:[/u]
supose that $ A,B\in R^{nxn} with A \neq\ B$ and they are in effect these circumstances:
$ A^3 \equal{} B^3$ and $ A^2B \equal{} AB^2$
Is the matrice $ A^2 \plus{} B^2$ invertible?

[u]problem 2:[/u]
If A and B are $ nxn$ matrices so $ A^2 \equal{} B^2 \equal{} (AB)^2 \equal{} I$ then is true $ AB \equal{} BA$

1. Compute $ (A\minus{}B)(A^2\plus{}B^2)$. You'll find that it's 0. If $ A^2\plus{}B^2$ is invertible it follows that $ A\equal{}B$. Contradiction.

2. $ ABAB\equal{}1$ multiply to the right with $ BA$. You get $ ABABBA\equal{}BA\Rightarrow AB\equal{}BA$

Thank you but i have some questions
1)If $ A^2 \plus{} B^2$ is invertible means that is not 0 so $ A \minus{} B \equal{} 0 \minus{} > A \equal{} B$
2)you mean $ ABAB \equal{} 1(one) or I$ , then where do we use $ A^2 \equal{} B^2$?
thanks again

1. A matrix $ X$ is invertible iff there exists another matrix $ Y$ so that $ XY\equal{}YX\equal{}I$ (the identity matrix).
If $ A^2\plus{}B^2$ would be invertible then we could multiply to the right with it's inverse and would get $ A\minus{}B\equal{}0\cdot (A^2\plus{}B^2)^{\minus{}1}\equal{}0$ so $ A\equal{}B$.

2. I meant, of course, $ I$ (the identity matrix).
well...
$ ABAB\equal{}I\Rightarrow ABABBA\equal{}BA\Rightarrow ABAB^2A\equal{}BA\Rightarrow ABAIA\equal{}BA\Rightarrow ABA^2\equal{}BA\Rightarrow AB\equal{}BA$.
Now you see where I used the that $ A^2\equal{}I\equal{}B^2$ ?


Prove that there does not exist a natural number $ n$,such that $ a,b\in\mathbb{N}$ are the divisiors of $ n$ and
$ a,b\in (\sqrt{n},\sqrt{n}\plus{}\sqrt[4]{n}\plus{}\frac{1}{2})$.

[quote="Erken"]Prove that there does not exist a natural number $ n$,such that $ a,b\in\mathbb{N}$ are the divisiors of $ n$ and
$ a,b\in (\sqrt {n},\sqrt {n} \plus{} \sqrt [4]{n} \plus{} \frac {1}{2})$.[/quote]
[hide]Since a,b are integers, clearly n must be a perfect square.
Let $ n\equal{}k^{2}$.
$ b\equal{}k\plus{}\sqrt{k}\plus{}.5$
Clearly in order for k to be an integer, it must be a perfect square, so let $ k\equal{}a^{2}$ where a is also an integer.
$ b\equal{}a^{2}\plus{}a\plus{}.5$ but this makes be a non-integer.[/hide]
This seemed too simple, perhaps I misinterpreted the problem?

[quote="MellowMelon"]Looks like an open interval and not a set to me. :wink:[/quote]
Yes,it is not set,it is an interval,excluding,$ \sqrt {n}$ and $ \sqrt {n} \plus{} \sqrt [4]{n} \plus{} \frac {1}{2}$.


Hi!

Anybody know the real nexus between the numbers on the single zero roulette wheel (french roulette)?

The great mathematician Blaise Pascal invented the roulette wheel, but Louis Blanc added later the "0" to achieve a house advantage.
So Pascal arranged only 1-36 numbers this way, but why? Why exactly this sequence? Why not put in order 1-36?
Only for harder to memorize wheel sectors or what?
I believe Pascal put every number the correct position, so it is not random, it is pure math.

Sorry guys if it is not the right topic for this.

Thanks,
Bronco

[quote="spix"]As far as I know numbers are placed such that the chances of wining on an even,odd,black,red,high or low bet are as balanced mathematically as they can be. I suppose this can be done by calculating probabilities along with taking into account neighbour squares and so on..  :ninja:[/quote]

I read somewhere when Pascal put the numbers to the wheel he put only numbers, after this the first number is red, the next  is black...
So I don't believe Pascal put just one red, one black,  one low, and so on.
But who knows

Bronco

[quote="Bronco"][quote="spix"]I didn't say that he put them in an alternating way, I just said that they are placed such that the probabilities for those 6 bets are as balanced as possible.[/quote]

Yes spix, you are right!  :wink:[/quote]

Maybe ... i was thinking about it.