On a $49\times 69$ rectangle formed by a grid of lattice squares, all $50\cdot 70$ lattice points are colored blue. Two persons play the following game: In each step, a player colors two blue points red, and draws a segment between these two points. (Different segments can intersect in their interior.) Segments are drawn this way until all formerly blue points are colored red. At this moment, the first player directs all segments drawn - i. e., he takes every segment AB, and replaces it either by the vector $\overrightarrow{AB}$, or by the vector $\overrightarrow{BA}$. If the first player succeeds to direct all the segments drawn in such a way that the sum of the resulting vectors is $\overrightarrow{0}$, then he wins; else, the second player wins. Which player has a winning strategy?

For a positive number $x$, let $f(x)=\lim_{n\to\infty} \sum_{k=1}^n \left|\cos \left(\frac{2k+1}{2n}x\right)-\cos \left(\frac{2k-1}{2n}x\right)\right|$ Evaluate \[\lim_{x\rightarrow\infty}\frac{f(x)}{x}\]

Hi guys, I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this: 1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though. 2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary. 3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions: A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh? B. Do NOT go back to the previous problem(s). This causes too much confusion. C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for. 4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving! Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D

$ ABCD$ is a cyclic quadrilateral. If $ \angle B \equal{} \angle D$, $ AC\bigcap BD \equal{} \left\{E\right\}$, $ \angle BCD \equal{} 150{}^\circ$, $ \left|BE\right| \equal{} x$, $ \left|AC\right| \equal{} z$, then find $ \left|ED\right|$ in terms of $ x$ and $ z$. $\textbf{(A)}\ \frac {z \minus{} x}{\sqrt {3} } \qquad\textbf{(B)}\ \frac {z \minus{} 2x}{3} \qquad\textbf{(C)}\ \frac {z \plus{} x}{\sqrt {3} } \qquad\textbf{(D)}\ \frac {z \minus{} 2x}{2} \qquad\textbf{(E)}\ \frac {2z \minus{} 3x}{2}$

For each positive integer $ n$, evaluate the sum \[ \sum_{k\equal{}0}^{2n}(\minus{}1)^{k}\frac{\binom{4n}{2k}}{\binom{2n}{k}}\]

From the identity $$ \int_{0}^{\pi \slash 2} \log \sin 2x \, dx = \int_{0}^{\pi \slash 2} \log \sin x \, dx + \int_{0}^{\pi \slash 2} \log \cos x \, dx +\int_{0}^{\pi \slash 2} \log 2 \, dx, $$ deduce the value of $\int_{0}^{\pi \slash 2} \log \sin x \, dx.$

Prove that \[\sum_{cyc}(x+y)\sqrt{(z+x)(z+y)} \geq 4(xy+yz+zx),\] for all positive real numbers $x,y$ and $z$.

Let $ABCD$ be a cyclic quadrilateral with $|AB|=26$, $|BC|=10$, $m(\widehat{ABD})=45^\circ$,$m(\widehat{ACB})=90^\circ$. What is the area of $\triangle DAC$ ? $ \textbf{(A)}\ 120 \qquad\textbf{(B)}\ 108 \qquad\textbf{(C)}\ 90 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ 80 $

For constant $k>1$, 2 points $X,\ Y$ move on the part of the first quadrant of the line, which passes through $A(1,\ 0)$ and is perpendicular to the $x$ axis, satisfying $AY=kAX$. Let a circle with radius 1 centered on the origin $O(0,\ 0)$ intersect with line segments $OX,\ OY$ at $P,\ Q$ respectively. Express the maximum area of $\triangle{OPQ}$ in terms of $k$. [i]2011 Tokyo Institute of Technology entrance exam, Problem 3[/i]

Find the area of a figure consisting of points whose coordinates satisfy the inequality $$(y^3 - arcsin x)(x^3 + arcsin y) \ge 0.$$

Find all polynomials $P(x)$ with real coefficients that satisfy \[P(x\sqrt{2})=P(x+\sqrt{1-x^2})\]for all real $x$ with $|x|\le 1$.

$(GBR 3)$ A smooth solid consists of a right circular cylinder of height $h$ and base-radius $r$, surmounted by a hemisphere of radius $r$ and center $O.$ The solid stands on a horizontal table. One end of a string is attached to a point on the base. The string is stretched (initially being kept in the vertical plane) over the highest point of the solid and held down at the point $P$ on the hemisphere such that $OP$ makes an angle $\alpha$ with the horizontal. Show that if $\alpha$ is small enough, the string will slacken if slightly displaced and no longer remain in a vertical plane. If then pulled tight through $P$, show that it will cross the common circular section of the hemisphere and cylinder at a point $Q$ such that $\angle SOQ = \phi$, $S$ being where it initially crossed this section, and $\sin \phi = \frac{r \tan \alpha}{h}$.

Let $ f(x)\equal{}x^{2}\plus{}|x|$. Prove that $ \int_{0}^{\pi}f(\cos x)\ dx\equal{}2\int_{0}^{\frac{\pi}{2}}f(\sin x)\ dx$.

The isosceles right triangle $ABC$ has right angle at $C$ and area $12.5$. The rays trisecting $\angle{ACB}$ intersect $AB$ at $D$ and $E$. What is the area of $\triangle{CDE}$? $\textbf{(A) }\frac{5\sqrt{2}}{3}\qquad\textbf{(B) }\frac{50\sqrt{3}-75}{4}\qquad\textbf{(C) }\frac{15\sqrt{3}}{8}\qquad\textbf{(D) }\frac{50-25\sqrt{3}}{2}\qquad\textbf{(E) }\frac{25}{6}$

Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$. [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

It is given the function $f:\mathbb{R} \to \mathbb{R}$ such that it holds $f(\sin x)=\sin (2011x)$. Find the value of $f(\cos x)$.

The number of real solutions to the equation \[ \frac{x}{100} = \sin x \] is $\text{(A)} \ 61 \qquad \text{(B)} \ 62 \qquad \text{(C)} \ 63 \qquad \text{(D)} \ 64 \qquad \text{(E)} \ 65$

Find all real numbers for which $3^x+4^x=5^x$.

The number of $ x$-intercepts on the graph of $ y \equal{} \sin(1/x)$ in the interval $ (0.0001,0.001)$ is closest to $ \textbf{(A)}\ 2900 \qquad \textbf{(B)}\ 3000 \qquad \textbf{(C)}\ 3100 \qquad \textbf{(D)}\ 3200 \qquad \textbf{(E)}\ 3300$

Find the area of the figure bounded by three curves $ C_1: y\equal{}\sin x\ \left(0\leq x<\frac {\pi}{2}\right)$ $ C_2: y\equal{}\cos x\ \left(0\leq x<\frac {\pi}{2}\right)$ $ C_3: y\equal{}\tan x\ \left(0\leq x<\frac {\pi}{2}\right)$.

Find the number of functions defined on positive real numbers such that $ f\left(1\right) \equal{} 1$ and for every $ x,y\in \Re$, $ f\left(x^{2} y^{2} \right) \equal{} f\left(x^{4} \plus{} y^{4} \right)$. $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{Infinitely many}$

Prove the following inequalities: (1) $ x\minus{}\sin x\leq \tan x\minus{}x\ \ \left(0\leq x<\frac{\pi}{2}\right)$ (2) $ \int_0^x \cos (\tan t\minus{}t)\ dt\leq \sin (\sin x)\plus{}\frac 12 \left(x\minus{}\frac{\sin 2x}{2}\right)\ \left(0\leq x\leq \frac{\pi}{3}\right)$

Prove that the number $\tan \frac{\pi}{3^n}$ is irrational for any natural $n$.

Find all ordered triples $ (x, y, z)$ of real numbers such that \[ 5 \left(x \plus{} \frac{1}{x} \right) \equal{} 12 \left(y \plus{} \frac{1}{y} \right) \equal{} 13 \left(z \plus{} \frac{1}{z} \right),\] and \[ xy \plus{} yz \plus{} zy \equal{} 1.\]

For real numbers $ a,\ b$ such that $ |a|\neq |b|$, let $ I_n \equal{} \int \frac {1}{(a \plus{} b\cos \theta)^n}\ (n\geq 2)$. Prove that : $ \boxed{\boxed{I_n \equal{} \frac {a}{a^2 \minus{} b^2}\cdot \frac {2n \minus{} 3}{n \minus{} 1}I_{n \minus{} 1} \minus{} \frac {1}{a^2 \minus{} b^2}\cdot\frac {n \minus{} 2}{n \minus{} 1}I_{n \minus{} 2} \minus{} \frac {b}{a^2 \minus{} b^2}\cdot\frac {1}{n \minus{} 1}\cdot \frac {\sin \theta}{(a \plus{} b\cos \theta)^{n \minus{} 1}}}}$