The squares of an $m\times n$ board are labeled from $1$ to $mn$ so that the squares labeled $i$ and $i+1$ always have a side in common. Show that for some $k$ the squares $k$ and $k+3$ have a side in common.

Prove that if $n$ is large enough, among any $n$ points of plane we can find $1000$ points such that these $1000$ points have pairwise distinct distances. Can you prove the assertion for $n^{\alpha}$ where $\alpha$ is a positive real number instead of $1000$?

Line segment $\overline{AB}$ is a diameter of a circle with $AB=24$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle{ABC}$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve? $\textbf{(A)} \text{ 25} \qquad \textbf{(B)} \text{ 38} \qquad \textbf{(C)} \text{ 50} \qquad \textbf{(D)} \text{ 63} \qquad \textbf{(E)} \text{ 75}$

Let $a>1$ be a positive integer and let $f(n)=n+[a\{n\sqrt{2}\}]$. Show that there exists a positive integer $n$, such that $f(f(n))=f(n)$, but $f(n) \neq n$.

Let $a, b$ be two real numbers that satisfy $a^3 + b^3 = 8-6ab$. Find the maximum value and the minimum value that $a + b$ can take.

Let $n^2-6n+1=0$. Find $n^6+\frac1{n^6}$.

In a non-obtuse triangle $ABC$, prove that \[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[ 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) \] is eventually constant. [The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \; (\mbox{mod} \; n)$ means the remainder which results from dividing $a_i$ by $n$.]

A particle moves along the $x$-axis. It collides elastically head-on with an identical particle initially at rest. Which of the following graphs could illustrate the momentum of each particle as a function of time? [asy] size(400); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,0)--(0,5)); draw((0,1.5)--(5,1.5)); label("$p$",(0,5),N); label("$t$",(5,1.5),E); label("$\mathbf{(A)}$",(2.5,-0.5)); draw((0,1.5)--(2.5,1.5)--(2.5,0.75)--(4,0.75),black+linewidth(2)); draw((0,3.5)--(2.5,3.5)--(2.5,4.25)--(4,4.25),black+linewidth(2)); draw((8,0)--(8,5)); draw((8,1.5)--(13,1.5)); label("$p$",(8,5),N); label("$t$",(13,1.5),E); label("$\mathbf{(B)}$",(10.5,-0.5)); draw((8,1.5)--(10.5,1.5)--(10.5,2.5)--(12,2.5),black+linewidth(2)); draw((8,3.5)--(10.5,3.5)--(10.5,4.5)--(12,4.5),black+linewidth(2)); draw((16,0)--(16,5)); draw((16,1.5)--(21,1.5)); label("$p$",(16,5),N); label("$t$",(21,1.5),E); label("$\mathbf{(C)}$",(18.5,-0.5)); draw((16,1.5)--(18.5,1.5)--(18.5,2.25)--(20,2.25),black+linewidth(2)); draw((16,3.5)--(18.5,3.5)--(18.5,2.75)--(20,2.75),black+linewidth(2)); draw((24,0)--(24,5)); draw((24,1.5)--(29,1.5)); label("$p$",(24,5),N); label("$t$",(29,1.5),E); label("$\mathbf{(D)}$",(26.5,-0.5)); draw((24,1.5)--(26.5,1.5)--(26.75,3.25)--(28,3.25),black+linewidth(2)); draw((24,3.25)--(26.5,3.25)--(26.75,1.5)--(28,1.5),black+linewidth(2)); draw((32,0)--(32,5)); draw((32,1.5)--(37,1.5)); label("$p$",(32,5),N); label("$t$",(37,1.5),E); label("$\mathbf{(E)}$",(34.5,-0.5)); draw((32,1.5)--(34.5,1.5)--(34.5,0.5)--(36,0.5),black+linewidth(2)); draw((32,3.5)--(34.5,3.5)--(34.5,2.75)--(36,2.75),black+linewidth(2)); [/asy]

How many solutions does the equation $ \frac{[x]}{\{ x\}} =\frac{2007x}{2008} $ have? [i]Mihail Bălună[/i]

Let eight points be given in a unit cube. Prove that two of these points are on a distance not greater than $1$.

([b]4[/b]) Find all $ y > 1$ satisfying $ \int^y_1x\ln x\ dx \equal{} \frac {1}{4}$.

$x,y,z\in\left(0,\frac{\pi}{2}\right)$ are given. Prove that: \[ \frac{x\cos x+y\cos y+z\cos z}{x+y+z}\le \frac{\cos x+\cos y+\cos z}{3} \]

For a natural number $n \ge 2$, consider all representations of $n$ as a sum of its distinct divisors, $n = t_1 + t_2 + ... + t_k, t_i| n$. Two such representations differing only in order of the summands are considered the same (for example, $20 = 10+5+4+1$ and $20 = 5+1+10+4$). Let $a(n)$ be the number of different representations of $n$ in this form. Prove or disprove: There exists M such that $a(n) \le M$ for all $n \ge 2$.

Determine all triples $(p,m,k)$ of positive integers such that $p$ is a prime number, $m$ and $k$ are odd integers, and $m^4+4^kp^4$ divides $m^2(m^4-4^kp^4)$.

Solve the equation $tan^2 2x+2 tan2x tan3x = 1$

Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following conditions: $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2.$ [i]Based on a problem of Romanian Masters of Mathematics[/i]

You are given a positive integer $n$. Find the smallest positive integer $k$, for which there exist integers $a_1, a_2, \ldots, a_k$, for which the following equality holds: $$2^{a_1} + 2^{a_2} + \ldots + 2^{a_k} = 2^n - n + k$$ [i]Proposed by Mykhailo Shtandenko[/i]

In nonnegative set of integers solve the equation: $$(2^{2015}+1)^x + 2^{2015}=2^y+1$$

For positive integers $ n$, denote by $ D(n)$ the number of pairs of different adjacent digits in the binary (base two) representation of $ n$. For example, $ D(3) \equal{} D(11_2) \equal{} 0$, $ D(21) \equal{} D(10101_2) \equal{} 4$, and $ D(97) \equal{} D(110001_2) \equal{} 2$. For how many positive integers $ n$ less than or equal to $ 97$ does $ D(n) \equal{} 2$? $ \textbf{(A)}\ 16\qquad \textbf{(B)}\ 20\qquad \textbf{(C)}\ 26\qquad \textbf{(D)}\ 30\qquad \textbf{(E)}\ 35$

Let $ f$ be a linear function for which $ f(6)\minus{}f(2)\equal{}12$. What is $ f(12)\minus{}f(2)$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 36$

Let $ABC$ be a triangle with $\angle ACB=90$ and $AC>BC.$ Let $\Omega$ be the circumcircle of $ABC.$ Point $ D $ is the midpoint of small arc $AC$ of $\Omega.$ Point $ M $ is symmetric with $ A$ with respect to $D.$ Point $ N$ is the midpoint of $MC.$ Line $AN$ intersects $\Omega$ in point $ P $ and line $BP$ intersects line $DN$ in point $Q.$ Prove that line $QM$ passes through the midpoint of $AC.$

Let $\alpha,\beta$ be the distinct positive roots of the equation of $2x=\tan x$. Evaluate \[\int_0^1 \sin \alpha x\sin \beta x\ dx\]

The sums of three whole numbers taken in pairs are $12$, $17$, and $19$. What is the middle number? $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 $

Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.