Let $A_1A_2A_3A_4A_5$ be a convex, cyclic pentagon with $\angle A_i + \angle A_{i+1} >180^{\circ}$ for all $i \in \{1,2,3,4,5\}$ (all indices modulo $5$ in the problem). Define $B_i$ as the intersection of lines $A_{i-1}A_i$ and $A_{i+1}A_{i+2}$, forming a star. The circumcircles of triangles $A_{i-1}B_{i-1}A_i$ and $A_iB_iA_{i+1}$ meet again at $C_i \neq A_i$, and the circumcircles of triangles $B_{i-1}A_iB_i$ and $B_iA_{i+1}B_{i+1}$ meet again at $D_i \neq B_i$. Prove that the ten lines $A_iC_i, B_iD_i$, $i \in \{1,2,3,4,5\}$, have a common point.

Find all real numbers a such that all solutions to the quadratic equation $ x^2 \minus{} ax \plus{} a \equal{} 0$ are integers.

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

For every $3$-digit natural number $n$ (leading digit of $n$ is nonzero), we consider the number $n_0$ obtained from $n$ eliminating all possible digits that are zero. For example, if $n = 207$, then $n_0 = 27$. Determine the number of three-digit positive integers $n$, for which $n_0$ is a divisor of $n$ different from $n$.

There are 2017 points in a plane. For each pair of these points we mark the middle of the segment they form when connected. What’s the least number of marked points?

Find all positive integers $n$ such that there exists a monic polynomial $P(x)$ of degree $n$ with integers coefficients satisfying $$P(a)P(b)\neq P(c)$$ for all integers $a,b,c$.

Beyond the Point of No Return is a large lake containing 2013 islands arranged at the vertices of a regular $2013$-gon. Adjacent islands are joined with exactly two bridges. Christine starts on one of the islands with the intention of burning all the bridges. Each minute, if the island she is on has at least one bridge still joined to it, she randomly selects one such bridge, crosses it, and immediately burns it. Otherwise, she stops. If the probability Christine burns all the bridges before she stops can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find the remainder when $m+n$ is divided by $1000$. [i]Evan Chen[/i]

A graph has $ n$ vertices and $ \frac {1}{2}\left(n^2 \minus{} 3n \plus{} 4\right)$ edges. There is an edge such that, after removing it, the graph becomes unconnected. Find the greatest possible length $ k$ of a circuit in such a graph.

Let $(a,b)$ be a pair of natural numbers. Henning and Paul play the following game. At the beginning there are two piles of $a$ and $b$ coins respectively. We say that $(a,b)$ is the [i]starting position [/i]of the game. Henning and Paul play with the following rules: $\bullet$ They take turns alternatively where Henning begins. $\bullet$ In every step each player either takes a positive integer number of coins from one of the two piles or takes same natural number of coins from both piles. $\bullet$ The player how take the last coin wins. Let $A$ be the set of all positive integers like $a$ for which there exists a positive integer $b<a$ such that Paul has a wining strategy for the starting position $(a,b)$. Order the elements of $A$ to construct a sequence $a_1<a_2<a_3<\dots$ $(a)$ Prove that $A$ has infinity many elements. $(b)$ Prove that the sequence defined by $m_k:=a_{k+1}-a_{k}$ will never become periodic. (This means the sequence $m_{k_0+k}$ will not be periodic for any choice of $k_0$)

The $ABC$ triangle is given, point $I_a$ is the center of an exscribed circle touching the side $BC$ , the point $M$ is the midpoint of the side $BC$, the point $W$ is the intersection point of the bisector of the angle $A$ of the triangle $ABC$ with the circumscribed circle around him. Prove that the area of the triangle $I_aBC$ is calculated by the formula $S_{ (I_aBC)} = MW \cdot p$, where $p$ is the semiperimeter of the triangle $ABC$. (Mykola Moroz)

Let $f : [a, b] \rightarrow [a, b]$ be a continuous function and let $p \in [a, b]$. Define $p_0 = p$ and $p_{n+1} = f(p_n)$ for $n = 0, 1, 2,...$. Suppose that the set $T_p = \{p_n : n = 0, 1, 2,...\}$ is closed, i.e., if $x \not\in T_p$ then $\exists \delta > 0$ such that for all $x' \in T_p$ we have $|x'-x|\ge\delta$. Show that $T_p$ has finitely many elements.

Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that: \[ a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}. \]

For $x^2+2x+5$ to be a factor of $x^4+px^2+q$, the values of $p$ and $q$ must be, respectively: $ \textbf{(A)}\ -2, 5\qquad\textbf{(B)}\ 5, 25\qquad\textbf{(C)}\ 10, 20\qquad\textbf{(D)}\ 6, 25\qquad\textbf{(E)}\ 14, 25 $

The area of the triangle whose altitudes have lengths $36.4$, $39$, and $42$ can be written as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Rhombus $PQRS$ is inscribed in rectangle $ABCD$ so that vertices $P$, $Q$, $R$, and $S$ are interior points on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$, respectively. It is given that $PB=15$, $BQ=20$, $PR=30$, and $QS=40$. Let $m/n$, in lowest terms, denote the perimeter of $ABCD$. Find $m+n$.

Let positive integers $p,q$ with $\gcd(p,q)=1$ such as $p+q^2=(n^2+1)p^2+q$. If the parameter $n$ is a positive integer, find all possible couples $(p,q)$.

Let \( BE \) and \( CF \) be the medians of \( \triangle ABC \), and \( G \) be their intersection point. On segments \( GF \) and \( GE \), points \( K \) and \( L \), respectively, are chosen such that \( BK = CL = AG \). Prove that \[ \angle BKF + \angle CLE = \angle BGC. \] [i]Proposed by Vadym Solomka[/i]

The sum of positive real numbers $ a,b,c,d$ is $ 1$. Prove that: $ \frac{a^2}{a\plus{}b}\plus{}\frac{b^2}{b\plus{}c}\plus{}\frac{c^2}{c\plus{}d}\plus{}\frac{d^2}{d\plus{}a} \ge \frac{1}{2},$ with equality if and only if $ a\equal{}b\equal{}c\equal{}d\equal{}\frac{1}{4}$.

A rectangular floor measures $ a$ by $ b$ feet, where $ a$ and $ b$ are positive integers with $ b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $ 1$ foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair $ (a,b)$? $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

Let $f : (0,+\infty) \to \mathbb R$ be a function having the property that $f(x) = f\left(\frac{1}{x}\right)$ for all $x > 0.$ Prove that there exists a function $u : [1,+\infty) \to \mathbb R$ satisfying $u\left(\frac{x+\frac 1x }{2} \right) = f(x)$ for all $x > 0.$

$ k$ is a given natural number. Find all functions $ f: \mathbb{N}\rightarrow\mathbb{N}$ such that for each $ m,n\in\mathbb{N}$ the following holds: \[ f(m)\plus{}f(n)\mid (m\plus{}n)^k\]

The prime factorization of $12 = 2 \cdot 2 \cdot 3$ has three prime factors. Find the number of prime factors in the factorization of $12! = 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1.$

Marvin really likes pancakes, so he asked his grandma to make pancakes for him. Every time Grandma sends pancakes, she sends a package of $32$. When Marvin is in the mood for pancakes, he eats half of the pancakes he has. Marvin ate $157$ pancakes for lunch today. At least how many times has Grandma sent pancakes to Marvin so far? Marvin does not necessarily eat an integer number of pancakes at once, and he is in the mood for pancakes at most once a day.

An arbitrary triangle $ABC$ is given. Construct a straight line passing through vertex $B$ and dividing it into two triangles, the radii of the inscribed circles of which are equal. (M. Volchkevich)

Let $k$ be a fixed even positive integer, $N$ is the product of $k$ distinct primes $p_1,...,p_k$, $a,b$ are two positive integers, $a,b\leq N$. Denote $S_1=\{d|$ $d|N, a\leq d\leq b, d$ has even number of prime factors$\}$, $S_2=\{d|$ $d|N, a\leq d\leq b, d$ has odd number of prime factors$\}$, Prove: $|S_1|-|S_2|\leq C^{\frac{k}{2}}_k$