Prove that

$1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$ is never an integer for $n>1$.

[quote="amirhtlusa"][hide="hint"]
$\displaystyle 1+\frac{1}{2}+(\frac{1}{3}+\frac{1}{4})+(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8})+....>1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+...$ ;) [/hide][/hide][/quote]

thats more used to prove that the series diverges, but i see what you are trying to say.

[quote="amirhtlusa"][hide="hint"]
$\displaystyle 1+\frac{1}{2}+(\frac{1}{3}+\frac{1}{4})+(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8})+....>1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+...$ ;) [/hide][/hide][/quote]

Umm... somehow I don't think that proves it... could you explain more? (Though it does prove that the series diverges)

i dont know either :D
it's quite hard to explaine what i meant.
basically the right side is an integer for an even number of $\frac{1}{2}$'s,...
so you all you need to show is that non of the expression in $()$ on the left side is equal to $\frac{1}{2}$, which is true, so it cant be an integer. ;)
(i know this is a horrible word choice, but it's really hard to explain the logic...)

do you have to group the $LHS$ like that? because if that is not the limitation, then i am sure we could find ways to contradict your reasoning.

I'm not sure if that proves it... but there is another way that is quite simple and elegant.

Assume that $1+\frac{1}{2}+\frac{1}{3}+\cdots \frac{1}{n}=\frac{q}{p}$ (p,q are coprime),.....

I will post later? ;)

amir's method is nonsense.

siuhochung posted a bunch of number theory problems in the pre-olympiad forum about a week back with solutions - this is one of them, and he included a proof.  However, this problem has probably been posted at least a half-dozen times over the last year and a bit or so.

one other thing to look at would be
IF, the sequence of Harmonic numbers DOESN'T start with $1$, then the sum would be always and odd number over an even number, therefore the sum can never be an integer :)

[quote="amirhtlusa"][quote="blahblahblah"]amir's method is nonsense.
[/quote]

i never said that was the right method. thats why i put an  ";) " in my hint.
now here is how i would prove it:
if $h_n=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$
Let $k$ be the largest power or $2$ less than or equal to $n$

Now if sum is $h_n=p$, multiply both side by $\frac{k}{2}*$(product of all odd numbers up to $n$ ) then except for the term corresponding to $\frac{1}{k}$ every other term would be an integer so: contradiction.
hope this helps... ;)[/quote]

This is basically it... just to clarify:

Let $S = 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$.

When we put it under a common denominator, all the terms in the numerator are even EXCEPT the $\frac{1}{k}$ term, where $k$ is the greatest power of $2$ less than or equal to $n$. Thus the sum in the numberator will be odd and the denominator will be even, so it cannot be an integer.

[quote="mkkool64"]Hey can this be extended to the series of reciprocal primes by any chance?  it seems like the proof might even be easier...[/quote]

If you start with $\frac{1}{2}$ you can use the same method... otherwise, I'm not sure yet.

I think I got proof for the reciprocal prime series:

[hide="Proof"]

Say you have a sequence of primes, p(1), p(2), ... p(n), and assume the sum of the reciprocals is equal to an integer n.

Then solve for the 1/p(1) term to get:

1/p(1) = n - 1/p(2)  - 1/p(3) - ... = (some integer) / (p(2)p(3)p(4)...)

So then multiply both sides by the prouct in the denominator and see that it is impossible because p(1) can't divide any of p(2), p(3), etc.
[/hide]

Yeah, now that I think about it, it's quite simple:

Again, let $S = \frac{1}{p_1}+\frac{1}{p_2}+...+\frac{1}{p_n}$.

We take the common denominator, $p_1p_2...p_n$. But then all the terms in the numerator are divisible by $p_1$ except for the first. Thus the numerator sum is not divisible by $p_1$ either, but the denominator is. So it can't be an integer.

$S = 1 + \frac{1}{3} + \frac{1}{5} + ... + \frac{1}{2n - 1} $

is that an integer? Hmm.. I don't think so.

if there is more than one such k, there are more than one number that is not an integer, and we cannot say the summation of two non-integers is not have to be non-integer.

e.g.; if k and 3k are both less than n...

You have to show that "another" largest powers exceed n.

[quote="Arne"]It isn't. There is a very short proof which uses the Bertrand postulate.[/quote]

I posted such a proof in this thread:
[url]http://www.artofproblemsolving.com/Forum/viewtopic.php?t=52930[/url]
a couple days ago.

Here's a different proof:
Suppose to the contrary $1 + \frac{1}{2} + \ldots + \frac{1}{n}$ is an integer then $1 + \frac{1}{2} + \ldots \frac{1}{n} = k$ for some $k$ integer. Then :
\[ n! + \frac{n!}{2} + \ldots + ( n - 1 ) ! = k \cdot n! \]
Let $p_n$ be the largest prime s.t $p_n \leqslant n$, then $p_n |n!$ and $p_n$ divides $\frac{n!}{j}$ for $j \neq p_n$, so it follows:
\[ \frac{n!}{p_n} = 0 ( \mbox{mod } p_n ) \]
Contradiction.
(EDIT: Obviously we must have $n \geq 2$)


function
calculus
derivative
analytic geometry
graphing lines
slope
algebra

Does anyone know the answer to this: Name three pairs of types of function that are inverses of each other.
Thanks so much in advance.

How about $ f(x)\equal{}x\plus{}a$ and $ g(x)\equal{}x\minus{}a$ there's infinite. Generally, any one-to one function on the plane has a unique inverse.

Actually, the inverse of $ f(x)\equal{}x\plus{}a$ is $ f^{\minus{}1}(y)\equal{}y\minus{}a$. More generally, the inverse of $ f(x)$ is the solution to the equation $ y\equal{}f(x)$ when solved for $ x$.
Graphically, the inverse of $ f(x)$ is obtained by reflecting $ f(x)$ across the line $ y\equal{}x$. Some properties of inverses include:

(i) The inverse of a function (if it exists) is unique.

(ii) If $ f^{\minus{}1}(x)$ is the inverse of $ f(x)$, then $ f(x)$ is the inverse of $ f^{\minus{}1} (x)$. This is called symmetry.

A continuous function $ f$ is a bijection if an only if it is strictly increasing or decreasing (with no local maxima or minima). It therefore can be derived that a continuous function is invertible only under these circumstances. Also, the inverse of a function is differentiable if and only if $ \frac{dy}{dx}\ne 0$. The derivative of the inverse of $ f(x)$ is:

$ \frac{d}{dy}[f^{\minus{}1}(y)]\equal{}\frac{1}{f'(f^{\minus{}1}(y))}$

I presume the OP is imagining functions from the reals to the reals, or some such.  In this case, I might state the example the previous two posters have given as, "the inverse of a linear function with slope 1 is another linear function with slope 1."  More generally, "the inverse of a nonconstant linear function is a nonconstant linear function" (and we can also say something here about the slopes of the functions, etc.)

[quote="JZambrano201"]Actually, the inverse of $ f(x) \equal{} x \plus{} a$ is $ f^{ \minus{} 1}(y) \equal{} y \minus{} a$.[/quote]  The functions $ f_1$ and $ f_2$ defined on the same domain by the formulas $ f_1(x) \equal{} x \minus{} a$ and $ f_2(y) \equal{} y \minus{} a$ are [i]the same function[/i].  The name you give the input to the function has nothing to do with the identity of the function.

Some of JZambrano's other comments deserve additional conditions on the domain and codomain/range to be explicitly mentioned.  For example:
[quote="JZambrano201"]A continuous function $ f$ is a bijection if an only if it is strictly increasing or decreasing (with no local maxima or minima).[/quote]  It would be more accurate to say, "A continuous function from an interval of real numbers (possibly all of $ \bf R$) to the real numbers is a bijection between its domain and range if and only if it is strictly monotone.  (In particular, being monotone does not imply that a function is a bijection between its domain and its codomain.)

[quote="i_thirst_for_knowledge"]Thanks to all these answers but i think the question asks about three pairs of functions. i don't quite understand that.[/quote]I suspect the question designer expects specific examples given in class/textbook, like exponential/logarithmic and quadratic/square root.
That's the type of question you can only answer if you were attending a specific class, or reading a specific textbook chapter. They will ask something like "list the three advantages of using renewable energy sources," and any energy expert who did not attend that class would be very puzzled.


calculus
integration
complex analysis
complex analysis unsolved

$h: \mathbb{R} \to  \mathbb{C}$ a bounded continuous function. Define $F(s)=\int_0^{\infty} h(u)e^{-su}du$ for all complex number $s$ such that $Re(s)>0$.
Suppose now that if $F$ has a continuous prolongation on $Re(s) \geq 0$. Then $F(0)=\int_0^{\infty} h(u)du$

Ah - a Tauberian theorem, with the the existence of an analytic continuation on a region which includes the closed right hand half plane as the Tauberian condition.

You wouldn't by any chance be trying to (eventually) prove the Prime Number Theorem?


Let a,b,c be positive real numbers ($ a,b,c>0$) and $ ab+bc+ca=3$
Prove that:
$ a^{3}+b^{3}+c^{3}+6abc\geq9$

Setting $ p = a+b+c$, $ q = ab+ac+bc = 3$ and $ r = abc$, we rewrite the inequality :

$ p^{3}-3pq+9r \geq 9 \Leftrightarrow p^{3}-9p+9r-9 \geq 0$

But from Schur we have :

$ r \geq \frac{4pq-p^{3}}{9}= \frac{12p-p^{3}}{9}$

Indeed, we have :

$ p^{3}-9p+9r-9 \geq 3p-9$

Now, we need to show that $ 3p-9 \geq 0 \Leftrightarrow p \geq 3$

But it's obvious because it's known we have $ p^{2}\geq 3q = 9$

Done.


linear algebra
matrix
counting
distinguishability

How many distinguishable 3 by 3 matrices exist such that each element is either a 0 or a 1 and the determinant of the matrix is 0?

Think about the vertices of the unit cube.  (You should probably also specify what you mean by "distinct":  up to row permutation?  Row and column permutation?  Similarity?)

[quote="t0rajir0u"]You should probably also specify what you mean by "distinct"[/quote]

Just like, don't look exactly the same in 3 by 3 form.


What are your goals for MathCounts Nationals this year? If you're aiming for something that's not on the poll, please select the next greater one. For example, if you want top 70, select top 100.

Well, I might not do MC this year (way too busy), but if I do, I just wanna make Nats. :P

I want to win for my coach, but I doubt I can get past the semifinals in Countdown. First, I must avoid making careless mistakes that cause me to lose 5 written points.

I would want to win for my old coach (she retired), but not for my new one.

I would want to win for myself. :P

Somebody put 1st place...I just want to say, guys, I want this to be realistic. If you're Bobby Shen(I know he has an account here) or someone else also smart, sure put that. But for people like me, please don't put that.

I'd be perfectly happy just making Nationals, but my goal is top 100, which I think is realistic for me. I'm not really sure, as I can't get my hands on a copy.

I already made the top 50, and this is my last year, so I have to aim high. My parents would be okay with top 12 though.

I figure that once I get to the Countdown, I should do more, because I've wanted this since 6th grade. If I don't make careless mistakes, this is realistic (clearly a stretch, but in range.)

Looks like most people are either aiming for first place or top 100. How ironic.

I put top 50, although my goal in my blog is top 57 :P  Meh, I think I can get that now considering that I make far fewer stupid mistakes (but still a lot) than last year. Also, hopefully the target won't be dominated by counting and probability again.

I think top 100 is the best thing I can hope for. I would already be thrilled just to make nationals.

Sadly, I cannot vote as I will not be participating in MATHCOUNTS this year. :(

Lol... I'm trying for top 50 (this year our top person had 90 something). Well, not failing at state... meh.

Everyone should be awesome like me and go for top 42.

For me, the tough part is making cd state and getting to 4th but if I make nats, I will probably get top 50 (you have to be good enough to get top 50 if you are going to get 4th in tx)

First, get on the school team(which will guarantee me to go to states).
Then make nats, which I would have to get top 4, since cd doesn't count.
Then get top 20, if I feel confident, maybe I may think about a possibility of reaching cd.

I am going to 8th grade, my final year.
I got 6th in Illinois as a seventh grader, and I wanna make top 20 at Nationals for 2009.
I wonder what I should do to help me study....

[quote="BOGTRO"]Next year will be my first  :lol:    I actually think I ahve a realistic shot at making Nats
CD, but I'm probably overconfident (no exp.)[/quote]

Meh.  You made ARML.


geometry
3D geometry
LaTeX
trigonometry
prism

1. Take a hexagonal pencil. Each side of the hexagon is $5 mm$ long. The pencil is $15 cm$ long,including the tip.

One end of the pencil is cut straight so that you can see the
hexagonal cross section with a neat circle at the center that is the lead.

The other end of the pencil is sharpened using a sharpener whose blade is at an angle of $10$ degrees to the axis of the pencil. Therefore, the pencil tip looks like a cone with an angle of $20$ degrees.

What is the volume of the pencil??

PS: how do you add degree symbol in latex?

2. Same as before, but now we have a triangular pencil. Each side is $7mm$ long. The pencil is $15 cm$ long up to the tip. And sharpened using the same sharpener.

What is the volume of this pencil?

I'm not sure...but here is an outline i think might work....

1) find the height of the hexagon...and therby diameter....
hexagon = 6 Equilateral triangles...i.e diameter is (3)^(1/2) x 5mm = 8.65 MM

2) finding the height of the conical portion would solve the problem... now for this look at it as a 2D figure.... it would be a sector of a circle with 20 degrees - the segment...the base of this triangle would be the diameter of the hexagon(not sure if that is what you call it!) from this u can find the radius if the sector and from that the height of the conical portion...then you can find the area....

Editing to Add image.... image edited from another post!

[hide]We have a hexagon-shaped "box" with side 5 and height $150-\frac{5}{\tan10^\circ}$.  So the area of the base is six equilateral triangles with side 5, or $\frac{75\sqrt{3}}{2}$.  So we have $\boxed{\left(\frac{75\sqrt{3}}{2}\right)\left(150-\frac{5}{\tan10^\circ}\right)}$ as the volume of the "box".
We also have a cone with base area of $25\pi$ and height $\frac{5}{\tan10^\circ}$.  So its volume is $\boxed{\frac13(25\pi)\left(\frac{5}{\tan10^\circ}\right)}$.
So the actual volume of the new pencil is the sum of the two boxed numbers.[/hide]

[quote="fizsle"]How did u get 5/tan 10???? can you explain this please?[/quote]
Well, the radius of the cone is $5$, so using basic triangle trig, we get $\tan10=\frac{5}{h}\Longrightarrow h=\frac{5}{\tan10}$.

[b]Of course, this answer is an approximation. If you sharpen a pencil, you'll see that the "yellow" part near the sharpened part is curved. It is not precisely a hexagonal prism attached to a cone.[/b]

Well yeah, that's true, LOTR.  I'm just giving an approximation.


Circles $ O_{1}$ and $ O_{2}$ are internally tangent to circle $ O$ at $ M$ and $ N,$ respectively. The internal tangents of circles $ O_{1}$ and $ O_{2}$ intersect circle $ O$ at $ X$ and $ Y,$ such that $ X,Y$ lie on the same side of the line connecting the centers of circles $ O_{1}$ and $ O_{2}.$ Prove that $ XY$ is parallel to an external common tangent of circles $ O_{1}$ and $ O_{2}.$

It was posted before: http://www.mathlinks.ro/viewtopic.php?t=15945


A pirate keeps $ \frac{2}{3}$ of his coins in a treasure chest and distributes the
rest equally among three crew members. If each crew member
receives 25 coins, how many coins did the pirate keep?

There are 3 crew members so $ 25*3\equal{}75$. $ 75$ is $ \frac{1}{3}$ of the pirate's treasure, so that means that the pirate kept $ 150$.


geometry
circumcircle
search
geometry proposed

For a triangle ABC, let I be the incenter. BI,CI meet CA,AB at D,E respectively.
line DE meets the circumcircle of triangle ABC at F,G.
Prove that the circumradius of triangle IFG is twice that of triangle ABC. (well known?)

Thank you for reading.

http://www.mathlinks.ro/Forum/viewtopic.php?search_id=400180822&t=86561

  darij


http://www.mathcounts.org/webarticles/articlefiles/127-December.pdf
That website was given to me by my teacher for practice,
i can do most of them but on Warm Up - 9 . I can't do # 7. Trapezoid Area somehow i'm not getting the answer
here's the answer [hide]800 radical 35[/hide]

[hide]Drop perpendiculars and then you have 2 rectangles and 2 congruent right triangles. In a rectangle, opposite sides are congruent so then the side is 2. Then the remaining lengths of that side are $\frac{6-2}{2}=2$. To find the height, use Pythag. $12^{2}-2^{2}=140$ then the height is $\sqrt{140}=2\sqrt{35}$.

The formula for the area is $A=h(b_{1}+b_{2})/2$.
Then it is $(2\sqrt{35)})(4)=8\sqrt{35}$.
Finally, scale it up to real size (multiply by $10^{2}$ for area) to get your answer of $\boxed{800\sqrt{35}}$[/hide]


modular arithmetic
number theory proposed
number theory

find all integer solutions to $ a^2\plus{}3b^2\equal{}c^2$ with $ a \equiv b \equiv 1 \pmod 2$

Use the 'factoring lemma': If $ ab\equal{}cd$ (all integers), than there are integers $ m,n,t,s$ such that: $ a\equal{}mn, b\equal{}ts, c\equal{}mt, d\equal{}ns$.
$ a^2\plus{}3b^2\equal{}c^2 \implies (c\minus{}a)(c\plus{}a)\equal{}3b b$.
Hence: $ c\plus{}a\equal{}mn, c\minus{}a\equal{}ts, 3b\equal{}mt, b\equal{}ns$. We get that $ 3ns\equal{}mt$, and by applying the lemma again: $ 3n\equal{}xy,s\equal{}zw,m\equal{}xz,t\equal{}yw$.
$ a\equal{}\frac{mn\minus{}ts}{2}\equal{}yz\frac{x^2\minus{}3w^2}{6}, b\equal{}\frac{yzxw}{3}, c\equal{}\frac{mn\plus{}ts}{2}\equal{}yz\frac{x^2\plus{}3w^2}{6}$
$ y,z,x,w$ must be odd. If we want to find the 'primitive' solutions ($ (a,b,c)\equal{}1$), we may assume that $ yz\equal{}1$ or $ yz\equal{}3$.
The solutions are: $ (\frac{x^2\minus{}3w^2}{6}, \frac{xw}{3}, \frac{x^2\plus{}3w^2}{6})$ and $ 3(\frac{x^2\minus{}3w^2}{6}, \frac{xw}{3}, \frac{x^2\plus{}3w^2}{6})$, with $ x \equiv w \equiv 1 \pmod 2$. We can put $ x\equal{}3t$ and get-
$ \lambda(\frac{3t^2\minus{}w^2}{2}, tw, \frac{3t^2\plus{}w^2}{2})$ and $ \lambda(\frac{x^2\minus{}3w^2}{2}, xw, \frac{x^2\plus{}3w^2}{2})$
$ \lambda \equiv x \equiv t \equiv w \equiv 1 \pmod 2$

It's easy to see that $ a$ is odd when $ t,w,x$ are odd (substitute $ x\equal{}2k\plus{}1$...)

See Method 2 in [url=http://www.artofproblemsolving.com/Forum/weblog_entry.php?t=197236]this post[/url].


Find $ det\left[\begin {array}{cccc} a & b & \cdots & b \\
c & \ddots & \ddots & b \\
\vdots & \ddots & \ddots & b \\
c & \cdots & c & a \end {array} \right] \equal{} ?$

[hide]Put $ D_n\equal{}det\left[\begin {array}{cccc} a & b & \cdots & b \\
c & \ddots & \ddots & b \\
\vdots & \ddots & \ddots & b \\
c & \cdots & c & a \end {array} \right]$
 Use : $ D_n\equal{}c(a\minus{}b)^{n\minus{}1}\plus{}(a\minus{}c)D_{n\minus{}1},D_1\equal{}a,D_2\equal{}a^2\minus{}bc$ [/hide]


geometry
conics
ellipse
geometry unsolved

1.What is the minimum number of   acute-angle triangles in wich we can divide an obtuse-angle triangle?
2.What is the maximum number of right-angle triangles with integer sides and the same area?
3.What is the maximum number of regions in wich n ellipses divide the plane?

ok vector. although I don't have enough time to spend without this anyway I gave some thought to (your) problems.

I think that for problem 1 the answer is that there no way to partition an obtuse angled triangle with acute angled triangle (I might be wrong, but it certainly seems so).

For the second problem I belive it is still an open problem, which falls beyond my math abilities for now.

Reference for it can be found in [url=http://www.amazon.com/exec/obidos/ASIN/0387942890/weisstein-20/102-0206035-8869769][this book][/url].

For your curiosity the quartet having smallest known common area is (111, 6160, 6161), (231, 2960, 2969), (518, 1320, 1418), (280, 2442, 2458), with area 341,880, so your number is greater or equal with 4.

For the last problem I suppose it's 2n+1-2, but however it is possible that this is an open problem too.

Well, in fact, for problem 2, the maximum is +oo, since we may prove the following result from Fermat which states that :
For each natural number n there exist n pythagorean triangles with different hypotenuses and the same area.

Pierre.

Well then it means that the problem is not open anymore :). Well I made a little more research and found out this (see attached file).

My source was :
W.Sierpinski, 'Pythagorean triangles', Published by the Graduate School of Science, Yeshiva university, New-York (1962).

Pierre.

For pb.1 :
A dissection in acute-angled triangles is possible for triangles as for any simple polygon.
It was Pb. E1406, A.M.M., vol.67 (1960), p.177-178.

I don't have it right here.

Pierre.

arggss :D I just found out that too. Nice picture. However I don't think it's that easy to prove that this is the minimal value :?

Actually, it is very easy.

Here is a sketch only.

1. Prove there is exactly 1 vertex inside of the triangle (use the fact that each such point accumulate at least 4 triangles).
2. Prove that there are two points insides sides of the triangle. Each such point accumulate at least 3 triangles, but some of them are shared. Anyway it gives 5+1+1 triangles.


[quote="canada"]If $ a$, $ b$ and $ c$ are positive reals then prove or disprove the following inequality:

$ \frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+a}}\ge\frac{1}{\sqrt{2}}(\sqrt{a}+\sqrt{b}+\sqrt{c})$[/quote]
See here:
http://www.artofproblemsolving.com/Forum/viewtopic.php?t=56071


which of the symphonies of $bethowen$ is the most interesting?

9 by far, though I'm certainly not as familiar with the rest as I should be to make a valid judgement.  I like 7 too.


Prove that sin squared x (1+ cot squared x)=1 this is what i did
sin squared x(csc squared x)=1
sin squared x              1
       1             sin squared x+1
sin squared x
sin squared x= 1

Prove that tan x (cot x + tan x)= sec squared x  this is what i did
tan x (cot x + tan x)= 1+tan squared x
tan x cot x+ tan squared x= 1+ tan squared x
1+ tan squared x= 1 + tan squared x
=1

Find the length of arc s, if r =10 and theta = pi/6 I got 5pi/3 for the answer

Find the area A of the sextor if theta = pi/4 and r =6 I got 9pi/2 for the answer

The proper way to type sin and cos in $\LaTeX$ is \sin and \cos.


Support geometry calculus derivative counting distinguishability function

Hi to all,

In this thread, I hope to discuss the many arguments relating to abortion. Before we proceed, I will define the abortion I intend to discuss, "medical termination of a pregnancy before the fetus has developed enough to survive outside the uterus."

I will begin by stating my thoughts. To begin, although I recognize that this issue is more moral than logical, I would like to point something out. Those who are pro life try to ban all abortion, making everyone happy except those who personally want an abortion (not pro choice). Those who are pro choice want to give everyone the option of having the abortion, which does not make anyone upset besides those who are bothered by the fact that there exist others in the world who are committing "murder". I feel that due to the serious and valid reasons that exist for people to have an abortion (mainly rape), for having the child could change one's life, pro choice is much more fair. No individual is [i]personally[/i] hurt by the fact that others are having abortions, or at least not nearly as much so as someone who is personally restricted from having an abortion.

Next, I feel that suicide, aside from the heavy financial impacts, is perfectly fine. A person should have the choice whether or not to end his or her own life, as long as he is not hurting anyone else physically. Hurting someone else emotionally is not necessarily up to he who commits suicide to decide, and so a generalization cannot be made in that regard. Just as one has the right to overeat (leading to health problems), take up dangerous sports, not sleep, refuse to see the doctor (potentially fatal), take medicine of any kind, or scratch themselves (killing skin cells), one should have the complete right to treat themselves in anyway they want. No one should forbid a person's choice about themselves, since no one has the right to control others. Now, I'm not against trying to advise someone against suicide, for suicide is usually an irrational and whimsical (exceptions being terminal illnesses) choice to make. As the popular saying goes, "a permanent solution to a temporary problem" suicide is usually a bad solution to something that has a difficult but better solution (confronting the problem). But after all, it is impossible to stop a determined person from commiting suicide, and by making it illegal, everything is just further and unncessarily complicated.

Now, with that said, I also believe that at any time, if the fetus is in the mother, the fetus is the mother, and not a separate being. This argument is obviously very difficult to support, for it depends on the drawing of a very fine line, the exact location of that line being impossible to determine. Let me give you another point on this plane on which the line must be drawn to give you an idea of what I'm talking about. When the fetus has no ability to survive outside the mother, the fetus should not be considered a separate being, for it is nothing more than any other living cell in the body. But even when the fetus can survive outside the mother, I feel that as long as it is in the mother, it should be the mother. If the value of something is determined by what it could potentially become, ("if the fetus can survive outside the mother, than it will become a person, and so killing a fetus is equivalent to killing a person"), then abstinence would be as bad as abortion. By allowing sperm cells and egg cells to die, instead of fulfilling their potential of human beings, you are killing thousands of lives. Also, the factors of likeliness or closeness are weak arguments. Fetuses are sperm and egg cells, if it is immoral to kill a fetus, it is immoral to kill a sperm or egg cell; just because a fetus is an "advanced" sperm and egg cell doesn't make it any more alive. Sperm and egg cells are also living.

The natural counter-argument to my equating fetuses with sperm and egg cells is, "just because the murder of life is inevitable doesn't mean we should kill more than we need to by having abortions". My response is that the happiness of a mother's (and hopefully, father's) lives is more important than the survival of the fetus. Besides, if a fetus is born into a family in which he or she is not wanted, he will not live a happy life anyways.

Of course the argument can be used outrageously to show that the murder of children by parents is also allowed, but of course this is another matter of drawing a line, and I would say that in a matter with so much opinion and potential generalization of any argument, the birth of a child is the clearest way to separate fetus from human.

I would like to know what the rest of you think.  :) .

I have several comments. First of all, I think the three options in your poll aren't broad enough to encompass the true range of possibility in people's opinion on this subject. People's opinion can be much more nuanced than either "pro-life", "pro-choice", or undecided. Some people are "pro-life" in some situations and "pro-choice" in some other situations. Some people are "pro-life or "pro-choice" personally, but have a different opinion legally.

(I also happen to really dislike the terms "pro-choice" and "pro-life." They're terms that were invented to win a propaganda war, but I don't think they faithfully represent the true heart of the disagreement.)

Now as for your argument I have a couple of bones to pick. First, with your definition of abortion: "medical termination of a pregnancy before the fetus has developed enough to survive outside the uterus." Who's to say when a fetus is old enough to survive outside of the uterus? What chance of survivability is sufficient? What if medical technology develops to the point where it becomes possible for [i]some[/i] very very young fetuses to survive outside of the womb - does that affect the definition of abortion for all fetuses? Does it change the legal definition?

Also, in your last sentence you say that birth is what makes someone human. How do you reconcile this with your definition of abortion?

Secondly, and more seriously, why should maximizing the happiness of the most people necessarily mean the best or the most moral solution to a problem? By a similar argument you could say that the abolition of slavery only directly affected the slave owners, since slaves weren't considered people (just like your definition doesn't consider fetuses people), and hence slavery should be allowed as the most fair option to the body of "free" people. The problem with this argument is that when it comes to questions of life and liberty - things which are [i]stringent[/i] human rights - even if something maximize the happiness of the majority of people, it can't be allowed if it overrides the stringent rights of other "human beings."

Human beings. That's the heart of the issue here. No matter who tries to distract you with other things, this is is the main question above all others in this debate. No one disagrees that mothers have a right over their bodies or that life is sacred - the question is whether unborn babies have the same [i]stringent right[/i] to life as the rest of humanity.

If you hinge the answer to that question on whether or not the baby is capable of surviving "on its own", then you run into some pertty peculiar questions. Like what about people in vegetative states? They can't survive on their own - do they have less stringent rights to life than the rest of us? If a baby had some kind of genetic disorder so that it needed its mother's milk, and only its mother's milk, to survive for the first year of its life - would it have less stringent rights to life than any other baby for that year? Why should independence be such a vital part of our definition of being human anyway?

Here's another question that I've always wondered about: why do we as people feel the most appalled when we hear about the death or murder of a child? And it seems like the younger the child is, the more agony we have over the news. Perhaps its because we realize that children are defenseless, and we have an innate sense of responsibility to protect them. Or perhaps its because, more than anything else, we value the [i]potentiality[/i] of life. When we hear of a child dying we're most hurt by the what-ifs of who they could have been and what they could have experienced in life. It seems that the potentiality of what we could do with our lives is what makes life most valuable!

Now imagine the irony of the fact that just a quirk of definition separates the most valuable stages of human life from the most invaluable stages of human life.

I feel that once the fertilized egg attaches to the uteran wall, its inevitable course towards life has begun and no one has the right to interfere with that course. From that moment, they have all the ingredients for life and a true and definitive future unless someone deliberately takes an action to stop it. This is different than the potentiality of an independent egg and a sperm - because without a definitive action to combine them they are not on a course towards life.

Simply put, I am pro-choice. I'm pro the choice of a fetus to decide what to do with its [i]own[/i] life someday. I believe it has a stringent right to life, and no matter how many people it makes uncomfortable or upset, its right to life should be protected.

Yes, whether you are pro choice or pro life isnt so 'black and white'.... I am in favor more of pro life generally.

If there is an illness that a mother has during pregnancy that has the slightest chance of killing her as a result of carrying the baby I think that the mother has the choice of abortion. This is, of course, excepting the chance that w/out sickness the mother could die in childbirth. Only with a sickness or something like that - like my sister has a friend who is i think 7 months pregnant now but she has gestational (is that the right word? i havent asked in a while) diabetes and other stuff as a result of carrying the baby and shes going all the way while in my opinion because the mother might die she could (and in that case, probably should) have an abortion.

If the baby is a result of a rape.. well... the baby isnt hers. She's only carrying in this case someone else's baby that she didnt consent to recieve. As a result.. unless she wants the baby - I think she should have an abortion.

On suicide - I believe that people should have the right to commit suicide, but ONLY in their homes or in a private area. Jumping off a bridge isnt right it draws attention and gets too many people involved in your own self-wallowing and mental perversion. Also, I think, suicide shouldnt be illegal... but thats because theres nothign they can do about it and theres a slight effect of reverse psychology which could work a little bit. If someone tells you you cant commit suicide - then you will want to more cuz 'hey hus gonna stop me' and theres the feeling of rebellion. People love to rebel, partly because it gets them so much attention and they can 'send their message' to the world or whatever they wanna do. If you tell someone they can commit suicide, yes, if they are disturbed enough they may commit suicide anyway, but when they know that its legal and nobody (besides relatives and friends) is gonna care that they commit suicide, then their suicide gets little meaning and they may be discouraged. Dunno, im not a psychologist. Another interesting thing - DESPERATION = A ROPE ENDS IT.

I do not think that people should have the option to have an abortion after they consented to um.. have the baby and its legal (legal age to consent and all). Like guass202 said - people mourn the loss of a child more, because of potentiallity. Why is a fetus any different? It has just as much potential as a 3 year old and yet some people are much more saddened by the loss of a 3 year old than of killing their own child. I really liked the reference to the abolition of slavery, and if you think about it, its really true. Slaves werent considered to be people, just as a lot of people today dont consider the fetus to be a person and so they feel just as well about killing a fetus as they do killing a slave! There really is no difference, looking at potentiallity a fetus has just as much if not more potentiallity and is just as alive as a slave. Lots of people dont realize it because slaves.. slaves could fight they were outspoken they MADE people realize that they were people. A fetus can't fight for its recognition, save for it's supporters who are anti-abortion. Its funny how in every generation, we fight the same battles. Same battles - different circumstances. Different people.

And, in my opinion, everyone has equal rights to life, vegetative, fetus, and healthy human alike. Who are we to judge on another innocent man's right to life? Someone who sits around all day in front of a tv and eats potato chips and does nothing else is just as productive as a vegetative person... no difference.. should we kill those people? No..

[quote="gauss202"]I think the three options in your poll aren't broad enough to encompass the true range of possibility in people's opinion on this subject. People's opinion can be much more nuanced than either "pro-life", "pro-choice", or undecided. Some people are "pro-life" in some situations and "pro-choice" in some other situations. Some people are "pro-life or "pro-choice" personally, but have a different opinion legally.

(I also happen to really dislike the terms "pro-choice" and "pro-life." They're terms that were invented to win a propaganda war, but I don't think they faithfully represent the true heart of the disagreement.)
[/quote]

I agree completely that the terms are more black and white than true life is, but I felt that it was the easiest way to convey the two sides, and by asking which "best describes", I would be asking "in most situations" and "personally". Being divided 50 - 50 is so highly unlikely that I didn't bother, but I do recognize that some people might feel both and be unwilling to put just one. If so, please choose "Undecided".

[quote="gauss202"]
First, with your definition of abortion: "medical termination of a pregnancy before the fetus has developed enough to survive outside the uterus." Who's to say when a fetus is old enough to survive outside of the uterus? What chance of survivability is sufficient? What if medical technology develops to the point where it becomes possible for [i]some[/i] very very young fetuses to survive outside of the womb - does that affect the definition of abortion for all fetuses? Does it change the legal definition?[/quote]

The restrictions some states put on abortion are centered around certain trimesters, due to the belief that once (I believe) the third trimester has been reached, abortion becomes illegal because of the chance the baby has to survive outside the mother. This is the "age" that I was referring to at which a fetus is old enough to survive outside of the uterus. As for the survivability, I doubt anyone would be able to answer that question to your satisfaction. Maybe a vague answer such as "significant" is what the state required. If medical technology develops to the point where it becomes possible for some very very young fetuses to survive, then I would say that does affect the definition of abortion. Maybe you could explain why you thought it might not.

[quote="gauss202"]
Also, in your last sentence you say that birth is what makes someone human. How do you reconcile this with your definition of abortion?[/quote]

Because I feel that birth is what makes someone human, I feel that abortion should be allowed at all times during the pregnancy.

[quote="gauss202"]
Secondly, and more seriously, why should maximizing the happiness of the most people necessarily mean the best or the most moral solution to a problem?[/quote]

Maximizing the happiness of the most people is obviously not always the best or most moral solution to a problem, but it is the fairest if you consider all lives equal. Being fair, or implementing "equality" is less subjective than being "moral". Fair means "same", moral means "good". Good and bad are relative to a perspective, and same is less so.

[quote="gauss202"]
By a similar argument you could say that the abolition of slavery only directly affected the slave owners, since slaves weren't considered people (just like your definition doesn't consider fetuses people), and hence slavery should be allowed as the most fair option to the body of "free" people. The problem with this argument is that when it comes to questions of life and liberty - things which are [i]stringent[/i] human rights - even if something maximize the happiness of the majority of people, it can't be allowed if it overrides the stringent rights of other "human beings."[/quote]

This "similar argument" which you refer to would only be similar if slaves weren't considered people. Since I do not believe this to be the case, your counter-argument doesn't really apply. What you are in essense saying is the generic argument that if group A are people and group B are not considered people, then the happiness of the majority of group A is less important than the human rights of group B. This generic argument is not false in all cases. I will prove by example, since I can set group B to be mosquitos, and group A to be humans.
Additionally, to show that my argument doesn't work, you have to assume that all fetuses are humans in the first place (since you are drawing the analogy to slaves, whom both of us agree are fully human). By assuming your point of view, you are not really proving what I've said to be wrong.

[quote="gauss202"]
Human beings. That's the heart of the issue here. No matter who tries to distract you with other things, this is is the main question above all others in this debate. No one disagrees that mothers have a right over their bodies or that life is sacred - the question is whether unborn babies have the same [i]stringent right[/i] to life as the rest of humanity.[/quote]

 :)  Agreed.

[quote="gauss202"]
If you hinge the answer to that question on whether or not the baby is capable of surviving "on its own", then you run into some pertty peculiar questions. Like what about people in vegetative states? They can't survive on their own - do they have less stringent rights to life than the rest of us? If a baby had some kind of genetic disorder so that it needed its mother's milk, and only its mother's milk, to survive for the first year of its life - would it have less stringent rights to life than any other baby for that year? Why should independence be such a vital part of our definition of being human anyway?[/quote]

I did not actually hinge the answer upon whether or not the baby is capable is surviving "on its own". First off, I felt that abortion should be an option to mothers in all stages of pregnancy - making the "survival on its own" irrelevant in the first place. I answered your argument earlier on just for clarification, not to neccessarily help my own. But to your point, I would agree with you. Everyone needs something to survive "on its own", and everyone is on a slightly different level of needs.

[quote="gauss202"]
Here's another question that I've always wondered about: why do we as people feel the most appalled when we hear about the death or murder of a child? And it seems like the younger the child is, the more agony we have over the news. Perhaps its because we realize that children are defenseless, and we have an innate sense of responsibility to protect them. Or perhaps its because, more than anything else, we value the [i]potentiality[/i] of life. When we hear of a child dying we're most hurt by the what-ifs of who they could have been and what they could have experienced in life. It seems that the potentiality of what we could do with our lives is what makes life most valuable![/quote]

I think the reason is that children have had the least time in the world to experience life. Others have enjoyed it for a longer period of time. And I guess we feel that equality is important - that everyone should have equal opportunity to enjoy and reap the benefits of life.

[quote="gauss202"]
Now imagine the irony of the fact that just a quirk of definition separates the most valuable stages of human life from the most invaluable stages of human life.

I feel that once the fertilized egg attaches to the uteran wall, its inevitable course towards life has begun and no one has the right to interfere with that course. From that moment, they have all the ingredients for life and a true and definitive future unless someone deliberately takes an action to stop it. This is different than the potentiality of an independent egg and a sperm - because without a definitive action to combine them they are not on a course towards life.[/quote]

What about deliberately and going out of one's way to protect oneself during sex. Is that deliberate action to stop something that will happen? Also, sperm cells are [i]alive[/i]. By not allowing them to enter the egg, you are killing them, for they will eventually die.

Thanks for comments!  :lol:

[quote="ToBe"]

I do not think that people should have the option to have an abortion after they consented to um.. have the baby and its legal (legal age to consent and all). Like guass202 said - people mourn the loss of a child more, because of potentiallity. Why is a fetus any different? It has just as much potential as a 3 year old and yet some people are much more saddened by the loss of a 3 year old than of killing their own child. I really liked the reference to the abolition of slavery, and if you think about it, its really true. Slaves werent considered to be people, just as a lot of people today dont consider the fetus to be a person and so they feel just as well about killing a fetus as they do killing a slave! There really is no difference, looking at potentiallity a fetus has just as much if not more potentiallity and is just as alive as a slave. Lots of people dont realize it because slaves.. slaves could fight they were outspoken they MADE people realize that they were people. A fetus can't fight for its recognition, save for it's supporters who are anti-abortion. Its funny how in every generation, we fight the same battles. Same battles - different circumstances. Different people.[/quote] Read my comments earlier about the argument regarding slavery. People who have consented to become pregnant rarely choose to have an abortion unless for example the mother becomes sick, in which case, as you said earlier, you would be for allowing them to have an abortion. If someone enters your house armed, it is perfectly legal to kill them, given that it is obvious enough that the intention was to hurt you. I feel that this law should be consistent with that of abortion. If the child will kill you or negatively impact your life, and it is quite certain, that it should be your choice whether or not to kill them. Otherwise, it's very much like saying that you may not kill someone who will hurt or kill you, due to the value of that person's life. In the analogy, house is analogous to body, and I feel that you should have much more control of your body than your house, legally and otherwise.

[quote="ToBe"]
And, in my opinion, everyone has equal rights to life, vegetative, fetus, and healthy human alike. Who are we to judge on another innocent man's right to life? Someone who sits around all day in front of a tv and eats potato chips and does nothing else is just as productive as a vegetative person... no difference.. should we kill those people? No..[/quote] No, these people you refer to should not be killed - unless they want to be, in which case we come to the suicide argument which you have already agreed to. But "agreement" includes writing something that says, "if I ever become vegetative, please end my life," and the like.

Thank you also, for commenting.  :lol:

[quote]If someone enters your house armed, it is perfectly legal to kill them, given that it is obvious enough that the intention was to hurt you. I feel that this law should be consistent with that of abortion. If the child will kill you or negatively impact your life, and it is quite certain, that it should be your choice whether or not to kill them. Otherwise, it's very much like saying that you may not kill someone who will hurt or kill you, due to the value of that person's life. In the analogy, house is analogous to body, and I feel that you should have much more control of your body than your house, legally and otherwise.[/quote]

I agree with you having the right to kill the intruder in your home, but it is not the unborn baby's intention to injure or kill you.  He or his future should not be punished for your own safety.  The law is supposed to be as fair as possible.  This is totally not fair for the unborn.

I think that all fetuses are perfectly human from the moment they are conceived.  Here's a very logical argument (this is a math forum):

How do they differ from any average human?  Four important differences are size, level of development, environment, and dependency.
Size - People of different heights are no more human than each other.
Level of development - An 8 year old is no more human than a 2 year old.
Environment - It doesn't matter where you are, you're still human.
Dependency - Like what you've already said, someone on a pacemaker or another medical device they need is no more human than someone without these.
If there are any other relevant differences, please mention them.

If you had asked me the same question a year ago, I would have given a qualified pro-choice statement along the lines of a fetus being unquestionably dependent on the mother, who should be entitled to her bodily freedoms. Further, if the conception was unintentional, then forcing the mother to cope with unintended results is at best undesirable. I also found it difficult to reconcile my own life as equivalent to that of a fetus.

However, more recently I have become more receptive to the pro-life stance. The validity of the argument from the perspective of the fetus seems a total product of one's own dogmas, so I have thought about the matter in another fashion. In particular, as I believe that the termination of a pregnancy is clearly an undesirable thing of itself, I cannot understand why [i]adoption[/i] does not present itself as the superior alternative in most circumstances. Indeed, there are many willing, would-be parents who desperately want a family of their own. The only potential conflict here would be between the family disowning their child and the adopting family. But, based on the hypothesis that both parties willingly agreed to the adoption, the only real potential conflict I see here is between the adopted child and the  parents (adopting and biological.) As for numbers in this regard, I do not precisely know what percentage of parents/adopted children develop mutual animosity stemming from this kind of transaction. Certainly, though, it is less than 100%.

That said, I do think it would be wrong to ban all adoptions, especially when there exist uncontrollable circumstances - medical complications, rape. etc.  The life entitlement of the mother, in my opinion, meets or equals that of the fetus, so if one or more lives are threatened, it makes sense that the mother should be able to choose. In an instance of rape, I think it would be undue burden for the victim to have to cope with the consequences of an unwanted pregnancy. (Financially, for example: the cost of maternity leave? What if the mother is single?)

Hence, my own views (currently) are pro-life excepting in the presence of extenuating circumstances beyond the mother's control.

By the way, a legal monkey wrench - With regard to outlawing abortion, it seems this would have to encompass any deliberate miscarriage, or by extension any intentional actions that make such a result inevitable. (Drugs, alcohol, malnutrition, etc.) How exactly does one accomplish this?

there's another discussion on this topic here: [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?t=32210]www.artofproblemsolving.com/Forum/viewtopic.php?t=32210[/url] which i previously commented on, so my views are there.

[quote="Brent Woodhouse"]
I agree with you having the right to kill the intruder in your home, but it is not the unborn baby's intention to injure or kill you.  He or his future should not be punished for your own safety.  The law is supposed to be as fair as possible.  This is totally not fair for the unborn.
[/quote]

Intentions are irrelevant, a drunkard who comes into your house armed is dangerous, like the clear thinking murderer.

[quote="Brent Woodhouse"]
I think that all fetuses are perfectly human from the moment they are conceived.  Here's a very logical argument (this is a math forum):

How do they differ from any average human?  Four important differences are size, level of development, environment, and dependency.
Size - People of different heights are no more human than each other.
Level of development - An 8 year old is no more human than a 2 year old.
Environment - It doesn't matter where you are, you're still human.
Dependency - Like what you've already said, someone on a pacemaker or another medical device they need is no more human than someone without these.
If there are any other relevant differences, please mention them.[/quote]

I would say that the fact that the fetus is in the womb of a human is a significance, making them part of a human, not fully human.

Also, many of your reasons are circular. Humans of different heights are no more human than each other, well if fetuses aren't humans, that reasoning doesn't apply.

It doesn't matter where you (a human) are, you are still human. Certainly, by definition, this is the case... etc.

[quote="MCrawford"]I don't plan to write a full analysis of my views here, but I rarely see an opinion on this topic that isn't based on religion, which is to say that opinions seem based on personally chosen axioms.  In that regard, I've never felt like laws on abortion could possibly based in anything other than religion -- which seems to be counter to a culture of seperation of church and state.

I also think that the debate on abortion is centered around moralism rather than practicality.  Teenage pregnancy rates and pregnancies out of long term relationships are far higher in areas with little or no sex education.  Teenage pregnancy and abortion rates in areas with the most comprehensive educational resources are far far lower than everywhere else.  We could solve the most dramatic portion of social and rights problems with an education-based approach to these problems.  If we prevent the need for abortion, we put to rest the religious and political dichotomization of the debate which I believe is fundamental to many other problems in our society.[/quote]

Fully agreed.

[quote="Mildorf"]
However, more recently I have become more receptive to the pro-life stance. The validity of the argument from the perspective of the fetus seems a total product of one's own dogmas, so I have thought about the matter in another fashion. In particular, as I believe that the termination of a pregnancy is clearly an undesirable thing of itself, I cannot understand why [i]adoption[/i] does not present itself as the superior alternative in most circumstances. Indeed, there are many willing, would-be parents who desperately want a family of their own. The only potential conflict here would be between the family disowning their child and the adopting family. But, based on the hypothesis that both parties willingly agreed to the adoption, the only real potential conflict I see here is between the adopted child and the  parents (adopting and biological.) As for numbers in this regard, I do not precisely know what percentage of parents/adopted children develop mutual animosity stemming from this kind of transaction. Certainly, though, it is less than 100%.
[/quote]

Very good point, I guess the only thing I would have to caution is that the children who grow up without the proper guidance will be more likely to grow up to lead a wayward and destructive life.

[quote="Mildorf"]
That said, I do think it would be wrong to ban all adoptions, especially when there exist uncontrollable circumstances - medical complications, rape. etc.  The life entitlement of the mother, in my opinion, meets or equals that of the fetus, so if one or more lives are threatened, it makes sense that the mother should be able to choose. In an instance of rape, I think it would be undue burden for the victim to have to cope with the consequences of an unwanted pregnancy. (Financially, for example: the cost of maternity leave? What if the mother is single?)

Hence, my own views (currently) are pro-life excepting in the presence of extenuating circumstances beyond the mother's control.

By the way, a legal monkey wrench - With regard to outlawing abortion, it seems this would have to encompass any deliberate miscarriage, or by extension any intentional actions that make such a result inevitable. (Drugs, alcohol, malnutrition, etc.) How exactly does one accomplish this?[/quote]

 :) True, ok.

[quote="gauss202"][quote="ToBe"]
If the baby is a result of a rape.. well... the baby isnt hers. She's only carrying in this case someone else's baby that she didnt consent to recieve. As a result.. unless she wants the baby - I think she should have an abortion.[/quote]

But if your argument for being against abortion is based on the belief that fetuses have a stringent right to life, then why should it matter whether the mother wants the baby in this case or not? It wasn't the fetuses choice for how it was conceived, so why should it be killed? All human life should be regarded equally, no matter how it was conceived don't you think?[/quote]

Well maybe not should. I probably should have rephrased that . - But the baby is a product of a crime - if a criminal forced you to let him hide in your house, what would you do? I know the baby isnt the criminal but it works on the same "intrusion".

All human life should be regarded equally, that is true. If someone commits a crime however (the baby didn't, I agree) he doesn't deserve to be regarded w/ equal rights, because he denies equal rights to others. What the mother should do with the baby then, I guess is choice. I would allow a rape victim to have an abortion then at the point where if the baby was taken out of her early and would be capable of surviving (even w/ an incubator) - so the baby would live, but she would not have to bear with it longer than she had to. And as always, she has the option of putting the baby up for adoption.

[quote="ToBe"][quote="gauss202"]
All human life should be regarded equally, that is true.[/quote][/quote]

From what axiom does this statement stem?

Does it also include the idea that cockroaches should be regarded equally?

If only humans, why?

I think here lies the problem with designating rights on some super-moral principle -- we get stuck trying to produce an axiom that justifies our personal feelings.

How about this approach to rights:  Rights are privileges afforded by the government of our society.  They are not as moralists would like to suggest [i]absolutes[/i] by the nature of the reality [i]our group[/i] believes in.  If they were, we would have them in a vacuum and so government would not be necessary to protect them.  Rights are, so far as they are based on something contractual, the privileges we would agree to before inhabiting this world.  In other words -- the rights that serve a utilitarian purpose are those that we would bargain our lives over.  If we did not have them, we would not serve the government.

Other rights that require subjugation to a religious standard are by definition theocratic.  I understand that by that standard most Americans believe in theocracy, but at the very least this belief in theocratic standards of rights should be made plain.  Otherwise, all such debates get stuck in a tangle of words designed to prey on emotions -- or confuse -- or erect a moral highground from which to attack the opposition.

On another note: Is Osama bin Laden's life equal to yours?  If not, how is it that we cobble together a statement as to where the values of human lives diverge?

[quote="banblink182"]"medical termination of a pregnancy before the fetus has developed enough to survive outside the uterus."[/quote]
60 year ago that would give the right to make abortion at the eight month. But today, they can live outside the uterus (with help from machines).

So that brings us to a question;
Is it the technology that's in charge, saying who's old enough to survive or is it the fetus (with the right equipment)?

Let's say I'm in a cave at 8000 m height with a two year old child. Obviously, this child will not be able to get down alive. So by applying the laws of abortion I'll have the right to eat the child so that I get enough strength to climb down.

Just a thought of mine.

I hate these topics... No light at the end of the tunnel.

Moreover, this is a very very touchy issue. It's one that can stir personal emotions... Not a very good debate topic.

I mean... debate... To uphold democracy, there must a separation of church and state... or something.

I've debated abortion many times in both debate club and as an issue brought up during other discussions, and all that resulted:

No consensus.

Tension.

Frustration.

Anger.

Aggression.

The dark side...

Not that last one... but close enough. I don't think this is something that should be discussed on this type of forum at all... leave it to lawmakers and politicians.  :|

[quote="1+1"]
Let's say I'm in a cave at 8000 m height with a two year old child. Obviously, this child will not be able to get down alive. So by applying the laws of abortion I'll have the right to eat the child so that I get enough strength to climb down.[/quote]

No, that is completely dissimilar to abortion. First, the child is not yours, secondly, it's a child, not a fetus.

[quote="white_horse_king88"]Not that last one... but close enough. I don't think this is something that should be discussed on this type of forum at all... leave it to lawmakers and politicians. [/quote]

I disagree. For the most part, I don't really trust politicians (except in specific cases.) I'd rather discuss the topic with well-informed and well-meaning AoPSers.

[quote="Spoon"][quote="1+1"]Let's say I'm in a cave at 8000 m height with a two year old child. Obviously, this child will not be able to get down alive. So by applying the laws of abortion I'll have the right to eat the child so that I get enough strength to climb down.
[/quote]

1) What does a cave have to do with height?
2) Neither of you would be able to breathe at 8000m (that's almost as high as Everest)
3) If there was some chance you COULD get down, why would you be able to get down and not the child... I don't see anything obvious about that.
Personal Edit: 4) Clearly, if you got this child up there, with no way to get down, and the child dies, then you are a stupid person, who merits professional help.
5) It's a 2 year old child, not a fetus.
6) WHAT ON EARTH does this have to do with abortion??[/quote]

The height assures that you must make a decision:

Eat the child and live, or you both die.

Scientists venture into caves which are 8000 feet high, but perhaps underground??? Ice caves.

The child is too young - he cannot climb that far down.

BLAHBLAH... Yeah. That rebuttal was rather meaningless...

Anyways, I'll have to disagree with the original metaphor. The fetus doesn't have to die in the case of abortion, but in the cave case, the child must die. He can't get down alive no matter what. So, that metaphor does not illustrate the case at hand. So, I agree with SPOON's point number 6. What does this have to do with abortion? (The topic at hand.)

8000 feet != 8000 m (which is what you originally posted), anyway...

So basically, the generalize your point:

Condition 1: The child dies and you die
Condition 2: The child dies

And then you want to know which would be chosen? Well... clearly I think the logical answer would be Condition 2. If the child dies regardless, then it seems irrational to sacrifice the other life just because. This relates to the issue of whether the mother is in danger when having the child. If it likely that the mother would die (and in the turn the baby would die also), then many more people support abortion. i.e. Some people who are against abortion in general think it's ok if the mother will likely die. So in general, I think Condition 2 is the accepted answer. I would support condition 2 under any circumstances, because that is the rational answer.

[quote="I"]
If those who are raped should be allowed to have abortions, so should anyone else, because the life of the fetus that was the result of the rape is as important as the life of any other fetus. If a raped woman can choose to kill her fetus, it would be unfair to not allow others the same choice (even if the fetus is a separate human). Again, the original intentions don't matter because we are strictly talking about whether or not to kill a fetus which itself does not have intentions. Any given fetus has equal rights to any other fetus, to those who take a pro-life stance the killing of any fetus should be equally bad, and so either rape victims should not be allowed to have abortions or all people should be given the choice (not the recommendation) to have abortions.[/quote]

This applies again - the life of a fetus who is "going to die" no matter what is not any less important than a fetus who has a chance of survival - since there is always the chance that "new technology" will be developed.

[quote="MCrawford"]Yes, quality of life is where I was hoping this "natural" discussion would lead -- that we individually judge value subjectively.  This is why I wanted to call into question any so-called objective standards for morality.[/quote]

This is what makes philosophy so difficult! Rhetorical argumentation is rarely free of subjectivity, even when it is used in the pursuit of seemingly logical ends. This is also why David Hume and his "empiricist wrecking ball" (the demand that every argument be based on human experience as perceived through the senses) were so effective when cast against allegedly "logical" and "universal" principles. Even a slight way around this was not offered until Immanuel Kant pointed out that such argumentation regarding life is a function of how we perceive it; hence, life itself is naturally subject to some degree of human interpretation. Though it is progress, this notion is hardly grounds for constructing assessments of serious questions such as what constitutes "morals."

As a rigor-conscious person, that whole concept is a bit unnerving to me.

In various developed countries abortion is routine and in some places, frequent, without having led to the demise of those societies.  (Gender-selection via abortion is a different story.)  In those Christian countries where abortion is an issue, including the US, probably most people in the anti-abortion crowd know one or more women who have had abortions and are not apparently genocidal criminals (they may not be aware that those women have had abortions, as the procedure is more often performed than publicized).   Actually, I suppose that some abortion opponents have [i]mothers[/i] who have had abortions without notifying the unaborted children of the fact.

But the reason I post this is to ask a question of the anti-abortionists who allow a rape/incest exemption.   Is there any reason to allow abortion in case of rape or incest that does not also justify abortion in many other (or all) cases?

[quote="fleeting_guest"]... the reason I post this is to ask a question of the anti-abortionists who allow a rape/incest exemption.   Is there any reason to allow abortion in case of rape or incest that does not also justify abortion in many other (or all) cases?[/quote]

For me, the reasons I would not oppose abortion in those instances are as follows:

1) The rape reflects a nonconcentual incident that should not force the mother to spend three-quarters of a year compromised both mentally and physically. The process does provide some progress towards closure. The key here is nonconcentual, and I generalize accordingly.

How can I phrase this daintily... I do not buy arguments that a person was too passionate to know what they were doing or even have a choice not to partake of the activity. Seriously, almost all of the time you have a choice.

2) I simply believe that incest is wrong, and that it even has its biological disadvantages. Many religions specifically forbid it as well. Although I prefer to offer reasons that are not dogmatic, I find my opinion on this one just reflects my core beliefs. Hence, I am not really able to generalize.

It seems to follow from definition that family trees should contain no cycles.

[quote="Mildorf"][quote]... a question [for] anti-abortionists who allow a rape/incest exemption.   Is there any reason to allow abortion in case of rape or incest that does not also justify abortion in many other (or all) cases?[/quote]

1) The rape reflects a nonconcentual incident that should not force the mother to spend three-quarters of a year compromised both mentally and physically. The process does provide some progress towards closure. The key here is nonconcentual, and I generalize accordingly. [/quote]

Without getting yet into whether consensuality is a supportible distinction here, it seems that various cases of malfunctioning contraception are "nonconsensual" impregnation as far as the woman is concerned, and in not all such cases can one escape that point by noting that she was merely taking her chances with imperfect technology.

[quote]2) I simply believe that incest is wrong, and that it even has its biological disadvantages. Many religions specifically forbid it as well. Although I prefer to offer reasons that are not dogmatic, I find my opinion on this one just reflects my core beliefs. Hence, I am not really able to generalize. [/quote]

Many pregnancies can occur from actions that are "wrong".  As for biological disadvantages, I see no principled arguments in that direction that do not also lead to justification of the proverbial leaving-of-defective-child-in-the-snow.

re: cycles in family trees, until recently almost all family trees had many short cycles (village life and all that).

fleeting_guest must be more popular than I am or something... or maybe he's just better than me at phrasing the same point  :)

Now that we finally get some answers to the question I've been "bringing across", I have this comment:

All of Mildorf's comments related to the mother, which is perfectly fine. But this shows that the reason many anti-abortionists feel that that rape victims should have the right to an abortion is due to the rights of the [i]mother[/i], not the [i]fetus[/i] - meaning that the [i]mother's[/i] rights outweigh the rights of the fetus when the mother has been mistreated. This supports the "pro-choice" argument that the life and happiness of the mother is more important than that of the fetus.

So what happened to the "pro-life" argument that all fetuses are human? Supposedly, isn't this unfortunate rape "fetus" as human as any other?

abortion needs to be legal in the rare case that a woman was raped and is now pregnant and she doesn't want the child. also, what if a fetus is certain to be born with a dangerous disease? is it worth putting the child through all that suffering when the child is born?


1) A disc of mass 10gmm is kept floating horizontally by throwing 10 marbles/sec against it from below.If the mass of each is 5gm,calculate the velocity with which the marbles are striking the disc.Assume that the marbles are striking the disc normally and rebound downward with the same speed.

2)A shot of mass m penetrates a thickness of a fixed plate of mass M.Prove that if M is free to move,the thickness penetrated is $\frac{Ms}{M+m}$

[quote="bus"]2.) Whoa? :huh: And what is $s$ ?[/quote]

Sorry,"s" is the trhickness it penetrates :D

And one of the [b]administrators[/b] said this site is all about helping people as well and he did mention homework questions.....I normally don't post such questions,just my first time :D

Well, going by your 'level', i suppose you dont need 'help' with these questions!  :D

Question 1 is easy - force created by the marbles is

$F=\frac{\Delta p}{\Delta t}$

so if you have 10 marbles / sec and each marble changes its momentum by $m\Delta v$ then

$F=10\text{/s}\times0.005\text{ kg}\times\Delta v$

and this must be equal to the gravitational force $Mg$:

$10\text{/s}\times0.005\text{ kg}\times\Delta v=0.010\text{ kg}\times9.81\text{ m/s}^2$

$\Delta v=\frac{9.81}{5}\text{ m/s}$.

Delta v is the change of velocity of each marble, which is twice its speed. (The velocity changes from v to -v.) The marbles strike the disc with velocity 0.981 m/s. (I hope I didn't make any mistakes.)

Unfortunately, I don't quite understand problem #2 :(. Do you say that the plate is $s$ thick? Or that the bullet gets $s$ deep into the plate? What is $\frac{Ms}{M+m}$ supposed to be again?


Here is a simple and nice problem involving a diophantine equation:

Find integers $ m$, $ n$, $ x$ and $ y$ such that the following equation holds:

$ 2x^{m}\minus{}7y^{n}\equal{}1$.

I don't know how to do it, but:
$ m\equal{}2$, $ n\equal{}1$, $ x\equal{}5$, $ y\equal{}7$
so
$ 2(5)^2\minus{}7(7)^1\equal{}50\minus{}49\equal{}1$


D,E,F are on BC,CA,AB resp and form a Miquel's triangle of P and A'B'C' is the pedal triangle of P.
2S(DEF)=2S(PDE)+2S(PEF)+2S(PFD)=PD.PE.sinC+PE.PF.sin A+PF.PD.sinB\geq PA'.PB'.sinC+PB'.PC'.sin A+PC'.PA'.sinB=2S(A'B'C')


calculus
videos
trigonometry
AMC
AMC 12
AIME

I have to do a survey on how much time it takes people to do their homework a day for my speech. So any help would be appreciated. Oh and this is for high school and middle school students only. Thx.

2.5-3 on a good night. 3-4.5 average. 5+ when there's a big test or a writing assignment.

Hmm...depends on whether you count the extra math I do as homework.  I usually don't have hw for every class every day.  But the usual amount of time when I do get hw for a class is as follow:

Calculus-don't do hw at home (even if its assigned-we have plenty of class time)

Stats-independent studying/taking the tests that the normal class takes so I study for about 2 hrs a week or about 20 min. a day

History-lots of reading, 1 hr a day when we have hw

Physics-don't do much of the assigned schoolwork at home but I do watch a few online video lectures; about 20 min a day on average

English-about 30 min a day when we have it

So, yeah, I would say that I do a little more than 2 hrs a day on average.

I usually do any homework I have during orchestra library, or right before the class that's due starts, so I usually don't do it at home.

According to my teachers, I'm supposed to spend an average of one hour a day on my homework. It takes me less time, though I spend around 45 mins practicing my instrument, if that counts as homework.

hmm, considering that I am "obesesive about hw as my friends call me it takes me about:
Math:20 minutes
English:10 minutes
History:30-45 minutes
Science:30 minutes
music (theory and practice):1 hour
so about 3 hours and 45 minutes a day but my friends call me obesive since I copy all my notes and stuff into either a spiral or type them onto a disk besides the stuff I turn in so I doubt I'm normal for the homwork I get assined.

I do my homework on the bus and at school. My parents think I have almost no homework :P .

[quote="eryaman"]About the same as yif man for me.[/quote]

about the same here too

[quote="BHorseMath"]Math:20 minutes
English:10 minutes
History:30-45 minutes
Science:30 minutes
music (theory and practice):1 hour[/quote]

Just wondering, if you're on this site quite a bit, wouldn't it take you less time to do your math homework than your english homework?

my english homework usually consists of looking up a word or making sure my notes are in order for the next day.  Math though is about 30-40 problems, and while they're easy to me my teacher requires all work shown and stuff so it takes me a fair amount of time to just do the work.  My english class isn't much at all we are reading a book and doing active discusions and I have already finished the book and so I just take notes in class and look up any words he tells us to.

Depends, I suppose. For example it took me around 2 hours just to memorize the main 10 main steps of muscle contraction and relaxation. For English essays I spend quite a lot of time on since my "epiphany;" i.e., I don't try and finish them within an hour. Fortunately, or unfortunately I suppose, I never have homework in my Trig or Pre-Calc classes, or if I do, it's easily done in roughly 15-30 minutes. My chemistry teacher is uninspired which makes me very angry as we have learned basically nothing in the class over the period of a semester, and thus, never have homework.

I put more than two hours of homework, but closely examining my workload, on average I'd have to say I do less than 2 hours.

I do nearly all of my homework at school also.

Thanks to those who voted   :) ! wrote up my results yesterday.

homework? what homework? *guilty grin*... well, when i get around to doing my calculus, it takes about half an hour. english varies between nothing and about an hour. we only have homework in AP bio on lab days... ditto for programming... theatre just requires some after school time in the scene shop... hm... well... i am SUPPOSED to read an hour a day for history... am i the only procrastinator on this site? surely NOT!

so that makes, oh, less than two hours, i suppose. but it can vary wildly, depending on what is going on.

Wow, I envy most of you people! I do at least 4 hours of homework every day . . . including Saturdays! Aaaahhh plus field & track (you can tell I'm a thrower lol) started last week so now I get home from school, do homework for an hour, go chuck spears for 2 hours, come home, do 3 or 4 more hours of homework and then lay in my bed, trying to do IMO problems, until I fall asleep (I actually solved one once, in about a week). Somehow I get to eat in there, too. Not to mention that my day starts at 6:00 AM (my first class is mathematics, obviously; otherwise there would be no way I'd get up at such an ungodly hour!). Ah, the joys of taking college classes in high school . . . :P

Hey me too, but so early?????

It seems so strange to have a college start that early.

[quote="aznness"]I absolutely hate english. I have B's in every single essay i wrote..[/quote]

Maybe you just have to work on English.

[quote="aznness"]I absolutely hate english. I have B's in every single essay i wrote..[/quote]
i know the feeling. i try hard at english and all then i'm getting B's. it'll be very hard for me to salvage an A- for the year.
and then guidence wanted to sign me up for a year of elective english. what were they thinking. but i got that changed to physics...shew

[quote="filletwho"]I like English. It's fun ( not as fun as math though ). I just took a really hard Literature test, but essays will raise my grade. That and lots of e.c.[/quote]
see, you get extra credit. i might like english more if i got extra credit and didn't have to worry about my grade all the time. this year, enlish is the only class i might get a B in and then not get the Board of Trustees award from the school district for getting all A's.

Well, in the good old days of easy middle school, I received As on most of essays, and did not pay too much attention to my writing.

Nowadays, I am in honors english at high school, and I struggle with writing essays. I have discovered that I cannot write a good english essay in 45 mins (in class writing), and thus my grade has sufferred. That, and the fact that now English is a lot of work, because we read 15 books a year and need to write essays on each one. With more and more standardized tests emphasizing timed writing, I feel that I should develop this necessary skill, and what better way to do this than to be motivated by mediocre grades?

My teachers have this stand on extra credit: Those that have the time to do it do not need it, because they have finished all their work and done it well. Those that do need extra credit do not have the time to do so, because they are not finished with their work.

Interesting.
Whenever I do timed writing, my handwriting goes out of control, and I sometimes scratch out entire paragraphs, or use arrows to move sentences around.... I have a funny feeling my teacher just gives me A's just b/c he can't read what I wrote. Anyway, half the people in my Honors class have A's (not counting the extra weight), so I suspect rampant grade inflation (lucky for me :))

Hey World History is like extremely hard for me this year. Consider my AMC12 score by doing # correct answer/25 (19/25= 76%), that's higher than the highest history test score I had this entire year! Wow my grade on a history paper got pretty close to my AIME percentage (9/15 = 60%, and I don't think I did good on AIME).

I did get A's last year though...

Wow. I really do envy you guys. I have 6 hours of homework per day on the average weekday, and 12 hours of homework per day on the average weekend day. This is not counting activities. My school delights in giving lots of work that combines being highly time-consuming with not actually teaching anything. An example: Outlining the book for APUSH. Worthless. I do better on tests where I just read the book and sleep well than on tests where I outline.

less than one hour a day... in the weekends, i normally don't do a thing until sunday-evening  :D
and i do all of my homework at night. i just can't seem to work during the day  :?  a good thing i only need 4-5 hours of sleep a day, so i have quite some time to do all sorts of other stuff  :)

[quote="filletwho"][quote="mcalderbank"][quote="sky9073"]Interesting.
 I have a funny feeling my teacher just gives me A's just b/c he can't read what I wrote. Anyway, half the people in my Honors class have A's (not counting the extra weight), so I suspect rampant grade inflation (lucky for me :))[/quote]

Wow, that makes zero sense, why would a teacher actively try and give you a better grade?  :?
Oh, and another thing, I hate how in my school, the system of grading is such that getting a 70/100 is a failure. This is suppossed to stop grade inflation, but it just squeezes the spectrum together.[/quote]

In my highly competitive school, 90 is  considered failing, by the students if not by the administration.[/quote]

wow, thats high. my old school had high standards but failing was at a 60...


[quote="dima ukraine"]$ x_1,x_2...x_n\ge0$, n>3, $ x_1 + x_2 + ... + x_n = 2$ Find min of

$ \frac {x_2}{1 + x^2_1} + \frac {x_3}{1 + x^2_2} + ... + \frac {x_1}{1 + x^2_n}$[/quote]
Here,we set two sequences, ${ \{x_{i + 1}(1 + x_i^2)}\}$ and $ \{\frac {x_{i + 1}}{{1 + x_i^2}}\}$.
Now, from Cauchy's Schwartz inequality we get,
\[ \left( \sum{x_{i + 1}(1 + x_i^2)} \right) \left( {\sum \frac {x_{i + 1}}{{1 + x_i^2}}\right ) \geq (\sum x_{i + 1})^2}
\\
\iff {\textbf{LHS}}\geq\frac {(\sum x_{i + 1})^2} {\sum (x_{i + 1} + x_i^3)} = \frac {4} {2 + \sum {x_i^3}}
\]
Here, equality iff, there exists some $ \lambda$ such that,
\[ \lambda = \frac {x_{i + 1}(1 + x_i^2)}{\frac {x_{i + 1}}{{1 + x_i^2}}} = (1 + x_i^2)^2 \\
\iff x_i = \sqrt {\sqrt {\lambda} - 1}
\]
i.e. all $ x_i$s are equal. So, we get, $ x_i = \frac {2}{n}$

Which gives, $ \min = \frac {4}{2 + \left (\frac {2}{n}\right )^3}$

[quote="can_hang2007"][quote="math_07"][quote="math_07"]$ \min \equal{} \frac {4}{2 \plus{} \left (\frac {2}{n}\right )^3}$[/quote]
I am not sure if $ n$ also matters. :maybe: But, we can also see that, $ n \to \infty \Longrightarrow \left(\frac {2}{n}\right )^3 \to 0$
So, the $ \min \to 2$[/quote]
The result must be $ \frac {3}{2}$. Try $ x_1 \equal{} x_2 \equal{} 1,x_3 \equal{} \cdots \equal{} x_n \equal{} 0$ ;)[/quote]
But, what's the mistake in the solution of math_07? And would you please post your solution?

[quote="Moonmathpi496"][/quote][quote="can_hang2007"][quote="math_07"][quote="math_07"]$ \min = \frac {4}{2 + \left (\frac {2}{n}\right )^3}$[/quote]
I am not sure if $ n$ also matters. :maybe: But, we can also see that, $ n \to \infty \Longrightarrow \left(\frac {2}{n}\right )^3 \to 0$
So, the $ \min \to 2$[/quote]
The result must be $ \frac {3}{2}$. Try $ x_1 = x_2 = 1,x_3 = \cdots = x_n = 0$ ;)[/quote][quote="Moonmathpi496"]
But, what's the mistake in the solution of math_07? And would you please post your solution?[/quote]

It's here

[quote="math_07"]Now, from Cauchy's Schwartz inequality we get,
\[ \left( \sum{x_{i + 1}(1 + x_i^2)} \right) \left( {\sum \frac {x_{i + 1}}{{1 + x_i^2}}\right ) \geq (\sum x_{i + 1})^2} \\
\iff {\textbf{LHS}}\geq\frac {(\sum x_{i + 1})^2} {\sum (x_{i + 1} + x_i^3)} = \frac {4} {2 + \sum {x_i^3}}
\]
[/quote]

It's must be
\[ LHS \ge \frac{{\left( {\sum {x_{i + 1} } } \right)^2 }}{{\sum {\left( {x_{i + 1}  + x_{i + 1} x_i^2 } \right)} }}
\]

[quote="Honey_S"]It's must be
\[ LHS \ge \frac {{\left( {\sum {x_{i \plus{} 1} } } \right)^2 }}{{\sum {\left( {x_{i \plus{} 1} \plus{} x_{i \plus{} 1} x_i^2 } \right)} }}
\]
[/quote]
OK, If we assume it to be a typo, it doesn't change anything. As, in the equality case all are equal.
So, What's the problem with it? :ninja:
Can anyone describe?

[quote="math_07"][quote="Moonmathpi496"][quote="Honey_S"]It's must be
\[ LHS \ge \frac {{\left( {\sum {x_{i \plus{} 1} } } \right)^2 }}{{\sum {\left( {x_{i \plus{} 1} \plus{} x_{i \plus{} 1} x_i^2 } \right)} }}
\]
[/quote]
OK, If we assume it to be a typo, it doesn't change anything.[/quote]
Right, It was just a typo :(

Can anyone say, what the problem is?[/quote]

Sorry but I don't understand clearly what you say, math07 and Moonmathpi496.  :(
But acording to your solution we will have, for example when $ n \equal{} 3$ (evently the given condition is $ n > 3$ then
\[ a^2 b \plus{} b^2 c \plus{} c^2 a \le \frac {8}{9}
\]
But it's obviously wrong. Try $ c \equal{} 0$  :wink: . Or because the true of the above is
\[ a^2 b \plus{} b^2 c \plus{} c^2 a \le \frac {4}{{27}}\left( {\sum a } \right)^3 \minus{} abc \le \frac {{32}}{{27}}
\]
The important things here is when does the equality of the inequality occurs and you are wrong here.

Another is for example, let $ a\plus{}b\plus{}c\equal{}3$ and find the minimum value of $ a^2 b \plus{} b^2 c \plus{} c^2 a$. By AM-GM we have
\[ a^2 b \plus{} b^2 c \plus{} c^2 a \ge 3abc
\]
with equality holds when $ a\equal{}b\equal{}c$. So I want to ask you, can you claim that the min is $ 3$ when $ a\equal{}b\equal{}c\equal{}1$.
Your solution didn't say anything, you just use the Cauchy Schwarz inequality and say when does the equality holds for yours.

Minimum must be $ \frac {3}{2}$ I killed it today by freezing vars (we assume $ a_3,a_4...,a_n$ - const and $ a_1 \equal{} 2 \minus{} a_2 \minus{} ... \minus{} a_n$ and derivates kill it) equality when $ n\minus{}2$ vars=0, and 2 neighbours(like $ a_1,a_2$)=1

@[b]Honey_S[/b] and [b]can_hang2007[/b]:
I know that the solution is not correct. As you have showed by example. But, the solution I made used CS inequality and it is true for all reals. And we know that equality is possible iff the respective terms of the sequence is proportional.And using this I showed the equality, (and we know that the left hand side will be minimum in the equality case).
So, I want the know the mistake in the argument and how we can find the minimum in this type of case.

Any help will be appreciated. Thanks in advance. :)

[quote="dima ukraine"]Minimum must be $ \frac {3}{2}$ I killed it today by freezing vars (we assume $ a_3,a_4...,a_n$ - const and $ a_1 \equal{} 2 \minus{} a_2 \minus{} ... \minus{} a_n$ and derivates kill it) equality when $ n \minus{} 2$ vars=0, and 2 neighbours(like $ a_1,a_2$)=1[/quote]
Would you show your solution?  :maybe: I am really looking forward to it....

[quote="math_07"]@[b]Honey_S[/b] and [b]can_hang2007[/b]:
So, I want the know the mistake in the argument and how we can find the minimum in this type of case.
[/quote]
Here
[quote="Honey_S"]
Another is for example, let $ a \plus{} b \plus{} c \equal{} 3$ and find the minimum value of $ a^2 b \plus{} b^2 c \plus{} c^2 a$. By AM-GM we have
\[ a^2 b \plus{} b^2 c \plus{} c^2 a \ge 3abc
\]
with equality holds when $ a \equal{} b \equal{} c$. So I want to ask you, can you claim that the min is $ 3$ when $ a \equal{} b \equal{} c \equal{} 1$.
Your solution didn't say anything, you just use the Cauchy Schwarz inequality and say when does the equality holds for yours.[/quote]

Don't you see that?  :|

[quote="dima ukraine"]$ x_1,x_2...x_n\ge0$, n>3, $ x_1 \plus{} x_2 \plus{} ... \plus{} x_n \equal{} 2$ Find min of

$ \frac {x_2}{1 \plus{} x^2_1} \plus{} \frac {x_3}{1 \plus{} x^2_2} \plus{} ... \plus{} \frac {x_1}{1 \plus{} x^2_n}$[/quote]
By AM-GM Inequality, we have that
\[ \frac{x_2}{x^2_1\plus{}1} \equal{}x_2 \minus{}\frac{x^2_1x_2}{x^2_1\plus{}1} \ge x_2 \minus{}\frac{1}{2} x_1x_2\]
Apply this for the similar terms and adding them up to obtain
\[ LHS \ge 2 \minus{}\frac{1}{2}(x_1x_2\plus{}x_2x_3\plus{}\cdots \plus{}x_nx_1)\]
Moreover, we can easily show that
\[ x_1x_2 \plus{}x_2 x_3\plus{}\cdots \plus{}x_nx_1 \le x_k (x_1\plus{}\cdots \plus{}x_{k\minus{}1}\plus{}x_{k\plus{}1}\plus{}\cdots \plus{}x_n) \le 1\]
for $ k$ is a number such that $ x_k \equal{}\max \{x_1,x_2,\ldots,x_n\}$.
Hence
\[ LHS \ge 2\minus{}\frac{1}{2} \equal{}\frac{3}{2}\]
:)

[quote="can_hang2007"][quote="dima ukraine"]$ x_1,x_2...x_n\ge0$, n>3, $ x_1 \plus{} x_2 \plus{} ... \plus{} x_n \equal{} 2$ Find min of

$ \frac {x_2}{1 \plus{} x^2_1} \plus{} \frac {x_3}{1 \plus{} x^2_2} \plus{} ... \plus{} \frac {x_1}{1 \plus{} x^2_n}$[/quote]
By AM-GM Inequality, we have that
\[ \frac {x_2}{x^2_1 \plus{} 1} \equal{} x_2 \minus{} \frac {x^2_1x_2}{x^2_1 \plus{} 1} \ge x_2 \minus{} \frac {1}{2} x_1x_2
\]
Apply this for the similar terms and adding them up to obtain
\[ LHS \ge 2 \minus{} \frac {1}{2}(x_1x_2 \plus{} x_2x_3 \plus{} \cdots \plus{} x_nx_1)
\]
Moreover, we can easily show that
\[ x_1x_2 \plus{} x_2 x_3 \plus{} \cdots \plus{} x_nx_1 \le x_k (x_1 \plus{} \cdots \plus{} x_{k \minus{} 1} \plus{} x_{k \plus{} 1} \plus{} \cdots \plus{} x_n) \le 1
\]
for $ k$ is a number such that $ x_k \equal{} \max \{x_1,x_2,\ldots,x_n\}$.
Hence
\[ LHS \ge 2 \minus{} \frac {1}{2} \equal{} \frac {3}{2}
\]
:)[/quote]
It is definitely sad that I got a big mistake in this solution. I am very sorry to everyone as I have wasted your time to read this proof.
For the proof of
$ x_1x_2 \plus{} x_2x_3 \plus{} \cdots \plus{} x_nx_1 \le 1.$
I change it to the following:
WLOG, we may assume that $ x_1 \equal{} \max\{x_1,\ldots, x_n\},$ then we will prove that
$ x_1x_2 \plus{} x_2x_3 \plus{} \cdots \plus{} x_nx_1 \le (x_1 \plus{} x_3)(x_2 \plus{} x_4 \plus{} \cdots \plus{} x_n).$
Indeed, it is clear that
$ x_1(x_2 \plus{} x_4 \plus{} \cdots \plus{} x_n) \ge x_1x_2 \plus{} x_4x_5 \plus{} \cdots \plus{} x_nx_1,$ (note: If $ n\equal{}4$, then $ x_5 \equiv x_1$)
and
$ x_3(x_2 \plus{} x_4 \plus{} \cdots \plus{} x_n) \ge x_2x_3 \plus{} x_3x_4.$
Therefore, the above statement is valid and apply it, we get
$ x_1x_2 \plus{} x_2x_3 \plus{} \cdots \plus{} x_nx_1 \le (x_1 \plus{} x_3)(x_2 \plus{} x_4 \plus{} \cdots \plus{} x_n) \le 1.$


What integer $ x$ satisfies $ \frac{1}{4}<\frac{x}{7}<\frac{1}{3}$?

If we make the equation 21/84<12x/84<28/84, we see that the only integer value that we can plug in is 2. Therefore the answer is 2.


1. How do you determine the least positive integer n such that
$n^2+n\equiv 2042 \mod4000$?
2. How about determining the smallest integer $k$ such that
$x^2=8169+16000k$?

Obviously, we are working in natural numbers here.

[hide="1"]
$n^2 + n = 2042 + 4000k$
$n^2 + n - 42 = 2000(2k+1)$
$(n+7)(n-6) = 2000(2k+1) = 2^4 5^3 (2k+1)$

Only one of $n+7, n-6$ can be divisible by $5$, so it must also be divisible by $5^3$.  Rewrite like so.

$(5^3 p)(5^3p \pm 13) = 2^4 5^3 (2k+1)$
$p(5^3 p \pm 13) = 2^4 (2k+1)$

Case 1:  $p$ is odd.  Then $2^4 | (5^3p \pm 13)$ so we write $13p \pm 13 \equiv 0 \bmod 16$ which gives $p \equiv \mp 1 \bmod 16$.  Write $p = 16q \mp 1$ and plug in.

$(16q \mp 1)( 125(16q) \mp 112) = 2^4 (2k+1)$
$(16q \mp 1) (125q \mp 7) = (2k+1)$

Which has its smallest (and only) solution (we choose the positive sign) when $q = 0, p = 1, n+7 = 125, n = \boxed{118}$.

Case 2:  $p$ is even.  Write $p = 2q$.

$q(250q \pm 13) = 2^3 (2k+1)$

Clearly $250q \pm 13$ is odd, so $2^3 | q$ and we write $q = 8a$.

$a(2000a \pm 13) = (2k+1)$

Which has its smallest solution (choose the negative sign) when $a = 1$, giving $p = 16, n+7 = 2000, n = 1993.$
[/hide]

[hide="2"]
Similar approach, I'll do it with the other factor this time.

$x^2 = 8169 + 16000k$
$(x+13)(x-13) = 8000(2k+1) = 2^6 5^3 (2k+1)$

It is clear that $x > 90$, so $x+13, x-13$ cannot share a factor of two larger than $2^1$.  Hence, one of them is divisible by $2^5$.  Write thus.

$(32p)(32p \pm 26) = 2^6 5^3 (2k+1)$
$p(16p \pm 13) = 5^3 (2k+1)$
$16p \pm 13 \equiv p \pm 3 \bmod 5$, so only one or the other can be divisible by $5$ and hence $5^3$.

Case 1:  $5^3 | p$.  Write $p = 125q$.

$q(2000q \pm 13) = 2k+1$ which has its smallest solution (choose the negative sign) when $q = 1, p = 125, x+13 = 4000, x = 3987$.

Case 2:  $5^3 | 16p \pm 13$.  The smallest solution (we choose the positive sign) is at $p = 7, x-13 = 224, x = \boxed{237}$.
[/hide]

[quote="1234567890"]1. How do you determine the least positive integer n such that
$n^2+n\equiv 2042 \mod4000$?[/quote]

[hide="1"]
$n^2+n-2042\equiv 0 \pmod {4000}$
${\displaystyle n \equiv \frac{-1\pm \sqrt{1+4(2042)}}{2}}$
${\displaystyle n \equiv \frac{-1\pm \sqrt{1+4(2042+4000\cdot 3)}}{2}}$
${\displaystyle n \equiv \frac{-1\pm 237}{2}}$
$n \equiv 118, -119 \pmod {4000}$
$n \equiv \boxed {118, 3881} \pmod {4000}$
(the least positive integer is 118)
[/hide]

Chess64, and t0rajir0u, beautiful solutions. :)
Ok, thanks. Got to factoring, but didn't know we had to do caseweork. Usually get intimitated by that on AIMES. :(

[quote="chess64"]
${\displaystyle n \equiv \frac{-1\pm \sqrt{1+4(2042+4000\cdot 3)}}{2}}$
[/quote]

Wait, hold on, didn't read this carefully enough. How did you know to add 4000*3? How did you know it'd be a perfect square?

[quote="Kalle"][quote="t0rajir0u"][hide="2"]
It is clear that $x > 90$, so $x+13, x-13$ cannot share a factor of two larger than $2^1$.
[/hide][/quote]
Why is this? I'm probably missing something obvious.[/quote]

Well, if one of them is divisible by four, then the other one is $\equiv \pm 26 \bmod 4 \equiv 2 \bmod 4$, so it's divisible by two but it can't be divisible by any higher powers of two.  (Actually, this argument doesn't require that $x > 90$, it only requires that $x+13, x-13$ don't differ by a number of the form $2^k (2^m - 1)$, and $26$ is not of such a form.)

For #2 you can also say $\ x^2 \equiv 8169\pmod {160000}$, then you can try and find a congruence on the RHS that is a perfect square, which is sort of guess and check but probably a little easier and quicker, although not as nice as t0rajir0u's.

[quote="chess64"][quote="1234567890"][quote="Elemennop"]Guess and check, most likely.[/quote]

But isn't that really inefficient? There are many numbers to guess and check with. For all we know, it could be 20*4000 or something.[/quote]

Luckily, it wasn't :D It could be difficult on a non-calculator competition such as the AIME though :([/quote]

Which is why I prefer my method - you would never have found $n = 1993$ by guess and check  :D

Mine is easier though. Your method scares me :D just kidding

[quote="1234567890"][quote="t0rajir0u"]
$(16q \pm 1)( 125(16q) \mp 112) = 2^4 (2k+1)$
$(16q \pm 1) (125q \mp 7) = (2k+1)$
[/quote]

You mean -+, not $\pm$ right?[/quote]

Right, sorry, fixed.  Then you choose the positive sign (second case) in that equation which means you choose the negative sign in the original equation so $n+7 = 125$.  Thanks for pointing that out  :)


trigonometry
trig identities
Law of Cosines

The numbers $\sqrt{2u-1}$, $\sqrt{2u+1}$ and $2\sqrt{u}$ are the side lengths of a triangle. How many degrees are in the measure of the largest angle?

I think this is #9 team but whatever.

[hide="Hint"]

Use pythagorean theorem
and then you will see something cool.[/hide]

Since $(\sqrt{2u-1})^2+(\sqrt{2u+1})^2=(2\sqrt{u})^2$, we know that this is a right triangle.  Therefore, since the largest angle in a right triangle is the right angle,  90 is the largest angle.

Lucky you all noticed that :D. I used the law of cosines and started banging my head once I realized that it was right. My teammates looked at me like I was crazy :ninja:.

I agree that you are crazy as well. :D  :P

But I am also crazy.

[quote="Ignite168"][hide]Plug in any number for u, let's say 1 to get the side lengths are $1, \sqrt{3}, 2$. Therefore, it must be a 30-60-90 triangle and the largest would be $\boxed {90}$.[/hide][/quote]
[hide]I just realized while my answer is correct, my solution is not. It is not always a 30-60-90 triangle but it is always a right triangle.[/hide]

[hide]Squaring all three of them, we see that the sum of the first two squares equals the square of the last side, so the triangle is right. 90 degrees.[/hide]


Hello everyone, this is my first post and I hope you will enjoy it.
I think that the Iran1996 is very popular with each of you because of it's beautiful.
And how about this ineuality?
This inequality is wrong or not?
$\frac{9(a^{2}+b^{2}+c^{2})}{4(ab+bc+ca)^{2}}\geq \frac{1}{(a+b)^{2}}+\frac{1}{(b+c)^{2}}+\frac{1}{(c+a)^{2}}$
(source: by myself)
With a,b,c >0
Let 's be fun with it.

timlaiminh

Here is the solution from one of my friends(detectivehien):
$\frac{9(a^{2}+b^{2}+c^{2})}{4(ab+bc+ca)^{2}}\geq \sum \frac{1}{(a+b)^{2}}$
$\Leftrightarrow \frac{9}{4}\sum a^{2}\geq \sum (\frac{ab+bc+ca}{a+b})^{2}$
 $\Leftrightarrow \frac{9}{4}.\sum a^{2}\geq \sum (c+\frac{1}{\frac{1}{a}+\frac{1}{b}})^{2}=\sum a^{2}+2\sum \frac{1}{\frac{1}{ab}+\frac{1}{ac}}+\sum \frac{1}{(\frac{1}{a}+\frac{1}{b})^{2}}$
$\Leftrightarrow 5\sum a^{2}\geq 8 \sum \frac{1}{\frac{1}{ab}+\frac{1}{ac}}+4\sum \frac{1}{(\frac{1}{a}+\frac{1}{b})^{2}}$

Have:
$a^{2}+b^{2}+c^{2}\geq ab+bc+ca=\frac{1}{\frac{1}{ab}}+\frac{1}{\frac{1}{ac}}+\frac{1}{\frac{1}{bc}}\geq 2 \sum \frac{1}{\frac{1}{ab}+\frac{1}{ac}}$
(Because $\sum \frac{1}{x}\geq 2\sum \frac{1}{x+y}$).

And : $\frac{1}{2}(a^{2}+b^{2})\geq\frac{(a+b)^{2}}{4}\geq \frac{4}{\frac{1}{a}+\frac{1}{b})^{2}}$
All above take us to the solution

COMMENT: It's really like your solution, but more easy and beautiful than yours. Are you ok?