Points $A_1,B_1,C_1$ are selected on the sides $BC$,$CA$,$AB$ respectively of an equilateral triangle $ABC$ in such a way that the inradii of the triangles $C_1AB_1$, $A_1BC_1$, $B_1CA_1$ and $A_1B_1C_1$ are equal. Prove that $A_1,B_1,C_1$ are the midpoints of the corresponding sides.

Each cell in a \( 2024 \times 2024 \) table contains the letter \( A \) or \( B \), with the number of \( A \)'s in each row being the same and the number of \( B \)'s in each column being the same. Alexandra and Boris play the following game, alternating turns, with Alexandra going first. On each turn, the player chooses a row or column and erases all the letters in it that have not yet been erased, as long as at least one letter is erased during the turn, and at the end of the turn, at least one letter remains in the table. The game ends when exactly one letter remains in the table. Alexandra wins the game if the letter is \( A \), and Boris wins if it is \( B \). What is the number of initial tables for which Alexandra has a winning strategy?

Given three distinct digits a, b and c, it is possible, by choosing two digits at a time, to form six two-digit numbers. Determine all possible sets {a, b, c} for which the sum of the six two-digits numbers is 484.

Let $ABCD$ be a convex quadrilateral. Ray $AD$ meets ray $BC$ at $P$. Let $O,O'$ be the circumcenters of triangles $PCD, PAB$, respectively, $H,H'$ be the orthocenters of triangles $PCD, PAB$, respectively. Prove that circumcircle of triangle $DOC$ is tangent to circumcircle of triangle $AO'B$ if and only if circumcircle of triangle $DHC$ is tangent to circumcircle of triangle $AH'B$.

The sequence $a_1,a_2,\dots,a_{13}$ is a geometric sequence with $a_1=a$ and common ratio $r$, where $a$ and $r$ are positive integers. Given that $$\log_{2015}a_1+\log_{2015}a_2+\dots+\log_{2015}a_{13}=2015,$$ find the number of possible ordered pairs $(a,r)$.

Circle $(O)$ and its chord $BC$ are given. Point $A$ moves on the major arc $BC$. $AL$ is the angle bisector in a triangle $ABC$. Show that the disctance from the circumcenter of triangle $AOL$ to the line $BC$ does not depend on the position of point $A$.

Find the numerator of \[\frac{1010\overbrace{11 \ldots 11}^{2011 \text{ ones}}0101}{1100\underbrace{11 \ldots 11}_{2011\text{ ones}}0011}\] when reduced.

Let $n > 10$ be an odd integer. Determine the number of ways to place the numbers $1, 2, \ldots , n$ around a circle so that each number in the circle divides the sum its two neighbors. (Two configurations such that one can be obtained from the other per rotation are to be counted only once.)

The decimal notation of three natural numbers consists of equal digits: $n$ digits $x$ for $a$, $n$ digits $y$ for $b$ and $2n$ digits $z$ for $c$. For every $n > 1$ find all the possible triples of digits $x,y,z$ such, that $a^2 + b = c$

Find all nonempty finite sets $X$ of real numbers such that for all $x\in X$, $x+|x| \in X$.

Two isosceles triangles with sidelengths $x,x,a$ and $x,x,b$ ($a \neq b$) have equal areas. Find $x$.

Anna and Ben decided to visit Archipelago with $2009$ islands. Some pairs of islands are connected by boats which run both ways. Anna and Ben are playing during the trip: Anna chooses the first island on which they arrive by plane. Then Ben chooses the next island which they could visit. Thereafter, the two take turns choosing an island which they have not yet visited. When they arrive at an island which is connected only to islands they had already visited, whoever's turn to choose next would be the loser. Prove that Anna could always win, regardless of the way Ben played and regardless of the way the islands were connected. [i](12 points for Juniors and 10 points for Seniors)[/i]

A circle is located on the plane. What is the smallest number of lines you need to draw so that, symmetrically reflecting a given circle relative to these lines (in any order a finite number of times), it could cover any given point of the plane?

Bob has $30$ identical unit cubes. He can join two cubes together by gluing a face on one cube to a face on the other cube. He must join all the cubes together into one connected solid. Over all possible solids that Bob can build, what is the largest possible surface area of the solid? [i]Proposed by Nathan Xiong[/i]

For each pair $(a,b)$ of positive integers, determine all non-negative integers $n$ such that \[b+\left\lfloor{\frac{n}{a}}\right\rfloor=\left\lceil{\frac{n+b}{a}}\right\rceil.\]

Find three consecutive natural numbers, each of which is divisible by the square of the sum of its digits. Prove that there are no five such numbers in a row.

Determine which of the two numbers $\sqrt{c+1}-\sqrt{c}$, $\sqrt{c}-\sqrt{c-1}$ is greater for any $c\ge 1$.

Does exist natural $n$, such that for any non-zero digits $a$ and $b$ \[\overline {ab}\ |\ \overline {anb}\ ?\] (Here by $ \overline {x \ldots y} $ denotes the number obtained by concatenation decimal digits $x$, $\dots$, $y$.) [i]V. Senderov[/i]

Let $ABC$ be a triangle with $AB < AC$. The tangent to the circumcircle of $\triangle ABC$ at $A$ intersects $BC$ at $D$. The angle bisector of $\angle BAC$ intersect $BC$ at $E$. Suppose that the perpendicular bisector of $AE$ intersect $AB, AC$ at $P,Q$, respectively. Show that $$\sqrt{\frac{BP}{CQ}} = \frac{AC \cdot BD}{AB \cdot CD}$$

Which of the following equations does NOT have a solution? $\textbf{ (A) }\:(x+7)^2=0$ $\textbf{(B) }\:|-3x|+5=0$ $\textbf{ (C) }\:\sqrt{-x}-2=0$ $\textbf{ (D) }\:\sqrt{x}-8=0$ $\textbf{ (E) }\:|-3x|-4=0 $

A tournament is arranged amongst a finite number of people. Every person plays every other person just once and each game results in a win to one of the players (there are no draws). Show that there must a person $X$ such that, given any other person $Y$ in the tournament, either $X$ beat $Y$ , or $X$ beat $Z$ and $Z$ beat $Y$ for some $Z$.

Let $p$ and $q$ be integers. Show that there exists an interval $I$ of length $1/q$ and a polynomial $P$ with integral coefficients such that \[ \left|P(x)-\frac pq \right| < \frac{1}{q^2}\]for all $x \in I.$

The angle and the point $K$ inside it are given on the plane. Prove that there is a point $M$ with the following property: if an arbitrary line passing through intersects the sides of the angle at points $A$ and $B$, then $MK$ is the bisector of the angle $AMB$.

Let $a$ and $b$ be real numbers, and let $r$, $s$, and $t$ be the roots of $f(x)=x^3+ax^2+bx-1$. Also, $g(x)=x^3+mx^2+nx+p$ has roots $r^2$, $s^2$, and $t^2$. If $g(-1)=-5$, find the maximum possible value of $b$.

Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ such that for all integers $x$, $y$, $$f(x-f(y))=f(f(y))+f(x-2y)$$ [i]Proposed by Ivan Chan Kai Chin[/i]