A class has $25$ students. The teacher wants to stock $N$ candies, hold the Olympics and give away all $N$ candies for success in it (those who solve equally tasks should get equally, those who solve less get less, including, possibly, zero candies). At what smallest $N$ this will be possible, regardless of the number of tasks on Olympiad and the student successes?

Triangle $ABC$ has $AB=10$, $BC=17$, and $CA=21$. Point $P$ lies on the circle with diameter $AB$. What is the greatest possible area of $APC$?

Find $k$ where $2^k$ is the largest power of $2$ that divides the product \[2008\cdot 2009\cdot 2010\cdots 4014.\]

There are many opposition societies in the city of N. Each society consists of $10$ members. It is known that for every $2004$ societies there is a person belonging to at least $11$ of them. Prove that the government can arrest $2003$ people so that at least one member of each society is arrested. [i]Proposed by V.Dolnikov, D.Karpov[/i]

Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from $1$ to $2024$ that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by $2024$ loses. Who wins if every player wants to win? [i]Proposed by Mykhailo Shtandenko[/i]

Let $ABCD$ be a cyclic quadrilateral. The lines $BC$ and $AD$ meet at a point $P$. Let $Q$ be the point on the line $BP$, different from $B$, such that $PQ=BP$. Consider the parallelograms $CAQR$ and $DBCS$. Prove that the points $C,Q,R,S$ lie on a circle.

Inside the equilateral triangle $ABC$, points $P$ and $Q$ are chosen so that the quadrilateral $APQC$ is convex, $AP = PQ = QC$ and $\angle PBQ = 30^o$. Prove that $AQ = BP$.

Let $ABC$ be an acute triangle with $AB=AC$, let $D$ be the midpoint of the side $AC$, and let $\gamma$ be the circumcircle of the triangle $ABD$. The tangent of $\gamma$ at $A$ crosses the line $BC$ at $E$. Let $O$ be the circumcenter of the triangle $ABE$. Prove that midpoint of the segment $AO$ lies on $\gamma$. [i]Proposed by Sam Bealing, United Kingdom[/i]

Given is a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $|f(x+y)-f(x)-f(y)|\leq 1$. Prove the existence of an additive function $g:\mathbb{R}\rightarrow \mathbb{R}$ (that is $g(x+y)=g(x)+g(y)$) such that $|f(x)-g(x)|\leq 1$ for any $x \in \mathbb{R}$

All the roots of polynomial $z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16$ are positive integers. What is the value of $B$? $\textbf{(A)}\ -88 \qquad\textbf{(B)}\ -80 \qquad\textbf{(C)}\ -64\qquad\textbf{(D)}\ -41 \qquad\textbf{(E)}\ -40$

Show that for any positive real numbers $a$, $b$, $c$ the following inequality takes place $$\frac{a}{\sqrt{7a^2+b^2+c^2}}+\frac{b}{\sqrt{a^2+7b^2+c^2}}+\frac{c}{\sqrt{a^2+b^2+7c^2}} \leq 1.$$

The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^{\circ}$ around the point $(1,5)$ and then reflected about the line $y=-x$. The image of $P$ after these two transformations is at $(-6,3)$. What is $b-a$? $\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9$

An infinite checkerboard is divided by a horizontal line into upper and lower halves as shown on the right. A number of checkers are to be placed on the board below the line (within the squares). A "move" consists of one checker jumping horizontally or vertically over a second checker, and removing the second checker. What is the minimum value of $n$ which will allow the placement of the last checker in row 4 above the dividing horizontal line after $n-1$ moves? Describe the initial position of the checkers as well as each of the moves. Picture: http://www.cms.math.ca/Competitions/IMTS/imts6.gif

Consider a triangle $XYZ$ and a point $O$ in its interior. Three lines through $O$ are drawn, parallel to the respective sides of the triangle. The intersections with the sides of the triangle determine six line segments from $O$ to the sides of the triangle. The lengths of these segments are integer numbers $a, b, c, d, e$ and $f$ (see figure). Prove that the product $a \cdot b \cdot c\cdot d \cdot e \cdot f$ is a perfect square. [asy] unitsize(1 cm); pair A, B, C, D, E, F, O, X, Y, Z; X = (1,4); Y = (0,0); Z = (5,1.5); O = (1.8,2.2); A = extension(O, O + Z - X, X, Y); B = extension(O, O + Y - Z, X, Y); C = extension(O, O + X - Y, Y, Z); D = extension(O, O + Z - X, Y, Z); E = extension(O, O + Y - Z, Z, X); F = extension(O, O + X - Y, Z, X); draw(X--Y--Z--cycle); draw(A--D); draw(B--E); draw(C--F); dot("$A$", A, NW); dot("$B$", B, NW); dot("$C$", C, SE); dot("$D$", D, SE); dot("$E$", E, NE); dot("$F$", F, NE); dot("$O$", O, S); dot("$X$", X, N); dot("$Y$", Y, SW); dot("$Z$", Z, dir(0)); label("$a$", (A + O)/2, SW); label("$b$", (B + O)/2, SE); label("$c$", (C + O)/2, SE); label("$d$", (D + O)/2, SW); label("$e$", (E + O)/2, SE); label("$f$", (F + O)/2, NW); [/asy]

Set $A=\{1,2,\cdots,n\}$. If there exists nonempty sets $B,C$, such that $B\cap C=\varnothing,B\cup C=A$. Sum of Squares of all elements in $B$ is $M$, Sum of Squares of all elements in $C$ is $N$, $M-N=2016$. Find the minimum value of $n$.

Let $(a_n)$ be defined by: $$ a_1 = 2, \qquad a_{n+1} = a_n^3 - a_n + 1 $$ Consider positive integers $n,p$, where $p$ is an odd prime. Prove that if $p | a_n$, then $p > n$.

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

The numbers from $1$ to $50$ are printed on cards. The cards are shuffled and then laid out face up in $5$ rows of $10$ cards each. The cards in each row are rearranged to make them increase from left to right. The cards in each column are then rearranged to make them increase from top to bottom. In the final arrangement, do the cards in the rows still increase from left to right?

Let $b\ge 2$ be an integer. Call a positive integer $n$ $b$-[i]eautiful[/i] if it has exactly two digits when expressed in base $b$ and these two digits sum to $\sqrt{n}$. For example, $81$ is $13$-[i]eautiful[/i] because $81 = \underline{6} \ \underline{3}_{13} $ and $6 + 3 = \sqrt{81}$. Find the least integer $b\ge 2$ for which there are more than ten $b$-[i]eautiful[/i] integers.

Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I.$ Let the internal angle bisectors of $\angle A,\angle B,\angle C$ meet $\Gamma$ in $A',B',C'$ respectively. Let $B'C'$ intersect $AA'$ at $P,$ and $AC$ in $Q.$ Let $BB'$ intersect $AC$ in $R.$ Suppose the quadrilateral $PIRQ$ is a kite; that is, $IP=IR$ and $QP=QR.$ Prove that $ABC$ is an equilateral triangle.

Let $P$ be a point in a square whose side are mirror. A ray of light comes from $P$ and with slope $\alpha$. We know that this ray of light never arrives to a vertex. We make an infinite sequence of $0,1$. After each contact of light ray with a horizontal side, we put $0$, and after each contact with a vertical side, we put $1$. For each $n\geq 1$, let $B_{n}$ be set of all blocks of length $n$, in this sequence. a) Prove that $B_{n}$ does not depend on location of $P$. b) Prove that if $\frac{\alpha}{\pi}$ is irrational, then $|B_{n}|=n+1$.

Determine all prime numbers $p$ for which $8p^4-3003$ is a positive prime number.

If two distinct integers from $1$ to $50$ inclusive are chosen at random, what is the expected value of their product? Note: The expectation is defined as the sum of the products of probability and value, i.e., the expected value of a coin flip that gives you $\$10$ if head and $\$5$ if tail is $\tfrac12\times\$10+\tfrac12\times\$5=\$7.5$.

Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that \[f(n+1)>\frac{f(n)+f(f(n))}{2}\] for all $n\in\mathbb{N}$, where $\mathbb{N}$ is the set of strictly positive integers.

Let $ABC$ be a triangle with symmedian point $K$. Select a point $A_1$ on line $BC$ such that the lines $AB$, $AC$, $A_1K$ and $BC$ are the sides of a cyclic quadrilateral. Define $B_1$ and $C_1$ similarly. Prove that $A_1$, $B_1$, and $C_1$ are collinear. [i]Proposed by Sammy Luo[/i]