Let $ABCD$ be a non-cyclic convex quadrilateral. The feet of perpendiculars from $A$ to $BC,BD,CD$ are $P,Q,R$ respectively, where $P,Q$ lie on segments $BC,BD$ and $R$ lies on $CD$ extended. The feet of perpendiculars from $D$ to $AC,BC,AB$ are $X,Y,Z$ respectively, where $X,Y$ lie on segments $AC,BC$ and $Z$ lies on $BA$ extended. Let the orthocenter of $\triangle ABD$ be $H$. Prove that the common chord of circumcircles of $\triangle PQR$ and $\triangle XYZ$ bisects $BH$.

Let $p$ be a prime number. For any $\sigma \in S_p$ (the permutation group of $\{1,2,...,p \}),$ define the matrix $A_{\sigma}=(a_{ij}) \in \mathcal{M}_p(\mathbb{Z})$ as $a_{ij} = \sigma^{i-1}(j),$ where $\sigma^0$ is the identity permutation and $\sigma^k = \underbrace{\sigma \circ \sigma \circ ... \circ \sigma}_k.$ Prove that $D = \{ |\det A_{\sigma}| : \sigma \in S_p \}$ has at most $1+ (p-2)!$ elements.

Omar has four fair standard six-sided dice. Omar invented a game where he rolls his four dice over and over again until the number 1 does not appear on the top face of any of the dice. Omar wins the game if on that roll the top faces of his dice show at least one 2 and at least one 5. The probability that Omar wins the game is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

The weight of $\frac 13$ of a large pizza together with $3 \frac 12$ cups of orange slices is the same as the weight of $\frac 34$ of a large pizza together with $\frac 12$ cup of orange slices. A cup of orange slices weighs $\frac 14$ of a pound. What is the weight, in pounds, of a large pizza? $\textbf{(A)}~1\frac45\qquad\textbf{(B)}~2\qquad\textbf{(C)}~2\frac25\qquad\textbf{(D)}~3\qquad\textbf{(E)}~3\frac35$

The function $f:\mathbb R^{\ge 0} \longrightarrow \mathbb R^{\ge 0}$ satisfies the following properties for all $a,b\in \mathbb R^{\ge 0}$: [b]a)[/b] $f(a)=0 \Leftrightarrow a=0$ [b]b)[/b] $f(ab)=f(a)f(b)$ [b]c)[/b] $f(a+b)\le 2 \max \{f(a),f(b)\}$. Prove that for all $a,b\in \mathbb R^{\ge 0}$ we have $f(a+b)\le f(a)+f(b)$. [i]Proposed by Masoud Shafaei[/i]

It is known that from segments of lengths $a$, $b$ and $c$, a triangle can be formed. Can it happen that from segments of lengths$$\sqrt{a^2 + bc}, \quad \sqrt{b^2 + ca}, \quad \sqrt{c^2 + ab}$$an obtuse triangle can be formed?

The fraction $\frac23$ can be eypressed as a sum of two distinct unit fractions: $\frac12 + \frac16$ . Show that the fraction $\frac{p-1}{p}$ where $p\ge 5$ is a prime cannot be expressed as a sum of two distinct unit fractions.

Let $L$ be a subset in the coordinate plane defined by $L = \{(41x + 2y, 59x + 15y) | x, y \in \mathbb Z \}$, where $\mathbb Z$ is set of integers. Prove that for any parallelogram with center in the origin of coordinate and area $1990$, there exist at least two points of $L$ located in it.

A pool table has the shape of a triangle whose angles are in a rational ratio. A ball positioned at an interior point of the table is hit by a stick. The ball reflects from the sides of the triangle according to the law of reflection. Prove that the ball will move only along a finite number of segments. (It is assumed that the ball does not reach the vertices of the triangle.)

Solve in $\Bbb{N}^*$ the equation $$ 4^a \cdot 5^b - 3^c \cdot 11^d = 1.$$

In a round-robin tournament with $n$ players in which there are no draws, the numbers of wins scored by the players are $s_1 , s_2 , \ldots, s_n$. Prove that a necessary and sufficient condition for the existence of three players $A,B,C$ such that $A$ beats $B$, $B$ beats $C$, and $C$ beats $A$ is $$s_{1}^{2} +s_{2}^{2} + \ldots +s_{n}^{2} < \frac{(2n-1)(n-1)n}{6}.$$

Knights, who always tell truth, and liars, who always lie, live on an island. They have been distributed into two teams $A$ and $B$ for a game of tennis, and team $A$ had more members than team $B$. Two players from different teams started the game, whenever a player loses the game, he leaves it forever and he is replaces by a member of his team (that has never played before). The team, all of whose members left the game, loses. After the tournament, every member of team $A$ was asked: "Is it true that you have lost to a liar in some game?", and every member of team $B$ was asked: "Is it true that you have won at least two games, in which your opponent was a knight?". It turns out that every single answer was positive. Which team won?

Let \(n\) be a non-negative integer and define \(a_n = 2^n - n\). Determine all non-negative integers \(m\) such that \(s_m = a_0 + a_1 + \dots + a_m\) is a power of 2.

In a school there are $2007$ male and $2007$ female students. Each student joins not more than $100$ clubs in the school. It is known that any two students of opposite genders have joined at least one common club. Show that there is a club with at least $11$ male and $11$ female members.

Let $M$ be a set of $n \ge 4$ points in the plane, no three of which are collinear. Initially these points are connected with $n$ segments so that each point in $M$ is the endpoint of exactly two segments. Then, at each step, one may choose two segments $AB$ and $CD$ sharing a common interior point and replace them by the segments $AC$ and $BD$ if none of them is present at this moment. Prove that it is impossible to perform $n^3 /4$ or more such moves. [i]Proposed by Vladislav Volkov, Russia[/i]

In triangle $ABC$ $(b \geq c)$, point $E$ is the midpoint of shorter arc $BC$. If $D$ is the point such that $ED$ is the diameter of circumcircle $ABC$, prove that $\angle DEA = \frac{1}{2}(\beta-\gamma)$

Let $a,b,c$ be the lengths of the sides of a triangle, and $\alpha, \beta, \gamma$ respectively, the angles opposite these sides. Prove that if \[ a+b=\tan{\frac{\gamma}{2}}(a\tan{\alpha}+b\tan{\beta}) \] the triangle is isosceles.

Maxi chose $3$ digits, and by writing down all possible permutations of these digits, he obtained $6$ distinct $3$-digit numbers. If exactly one of those numbers is a perfect square and exactly three of them are prime, find Maxi’s $3$ digits. Give all of the possibilities for the $3$ digits.

None of the $n$ (not necessarily distinct) digits selected are equal to $0$ or $7$. It turns out that every $n$-digit number formed using these digits is divisible by $7$. Prove that $n$ is divisible by $6$.

Solve in positive integers the following equation: \[{1\over n^2}-{3\over 2n^3}={1\over m^2}\] [i]Proposed by A. Golovanov[/i]

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

Consider the infinite polynomial $G(x) = F_1x+F_2x^2 +F_3x^3 +...$ defined for $0 < x <\frac{\sqrt5 -1}{2}$ where Fk is the $k$th term of the Fibonacci sequence defined to be $F_k = F_{k-1} + F_{k-2}$ with $F_1 = 1$, $F_2 = 1$. Determine the value a such that $G(a) = 2$.

Prove that for all integers $n > 1$, $$\frac{1}{n}+\frac{1}{n + 1}+ ...+\frac{1}{n^2- 1}+\frac{1}{n^2} > 1$$

Let $\vartriangle ABC$ be a right triangle with legs $AB = 6$ and $AC = 8$. Let $I$ be the incenter of $\vartriangle ABC$ and $X$ be the other intersection of $AI$ with the circumcircle of $\vartriangle ABC$. Find $\overline{AI} \cdot \overline{IX}$.

Let a,b,c be three distinct positive numbers. Consider the quadratic polynomial $P (x) =\frac{c(x - a)(x - b)}{(c -a)(c -b)}+\frac{a(x - b)(x - c)}{(a - b)(a - c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}+ 1$. The value of $P (2017)$ is (A): $2015$ (B): $2016$ (C): $2017$ (D): $2018$ (E): None of the above.