geometry
3D geometry
sphere
Euler
graph theory

I'm taking a course on graph theory, and was given this problem:

We consider maps on two spheres, say the earth and the moon.  Each region on the earth is a country, and has corresponding region or colony on the moon.  We desire to color the countries and colonies so that:
   1) each country recieves the same color as its colony, and
   2) If two countries or colonies share a common boarder, then they receive distinct colors.
What is the minimum number of colors needed to color all earth-moon maps?

I want to guess 7 very badly before doing any work.

Here is as far as I have gotten:

[hide]You can get 8 colors with eight states, any one adjacent to any other on either the earth or the moon.  I read somewhere that the maximum was 12, based on a formula made by Euler used to prove the six-color planar graph.[/hide]

Ok I'll tell you why I guessed 7 then (in spoiler). [hide]For a plane, you need 4 colors. For a torus, you need 7. Topologically, a torus is the direct sum of two circles. Also, as far as topological properties go for this problem, a plane is the same as a sphere (puncture a hole in one of the countries and then perform stereographic projection using that point as the north pole). I don't think it's a proof, but I think the answer is right. When we were asked in differential geometry whether a sphere and a torus are diffeomorphic, one of my friends used the coloring theorems to show that they weren't. I thought that was neat.[/hide]

Remember that you have two sets, with each country having a colony on the other sphere

I managed to get 8 with 9 countries from the first try. haven't really been working on it though.


a wave propagates by compressions and rarefactions that is by compressing and rarefacting the particles of a medium.so it moves by disturbing the particles

This is for sound wave ( could also refer to longitudinal waves in a coil...).
But it should be extended to any wave; for instance, electromagnetic waves, where the field ( electric or magnetic, or both) is disturbed.
So, the disturbance of any physical characteristic in a point or region of space, propagates to another region through wave phenomenon.

Important: disturb here means modify, change.


analytic geometry
limit
invariant
algebra
polynomial
function
geometry

Let $a_1,b_1,c_1 >0$ such that $a_1+b_1+c_1=1$. Define the sequences $(a_n)_n,(b_n)_n,(c_n)_n,$ as follows: $a_{n+1}=a_n^2+2b_nc_n, b_{n+1}=b_n^2+2a_nc_n, c_{n+1}=c_n^2+2a_nb_n,$. Prove that the sequence $(a_n)_n,(b_n)_n,(c_n)_n,$ converges and find it's limits

[quote="fleeting_guest"]One can think of this as a transformation of points in a triangle, converging to the centroid (or possibly the vertices).  Therefore, one can hope for some geometric interpretation.    Otherwise it is a standard type of problem.[/quote]
Fleeting_guest, could you be more explicit, why this transformation converges to the centroid(the limit must be $\frac{1}{3}$) and how exactly should we consider our numbers and relations as a transformation
Sorry, maybe it is evident how to get through my questions, but i i didn't manage from the first glance

Note that $\Sigma a_n = 1$ so that these are barycentric coordinates of a point in a triangle.  Hence the hope for a geometric statement of the problem.   If there is convergence it must be either to the centroid $(1/3, 1/3, 1/3)$ or (maybe) the vertices such as $(1,0,0)$.  To determine if the vertices are possible limits one can use coordinates $(1 - x - y, x,y)$ and check the behavior for small positive $(x,y)$ near $(0,0)$.  If the vertices are not possible, then the limit is the centroid, and to prove this the usual idea is to prove that some expression like $\Sigma (a-b)^2$ or $\Sigma |a-b|$ is decreasing at each step.

Here's a very clean algebraic approach. Let $u_n=a_n^2+b_n^2+c_n^2$. Using the fact that $a_n+b_n+c_n=1$ for all $n$, we can show that
$u_n-u_{n+1}= 2(a_nb_n+b_nc_n+c_na_n)(a_n^2+b_n^2+c_n^2-a_nb_n-b_nc_n-c_na_n)$
$=2\left(\frac{1-u_n}{2}\right)\cdot\left(\frac{3u_n-1}{2}\right)=\frac{(1-u_n)(3u_n-1)}2$
This can be rearranged to $\left(u_{n+1}-\frac13\right)=\frac32\left(u_n-\frac13\right)^2$
Since $\frac13\le u_1<1$, clearly $\lim u_n=\frac13$ and $\lim a_n=\lim b_n=\lim c_n=\frac13$.

[quote="jmerry"]Let $u_n=a_n^2+b_n^2+c_n^2$ [/quote]

Note that $\Sigma a^2$, $\Sigma (a-b)^2$, $\Sigma (a - \frac{1}{3})^2$, and $\Sigma (-ab)$ are all equivalent here; one is decreasing if and only if all the others are.  Because $(a+b+c)$ and $(a+b+c)^2$ are fixed, all the cyclically invariant polynomials of degree at most $2$ are linearly expressible in terms of $ab+bc+ca$, and so in terms of each other.

For example, the $3u_n - 1$ in your solution is equal to $q_n = \Sigma (a-b)^2$, and $q_{n+1} = {\frac{1}{2}} {q_n}^2$, so ${q_n}/2$ is squared at each step and tends to 0.  Or more directly, $u_n- 1/3$ equals  $\Sigma (a - 1/3)^2 $, so once you show this goes to zero, no more is needed.

Concerning the geometric interpretation, introduce coordinates with origin at the centroid, $(a,b,c) = ({\frac{1}{3} + p},{\frac{1}{3}}+q,{\frac{1}{3}}+r)$.  Then $p+q+r = 0$ and the transformation has the same formula, $F(p,q,r) = (p^2 + qr, q^2 + rp, r^2 + pq)$.  This has some geometric implications:

Considered as a map of the triangle, $F(P) = F(P')$ when $P$ and $P'$ are symmetrical with respect to the centroid;

$F$ sends lines through the centroid to lines through the centroid;

On each line, $F$ acts as $t \to t^2$;

On the $P^1$ of lines through the centroid, $F$ acts as a 2-to-1 rational function (it is of degree 2 after all).  So one has some hope of identifying the 2 additional points with the same image as a given point and its reflection with respect to the centroid.

I have a geometric interpretation for the easier case of the transformation $(a,b,c) \to {\frac{1}{2}}(a+b,b+c,c+a)$ and for some quadratic cases.

[quote="fleeting_guest"]For example, the $3u_n - 1$ in your solution is equal to$ q_n = \Sigma (a-b)^2$[/quote]
That's exactly why I guessed it as a factor.

I don't think the geometric interpretation really helps here- the problem is simply algebraic.

Here's the work I didn't show before to get my answer:

Let $x=a_n,\,y=b_n,\,z=c_n$. We guess that $u_n=a_n^2+b_n^2+c_n^2$ will be a useful monovariant, since the obvious point to converge to is the center where $u$ is minimized. $u_n=x^2+y^2+z^2$ and $u_{n+1}=(x^4+4x^2yz+4y^2z^2)+\cdots$. (Of course, everything here will be symmetric polynomials)

We want to look at $u_n-u_{n+1}$. First, we homogenize by using $u_n=(x^2+y^2+z^2)(x+y+z)^2=x^4+2x^3y+2x^2y^2+2x^2yz+\cdots$.
Subtracting, $u_n-u_{n+1}=2x^3y+2x^3z-2x^2y^2-2x^2yz+\cdots$.

We know that $u_n=u_{n+1}$ if $x=y=z$, so we guess that $(x-y)^2+(y-z)^2+(z-x)^2=2(x^2+y^2+z^2-xy-yz-zx)$ is a factor. It is, and the other factor is $xy+yz+zx$. There aren't very many different homogenous symmetric polynomials, so this is easy to check. The rest of the algebra is in my previous post.

Here is an interpretation, using complex numbers.

Any complex $z$ can be uniquely written as $a + b \omega + c \overline{\omega}$ where $\omega$ is a cube root of unity, real numbers $a,b,c$, and $a+b+c = 1$.  Clearly if $z$ and $w$ are of this form then $zw$ can be expressed by simply multiplying algebraically (the sum = 1 condition is preserved).   Then:

$\overline{z^2} = (a^2 + 2bc) + (b^2+2ac) \omega + (c^2 + 2ab) \overline{\omega}$.

The complex norm $|z^2|$, expressed in these coordinates, is not $\Sigma a^2$ but $a^2 + b^2 + c^2 - ab -bc -ca$.  However, this is equal to $(1/2) \Sigma (a-b)^2$, which is the now-familiar ${q_n} / 2$.

This explains, in particular, why ${q_n}/2$ is squared at each step; the norm is multiplicative.

The geometric interpretation, then, is: take the point with barycentric coordinates $(a,b,c)$ in the triangle with vertices the complex cube roots of unity, and repeatedly square and conjugate it.  Clearly this tends to zero.

Note that we could have required $a+b+c = 0$ instead (coordinates relative to the centroid) and the above statements would remain correct, as all formulas continue to hold, and $\Sigma=0$ is also preserved under multiplication.

I am still hoping for an interpretation that works in an arbitrary triangle.


Has anyone seen this or read this in chinese?  It's a wuxia story.  Very good story it's "chinese mythology" kinda lol.  Anyone seen it?

elvenchamp777

I have read it.... ya Jinyong wuxia books are pretty good, although Heaven Swords and Dragon Sabre is not as good as my favorite one (State of Divinity, or Xiao Ao Jiang Hu)

It's a good idea to post it in the Chinese forum...


linear algebra
matrix
linear algebra unsolved

Prove that $\text{tr}(AB)=\text{tr}(BA)$
where $A$ and $B$ are square matrices.

$\text{tr}(X)$ denotes the trace of the square matrix $X$.

It is not hard but i think it has its own beauty!
 :)

This is just proving that \[ \sum_{i=1}^n \sum_{k=1}^n a_{ik} b_{ki} = \sum_{i=1}^n \sum_{k=1}^n b_{ik} a_{ki} .  \]

We can interchange the sum signs in the second double sum, which turns out to be equal to the first.

[quote="yannick"][quote="Cezar Lupu"]This nice identity follows from the fact that \[ P_{AB}(X)=P_{BA}(X).  \] :)  ;)[/quote]

What is P?[/quote]

same question..

This is the characteristic polynomial. :)  ;)


calculus
integration
Asymptote
function
LaTeX
calculus computations

This problem came up on a quiz as extra credit for our class
[img]http://img84.imageshack.us/img84/8716/infproblem1mi1.gif[/img]
I saw it was improper because of the asymptote at x = 2/3 so split the integral and took the limit
[img]http://img300.imageshack.us/img300/8418/infproblem2fo0.gif[/img]
After integrating and computation, I end up with the following
[img]http://img146.imageshack.us/img146/7397/infproblem3cz2.gif[/img]

At first glance, I was thinking divergent because of the infinity - infinity, but I was thinking since the functions that are left are the same and with opposite signs that for every x the y values would become 0. So would the answer be divergent or (1/3) ln(2)

Sorry for the lack of LaTeX, I'm not used to it

Edit: Oh sorry it is supposed to be 2/3...1 it was a typo

You split the integral incorrectly; the second one should have limits $\frac23\dots 1$. Also, the two integrals should be treated separately. You start by messing with the first one, and as soon as you see that it's divergent, the game is over.


Support search articles email Yale college videos

For all those pplz out there who arent 18, assume you are.  Would you vote for Kerry or Bush in the election? and why?

i'm not that sure myself.  I would like to go for Kerry now, but yesterday i supported bush.
the nation's lucky i'm not eighteen yet.

noooooooooooooooooooooo, no more! no more! :lol: look at the logs of The Round Table....

i dun look at the round table.  i jus spam.

[quote="JBL"]We actually already had this post on this forum a little while ago.[/quote]

Well, there are new members of AoPS all the time, and the election hasn't happened yet. (After edit:) And it looks like the pruning of posts is pretty vigorous here on F&G now, so that any old thread on exactly this question seems to have already been poofed.

I'm just glad I'm over eighteen, and thus outside the scope of the poll   :wink:

[quote="ComplexZeta"]You aren't outside the scope of the poll. The first comment means that if you aren't at least 18, then you should pretend you are. (At least that was the way I interpreted it.)[/quote]

I think your reading is plausible, but I am gonna read the request to reply the other way, so you won't see my vote here.  :D

I will want to vote, somehow, in November, but that is November's task. My United States representative and state representative both enjoy "safe" districts, and I'm not aware of any local issues that depend on my vote, so my main reason to vote will be to influence the presidential election. At least it's refreshing for Minnesota to be "in play" this year in the presidential election--that is very unusual for this state, and is a sign how much people moving here during the 1990s changed the political composition of the state.

[quote="tokenadult"]And it looks like the pruning of posts is pretty vigorous here on F&G now, so that any old thread on exactly this question seems to have already been poofed.
[/quote]

I don't think the auto-pruning is necessarily being any more vigorous than before. However, as you may have noticed, many of the older polls that do not get auto-pruned are now being manually deleted by moderators (mainly so that we do not have an issue with "reawakening" old threads) and thus the number of threads/posts has ben significantly decreased. The last thread on this topic was more than a month ago so it's definitely gone by now.

[quote="tokenadult"]At least it's refreshing for Minnesota to be "in play" this year in the presidential election[/quote]

Isn't Minnesota one of the "battleground" states?...It seems like Kerry's speaking over there all the time...

DARNIT bush is winning...
i dunno... i like republican ideas but i always support the democrats, dun kno why.

and yes i think adults shoud vote in this thing too.

[quote="starrynightskies"]YAY bush is winning...
[/quote]

yea...when'd that happen?...last time I checked, Kerry was ahead by like 5 votes...

[quote="BHorseMath"]I dunno when it happened but I like it.  lol.

I can't vote this election but I will be able to vote next election.  Lol...My B-Day is on election day lol.[/quote]

How old are you?

If you turn 18 on election day 2008, you still can't vote...

Yes I can.  theres rules here that say I could.  I will turn 14 this year.

I think both Bush and Kerry flip-flop, along with many other politicians.  It's just that the Bush people are concentrating on it a whole lot more, making people think that Kerry is the big flip-flopper.

This debate seems really weird to me, since supposedly, 80% of they're views are the same.

William Saletan of Slate Magazine takes a look at the RNC video (to which confuted links) [url=http://slate.msn.com/id/2105096/]here[/url].  Kerry's not perfect, but the charge of "flip-flopper" is definitely overstated.

[quote="takuanitromars36"]Lol. I think Bush is better. I've been analyzing past stuff that Kerry said; he's been inconsistent.[/quote]

Care to post your analysis?

[quote="takuanitromars36"]I don't think that we want a wavering president--we need one that's steadfast in his values.[/quote]

I [i]strongly[/i] encourage you to check out [url=http://www.americanprogress.org/site/pp.asp?c=biJRJ8OVF&b=42263]the link posted above, which I reproduce here[/url]. Do not believe the propoganda that you hear from either campaign--trust the facts.

[quote="takuanitromars36"]Besides, if Bush can prevent this supposedly coming up terrorist attack (another 9/11), though human dignity (right to privacy) will be lowered, human lives will be preserved (much more important).[/quote]

Do you mean that you believe that the Patriot Act will prevent a future terrorist attack? If so, find out how [url=http://www.johnkerry.com/pdf/pr_2004_0525b_a.pdf]John Kerry will strengthen the Patriot Act[/url].

no use, Jrosen, most bush voters just don't [color=red]want[/color] to know about Kerry's opinions...

[quote="Peter VDD"]Awkward poll... it says 29 out of 100 votes = 28 % :D [/quote]  Hmm.. is the above  counting done in Florida by any chance?

No, the server is in Romania I think. Unless that changed when merging with aops, then it may be in Florida indeed :)

go Nadir and kerry! :clap:  :clap2:  :heli:  :jump:

sorry, the election was over ages ago  ;)

Well... it's never too late for anti-bush support :D

hey i'd prefer bush anyday for invading iraq and spending billions and billions of dollars on this patriotic and necessary act

i also like him for his great, politically sound strategems on how to deal with governments of other nations

don't forget his amazing ability to bring the dollar down in international value by like 20%

yup...bush is wonderful  :rotfl:

Come on folks, stop reviving topics. If one's been down for a long time and someone does revive, don't reply. -.-

Locked.


So, the AMCs are tomorrow. I'm taking the 10 this time and the 12 in the B part.

I think the cutoff for AIME qualification with the AMC 10 is too high. I think I have to get 20/25 questions and that allows one mistake? It's the 10, but I hope I can get some of the 20-25 questions to cement my qualification...

My teacher only administers AMC 10 tomorrow and AMC 10B on February 21.

[quote="PenguinIntegral"]I think the cutoff for AIME qualification with the AMC 10 is too high. I think I have to get 20/25 questions and that allows one mistake? It's the 10, but I hope I can get some of the 20-25 questions to cement my qualification...
[/quote] If you get 20 questions right, you can make 5 mistakes! (6*20=120) Last year I missed 3 questions on the 10A but still got 20 right.

Even though most people say it's harder to qualify via the 10, it's easier for me to get 120 on the 10 than 100 on the 12 (since I can make more mistakes on the 10), although both are no problem. The big thing is getting 8+ on the AIME.


How many positive perfect squares less than 300 are multiples of 9?

[hide]To be a multiple of 9, the prime factorization of the square root has to include at least one 3.

There are 5 square roots less than or equal than 16 that fit the description.

5[/hide]

[hide]
9
36
81
144
225

squares of 3,6,9,12, and 15

confusing, right? :D

this is because if the prime factorization is $3.........$ and if that's multiplied by itself, the 3's make an even power (multiple of 9), and that always makes something times a multiple of 9

[/hide]
-jorian

In order for it to be a multiple of 9, its sqrt must have a factor of 3.

$\sqrt{300}\approx 17.320$.

So, the square roots are $x\in\{3^{2}, 6^{2}, 9^{2}, 12^{2}, 15^{2}\}$


Let $ a, b, c$, be distinct non-negative real numbers. Prove that

$ \frac{a^2\plus{}b^2}{(b\minus{}c)^2}\plus{}\frac{b^2\plus{}c^2}{(c\minus{}a)^2}\plus{}\frac{c^2\plus{}a^2}{(a\minus{}b)^2}\geq(\frac{a^2\plus{}b^2\plus{}c^2}{ab\plus{}bc\plus{}ca})^2\minus{}\frac{a^2\plus{}b^2\plus{}c^2}{ab\plus{}bc\plus{}ca}\plus{}4$

[quote="Ligouras"]Let $ a, b, c$, be distinct non-negative real numbers. Prove that

$ \frac {a^2 \plus{} b^2}{(b \minus{} c)^2} \plus{} \frac {b^2 \plus{} c^2}{(c \minus{} a)^2} \plus{} \frac {c^2 \plus{} a^2}{(a \minus{} b)^2}\geq(\frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca})^2 \minus{} \frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca} \plus{} 4$[/quote]

try   $ a\equal{}2,b\equal{}\frac{1}{30},c\equal{}1$

[quote="hedeng123"][quote="Ligouras"]Let $ a, b, c$, be distinct non-negative real numbers. Prove that

$ \frac {a^2 \plus{} b^2}{(b \minus{} c)^2} \plus{} \frac {b^2 \plus{} c^2}{(c \minus{} a)^2} \plus{} \frac {c^2 \plus{} a^2}{(a \minus{} b)^2}\geq(\frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca})^2 \minus{} \frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca} \plus{} 4$[/quote]

try   $ a \equal{} 2,b \equal{} \frac {1}{30},c \equal{} 1$[/quote]

Sorry my friends......

$ \frac {a^2 \plus{} b^2}{(b \minus{} c)^2} \plus{} \frac {b^2 \plus{} c^2}{(c \minus{} a)^2} \plus{} \frac {c^2 \plus{} a^2}{(a \minus{} b)^2}\geq\frac {1}{2}(\frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca})^2 \minus{} \frac {a^2 \plus{} b^2 \plus{} c^2}{2(ab \plus{} bc \plus{} ca)} \plus{} 2$

[quote="Ligouras"]Let $ a, b, c$, be distinct non-negative real numbers. Prove that

$ \frac {a^2 \plus{} b^2}{(b \minus{} c)^2} \plus{} \frac {b^2 \plus{} c^2}{(c \minus{} a)^2} \plus{} \frac {c^2 \plus{} a^2}{(a \minus{} b)^2}\geq(\frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca})^2 \minus{} \frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca} \plus{} 4$[/quote]

Correct:

$ \frac{a^{2}\plus{}b^{2}}{(b\minus{}c)^{2}}\plus{}\frac{b^{2}\plus{}c^{2}}{(c\minus{}a)^{2}}\plus{}\frac{c^{2}\plus{}a^{2}}{(a\minus{}b)^{2}}\geq\frac{1}{2}(\frac{a^{2}\plus{}b^{2}\plus{}c^{2}}{ab\plus{}bc\plus{}ca})^{2}\minus{}\frac{a^{2}\plus{}b^{2}\plus{}c^{2}}{2(ab\plus{}bc\plus{}ca)}\plus{}2$

[quote="Ligouras"][quote="Ligouras"]Let $ a, b, c$, be distinct non-negative real numbers. Prove that

$ \frac {a^2 \plus{} b^2}{(b \minus{} c)^2} \plus{} \frac {b^2 \plus{} c^2}{(c \minus{} a)^2} \plus{} \frac {c^2 \plus{} a^2}{(a \minus{} b)^2}\geq(\frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca})^2 \minus{} \frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca} \plus{} 4$[/quote]

Correct:

$ \frac {a^{2} \plus{} b^{2}}{(b \minus{} c)^{2}} \plus{} \frac {b^{2} \plus{} c^{2}}{(c \minus{} a)^{2}} \plus{} \frac {c^{2} \plus{} a^{2}}{(a \minus{} b)^{2}}\geq\frac {1}{2}(\frac {a^{2} \plus{} b^{2} \plus{} c^{2}}{ab \plus{} bc \plus{} ca})^{2} \minus{} \frac {a^{2} \plus{} b^{2} \plus{} c^{2}}{2(ab \plus{} bc \plus{} ca)} \plus{} 2$[/quote]

stronger

$ \frac {(a^2 \plus{} b^2 \plus{} c^2)^2}{\sum a^2(b \minus{} c)^2} \plus{} \frac {(a \plus{} b \plus{} c)^2}{\sum (b \minus{} c)^2} \geq \frac {1}{2}(\frac {a^{2} \plus{} b^{2} \plus{} c^{2}}{ab \plus{} bc \plus{} ca})^{2} \minus{} \frac {a^{2} \plus{} b^{2} \plus{} c^{2}}{2(ab \plus{} bc \plus{} ca)} \plus{} 2$


geometry
trigonometry
analytic geometry
graphing lines
slope

Show that there is no triangle whose vertices are lattice points and the measure of one of whose angles is $ 60^{\circ}$.

[hide="Comment"]This is in fact a generalization of the classic problem that there is no equilateral triangle all of whose vertices are lattice points.[/hide]

[hide]
you can use sum of tangents to get a contradiction where tan(60) is rational.[/hide]

[hide="Solution"]If there is such a triangle, it has area $ \frac{1}{2}ab\sin 60^{\circ}$, but by Pick's Theorem, it has area $ k/2$ for integer $ k$.  Therefore, $ ab\sin 60^{\circ}\equal{}k$, or equivalently $ 3a^{2}b^{2}\equal{}4k^{2}$.

Now since $ a,b$ are lengths of segments with endpoints at lattice points, we can write $ a^{2}\equal{}x^{2}\plus{}y^{2}, b^{2}\equal{}z^{2}\plus{}w^{2}$, for integers $ x,y,z,w$.  Now we have

$ 3(x^{2}\plus{}y^{2})(z^{2}\plus{}w^{2})\equal{}4k^{2}$.

Since squares are 0 or 1 mod 3, if either of the factors on the left side of the equation are divisible by 3, say $ x^{2}\plus{}y^{2}$, then $ x$ and $ y$ are both divisible by 3.  Suppose $ 3^{n}\parallel{}x$ and $ 3^{m}\parallel{}y$, and WLOG assume $ n\le m$.  Then $ x^{2}\plus{}y^{2}\equal{}(3^{n}x')^{2}\plus{}(3^{n}y')^{2}\equal{}3^{2n}(x'^{2}\plus{}y'^{2})$.  Since $ x'$ is not divisible by 3, $ x'^{2}\plus{}y'^{2}$ is not divisible by 3, so 3 divides $ x^{2}\plus{}y^{2}$ exactly $ 2n$ times.  Therefore, 3 divides each of the factors on the left side of the equation above an even number of times, but then it divides the entire left side an odd number of times, a contradiction because the right side is a square.[/hide]

[hide="Note"]You could also say that the side length of the triangle would have to be the square root of some integer $ k$.  So the area of the triangle is $ \frac{k\sqrt{3}}{4}$, which is irrational.[/hide]

[quote="lingomaniac88"][hide="Note"]You could also say that the side length of the triangle would have to be the square root of some integer $ k$.  So the area of the triangle is $ \frac {k\sqrt {3}}{4}$, which is irrational.[/hide][/quote]

True, but that only solves the equilateral triangle problem, not the more generalized version here.

I saw $ 60^\circ$ and I just thought, "equilateral triangle."  Oh well...

[b]Follow-up:[/b]  Describe the possible angles that can occur in a lattice polygon that are rational multiples of $ \pi$.

[hide="An approach to t0rajir0us follow up"]Using my idea for the original problem in this thread I could only show that if $ \theta$ is an angle of such a polygon then $ \tan\theta$ must be rational. So we can reduce our problem into finding all possible values of $ \theta$ such that $ \theta$ (in degrees) and $ \tan\theta$ are both rational. Can we do something from here? :maybe:

My guess is that $ \theta\equal{}\frac{n\pi}{4}$ are all the possible values, but I can't prove it.[/hide]

Sorry for reviving this old topic, but I believe I've found a solution to t0rajir0u's follow-up.

[b]Lemma.[/b] We call a straight line [i]good[/i] if it contains two lattice points. Let two lines intersect at an angle $ \theta$ at a lattice point. If one of the lines is good, then the other one is also good iff $ \tan\theta$ is rational or undefined.

[b]Proof.[/b] Verify that a line is good iff it contains a lattice point and has a rational or undefined slope. First assume that both lines are good, and let their slopes be $ m_1$ and $ m_2$. If at least one of $ m_1$ and $ m_2$ is undefined, WLOG $ m_1$, then $ m_2 = \cot\theta$ which is rational or undefined, implying $ \tan\theta$ is rational or undefined. If none of $ m_1$ and $ m_2$ is undefined, it follows that both $ m_1$ and $ m_2$ are rational. Hence
\begin{align}\tan\theta = \frac {m_1 - m_2}{1 + m_1m_2}\end{align}
is rational or undefined, proving the first part of the lemma. Now suppose that the line with slope $ m_1$ is good (hence $ m_1$ is rational or undefined), and $ \tan\theta$ is rational or undefined. Therefore from $ (1)$ we conclude that $ m_2$ is rational or undefined. And as the intersection point of the two lines is a lattice point, the second line must be good as well. $ \blacksquare$

Now if $ \theta$ is an angle of a lattice polygon, $ \tan\theta$ must be rational or undefined. Hence we need to find all $ \theta$ that are rational multiples of $ \pi$ such that $ \tan\theta$ is rational or undefined. If it is undefined, then $ \theta = \pi/2$. Otherwise, using [url=http://www.lacim.uqam.ca/~plouffe/articles/plouffe.pdf]Margolius' theorem[/url], we have that $ \tan\theta\in\mathbb Q - \{0,\pm 1\}\Rightarrow \theta/\pi\not\in\mathbb Q$. Therefore we must have $ \tan\theta\in\{0,\pm 1\}$ from which we find the only possibilities $ \theta\in\{\pi/4,3\pi/4\}$. To conclude, the possible angles that can occur in a lattice polygon that are rational multiples of $ \pi$ are $ n\pi/4,\,n = 1,2,3$.

Anything wrong with this sol:

We know that using shoelace's theorem, the area will be rational. However, using the formula for the area of the equilateral triangle in terms of side length, we have that the area is irrational. Contradiction.

seems too easy

The triangle in this problem is not specified to be equilateral.

(That being said, this approach can be modified to work; see [b]matt276eagles[/b]'s solution.)


For how many values of positive integer $n$ less than $100$ is $\frac{n(n+21)}{2n+1}$ not in reduced form?

ok, but give an explanation. this is the type of problem that anyone could do with enough time and a computer, but the solution is what counts.

I only found two, 0 and 20.

zero is obvious because 0(21)/1 = 0/1 = 0.

for the second n+21 = 2n+1
n = 20.

Everything else is just giving me more and more negative numbers which I don't know if they should even be included.


This is just another one of my famous Games (okay, maybe not THAT famous)

The goal of this game is to win as many battles as possible. Or be the Last one standing.

When an enemy comes in sight, I will roll a die, and if it lands on an even number, you lose the battle and you die. If it lands on an odd number, you win the fight.

This goes in turns, and Send me a private message if you want to play.

NOTE: if you do not respond  within 2 days when there is a battle, you are captured (in other words, you lost)

Have fun! :)

NOTE II: you have two choices when in battle, fight or escape.

Fight enables dice roll,

escape takes up 1/3 of your energy. ENERGY DOES NOT RECHARGE. so you can only escape 3 times before dying.

to state whether you are using escape of Fight at the time, say Escape, or Fight, in [b]Bolded Text[/b]

And remember! There are 20 places for sign ups, and sign up using Private message!

Isn't this exactly the same is the DICE game?

err, dice game was moved to FF. And plus, this one is more confusing :D :lol:

Mods, can you lock or delete this?


$\mathbf{Theorem.}\; \mathbf{(Yet\;Another\;Generalization\;Of\;Schur)}$ $\mathrm{Suppose\;that}$ $a_{1},a_{2},\ldots,a_{n}$ $\mathrm{are\;positive\;real\;numbers\;such\; that}$ $a_{1}+a_{2}+\cdots+a_{n}=1.\;$ $\mathrm{\textnormal{Then\;the\;following\;inequality\;is\;true: }}$
\[\frac{n-2+n^{n-1}a_{1}a_{2}\cdots a_{n}}{2n-2}\geq\sum_{1\leq i<j\leq n}a_{i}a_{j}.\]

$\mathrm{\textnormal{Why is this a generalization of Schur?}}$

$\mathrm{\textnormal{As }}a^{3}+b^{3}+c^{3}=3abc+(a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ca)$ $=3abc+1-3(ab+bc+ca)\mathrm{\textnormal{ according to Schur}}$\[0\leq a^{3}+b^{3}+c^{3}+6abc-(a+b+c)(ab+bc+ca)=9abc+1-4(ab+bc+ca)\]$\mathrm{\textnormal{which is exactly what YAGOS states for }}n=3\mathrm{\textnormal{.}}$

It is equivalent to Vasc's one http://www.mathlinks.ro/Forum/viewtopic.php?p=276196#276196 post 3
(quoted as (6) [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=110149]here[/url])
not proved by Vasc nor by Harazi  :maybe:

Note that it is stronger than the one by zaimi marku, quoted as (1b). Yours is equivalent to

$(n-1)p_{2}^{2}+{p_{0}^{n}}{p_{1}^{2-n}}\ge np_{1}^{2}$, zaimi marku's one ($k=2$) is equivalent to

$(n-1)p_{2}^{2}+{p_{0}^{n}}{p_{2-n}^{2-n}}\ge np_{1}^{2}$. Now $p_{1}\ge p_{2-n}$ and $2-n\le0$ :P


For $ x \leq 1 ,x\plus{}y \geq 3$
Find min :A= $ 3x^2 \plus{}y^2 \plus{}3xy$

$ x\leq 1\Longrightarrow y\geq 3 \minus{} x\geq 2$,

$ A\geq 3x^2 \plus{} (3 \minus{} x)^2 \plus{} 3x(3 \minus{} x) \equal{} x^2 \plus{} 3x \plus{} 9$

$ \equal{} \left(x \plus{} \frac 23\right)^2 \plus{} \frac {27}{4}$, yielding $ A_{min} \equal{} \frac {27}{4}\ \left(x \equal{} \minus{} \frac 32,\ y \equal{} \frac 92\right)$

Hi

I don't know how to show $ A\geq 3x^{2}\plus{}(3\minus{}x)^{2}\plus{}3x(3\minus{}x)$

The first two terms of A are clearly larger than the first two terms of the right hand side, but the third term could be smaller when $ x$ is negative. And I'm stuck here.

kunny just used $ y\ge 3\minus{}x$ , which is true from one of the conditions .

Hm. I still don't get it

when $ x$ is negative

$ 3xy\leq 3x(3\minus{}x)$

So I think kunny did it not by comparing terms?

[quote="yaa"]For $ x \leq 1,x \plus{} y \geq 3,$ find the minimum of $ A\equal{}3x^2 \plus{} 3xy\plus{} y^2.$[/quote]$ 3x^2 \plus{} 3xy\plus{} y^2\equal{}\frac{3}{4}(x\plus{}y)^2\plus{}\frac{1}{4}(3x\plus{}y)^2\geq \frac{3}{4}\times 3^2\plus{}0\equal{}\frac{27}{4},$

with equality if and only if $ x\equal{}\minus{}\frac{3}{2},y\equal{}\frac{9}{2}.$

Hello !!

But i find it $ \equal{}1$

proof:

we know that $ y\ge 2$ so $ A\ge 3x^{2}\plus{}6x\plus{}4$

and now , let $ f(x)\equal{}3x^{2}\plus{}6x\plus{}4$ $ \rightarrow$ $ f'(x)\equal{}6x\plus{}6$

so f have a minimum value which is equal to $ 1$ , it will be equal when $ y\equal{}2$ and $ x\equal{}\minus{}1$

is that true ??


Let $ a,b,c>0$ and $ a^2\plus{}b^2\plus{}c^2\equal{}3$ prove that

\[ (\sqrt{3}\minus{}a)(\sqrt{3}\minus{}b)(\sqrt{3}\minus{}c)\ge (\sqrt{3}\minus{}1)^3abc\]

[quote="M.A"]Let $ a,b,c > 0$ and $ a^2 \plus{} b^2 \plus{} c^2 \equal{} 3$ prove that
\[ (\sqrt {3} \minus{} a)(\sqrt {3} \minus{} b)(\sqrt {3} \minus{} c)\ge (\sqrt {3} \minus{} 1)^3abc
\]
[/quote]

$ \Leftrightarrow 3 \plus{} ab \plus{} bc \plus{} ca \minus{} \sqrt {3}(a \plus{} b \plus{} c) \plus{} abc(3\sqrt {3} \minus{} 6)\ge 0$

setting $ F(a,b,c) \equal{} 3 \plus{} ab \plus{} bc \plus{} ca \minus{} \sqrt {3}(a \plus{} b \plus{} c) \plus{} abc(3\sqrt {3} \minus{} 6)$

assume $ a \equal{} max(a,b,c)$

$ F(a,b,c) \minus{} F(a,\sqrt {\frac {b^2 \plus{} c^2}{2}},\sqrt {\frac {b^2 \plus{} c^2}{2}}) \equal{} (b \minus{} c)^2[\frac { \sqrt {3} \minus{}a }{b \plus{} c \plus{} \sqrt {2(b^2 \plus{} c^2)}} \minus{} \frac {1 \plus{} a(3\sqrt {3} \minus{} 6)}{2}]$

$ a \equal{} max(a,b,c) \Rightarrow a\ge 1, b \plus{} c \le2$ and $ b^2 \plus{} c^2\le 2$

$ \Rightarrow F(a,b,c)\ge F(a,\sqrt {\frac {b^2 \plus{} c^2}{2}},\sqrt {\frac {b^2 \plus{} c^2}{2}})$

thus, by Mixing Variable Theorem we need prove that $ F(\sqrt {3 \minus{} 2t^2},t,t)\ge 0$ (a variable --> easy :wink: )

Yes, I agree with you, bunhiacovski.
$ F(a,b,b)\geq0\Leftrightarrow F\left(\sqrt {3 \minus{} 2b^2},b,b\right)\geq0\Leftrightarrow$
$ \Leftrightarrow3 \minus{} 2\sqrt3b \plus{} b^2\geq\left((6 \minus{} 3\sqrt3)b^2 \minus{} 2b \plus{} \sqrt3\right)\sqrt {3 \minus{} 2b^2}\Leftrightarrow$
$ \Leftrightarrow(21 \minus{} 12\sqrt3)b^6 \minus{} (8 \minus{} 4\sqrt3)b^5 \plus{} (22\sqrt3 \minus{} 36)b^4 \minus{} (8\sqrt3 \minus{} 12)b^3 \plus{} (11 \minus{} 6\sqrt3)b^2\geq0\Leftrightarrow$
$ \Leftrightarrow b^2(b \minus{} 1)^2\left((21 \minus{} 12\sqrt3)b^2 \minus{} 2(10\sqrt3 \minus{} 17)b \plus{} 11 \minus{} 6\sqrt3\right)\geq0,$ which is true.

Nice arqady and bunhiacovski :lol:
Also There is a nice solution using CAUCHY-SHWARTZ :wink:

I found a similar type of problem interlinked with this inequality, I mean when one is proved other is proved automatically. I post the complete problem:
Let $ a,b,c>0$ $ a^2\plus{}b^2\plus{}c^2\equal{}3$ prove that
i)$ (\sqrt{3}\minus{}a)(\sqrt{3}\minus{}b)(\sqrt{3}\minus{}c)\geq(\sqrt{3}\minus{}1)^3{abc}$
ii)$ (\sqrt{3}\plus{}a)(\sqrt{3}\plus{}b)(\sqrt{3}\plus{}c)\leq(\sqrt{3}\plus{}1)^3{abc}$
Can anybody post an independent solution for the latter inequality?

Let $ a\equal{}0,b\equal{}0$,then

$ (\sqrt {3} \plus{} a)(\sqrt {3} \plus{} b)(\sqrt {3} \plus{} c)>(\sqrt {3} \plus{} 1)^3{abc}.$

[quote="Sunkern_sunflora"]
Let $ a,b,c > 0$ $ a^2 \plus{} b^2 \plus{} c^2 \equal{} 3$ prove that
ii)$ (\sqrt {3} \plus{} a)(\sqrt {3} \plus{} b)(\sqrt {3} \plus{} c)\leq(\sqrt {3} \plus{} 1)^3{abc}$
Can anybody post an independent solution for the latter inequality?[/quote]
I think, it's wrong. Try $ a\rightarrow0^\plus{}.$ :wink:

The inequality

$ (\sqrt {3} \plus{} a)(\sqrt {3} \plus{} b)(\sqrt {3} \plus{} c)\leq(\sqrt {3} \plus{} 1)^3{abc}$

is inversely holds!

$ (\sqrt {3} \plus{} a)(\sqrt {3} \plus{} b)(\sqrt {3} \plus{} c)\geq{(\sqrt {3} \plus{} 1)^3{abc}}$

<=>

$ [a^2 \plus{} b^2 \plus{} c^2 \plus{} ab \plus{} bc \plus{} ca]^2(a^2 \plus{} b^2 \plus{} c^2)$

$ \minus{} [(a \plus{} b \plus{} c)(a^2 \plus{} b^2 \plus{} c^2) \minus{} (9 \plus{} 6\sqrt {3})abc]^2$

$ \equal{} \sum{a^2(b \minus{} c)^2(a \minus{} b)(a \minus{} c)}$

$ \plus{}2 \sum{b^2c^2(a \minus{} b)(a \minus{} c)}$

$ \plus{} (22 \plus{} 12\sqrt {3})\sum{a^2bc(a \minus{} b)(a \minus{} c)}$

$ \plus{} (42 \plus{} 24\sqrt {3})\sum{abc(b \plus{} c)(a \minus{} b)(a \minus{} c)}\ge0$

[quote="arqady"]fjwxcsl,  are you hacker?  :D  :)[/quote]

He is an inequailty expert. :)


The Fibonacci numbers are defined by $ F_0 \equal{} 1, F_1 \equal{} 1, F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n$. The positive integers $ A, B$ are such that $ A^{19}$ divides $ B^{93}$ and $ B^{19}$ divides $ A^{93}$. Show that if $ h < k$ are consecutive Fibonacci numbers then $ (AB)^h$ divides $ (A^4 \plus{} B^8)^k$

Fix a prime $p\mid AB$, let $v_p(A)=a$ and $v_p(B)=b$. From the given condition, we have
\[
\frac{19}{93}\le \frac{a}{b}\le \frac{93}{19}.
\]
Now, assume $a\ge 2b$.Then $v_p(A^4+B^8)\ge 8b$, hence it suffices to verify $8bk \ge (a+b)h$. As $a\le 93b/19$, we have $(a+b)h\le 112bh/19$. Since $k>h$ and $112/19<8$, we are done in this case. We now assume $a<2b$, which is more challenging. In this case, we have $19/93\le a/b<2$ and $v_p(A^4+B^8)\ge 4a$. Hence, it suffices to verify
\[
k \cdot 4a \ge h\cdot (a+b) \iff \frac{k}{h}\ge \frac14+\frac{b}{4a}.
\]
We will actually prove $k/h\ge 3/2$ from which the conclusion will follows immediately. Let $k=F_{n+1}$ and $h=F_n$. Then,
\[
\frac{k}{h} = \frac{F_{n+1}}{F_n} = 1+\frac{F_{n-1}}{F_n}>\frac32,
\]
as $F_n = F_{n-1}+F_{n-2}<2F_{n-1}$.


algebra
polynomial
trigonometry
calculus
calculus computations

If anyone could show me how the following problem is done, I would be very grateful!  There are other problems like this in my textbook, so I'm looking for an example to see how this type of question is answered.  Thanks in advance!

Okay so here's the problem:

Without actually constructing a Maclaurin polynomial for cos(x), you will find out how long it would need to be.

a) Use Taylor's theorem to determine the minimum degree of a Maclaurin polynomial that will approximate cos(2) to within .0001 of its true value.

b) Would that polynomial also approximate cos(3) to within .0001 of its actual value?  Adapt your work in a to justify your answer.

c) Determine to two decimal places, the greatest value of x for which we can be certain that the same polynomial will approximate cos(x) to within .0001 of its true value.

[quote="Yury"]Hint: use Taylor's theorem with the Lagrange form of the remainder.[/quote]

I haven't learned about Lagrange form of the remainder yet.  Is there another way to do it?

You know [i]some[/i] form of the remainder/error term. Use it, and remember that sines and cosines are bounded by $\pm 1$.


algebra
polynomial
Ring Theory
superior algebra
superior algebra unsolved

I am trying to understand products of ideals.

Suppose we have two ideals $ (f_1,\ldots,f_m)$ and $ (g_1,\ldots,g_n)$ of a polynomial ring $ k[X]$ where $ f_i$ and $ g_i$ are polynomials.

What is the explicit map $ (f_1,\ldots,f_m)\times(g_1,\ldots,g_n)\stackrel{\cong}{\rightarrow}(f_1g_1,\ldots,f_1g_n,\ldots,f_mg_1,\ldots,f_mg_n)$?

In other words, where do the explicit $ \left(\sum_{i\equal{}1}^m a_if_i,\sum_{i\equal{}1}^n b_ig_i\right)$ where $ a_i,b_i\in k[X]$ map to? How is this injective and surjective?

Make sure you really understand what the product of two ideals $ A,B$ is.
It is _not_ the coordinatewise product of $ A$ and $ B$. It's not even isomorphic to this, in fact, an isomorphism of what should this be, groups/moduls¿

The product of two ideals $ A,B$ is the ideal generated by the elements of type $ ab$ for $ a \in A$ and $ b\in B$. Nothing more, nothing less. There are no "isomorphisms", nor injective/surjective maps of any reasonable sense.


algebra
polynomial
search
linear algebra
linear algebra unsolved

A,B are nxn matrices with whole entries and A+kB has reverse also with whole entries for k=0,1,...2n. find detB.

Since $A+kB$ has whole entries and it's invers has also whole entries $\det(A+kB)\in \{-1,1\}$.
Let $p,q\in Z[X],p(x)=\det(A+xB)-1,q(x)=\det(A+xB)+1$. So at least one of these polynomials has $n+1$ roots, but it's degree is at most $n$ so $p$ or $q$ is $0$ . this means that $\det(B)=0$.
edit: also I think you should consider using the search function.

does this help doing such problem, i mean using polynomials

problem:A i nxn, p>2 is prime. prove that if A-E has enties all divisible by p and A^k=E for some k<p then A=P.


$ 4.$ How many $ 6 \minus{} tuples$ $ (a_1,a_2,a_3,a_4,a_5,a_6)$ are there such that each of $ a_1,a_2,a_3,a_4,a_5,a_6$ is from the set $ \{1,2,3,4\}$ and the six expressions
\[ a_j^2 \minus{} a_ja_{j \plus{} 1} \plus{} a_{j \plus{} 1}^2\]
for $ j \equal{} 1,2,3,4,5,6$ (where $ a_7$ is to be taken as $ a_1$) are all equal to one another?

Please post the answer for this one. I got 52 as answer :?:

Problems should hav been posted on Olympiad Section so that everyone could discuss, nt just indians.

Im getting my answer 40. I mad a slight error to get 32 bt the answer is 40.

i got the answer as 40 and as far as i checked it seemed right

@bnarsim3:

Could you post all those 40 tuples???
I got 24!!! :(

even i have also got 40

the approach is

we get all $ {a_i}$s in order of such from

$ {p,q,p,q,p,q}$ (where $ {p \equal{} q}$}is also possible so we get 16 cases from here keeping in mind that we have to choose from 1,2,3,4 only

other cases give

the nos.

$ {p,q,p,q,p,p \minus{} q}$ it gives 6 cases
$ {p,q,p,q,p \minus{} q,q}$ it gives 6 cases

similarly other 2 cases(cyclic permutation of p-q) also give 6 cases each thus we get

16+6+6+6+6=40 cases in all

qed


Does anybody know the exact date of Mathcounts District? Our teacher told us it was Feb 11 but some other school annouced it to be the 25th. I really need to konw because I have to decide if I'll be able to go. thank you.

**EDIT**
nvm i got it, we were different districts. is there anybody here going to be at mathcounts? am i the only middle schooler now?

and you are?

btw if anybody's wondering, here's the results from chapter

eugene (seventh grader from hoover) 1st
kenneth liu - pizitz - 2nd
theo roth - pizitz - 3rd
me - pizitz - 4th
johnny wei - pizitz - 5th

that's the top 5. all 8 people from our school (pizitz) is going

i did really bad today. it was depressing.

yea... i think i saw you during lunch.

kenneth did exceptionally good. he usually isn't great a countdown.

i was way to tense to read any of the questions.

hope ill do better at state.

You guys lost to Simmons by I think .75 points. :/

[quote="LuCy4EvA"]and you are?[/quote]

One of the high school people there... Blue jacket, not Chinese or Russian

yea. was it scored so that it's only team? or does the individual count? they only had 2 people up in the top 10, we had 7... so i dunno how it works.

i don't really remember u specifically... but i know i saw you.

(sum of individual sprint + 2*target ) / 4 + team*2

You guys were up in target but down in sprint and team.

oh ok thank you. i didn't know that.

do you go to vestavia? or hoover?

The three of us that were there go to Vestavia.

I'm going to have to remove a part of this thread.  Remember that not all chapter competitions have taken place and MATHCOUNTS asks us to help prevent discussion of even scores and difficulty the events until all of them have taken place.

I'm guessing listing those who placed isn't going to hurt anything, so I'll leave the rest.

AoPS Historical Note -- the removed post was between Lucy's consecutive posts above.  She's not just talking to herself.  Elemennop was just giving his take on the level of some of the problems.

Congrats to the students who advanced to state.  I'm sure that was the easy part for you, so good luck at state.

Well thank you SO much for the note, I was worried the poor girl'd gone mad!
 ;)


algebra
function
domain
algebra unsolved

The domain and image of you function are wrong

Example: Put x=0 and then we have f(0) = f(0) + 1

Still looks wrong.

Put $ y \equal{} 0$ in to get $ f(xf(0)) \equal{} f(0) \plus{} 1$. Suppose $ f(0) \equal{} 0$, then $ f(xf(0)) \equal{} f(0) \equal{} 0 \neq f(0) \plus{} 1$, contradiction. If $ f(0) \neq 0$, then $ xf(0)$ can take all nonzero values since $ x$ is any nonzero real, so $ f(x) \equal{} f(0) \plus{} 1$ for all nonzero reals $ x$. Plug any nonzero $ x,y$ into the original equation to get the RHS to be $ f(xy) \plus{} 1 \equal{} f(0) \plus{} 2$. The other side is $ f$ of some number, and $ f(0) \plus{} 2$ is not in the image of $ f$. Contradiction.

Is the function supposed to be on the positive reals?

[quote="FOURRIER"]*$ f$: $ \mathbb{R} \rightarrow \mathbb{R}$
*$ f(xf(y)) \equal{} f(xy) \plus{} 1$ ($ x\neq 0$)

[hide="HINT"]After putting $ x \equal{} 1$ we have $ f o f(y) \equal{} f(y) \plus{} 1$ thus $ f(x) \equal{} x \plus{} 1$  $ \forall x \in f(\mathbb{R})$ But we still have to show $ f(\mathbb{R}) \equal{} \mathbb{R}$ wich  I can't prove it [/hide][/quote]
I think must be
$ f(xf(y))\equal{}f(xy)\plus{}x$


I really want to excel in math this year, but I constantly find myself wasting time...can somebody help motivate me to do more math?

Motivation comes from within. Why do you want to excel? The answer to this question should be enough to motivate you; if its not, then you don't truly want to excel.

[url=http://www.artofproblemsolving.com/Forum/viewtopic.php?t=280673]BEWARE: This is highly motivational[/url]

That's what made me motivated to do math. I always read it when I start getting off track and it really helps.

Set a goal for a checkpoint with determination. Never quit and even if everything fails, don't be gloomy.


Does there exists natural number $n$ such that $5^n$ is begining with $2$ and $2^n$ is beginig with $5$. They are represented in decimal notation.

yes,exist.
(1) $5^n=2\dots %Error. "Longarrow" is a bad command.
a+lg(2)<nlg(5)<a+lg(3),$
(2) $2^n=5\dots \Longrightarrow b+lg(5)<nlg(2)<b+lg(6)$
Because lg5/lg2 is irrational exist
$p>2max(\frac{1}{lg3-lg2},\frac{1}{lg6-lg5})=\frac{2}{lg6-lg5}:$
$|{plg5}|<1/\sqrt(p),|{plg2}|<1/\sqrt(p)$
It give n, satisfied (1),(2).


search geometry function calculus integration

I started randomly talking about this in a different thread, but it got pretty off-topic so I decided to carry it here.

Basically, what do YOU consider the difference between being Atheist, Theist, and Agnostic? I think atheist means believing completely that there is definitely no god, theism is the definite belief in a god, and agnosticism is not really being sure. Finally, not being any of the three, "other", just means having no beliefs whatsoever or having completely different beliefs that are still supernatural but don't necessarily include a god.

I consider myself "other". I believe in some sort of greater spirit/being/god floating out there beyond the universe, watching our every move, but not really having an influence in out lives. But then, I also believe in a twisted version of Karma (every time something good happens to you, something bad will follow, and vice versa) and have a very strange view of life after death. (It's a combination of Bhuddism's reincarnation and Christianity's heaven.)

Science is explaining what you see. Religion is explaining what you can't see.
I have some ideas on beliefs (every time you die you go into some sort of heaven for a period and then you reincarnate, sometimes you have a choice, sometimes you don't) but all in all I have no real beliefs about the next world (if there is in fact a next world). Technically I'm in the middle of agnosticism and atheism, with a little dash of other. When served with butter, I'm part of a complete breakfast! :)
I have morals and virtues (courage, honesty, compassion, etc.), but these are all concerning the current world. I also believe that in the current world we should be focusing on the current and not the next world.

[quote="1=2"]Religion is explaining what you can't see.[/quote]  Yes, because look at all the useful work theologians have done in developing germ theory, an understanding of radiation, quantum physics, relativity, and so on  :roll:  :wink:

Discussions of this sort (at least on the internet) often get bogged down in annoying discussions of linguistic details that don't shed much light at all on the underlying question.  For example, what precisely is the distinction between "not believing X exists" and "believing that X does not exist"?  As a practical matter, these two seem pretty interchangable (at least if, like me, you believe that religion is significant primarily in the ways it causes people to act), but I've seen plenty of arguments premised on the notion that there is some deep and important difference between them.

@JBL: Yes, I know these sort of discussions aren't exactly ideal, but then, religion is a very vague topic, which is what I like about it. There really isn't an "underlying question", or at least I wasn't trying to start one. I just kind of wanted to hear opinions. So you could say this thread really doesn't have a point.  :)

But on another note, I actually DO think there's a difference between not believing X exists and believing X doesn't exist. Simply not believing X exists means not aknowledging X's existence. But believing X doesn't exist means actually denying X's existence. That makes very little sense, but it has to do with my view of "existence".

I think that you only "exist" when someone or something aknowledges your existence and KNOWS you exist. For instance, if you spend your entire life on the street and no one, I mean NO ONE, ever aknowledges your existence and simply pass you by, you don't exist. Physically you exist, but I think "existing" is a matter of perception. Just like good and evil.

Another example: a piece of dirt doesn't exist to YOU until you notice it. Because until that moment that you aknowledged its existence, you didn't know it existed, and therefore, you did not believe it existed! However, even if you know about that piece of dirt but refuse to believe it exists, you believe it does not exist. Gah, that makes no sense whatsoever. Oh well.  :|

~~~~~~~~~~~~~~~~~~~~~~~~~~~~

@1=2: You said "Science is explaining what you see. Religion is explaining what you can't see." (I wish I knew how to quote posts.) I agree. I think religion is just an "excuse" (dang, that's such a harsh word) for explaining what we don't know. But then, that means that all my views of reincarnation and whatnot are just excuses for what I don't know. And it IS just an excuse. When there is sufficent proof that reincarnation does NOT exist, I probably won't believe in reincarnation anymore. But until then, I can keep dreaming, since in the end, "right" and "wrong" are simply a matter of perception.  :)

You could say that my one opinion that will never change is "humans are stupid, selfish, greedy creatures". I don't hate humanity, I just think we're all stupid and greedy and selfish (I most certainly am).

Haha, I probably just got off-topic on a thread that I created!

@fedja: I see the button, but I don't know how to use it! (do you copy+paste or something?!?)

Gah. I am lost.  :|

EDIT: nevermind, I figured it out.  :D

[quote="dreambold22"]
EDIT: nevermind, I figured it out.  :D[/quote]
Now you are a true believer in the "quote" button! I wish it would work this way with God...
(Unfortunately, this pun is more serious than it seems. :( But, nevermind, do continue with the argument and don't pay too much attention to these two posts of mine).

I am an atheist. I know that there is no god to the extent that I know anything. If the concept of agnosticism is just that you can not know everything, then it seems pointless to me.

[quote="fedja"][quote="dreambold22"](I wish I knew how to quote posts.)[/quote]
Just start believing that there exists a "quote" button in the right lower corner of each post. The effect will be immediate and overwhelming :P.[/quote][quote="fedja"][quote="dreambold22"]
EDIT: nevermind, I figured it out.  :D[/quote]
Now you are a true believer in the "quote" button! I wish it would work this way with God...
(Unfortunately, this pun is more serious than it seems. :()[/quote]  Awesome.

(And yes, this is spam.  Moderator's perogative  :P )

Edit: okay, a tiny bit of content.  It seems to me that when you write [quote="dreambold77"]I think that you only "exist" when someone or something aknowledges your existence and KNOWS you exist. For instance, if ... no one, I mean NO ONE, ever aknowledges your existence ... you don't exist.
...
However, even if you know about that piece of dirt but refuse to believe it exists, you believe it does not exist.[/quote] you are actually agreeing with me: you've just argued for the equivalence of "not believing" and "believing not," even though you claimed to be arguing for their inequivalence.

Generally, it seems to me that for purposes of believing in religious things, agnostics are just atheists who want to not be too disagreeable.  Or, perhaps, who want to reserve the right to continue believing in silly things and to call themselves "spiritual" without having to align themselves with any particular institution.

@Kalle:

[i]"Before God, we are equally wise - and equally foolish."[/i]
- Albert Einstein

I don't believe in God because there is no reason to.

As for agnosticism, you could include 85% of the worlds people (in your definition) because of all the people who are not completely sure or have slight nagging doubts. I would say someone is agnostic if they think it is 50-50 that god exists or not.

Can we know God in any other way than by our natural reason?
Besides knowing God by our natural reason, we can also know Him from supernatural revelation, that is, from the truths found in Sacred Scripture and in Tradition, which God Himself has revealed to us.

(a) Supernatural revelation is the communication of some truth by God to a creature through means that are beyond the ordinary course of nature. Some revealed truths, for example, the mystery of the Blessed Trinity, are strictly beyond the power of the human mind. We could never know such truths unless God revealed them. Other truths, for example, the immortality of the soul, while not beyond the power of the human mind, are objects of revelation because God has revealed them in a supernatural way. Although these latter truths could be known without revelation, they are grasped with greater ease and certainty once God has revealed them.

(b) God's public revelation of truths to men began with Adam and Eve and ended at the death of Saint John the Apostle.

(c) Divine revelation contained in the Old Testament is called pre-Christian. It can be divided into:

first, Primitive revelation, made to Adam and Eve
second, Patriarchal revelation, made to the patriarchs, for example, to Abraham and Lot;
third, Mosaic revelation, made to Moses and the prophets.

(d) Christian revelation contains the truths revealed to us by Jesus Christ, either directly or through His apostles.

(e) The Church does not oblige the faithful to believe private revelations given, at certain times, to individuals. For our edification, however, the Church permits the publication of some private revelations. Those to whom private revelations are given are obliged to believe them when they are certain that the revelations are from God.

(f) Sacred Scripture, or the Bible, is the word of God written by men under the inspiration of the Holy Ghost and contained in the books of the Old and the New Testament.

(g) Inspiration is the act by which God moves and directs the sacred writers faithfully to commit to writing all those things and only those things that He wishes them to write. The sacred writers act as free instruments of God, who is the principal author of Sacred Scripture.

(h) Tradition is the unwritten word of God--that body of truths revealed by God to the apostles, and not committed by them to writing but handed down by word of mouth. These truths, which were later committed to writing, particularly by the Fathers of the Church, have been preserved and handed down to the present day.

_____________________________________________________

@Ihatepie: Sometimes, people prefer that 50-50 way since they feel less of a burden. They feel that they don't have to follow the rules of the religion or anything this way.  They might also think that, if there isn't a God, oh well nothing wasted and if there is a God, well good I never said he didn't exist. But, that's not how God sees it. It's faith that he wants to see. People say that there is no proof, but that's part of the whole idea. It's one of God's mysteries.

@all This is NOT an argument about religion. We've had enough of those. As dreambold said, this is what you think the definitions of the words theist, agnostic, and atheist mean.

@dream if you actually did want this to be an argument on religion, sorry for misinterpreting your post.

The definitions of atheist, theist, and agnostic are pretty clear-cut.  I mean, look in the dictionary.  "The-" means god, "a-" is privative, and "gno" means know.

Yes, the actual definitions are pretty clear cut. But how people use them is a very different matter. For example, irony is when something happens the opposite of how it would seem. However, people commonly use it as any weird or strange occurrence.

[quote="matt276eagles"][quote="dreambold22"]what about the theists? This is something I never really understood. Did you simply follow the book and believe everything your holy book told you, or did you set your own beliefs? Why do you follow those beliefs? Do you REALLY believe in them, deep down?[/quote]

I really enjoyed reading your explanation of why you believe what you believe, but I have trouble seeing how you could be convinced that the actual truth is such a mix of different ideas.  If your faith is possibly different from anyone else's, how likely is it that it's correct? (Sorry if I offend anyone who believes more than one system of beliefs can be "correct")[/quote]

Well, first of all, I believe what I believe simply because to me, it makes logical sense. It's not some random crap I randomly threw together to be unique. I actually came to the conclusion of my belief through a series of conflicts and arguments in my mind. First, I started to believe there was more to death than eternal nothing-ness, so I started wondering what could happen after death. Eventually, I reached the conclusion that maybe after you die, you get reincarnated, but you don't know it. That was actually a theory I half-believed ever since like, first grade. Then, once I told myself that reincarnation did indeed exist, I wondered how you could get reincarnated if your physical body just slowly rots away in a grave. Zombies rising from cemeterys wasn't a very realistic thought, so I decided that only your soul gets reincarnated, the nonphysical part of your body that contains your personality and emotions.

Secondly, I do not consider my belief a "faith". I am not "faithful" to it. If someone else asserts a very logical (possibly scientific) argument on why my theory can't possibly be true, I will most probably stop believing in my theory.

I can believe in my theory even when nobody else does because: one, I hate going with the flow (most of the time). I don't care what other people think. Unless they can make a reasonable argument against my theory, they have no right to tell me I am wrong. Two, I believe religion is all a matter of perception. If you say God exists, then God does indeed exist to YOU. If that person believes god doesn't exist, then god doesn't exist to THAT PERSON. If I believe in reincarnation, then I will be reincarnated. If that person believes in eternal nothing-ness after death, then that person will experience eternal nothing-ness.

Okay, I know the last part of the above paragraph doesn't seem like a very logical argument, especially since I said I believe in reincarnation, but it's as Confucius said: "How can we know about death if we do not yet know about life?" In other words, quit worrying about what's beyond death, and worry about the present. We'll worry about death when it gets here. As I said in an earlier post, even though on the outside I believe in reincarnation, on the inside I'm still Agnosist(sp?). I don't KNOW what's beyond life. "If you only look forward, you will trip over the stones at your own feet." Whoever said that was very smart. Death is still (hopefully) a long ways off; enjoy your life now, worry about death when you get there. (that sounds weird....)

[quote="dreambold22"]so I decided that only your soul gets reincarnated, the nonphysical part of your body that contains your personality and emotions.[/quote]Well I'll start here. Emotions and personality are caused by neurons in your brain firing, not a nonphysical part of your body. It has been shown that certain areas of the brain fire neurons when exposed to certain emotions. Your genes also have control over your personality. They determine to an extent how your brain reacts to certain situations.

[quote="dreambold22"]Secondly, I do not consider my belief a "faith". I am not "faithful" to it. If someone else asserts a very logical (possibly scientific) argument on why my theory can't possibly be true, I will most probably stop believing in my theory.[/quote]You don't consider your belief a faith yet you're believing in something unprovable?

[quote="dreambold22"]I can believe in my theory even when nobody else does because: one, I hate going with the flow (most of the time). I don't care what other people think. Unless they can make a reasonable argument against my theory, they have no right to tell me I am wrong. Two, I believe religion is all a matter of perception. If you say God exists, then God does indeed exist to YOU. If that person believes god doesn't exist, then god doesn't exist to THAT PERSON. If I believe in reincarnation, then I will be reincarnated. If that person believes in eternal nothing-ness after death, then that person will experience eternal nothing-ness.[/quote]So if I don't think gravity exists, it doesn't exist to me? If fundamentalist Christians don't think evolution exists, it doesn't exist to them?

[quote="dreambold22"]
Eventually, I reached the conclusion that maybe after you die, you get reincarnated, but you don't know it.[/quote]
Is it possible for new "souls" to be created?  If not, how do you explain the fact that the world's population is always growing?  And assuming reincarnation holds implies it is impossible for the human species to ever become extinct.  Do you believe this too?

[quote="crazypianist1116"][quote="dreambold22"]so I decided that only your soul gets reincarnated, the nonphysical part of your body that contains your personality and emotions.[/quote]Well I'll start here. Emotions and personality are caused by neurons in your brain firing, not a nonphysical part of your body. It has been shown that certain areas of the brain fire neurons when exposed to certain emotions. Your genes also have control over your personality. They determine to an extent how your brain reacts to certain situations.[/quote]
I'm not saying I don't believe in the brain. I believe your brain is what allows you to THINK, to make decisions. But that warm, fuzzy feeling you get when you're with a best friend, or the undescribable rage inside you when you really hate someone's guts, these types of feelings just aren't the type of thing some big blob of neurons can create. And I'm not just talking about "personality" in the ways you act to your surroundings. The ways you act to certain situations is from your brain, for sure. But the way you ACT is just your outwards personality. Your inner, soul personality is how you FEEL. Well, not really feel, but just... well, I can't really describe it.

My "belief" is really probably too philosophical and abstract for a forum full of mathematicians who need proof for everything, but, well... we're all allowed to believe something, aren't we?

[quote="crazypianist1116"][quote="dreambold22"]Secondly, I do not consider my belief a "faith". I am not "faithful" to it. If someone else asserts a very logical (possibly scientific) argument on why my theory can't possibly be true, I will most probably stop believing in my theory.[/quote]You don't consider your belief a faith yet you're believing in something unprovable?[/quote]
no, it's just an "excuse", as you might say, a "placeholder" to make myself satisfied until someone proves otherwise. (Hey, you can't PROVE to me that there's eternal nothingness beyond death, just as we can't PROVE there is no God. While there is not proof FOR it, there is also no proof AGAINST it. Therefore, in that respect, anything is still possible.) Hey, every statement ever made is a theory. But then again, the fact that everything is a theory is a theory too.  :D

[quote="crazypianist1116"][quote="dreambold22"]I can believe in my theory even when nobody else does because: one, I hate going with the flow (most of the time). I don't care what other people think. Unless they can make a reasonable argument against my theory, they have no right to tell me I am wrong. Two, I believe religion is all a matter of perception. If you say God exists, then God does indeed exist to YOU. If that person believes god doesn't exist, then god doesn't exist to THAT PERSON. If I believe in reincarnation, then I will be reincarnated. If that person believes in eternal nothing-ness after death, then that person will experience eternal nothing-ness.[/quote]So if I don't think gravity exists, it doesn't exist to me? If fundamentalist Christians don't think evolution exists, it doesn't exist to them?[/quote]
Yeah, pretty much. That's about it.  :) But seriously, think about it: If you genuinely DID believe gravity didn't exist and you thought some supernatural force was holding us down, then if I told you that gravity exists, you would most certainly deny it!

[quote="eryaman"][quote="dreambold22"]Eventually, I reached the conclusion that maybe after you die, you get reincarnated, but you don't know it.[/quote]Is it possible for new "souls" to be created? If not, how do you explain the fact that the world's population is always growing? And assuming reincarnation holds implies it is impossible for the human species to ever become extinct. Do you believe this too?[/quote]
First, no, I don't believe souls can be created. However, I do believe souls can "divide", just like a cell. It's weird, but I can't figure out any other way to describe it. If a single soul becomes so diverse that it almost has enough personality for two souls, then it splits. And not just into two, sometimes into three or four, if they can. Yeah, really weird.

Second, it is possible for humans to go extinct. Because I believe animals also have souls. Human and animal souls are essentially the same; you could easily be reincarnated as an ant in your next life. In fact, it's rather unlikely you'll be reincarnated as a human twice in a row. And in the event that the earth explodes and everything on it dies, well.... I haven't gotten that far yet.  :oops:

[quote="dreambold22"][quote="crazypianist1116"][quote="dreambold22"]I can believe in my theory even when nobody else does because: one, I hate going with the flow (most of the time). I don't care what other people think. Unless they can make a reasonable argument against my theory, they have no right to tell me I am wrong. Two, I believe religion is all a matter of perception. If you say God exists, then God does indeed exist to YOU. If that person believes god doesn't exist, then god doesn't exist to THAT PERSON. If I believe in reincarnation, then I will be reincarnated. If that person believes in eternal nothing-ness after death, then that person will experience eternal nothing-ness.[/quote]So if I don't think gravity exists, it doesn't exist to me? If fundamentalist Christians don't think evolution exists, it doesn't exist to them?[/quote]
Yeah, pretty much. That's about it.  :) But seriously, think about it: If you genuinely DID believe gravity didn't exist and you thought some supernatural force was holding us down, then if I told you that gravity exists, you would most certainly deny it![/quote]  Certainly, but this hypothetical me would [i]be completely wrong[/i].   :wink:

[color=green]Moderator says: there is never a good reason to quote the entire post immediately preceding your own.[/color]

Maybe you'd tell the counter-you that they were wrong, but to the counter-you they are still right and you are wrong. It's all a matter of perception. After all, how can you say for sure that all those wonderful scientific theories are right? they are, in the end, only theories. what's "right" isn't actually the "truth of the universe", but merely society's veiw of what is "right".

There is no "good" or "evil" in this world, and neither is there "right" or "wrong". There is simply what is and what isn't. (I'm positive that turtle dude in Kung Fu Panda said that... hmm... whatever.)

If you take as an axiom that there is no such thing as an objective physical world whose properties we can observe, then I agree that this axiom may be used to invalidate any argument about any actual properties that the world may have.  However, I strongly suspect that you do not actually believe in this axiom -- in particular, your penultimate statement appears to directly contradict it.

In more detail: if "there is what is," then either there is gravity or there isn't.  It follows immediately that either I am wrong or the hypothetical me who claims there is no gravity is wrong.  However, you also assert the negation of this statement on the basis that it is theoretically possible to assert both the two statements "there is gravity" and "there is not gravity."  (Note that this contradiction has nothing to do with the effectiveness of science in determining the nature of the world, nor with the validity of any particular scientific theory.*) Perhaps you should work out this contradiction in your thinking before trying to push this too much further :).

Two further notes:
(1) No matter how you resolve this conflict, I still maintain that the hypothetical me is wrong.  Of course, he still maintains that I am wrong -- the important difference is that I am correct and he is not.
(2) In making this argument, I do not claim to say anything about whether it extends to moral questions, i.e., whether there is an objective moral framework.

* I would like to note, as an aside, that the disparagement of scientific results as "just theory" is a dishonest twisting of two different meanings of the same word, one formal (what a scientist means by the phrase "theory of gravity") and one informal (what a person means by the colloquialism "my theory is ..." or the like).  This rhetorical technique is most commonly practiced by those who wish to mislead others about the nature of the world for religio-political reasons.  It is, in many cases, a signifier of either the ignorance or intellectual dishonesty of the person employing it.  I strongly suggest that you avoid it in the future if this is not a persona you wish to promote for yourself.

I'm buddhist and really proud of my religion  :) . So , Does  the term Atheism only apply on christianism ?

[quote="dreambold22"] After all, how can you say for sure that all those wonderful scientific theories are right? they are, in the end, only theories. [/quote]
The main test is the predictive value. You describe an experiment that hasn't been staged yet and you predict what will happen. Then you stage it. If the predicted result happens, your theory gets a point, if not, it is refuted. Once the theory gets sufficiently many points, it is promoted to the status of an (approximate) truth. The problem with both religion and witchcraft is that their predictive value is currently zero. If you can change that, then I'll be the first to promote their study in schools on par with traditional sciences. If not, there is no point in questioning the things that do work and bring you everything from potatoes to satellite TV's.

[b]fedja[/b], I'm not exactly sure what you're trying to prove. I strongly doubt anyone here is proposing religion to be taught in school as being "on par" with science, in the sense you're using it. And the reason for that is pretty obvious, really -- we can stipulate that:

[b](1)[/b] Science is, by construction, a body of statements developed with the intention of having predictive value.
[b](2)[/b] Religion explains what we believe to be out of the plausible scope of human observation; hence the term "faith". Obviously, you can't measure predictive value of a theory which models that which you cannot observe. The exception to this is revelation and divine intervention, which still does not qualify as science because it is not generally reproducible.

I disagree with your statement that this is a [i]problem[/i] with religion. There is a problem if and only if someone, having a fundamental misunderstanding of what science and/or religion is, mixes up their roles, and then acts on this misunderstanding.

While the Universe can, apparently, be (very well) modeled as a closed, autonomous system, this is not necessarily the most insightful perspective. The only advantage of modeling the Universe this way is that it is the only perspective that will give us a reasonable level of predictive power, due to our highly limited capacity to observe.

Admitted, at our level, this has tremendous [i]practical[/i] value, and can lead to human intellectual satisfaction. Perhaps this is why it is fashionable among intellectuals to hold science as being fundamentally superior to religion. However, I think it is important to have the humility to realize that, in a macroscopic sense, this itself does not make the scientific model of the Universe fundamentally superior.

If you have issues with particular systems of belief being self-contradictory, or particular people doing absurd things in the name of religion, that's an entirely different issue.

[quote="JBL"]In more detail: if "there is what is," then either there is gravity or there isn't.  It follows immediately that either I am wrong or the hypothetical me who claims there is no gravity is wrong.  However, you also assert the negation of this statement on the basis that it is theoretically possible to assert both the two statements "there is gravity" and "there is not gravity."  (Note that this contradiction has nothing to do with the effectiveness of science in determining the nature of the world, nor with the validity of any particular scientific theory.*) Perhaps you should work out this contradiction in your thinking before trying to push this too much further.

Two further notes:
(1) No matter how you resolve this conflict, I still maintain that the hypothetical me is wrong.  Of course, he still maintains that I am wrong -- the important difference is that I am correct and he is not.
(2) In making this argument, I do not claim to say anything about whether it extends to moral questions, i.e., whether there is an objective moral framework.[/quote]

Okay, you were right about that first part... about my contradictory statement "There is simply what is and what isn't". I don't really have anything to back up that statement. It just kind of popped into my head randomly. Infact, I'm not really sure what it means.  :)

But on your first "further note", you say the important difference is that you are correct and he is not. But he would say the same thing about himself. How can you know he's "right" and you're "wrong"? The thing is, you really can't. Sure, you've got all this scientific evidence that says gravity exists, but before the concept of gravity was discovered, hundreds of years ago, the counter-you's thinking might of been "right". I'm not talking about mathematics, where everything has a "right" and "wrong" answer; I'm talking about life, religion, and everything. It just doesn't HAVE a "right" or "wrong" answer. The universe isn't that simple.

On a different note, I still don't think any amount of arguing will get either of us to really grasp what the other is saying... so I give up. Meh. :|

~~~~~~~~~~~~~~~~~~~~~~~~~

[quote="Tiger100"]I'm buddhist and really proud of my religion. So , Does the term Atheism only apply on christianism ?[/quote]

no, it doesn't only apply to christianism. "Theism" is the belief that there IS a god, "atheism" is simply the belief that there ISN'T a god. so, it can apply to any religion that has a "god".

Some people say that Buddhism is "theism", because the Buddha is basically a "god", but I'd consider it in the category "other", since for me the Buddha is not a "god", but Atheism usually implies the nonbelief in most other supernatural goings-on like reincarnation and souls, so it's really neither "theism" or "atheism". (At least, that's how I see it. A lot of people define the two terms differently.)

[quote="TZF"] I'm not exactly sure what you're trying to prove.[/quote]
Not much :) Merely that the scientific theories are a bit more than just elaborate exercises in wild imagination (though, I should admit that some of them seem much crazier than anything ever proposed by the priests and prophets).
[quote]
Religion explains what we believe to be out of the plausible scope of human observation; hence the term "faith". Obviously, you can't measure predictive value of a theory which models that which you cannot observe.
[/quote]
If you replace "explains" and "models" by "makes wild guesses about", I would agree 100% :).
Both the words you used require direct comparison with observations almost by their definitions (especially the word "model").
[quote]
The exception to this is revelation and divine intervention, which still does not qualify as science because it is not generally reproducible.
[/quote]
Non-reproducibility would be only half of a problem. But who has ever observed any of the two? I can easily tell you that I once saw a scratched commandment on my office door that disappeared five minutes later when I came back to show it to my friend. I can even tell you the message. It consisted of just one word: Evolve! (which, by itself, is not such a bad commandment, is it?) Will you believe me? If yes, why wouldn't you believe every crazy story I choose to present to you? If no, why do you believe the stories about saints and prophets? How can you tell the truth from a lie if you not only do not currently have any means to verify the statement but believe that no such means can be developed in principle?
[quote]
I disagree with your statement that this is a [i]problem[/i] with religion. There is a problem if and only if someone, having a fundamental misunderstanding of what science and/or religion is, mixes up their roles, and then acts on this misunderstanding.
[/quote]
I believe that the idea that religion explains and models anything is exactly an instance of such mix-up. The religion is a pure belief, not based on any observable reality. It is just a good exercise in creating an imaginary world and trying to live by its rules. We all played such games in our childhood and continue to play them all our life. There is absolutely nothing wrong with any such game as long as you remember that it is just a game and stop it when somebody gets hurt for real.
[quote]
The only advantage of modeling the Universe this way is that it is the only perspective that will give us a reasonable level of predictive power, due to our highly limited capacity to observe.
[/quote]
Erm... Has modeling ever had any other purposes than prediction and control?
[quote]
However, I think it is important to have the humility to realize that, in a macroscopic sense, this itself does not make the scientific model of the Universe fundamentally superior.
[/quote]
The problem is that the religion doesn't offer [b]any[/b] model of the Universe, not by today's standards, at least. It is not enough to postulate that God created this and that and tell the rough timeframe. If you want to model the creation, you should tell us every detail of how it was done. Nobody demands you to do all of it right now. Take your time and add the corresponding chapters to the Bible as you figure things out (like we do with our physics textbooks). The problem is not that the Book of Genesis is hard to believe (it is much easier to believe than quantum mechanics) but that there is nothing to believe there except some generalities that wouldn't satisfy even the curiosity of a child, not mentioning the one of a modern scientist. The whole story desperately lacks technical details.
[quote]
If you have issues with particular systems of belief being self-contradictory, or particular people doing absurd things in the name of religion, that's an entirely different issue.
[/quote]
No, unfortunately, I have issues with religion as a lore. The main problem is that it pretends to add something to my understanding of the world surrounding me but fails at that completely. At its best, it is just like poetry, only less beautiful. We all remember these lines:
[i]Our birth is but a sleep and a forgetting;
The Soul that rises with us, our life's Star,
Hath had elsewhere its setting.
And cometh from afar. ...[/i]
These words make a lot of sense to me and have absolutely no predictive value. But they do not try to "explain" or "model" something unobservable either. And nobody will ever think of starting a bloody war over whether the fourth word here should be "but" or "just". We just share the poem's beauty without an attempt to bitterly argue about its meaning and just let the words transform ourselves in some ways we don't even try to understand. I wish it worked the same way with the Bible but the history is too hard to forget...

[b]fedja[/b], I can see you're quite invested in your beliefs :) Before I respond, I just want to clarify that by religion, I am referring to a system of belief that does not explain the Universe as an autonomous function of itself, but  rather as one that has the potential to be affected by external influence. In the latter case, we can collectively call all entities with the power to influence the Universe "God". I am [i]not[/i] referring to institutional religion or  religion as a social phenomenon.

I understand that the two are highly intertwined at the practical level (for most people), but I think separating them has enough value that it's worth trying. In particular, if non-spiritual atheism is indeed the result of one's logical thinking (as many atheists claim) and not the result of a social phenomenon, then these atheists must have rejected religion in principle, and not just religion as a social institution.

[quote]Both the words you used require direct comparison with observations almost by their definitions (especially the word "model").[/quote]

If it bothers you, then replace "model" with "explain"  :) . And there are certainly observations that can you can make that religion can potentially explain in a way that is consistent with everything we know. For example, I observe  that the Universe exists. I observe that I am a conscious being. Etc.

[quote]Non-reproducibility would be only half of a problem. But who has ever observed any of the two?[/quote]

This is why reproducibility is important to science in the first place -- any scientist can say any random thing, but if the experiment is reproducible, scientists are more likely to believe it. Also, a scientist is more likely to be honest/serious if s/he knows that his/her work will be put to the test of reproducibility before being accepted.

Of course, this is not something that we have with religion. And, that's kind of the point. As with science, it is the individual's choice what to believe, and they have their own knowledge and intuition to help them decide, rather than observation and reproducibility.

Of course, some religions argue that the ability of people to make such judgments effectively is evidence of their beliefs in the first place, which has the danger of leading the believer into circular logic. But this isn't necessary.

[quote]I believe that the idea that religion explains and models anything is exactly an instance of such mix-up. The religion is a pure belief, not based on any observable reality.[/quote]

There's a rather large difference between religion/God not being directly observable and religion having nothing to do with observable reality. Otherwise, we would not have conceived of it in the first place. Religion attempts to explain those unobservable things that influence the observable world, and how they do so.

In the sense that religion makes claims about unobservable things, yes, it certainly is pure belief. But it being a "pure belief" doesn't make it any less of an explanation. More likely, what you meant is that mainstream religion does not provide [i]you[/i] with a [i]satisfactory[/i] explanation.

[quote][quote]The only advantage of modeling the Universe this way is that it is the only perspective that will give us a reasonable level of predictive power, due to our highly limited capacity to observe.
[/quote]
Erm... Has modeling ever had any other purposes than prediction and control?[/quote]

Good catch. I would answer that by differentiating between observable predictive power and actual predictive power. Suppose Ajdef subscribes to Christianity (I'm not a Christian, so hopefully this is unbiased). Christianity predicts what sort of lifestyle will lead a person to Heaven. Ajdef follows such a lifestyle, and ends up in Heaven. Christianity has provided Ajdef with predictive power. However, from the point of view of the rest of the world, the ultimate consequence Ajdef's lifestyle is not observable, and one, such as yourself, might perceive that the predictive value of Christianity is "absolutely zero".

[quote]The problem is that the religion doesn't offer [b]any[/b] model of the Universe, not by today's standards, at least. It is not enough to postulate that God created this and that and tell the rough timeframe. If you want to model the creation, you should tell us every detail of how it was done. Nobody demands you to do all of it right now. Take your time and add the corresponding chapters to the Bible as you figure things out (like we do with our physics textbooks). The problem is not that the Book of Genesis ...[/quote]

The fact that mainstream religion has a particularly skimpy section on the creation of the Universe does not demerit religion in general. It is more a testament to the perceived short attention span of the average working class person.

Nice, Wordsworth :) The Bible indeed has a bloody past, but again, I'm not really interested in the Bible, or mainstream religion in general, here (and I apologize for not clarifying that earlier).

[quote="dreambold22"][quote="JBL"]No matter how you resolve this conflict, I still maintain that the hypothetical me is wrong.  Of course, he still maintains that I am wrong -- the important difference is that I am correct and he is not.[/quote]

But on your first "further note", you say the important difference is that you are correct and he is not. But he would say the same thing about himself. How can you know he's "right" and you're "wrong"? The thing is, you really can't. Sure, you've got all this scientific evidence that says gravity exists, but before the concept of gravity was discovered, hundreds of years ago, the counter-you's thinking might of been "right". I'm not talking about mathematics, where everything has a "right" and "wrong" answer; I'm talking about life, religion, and everything. It just doesn't HAVE a "right" or "wrong" answer. The universe isn't that simple.

On a different note, I still don't think any amount of arguing will get either of us to really grasp what the other is saying.[/quote]  Since (I presume) you [i]agree with me[/i] that gravity exists, it's unclear to me what point you think you're making by repeatedly pointing out that "hypothetical me" asserts the opposite.  I agree that "hypothetical me" asserts the opposite -- but you and I both agree that he's wrong.  Now, I can't prove this in a way that doesn't rely on statements about and observations of the nature of the physical world, and if "hypothetical me" rejected all arguments based on, say, physical observations, I would never be able to convince him of the correctness of my position.  All this means is that "hypothetical me" is stubbornly wrong.  (And, as you may note, this last statement is again merely an assertion -- but again, it's an assertion you agree with!)

The fact that I can't necessarily convince someone who's stubbornly wrong does not, in fact, mean that they are not stubbornly wrong.  And "hypothetical me" could write this same post, of course, making the same assertions.  The difference is that [i]the universe exists[/i] ("there is what is") and we can make an appeal to the objective features of the universe, and so it is absolutely certain that at most one of me and "hypothetical me" is right.  And on this last point, I agree with "hypothetical me" -- we both happily admit that at most one of us may be correct, and that your belief that we could both be correct is incompatible with the notion of an objective physical world.  And we both agree that you also probably agree that there is an objective physical world, and have staked out your current position mostly because you wandered into a very bad analogy between religious claims (which tend to be either unverifiable, trivially consistent with the nature of the universe (something TZF seems to find impressive, for some reason), or wrong) and actual claims about the nature of the physical universe (which are either right or wrong, as they refer to some very particular object possessing very particular properties).

With respect to the last statement I've quoted: I think I understand perfectly well what you're saying, I just think that (1) it's wrong, and (2) you don't actually believe the logical implications of some of the things you're asserting.  You seem to be under the misconception that by pointing out that some of my statements are unproven (with reference to a system where one of the axioms might be "there are no such thing as physical laws") assertion, you will undermine my strength of belief in them.  But I already know that I'm making assertions that a stubborn "hypothetical me" wouldn't agree with -- this doesn't say anything at all about the truth value of my statements, only about the possibility of the existence of a person to honestly believe something that is completely wrong.  (Of course, you're perfectly welcome to not respond to this, as I am continuing to jump up and down after you've said you're not playing any more :).)

@TZF:  Off-topic, but your opening sentence caught my eye: [quote="TZF"][b]fedja[/b], I can see you're quite invested in your beliefs :)[/quote]  It reminds me of something that struck me oddly before -- see the last paragraph of [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1472565#1472565]this post[/url].  I'm a bit curious why you included this statement in your post and what purpose you think it serves.

It's about time I address the OP's question (since I posted [i]twice[/i] without doing so! Terrible me).

By definition, those are what the three terms mean. However, I don't think that's the most useful way of categorizing people's beliefs. I am more inclined to distinguish people as being spiritual, apathetic, or non-spiritual.

Theist is pretty straightforward.

I don't think Buddhism is really an atheist religion, even though some people describe it as being so. Buddha himself said that one can't know what the origin of the Universe is, and such. Some schools of Buddhist thought simply exalt the human spirit and soul, while others additionally worship the Buddha. However, Buddhist thought is certainly spiritual.

Then there's the agnostics. Agnostics can be either apathetic (ie, "I don't claim or care to know anything about God and spirituality, so I'll just keep on living my life") or spiritual. I think most people in the latter category are unsure of what "God" means in the first place (what powers God would have, etc), and for simplicity, just keep God out of the picture of their spirituality. One can take everything beyond the physical realm that has the power to influence it and collectively call it God -- confusion solved.

Then there's atheists. Atheists believe that God does not exist. There's the kind of Atheists that summarily reject spirituality (for widely varied reasons, some making much less sense than others), and there's the kind that claim to be spiritual despite being atheists. The latter, I think, is a bit clumsy, because on one hand, the person is believes that there is something beyond the physical realm of the observable Universe, that plays an integral role in our lives, but on the other hand, believes that there is no external power.

In terms of the atheist/agnostic/theist classification scheme, I think a lot of self-proclaimed atheists are really agnostics. They reject mainstream religion because of lack of compelling evidence for any of them. They also feel that spirituality has no practical value in helping them go about their lives. So, they leave it at that, and live a secular life.

[quote="JBL"]@TZF:  Off-topic, but your opening sentence caught my eye: [b](quote)[/b] It reminds me of something that struck me oddly before -- see the last paragraph of [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1472565#1472565]this post[/url].  I'm a bit curious why you included this statement in your post and what purpose you think it serves.[/quote]

JBL, no doubt you are a brilliant debater and mathematician -- I would be most honored to see your math proofs some day. I also Facebook stalk you regularly* (because I practically worship you), and am amazed by how really, really, incredibly good looking you are.

That said, I don't see the connection between my statement and the one you pointed out (apart from the fact that they both caught your eye). Math proofs have nothing whatsoever to do with proposition 8, whereas the nature of fedja's beliefs are at least somewhat related to a discussion about atheism (which he subscribes to).

And it's a true statement, he [i]is[/i] invested in his beliefs. I basically opened by saying, "I'm going to respond to your post, even though I don't expect you to change your mind." It was also an excuse to include another smiley in my post. It didn't really have any other purpose (although this is enough to make it more purposeful than many other statements made at the Round Table).

In direct response to the paragraph you linked me -- from my experience, people often use unrelated praise to seem more amiable, to keep the debate looking civil, and to prevent it from getting overheated. Not done properly, standalone criticism can come across as being aggressive, and some people use these means to prevent being perceived that way. No wonder you see it used all the time in political debates! Personally, I'd rather say, "I respectfully disagree," than, "you're super amazing in all that, but I disagree with you on this."

[size=75][i]* - despite not knowing your name.[/i][/size]


Determine wether or not it is possible to put the numbers 0,1,2...9 on a circle in such a way that the difference between consecutive placed numbers is 3,4,5.

Answer the same with the numbers 0,1,2,3,....13.

Consider the set (0;1;7;8;9). None of the elements of this set can be consecutive. Hence every two consecutive numbers on the circle must contain exactly one element of the set. Consider integer 2. 2 must be put between the elements of the set, as none of these elements can be consecutive and there are 10 numbers and the set contains 5 elements. But only integers 2 and 7 can be consecutive, and none other integer from the set can be consecutive with two. On the other hand, two elements of the set must be consecutive with 2, as we assumed. Hence the given integers can not be writen on the circle having the desired property.

For the second part of the question the answer is positive. It may be shown by this sequence example (the last and the first elements are consecutive, as the sequence is writen around the circle):
4, 0, 3, 7, 11, 8, 12, 9, 13, 10, 6, 2, 5, 1.

I hope you could understand something, as my English is not very well...


Evaluate cos 36 - cos 72 using trigonometric formulas and identities.

Note: Angle measurements are in degrees.

[hide="Hint"]Obviously, you just need to compute one of them, since you can then call on a half/double-angle formula for the other.  To compute $ \cos{72}$, try coming up with a more generalized multi-angle formula; i.e., $ \cos(3\theta)$, $ \cos(4\theta)$,... and then use this to create a polynomial, one of whose roots is $ \cos{72}$.[/hide]


1. Let p(x) be a polynomial such that p(cos(x))= cos(3x) for all x, which of the following is a factor for p(x)?
a. x-(6/pi)     b. x-(3/pi)   c. x-((2√2)/2)   d. x-((2√3)/2)   e. x-(pi/2)

2. let "a" be a positive number, what values of "a" have a solution x fpr the equation a^x=x?
a. 0to 0.5  b. 0to0.71  c. 0 to 1  d. 0 to 1.41  e 1 to 1.44

3. Using pennies, nickels, dimes, and quarters, what is the largest possible total valve of coins you can have without having exact change for a dollar?
a.99cents   b. 101 cents    c. 104 cents   d. 109 cents   e. none of these

4.  First  x is chosen at radom from the set ( 1,2,3,...99,100), and then y is chosen at random from the same full set. The possiblity that the integer 3x+7y has untis digit 8 is?

thx for helping me.

[hide="Number 3 answer"] 3 is e since 3 quarters, 4 dimes, and 4 pennies is 119 cents, which is greater than the largest answer.[/hide]

the answer is e for num.3..... actually i cannot understand what it means..-.-

I know, I changed it about 10 seconds after

You see, for the combination of coins I posted, try to make exactly a dollar using some of them. You can't. They're asking what's the most money you can have so you can't make a dollar.

but since they have pennies, i can make whatever i want. like 3 quarters, 2 nickels, i dimes and 5 pennies, and this is a dollar.. sorry, i am ESL..asking stupid questions.. thx for answering..

[hide="Number 4 Solution"]
A units digit of 8 can only result if the original pair of numbers ends in the units digit combinations of
0 and 8
1 and 7
2 and 6
3 and 5
4 and 4
9 and 9

It is clear to see a pattern when finding that "3x" could be:
30 60 90
18 48 78
21 51 81
27 57 87
12 42 72
33 63 93
15 45 75
24 54 84
 9  39 69 99
 6  36 66 96

and in a similar pattern, "7y" could be:
7
70
28
21
77
42
63
35
84
14
49
56

Working through the messy conditional prob.,
The probabilty of 3x + 7y having a units digit of 8 is [b]19/176[/b]

I am probably overlooking a simpler way than semi-brute-forcing this, but it works for these small numbers from 1 to 100.[/hide]
Also, For problem 1, it could be said that P(x) = 4x^3 - sqrt(1 - x^2)

[hide="#4"]
Multiples of 3 (mod 10) includes all ten digits. Same goes for multiples of 7 (mod 10).

That means for either $ x$ or $ y$, there's an equal chance to choose a number of any units digit.

We're adding two numbers to get a units digit of $ 8$. Let's consider the units digit of either of the two numbers summed:

$ 0\plus{}8\equal{}8$
$ 1\plus{}7\equal{}8$
.
.
.
$ 8\plus{}0\equal{}8$

That's $ 9$ ways.

$ 8\plus{}10\equal{}18\equal{}8\mbox{ (mod 10)}$
$ 9\plus{}9\equal{}18\equal{}8\mbox{ (mod 10)}$
$ 10\plus{}8\equal{}18\equal{}8\mbox{ (mod 10)}$

That's $ 3$ ways.

So $ 12$ ways out of $ 10^2\equal{}100$ total combinations,

that's $ \frac{12}{100}\equal{}\boxed{\frac{3}{25}}$.

Hopefully I didn't oversimplify this thing :|
[/hide]