Triangle $ABC$ is given. On its sides $AB$, $BC$ and $CA$, respectively, points $X, Y, Z$ are chosen so that $$AX : XB =BY : YC = CZ : ZA = 2:1.$$ It turned out that the triangle $XYZ$ is equilateral. Prove that the original triangle $ABC$ is also equilateral.

Find all pairs $(m, n)$ of nonnegative integers for which $ m ^ { 2 } + 2 \cdot 3 ^ { n } = m \left( 2 ^ { n + 1 } - 1 \right) $.

Inside the parallelogram $ABCD$, choose a point $P$ such that $\angle APB+ \angle CPD= \angle BPC+ \angle APD$. Prove that there exists a circle tangent to each of the circles circumscribed around the triangles $APB$, $BPC$, $CPD$ and $APD$.

Does there exist a natural number $ n$ such that $ n>2$ and the sum of squares of some $ n$ consecutive integers is a perfect square?

Let $ABCD$ be a convex quadrilateral, and $M$ and $N$ the midpoints of the sides $CD$ and $AD$, respectively. The lines perpendicular to $AB$ passing through $M$ and to $BC$ passing through $N$ intersect at point $P$. Prove that $P$ is on the diagonal $BD$ if and only if the diagonals $AC$ and $BD$ are perpendicular.

Find the strictly monotone functions $ f:\{ 0\}\cup\mathbb{N}\longrightarrow\{ 0\}\cup\mathbb{N} $ that satisfy the following two properties: $ \text{(i)} f(2n)=n+f(n), $ for any nonnegative integers $ n. $ $ \text{(ii)} f(n) $ is a perfect square if and only if $ n $ is a perfect square.

Find all positive integers $n$ for which it is possible to partition a regular $n$-gon into triangles with diagonals not intersecting inside the $n$-gon such that at every vertex of the $n$-gon an odd number of triangles meet.

Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQR$? $\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}$

There are some real numbers on the board (at least two). In every step we choose two of them, for example $a$ and $b$, and then we replace them with $\frac{ab}{a+b}$. We continue until there is one number. Prove that the last number does not depend on which order we choose the numbers to erase.

Let $x,y,z,t$ be real numbers with $x,y,z$ not all equal such that \[x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}=t.\] Find all possible values of $ t$ such that $xyz+t=0$.

Let $a,b$ and $c$ be the side lengths of a triangle. Prove that: \[\frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a}<\frac{1}{16}\]

Let $n\geq 2$ be a given integer. Initially, we write $n$ sets on the blackboard and do a sequence of moves as follows: choose two sets $A$ and $B$ on the blackboard such that none of them is a subset of the other, and replace $A$ and $B$ by $A\cap B$ and $A\cup B$. This is called a $\textit{move}$. Find the maximum number of moves in a sequence for all possible initial sets.

Let $k$ be a given even positive integer. Sarah first picks a positive integer $N$ greater than $1$ and proceeds to alter it as follows: every minute, she chooses a prime divisor $p$ of the current value of $N$, and multiplies the current $N$ by $p^k -p^{-1}$ to produce the next value of $N$. Prove that there are infinitely many even positive integers $k$ such that, no matter what choices Sarah makes, her number $N$ will at some point be divisible by $2018$.

A square is drawn on a sheet of grid paper on the sides of the cells $ABCD$ with side $8$. Point $E$ is the midpoint of side $BC$, $Q$ is such a point on the diagonal $AC$ such that $AQ: QC = 3: 1$. Find the angle between straight lines $AE$ and $DQ$.

Inside the triangle $ ABC $ a point $ P $ is given; find a point $ Q $ on the perimeter of this triangle such that the broken line $ APQ $ divides the triangle into two parts with equal areas.

Note that there are exactly three ways to write the integer $4$ as a sum of positive odd integers where the order of the summands matters: \begin{align*} 1+1+1+1&=4,\\ 1+3&=4,\\ 3+1&=4. \end{align*} Let $f(n)$ be the number of ways to write a natural number $n$ as a sum of positive odd integers where the order of the summands matters. Find the remainder when $f(2008)$ is divided by $100$.

Raquel painted $650$ points in a circle with a radius of $16$ cm. Shows that there is a circular crown with $2$ cm of inner radius and $3$ cm of outer radius that contain at least $10$ of these points.

The map below shows an east/west road connecting the towns of Acorn, Centerville, and Midland, and a north/south road from Centerville to Drake. The distances from Acorn to Centerville, from Centerville to Midland, and from Centerville to Drake are each 60 kilometers. At noon Aaron starts at Acorn and bicycles east at 17 kilometers per hour, Michael starts at Midland and bicycles west at 7 kilometers per hour, and David starts at Drake and bicycles at a constant rate in a straight line across an open field. All three bicyclists arrive at exactly the same time at a point along the road from Centerville to Midland. Find the number of kilometers that David bicycles. [center][img]https://i.snag.gy/Ik094i.jpg[/img][/center]

A circle has radius $52$ and center $O$. Points $A$ is on the circle, and point $P$ on $\overline{OA}$ satisfies $OP = 28$. Point $Q$ is constructed such that $QA = QP = 15$, and point $B$ is constructed on the circle so that $Q$ is on $\overline{OB}$. Find $QB$. [i]Proposed by Justin Hsieh[/i]

In the Euclidean plane, let be a point $ O $ and a finite set $ \mathcal{M} $ of points having at least two points. Prove that there exists a proper subset of $ \mathcal{M}, $ namely $ \mathcal{M}_0, $ such that the following inequality is true: $$ \sum_{P\in \mathcal{M}_0} OP\ge \frac{1}{4}\sum_{Q\in\mathcal{M}} OQ $$

[asy]draw((0,0)--(0,6)--(8,6)--(8,3)--(4,3)--(4,0)--cycle); label("6",(0,3),W); label("8",(4,6),N);[/asy] Given that all angles shown are marked, the perimeter of the polygon shown is \[ \textbf{(A)}\ 14 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ \text{cannot be determined from the information given} \qquad \]

The circumference of a circle is $100$ inches. The side of a square inscribed in this circle, expressed in inches, is: $ \textbf{(A) }\frac{25\sqrt{2}}{\pi} \qquad\textbf{(B) }\frac{50\sqrt{2}}{\pi}\qquad\textbf{(C) }\frac{100}{\pi}\qquad\textbf{(D) }\frac{100\sqrt{2}}{\pi}\qquad\textbf{(E) }50\sqrt{2} $

$n>1$ and distinct positive integers $a_1,a_2,\ldots,a_{n+1}$ are  given. Does there exist a polynomial $p(x)\in\Bbb{Z}[x]$ of degree  $\le n$ that satisfies the following conditions? a. $\forall_{1\le i < j\le n+1}: \gcd(p(a_i),p(a_j))>1 $ b. $\forall_{1\le i < j < k\le n+1}: \gcd(p(a_i),p(a_j),p(a_k))=1 $ [i]Proposed by Mojtaba Zare[/i]

Does there exist a set $S$ of $100$ points in a plane such that the center of mass of any $10$ points in $S$ is also a point in $S$?

Let $a, b, c$ be integers. Prove that there are integers $p_1, q_1, r_1, p_2, q_2, r_2$ such that \[a = q_1r_2 - q_2r_1, b = r_1p_2 - r_2p_1, c = p_1q_2 - p_2q_1.\]