Let $ABC$ be a triangle where $AB > BC$, and $D$ and $E$ be points on sides $AB$ and $AC$ respectively, such that $DE$ and $AC$ are parallel. Consider the circumscribed circumference of triangle $ABC$. A circumference that passes through points $D$ and $E$ is tangent to the arc $AC$ that does not contain $B$ at point $P$. Let $Q$ be the reflection of point $P$ with respect to the perpendicular bisector of $AC$. The segments $BQ$ and $DE$ intersect at $X$. Prove that $AX = XC$.

For each positive integer $k$, we define the polynomial $S_k(x)=1+x+x^2+x^3+...+x^{k-1}$ Show that $n \choose 1$ $S_1(x) +$ $n \choose 2$ $S_2(x) +$ $n \choose 3$ $S_3(x)+...+$ $n \choose n$ $S_n(x) = 2^{n-1}S_n\left(\frac{1+x}{2}\right)$ for every positive integer $n$ and every real number $x$.

Find all functions $f$ $:$ $\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$ : $$(f(x)+y)(f(y)+x)=f(x^2)+f(y^2)+2f(xy)$$

Let $n$ be a positive integer that is not a perfect cube. Define real numbers $a,b,c$ by \[a=\root3\of n\kern1.5pt,\qquad b={1\over a-[a]}\kern1pt,\qquad c={1\over b-[b]}\kern1.5pt,\] where $[x]$ denotes the integer part of $x$. Prove that there are infinitely many such integers $n$ with the property that there exist integers $r,s,t$, not all zero, such that $ra+sb+tc=0$.

Put $\mathbb{A}=\{ \mathrm{yes}, \mathrm{no} \}$. A function $f\colon \mathbb{A}^n\rightarrow \mathbb{A}$ is called a [i]decision function[/i] if (a) the value of the function changes if we change all of its arguments; and (b) the values does not change if we replace any of the arguments by the function value. A function $d\colon \mathbb{A}^n \rightarrow \mathbb{A}$ is called a [i]dictatoric function[/i], if there is an index $i$ such that the value of the function equals its $i$th argument. The [i]democratic function[/i] is the function $m\colon \mathbb{A}^3 \rightarrow \mathbb{A}$ that outputs the majority of its arguments. Prove that any decision function is a composition of dictatoric and democratic functions.

In $\triangle ABC$, $AB = 4$, $BC = 5$, and $CA = 6$. Circular arcs $p$, $q$, $r$ of measure $60^\circ$ are drawn from $A$ to $B$, from $A$ to $C$, and from $B$ to $C$, respectively, so that $p$, $q$ lie completely outside $\triangle ABC$ but $r$ does not. Let $X$, $Y$, $Z$ be the midpoints of $p$, $q$, $r$, respectively. If $\sin \angle XZY = \dfrac{a\sqrt{b}+c}{d}$, where $a, b, c, d$ are positive integers, $\gcd(a,c,d)=1$, and $b$ is not divisible by the square of a prime, compute $a+b+c+d$. [i]Proposed by Michael Tang[/i]

We say that a quadrilateral $Q$ is [i]tangential [/i] if a circle can be inscribed into it, i.e. there exists a circle $C$ that does not meet the vertices of $Q$, such that it meets each edge at exactly one point. Let $N$ be the number of ways to choose four distinct integers out of $\{1, . . . , 24\}$ so that they form the side lengths of a tangential quadrilateral. Find the largest prime factor of $N$.

A regular $2017$-gon is partitioned into triangles by a set of non-intersecting diagonals. Prove that among those triangles only one is acute-angled.

For each positive integer $n\ge 2$, define $k\left(n\right)$ to be the largest integer $m$ such that $\left(n!\right)^m$ divides $2016!$. What is the minimum possible value of $n+k\left(n\right)$? [i]Proposed by Tristan Shin[/i]

We consider triangle $ABC$ with $\angle BAC = 90^{\circ}$ and $\angle ABC = 60^{\circ}.$ Let $ D \in (AC) , E \in (AB),$ such that $CD = 2 \cdot DA$ and $DE $ is bisector of $\angle ADB.$ Denote by $M$ the intersection of $CE$ and $BD$, and by $P$ the intersection of $DE$ and $AM$. a) Show that $AM \perp BD$. b) Show that $3 \cdot PB = 2 \cdot CM$.

Dexter's Laboratory has $2024$ robots, each with a program setup by Dexter. One day, his naughty sister Dee Dee intrudes and writes an integer in $\{1, 2, \dots, 113\}$ on each of the robot's forehead. Each robot detects the numbers on all other robots' foreheads, and guess its own number base on its program, individually and simultaneously. Find the largest positive integer $k$ such that Dexter can setup the programs so that, no matter how the numbers distribute, there are always at least $k$ robots who guess their numbers right. [i]Proposed by sn6dh[/i]

Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. [i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]

Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\] Determine all possible values of $P(0)$. [i]Proposed by Belgium[/i]

Let $O $, $O_1$, $O_2 $, $O_3$, $O_4$ be points such that $O_1$, $O$, $O_3$ and $O_2$, $O$, $O_4$ are collinear in that order, $OO_1 =1$, $OO_2 = 2$, $OO_3 =\sqrt2$, $OO_4 = 2$, and $\angle O_1OO_2 = 45^o$. Let $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$ be the circles with respective centers $O_1$, $O_2$ , $O_3$, $O_4$ that go through $O$. Let $A$ be the intersection of $\omega_1$ and $\omega_2$, $B$ be the intersection of $\omega_2$ and $\omega_3$, $C$ be the intersection of $\omega_3$ and $\omega_4$, and $D$ be the intersection of $\omega_4$ and $\omega_1$ with $A$, $B$, $C$, $D$ all distinct from $O$. What is the largest possible area of a convex quadrilateral $P_1P_2P_3P_4$ such that $P_i$ lies on $O_i$ and that $A$, $B$, $C$, $D$ all lie on its perimeter?

Does there exist a right angled triangle, which hypotenuse is $2016^{2017}$ and two other sides positive integers.

Let $ABCD$ be cyclic quadrilateral and $E$ the midpoint of $AC$. The circumcircle of $\triangle CDE$ intersect the side $BC$ at $F$, which is different from $C$. If $B'$ is the reflection of $B$ across $F$, prove that $EF$ is tangent to the circumcircle of $\triangle B'DF$. [i]Proposed by Nikola Velov[/i]

Find the largest value for $\frac{y}{x}$ for pairs of real numbers $(x,y)$ which satisfy \[(x-3)^2 + (y-3)^2 = 6.\] $\textbf{(A) }3 + 2 \sqrt 2\qquad \textbf{(B) } 2 + \sqrt 3\qquad \textbf{(C ) }3 \sqrt 3\qquad \textbf{(D) }6\qquad \textbf{(E) }6 + 2 \sqrt 3$

Let $A,B,C$ be real square matrices of the same order, and suppose $A$ is invertible. Prove that \[ (A-B)C=BA^{-1}\implies C(A-B)=A^{-1}B \]

Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$ Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$

Determine all functions $f : R_{\ge 0} \to R_{\ge 0}$ with the following properties: (a) $f(1) = 0$, (b) $f(x) > 0$ for all $x > 1$, (c) For all $x, y\ge 0$ with $x + y > 0$ holds $$f(xf(y))f(y) = f\left( \frac{xy}{x + y}\right)$$

Starting with an empty string, we create a string by repeatedly appending one of the letters $H$, $M$, $T$ with probabilities $\frac 14$, $\frac 12$, $\frac 14$, respectively, until the letter $M$ appears twice consecutively. What is the expected value of the length of the resulting string?

In a group of $n$ persons, (i) each person is acquainted to exactly $k$ others, (ii) any two acquainted persons have exactly $l$ common acquaintances, (iii) any two non-acquainted persons have exactly $m$ common acquaintances. Prove that $m(n-k -1) = k(k -l -1)$.

Let $ f:[0,\infty )\longrightarrow\mathbb{R} $ a bounded and periodic function with the property that $$ |f(x)-f(y)|\le |\sin x-\sin y|,\quad\forall x,y\in[0,\infty ) . $$ Show that the function $ [0,\infty ) \ni x\mapsto x+f(x) $ is monotone.

Given is an array $A$ of $2n$ numbers, where $n$ is a positive integer. Give an algorithm to create an array $prod$ of length $2n$ where $$prod[i] \, = \, A[i] \times A[i+1] \times \cdots \times A[i+n-1],$$ ($A[x]$ means $A[x \ \text{mod}\ 2n]$) in $O(n)$ time [b]withou[/b]t using division. Assume that all binary arithmetic operations are $O(1)$

We are given a chessboard with 100 rows and 100 columns. Two squares of the board are said to be adjacent if they have a common side. Initially all squares are white. a) Is it possible to colour an odd number of squares in such a way that each coloured square has an odd number of adjacent coloured squares? b) Is it possible to colour some squares in such a way that an odd number of them have exactly $4$ adjacent coloured squares and all the remaining coloured squares have exactly $2$ adjacent coloured squares? c) Is it possible to colour some squares in such a way that an odd number of them have exactly $2$ adjacent coloured squares and all the remaining coloured squares have exactly $4$ adjacent coloured squares?