In a foreign island $5781$ sheep are sacrificed every year for the two deities of the island, Alice and Bob. Every deity wants as many sheep as he can to be sacrificed for him, and not for the other deity. Initially all $5781$ sheep are arranged around a circle with equal distances. At the first step, Alice puts one magic stone between several pairs of neighboring sheep, so that the total number of magic stones is odd. After that, Bob sacrifices one of the sheep for himself and replaces it by a food bucket. At the third step, Alice chooses a pair of neighboring sheep (not including the two which are separated by the bucket) and puts a border between them (the border may be at the same place as a magic stone). After that, all remaining sheep walk through an arc of the circle to the food bucket without crossing the border (so that there is only one possible route). Every sheep which walks on an odd number of magic stones is sacrificed for Alice, and every other sheep is for Bob. What is the maximal number of sheep which Alice can sacrifice for herself in a certain year, regardless of Bob's action?

The incircle of a triangle $ABC$ touches its sides $BC$, $CA$, $AB$ at the points $X$, $Y$, $Z$, respectively. Let $X^{\prime}$, $Y^{\prime}$, $Z^{\prime}$ be the reflections of these points $X$, $Y$, $Z$ in the external angle bisectors of the angles $CAB$, $ABC$, $BCA$, respectively. Show that $Y^{\prime}Z^{\prime}\parallel BC$, $Z^{\prime}X^{\prime}\parallel CA$ and $X^{\prime}Y^{\prime}\parallel AB$.

At least how many weighings of a balanced scale are needed to order four stones with distinct weights from the lightest to the heaviest? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 8 $

Let $N$ be a positive integer, and consider an $N \times N$ grid. A [i]right-down path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A [i]right-up path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence. Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths. [asy] size(4cm); draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin); draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin); draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin); draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin); draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin); draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin); draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin); [/asy] [i]Proposed by Zixiang Zhou, Canada[/i]

A set of lattice points is called [i]good[/i] if it does not contain two points that form a line with slope $-1$ or slope $1$. Let $S = \{(x, y)\ |\ x, y \in \mathbb{Z}, 1 \le x, y \le 4\}$. Compute the number of non-empty good subsets of $S$. [i]Proposed by Lewis Chen[/i]

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that \[ f(a-b)f(c-d) + f(a-d)f(b-c) \leq (a-c)f(b-d) \] for all real numbers $a, b, c,$ and $d$.

In the quadrilateral $ABCD$, angles $B$ and $C$ are equal to $120^o$, $AB = CD = 1$, $CB = 4$. Find the length $AD$.

Points $A, B, C,$ and $D{}$ lie on the surface of a sphere with diameter 1. Determine the maximum possible volume of tetrahedron $ABCD.$ [i]Proposed by Fredy Yip[/i]

Suppose, instead, that all collisions are instantaneous and perfectly inelastic. After a long time, which of the following is true? (A) The center block is moving to the left. (B) The center block is moving to the right. (C) The center block is at rest somewhere to the left of its initial position. (D) The center block is at rest at its initial position. (E) The center block is at rest somewhere to the right of its initial position.

Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.

Let $p$ be a real number and $f(x)=x^p-x+p$. Prove that: (a) Every root $\alpha$ of $f(x)$ satisfies $|\alpha|<p^{\frac1{p-1}}$; (b) If $p$ is a prime number, then $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients.

Let $ABC$ be an acute angled triangle and let $D$ be an interior point of the segment $BC$. Let the circumcircle of $ACD$ intersect $AB$ at $E$ ($E$ between $A$ and $B$) and let circumcircle of $ABD$ intersect $AC$ at $F$ ($F$ between $A$ and $C$). Let $O$ be the circumcenter of $AEF$. Prove that $OD$ bisects $\angle EDF$.

Two towns, $A$ and $B$, are $100$ miles apart. Every $20$ minutes, (starting at midnight), a bus traveling at $60$ mph leaves town $A$ for town $B$, and every $30$ minutes (starting at midnight) a bus traveling at $20$ mph leaves town $B$ for town $A$. Dirk starts in Town $A$ and gets on a bus leaving for town $B$ at noon. However, Dirk is always afraid he has boarded a bus going in the wrong direction, so each time the bus he is in passes another bus, he gets out and transfers to that other bus. How many hours pass before Dirk finally reaches Town $B$?

Lizzie and Alex are playing a game on the whiteboard. Initially, $n$ twos are written on the board. On a player’s turn they must either 1. change any single positive number to 0, or 2. subtract one from any positive number of positive numbers on the board. The game ends once all numbers are 0, and the last player who made a move wins. If Lizzie always plays first, find all $n$ for which Lizzie has a winning strategy.

$a_1,a_2,\dots ,a_n>0$ are positive real numbers such that $\sum_{i=1}^{n} \frac{1}{a_i}=n$ prove that: $\sum_{i<j} \left(\frac{a_i-a_j}{a_i+a_j}\right)^2\le\frac{n^2}{2}\left(1-\frac{n}{\sum_{i=1}^{n}a_i}\right)$

Given is a triangle $ABC$. A line $\ell$ passes through reflection wrt $BC$ changes into the line $\ell'$, $\ell'$ changes into $\ell''$ through reflection wrt $AC$ and $\ell''$ through reflection wrt $AB$ changes into $\ell'''$. Construct the line $\ell$ given that $\ell'''$ coincides with $\ell$.

Dunno wrote several different natural numbers on the board and divided (in his head) the sum of these numbers by their product. After this, Dunno erased the smallest number and divided (again in his mind) the amount of the remaining numbers by their product. The second result was $3$ times greater than the first. What number did Dunno erase?

Define the sequence of positive prime numbers. $p_1,p_2,p_3,...$. Let set $A$ be the infinite set of positive integers whose prime divisor does not exceed $p_n$. How many at least members must be selected from the set $A$ , such that we ensures that there are $2$ numbers whose products are perfect squares? [i](PP-nine)[/i]

How many ways to fill the board $ 4\times 4$ by nonnegative integers, such that sum of the numbers of each row and each column is 3?

For any finite non empty set $X$ of integers, let $\max (X)$ denote the largest element of $X$ and $|X|$ denote the number of elements in $X$. If $N$ is the number of ordered pairs $(A, B)$ of finite non-empty sets of positive integers, such that $$ \begin{aligned} & \max (A) \times|B|=12 ; \text { and } \\ & |A| \times \max (B)=11 \end{aligned} $$ and $N$ can be written as $100 a+b$ where $a, b$ are positive integers less than 100 , find $a+b$.

Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.

Alice thinks of four positive integers $a\leq b\leq c\leq d$ satisfying $\{ab+cd,ac+bd,ad+bc\}=\{40,70,100\}$. What are all the possible tuples $(a,b,c,d)$ that Alice could be thinking of?

Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.

Let $a, b$ and $c$ be positive reals such that $a + b + c \ge abc$. Prove that $a^2 + b^2 + c^2 \ge \sqrt3 abc$.

We have $n$ integers $a_1, a_2,. . . , a_n$, not necessarily distinct, with sum $2S.$ An integer $k$ is called [i]separator[/i] if $k$ of the numbers can be chosen with sum equal to $S.$ What is the maximum possible number of separators?