The polynomial $ W(x) = x^4 + ax^3 + bx + cx + d $ is given. Prove that if the equation $ W(x) = 0 $ has four real roots, then for there to exist $ m $ such that $ W(x+m) = x^4+px^2+q $, it is necessary and it is enough that the sum of certain two roots of the equation $ W(x) = 0 $ equals the sum of the remaining ones.

Consider an $n\times n$ matrix whose entry at the intersection of the $i$-th row and the $j-$th column equals $i+ j -1$. What is the largest possible value of the product of $n$ entries of the matrix, no two of which are in the same row or column?

Let $I$ be the set of points $(x,y)$ in the Cartesian plane such that $$x>\left(\frac{y^4}{9}+2015\right)^{1/4}$$ Let $f(r)$ denote the area of the intersection of $I$ and the disk $x^2+y^2\le r^2$ of radius $r>0$ centered at the origin $(0,0)$. Determine the minimum possible real number $L$ such that $f(r)<Lr^2$ for all $r>0$.

In trapezoid $ABCD$ the sides $BC$ and $AD$ are parallel, $AC = BC + AD$, and the angle between the diagonals is equal to $ 60^o$. Prove that $AB = CD$. (Stanislav Smirnov, St Petersburg)

[b]p1.[/b] At a conference a mathematician and a chemist were talking. They were amazed to find that they graduated from the same high school. One of them, the chemist, mentioned that he had three sons and asked the other to calculate the ages of his sons given the following facts: (a) their ages are integers, (b) the product of their ages is $36$, (c) the sum of their ages is equal to the number of windows in the high school of the chemist and the mathematician. The mathematician considered this problem and noted that there was not enough information to obtain a unique solution. The chemist then noted that his oldest son had red hair. The mathematician then announced that he had determined the ages of the three sons. Please (aspiring mathematicians) determine the ages of the chemists three sons and explain your solution. [b]p2.[/b] A square is inscribed in a triangle. Two vertices of this square are on the base of the triangle and two others are on the lateral sides. Prove that the length of the side of the square is greater than and less than $2r$, where $r$ is a radius of the circle inscribed in the triangle. [b]p3.[/b] You are given any set of $100$ integers in which none of the integers is divisible by $100$. Prove that it is possible to select a subset of this set of $100$ integers such that their sum is a multiple of $100$. [b]p4.[/b] Find all prime numbers $a$ and $b$ such that $a^b + b^a$ is a prime number. [b]p5.[/b] $N$ airports are connected by airlines. Some airports are directly connected and some are not. It is always possible to travel from one airport to another by changing planes as needed. The board of directors decided to close one of the airports. Prove that it is possible to select an airport to close so that the remaining airports remain connected. [b]p6.[/b] (A simplified version of the Fermat’s Last Theorem). Prove that there are no positive integers $x, y, z$ and $z \le n$ satisfying the following equation: $x^n + y^n = z^n$. PS. You should use hide for answers.

Let $k$ and $m$ be integers greater than $1$. Consider $k$ pairwise disjoint sets $S_1,S_2, \cdots S_k$; each of these sets has exactly $m+1$ elements, one of which is red and the other $m$ are all blue. Let $\mathcal{F}$ be the family of all subsets $F$ of $S_1 \bigcup S_2\bigcup \cdots S_k$ such that, for every $i$ , the intersection $F \bigcap S_i$ is monochromatic; the empty set is also monochromatic. Determine the largest cardinality of a subfamily $\mathcal{G} \subseteq \mathcal{F}$, no two sets of which are disjoint. [i]Proposed by Russia, Andrew Kupavskii and Maksim Turevskii[/i]

Two circles in the plane, $\gamma_1$ and $\gamma_2$, meet at points $M$ and $N$. Let $A$ be a point on $\gamma_1$, and let $D$ be a point on $\gamma_2$. The lines $AM$ and $AN$ meet again $\gamma_2$ at points $B$ and $C$, respectively, and the lines $DM$ and $DN$ meet again $\gamma_1$ at points $E$ and $F$, respectively. Assume the order $M$, $N$, $F$, $A$, $E$ is circular around $\gamma_1$, and the segments $AB$ and $DE$ are congruent. Prove that the points $A$, $F$, $C$ and $D$ lie on a circle whose centre does not depend on the position of the points $A$ and $D$ on the respective circles, subject to the assumptions above. [i]***[/i]

An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ [i]Shiny[/i] if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have $$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$ Find the largest constant $K = K(n)$ such that $$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$ holds for every Shiny $n$-tuple $(x_1, x_2, \dots, x_n)$.

[b]a)[/b] Let be an even number $ n\ge 4 $ and $ n $ positive real numbers $ x_1,x_2,\ldots ,x_n. $ Prove that: $$ \min_{1\le i\le n/2} \frac{x_i}{x_{i+n/2}}\le \frac{x_1+x_2+\cdots +x_{n/2}}{x_{1+n/2}+ x_{2+n/2} +\cdots + x_n}\le \max_{1\le i\le n/2} \frac{x_i}{x_{i+n/2}}$$ [b]b)[/b] Let be $ m\ge 1 $ pairwise distinct natural numbers $ a,b,\ldots ,c. $ Show that: $$ \frac{ab\cdots c}{a+b+\cdots +c}\ge (m-1)!\cdot\frac{2}{m+1} $$ [i]M. Tetiva[/i]

In a class, there are $n \geqslant 3$ students and a teacher with $M$ marbles. The teacher then play a [i]Marble distribution[/i] according to the following rules. At the start, the teacher distributed all her marbles to students, so that each student receives at least $1$ marbles from the teacher. Then, the teacher chooses a student , who has never been chosen before, such that the number of marbles that he owns in a multiple of $2(n-1)$. That chosen student then equally distribute half of his marbles to $n-1$ other students. The same goes on until the teacher is not able to choose anymore student. Find all integer $M$, such that for some initial numbers of marbles that the students receive, the teacher can choose all the student(according to the rule above), so that each student receiving equal amount of marbles at the end.

Let $ ABCD$ be an isosceles trapezium. Determine the geometric location of all points $ P$ such that \[ |PA| \cdot |PC| \equal{} |PB| \cdot |PD|.\]

The two circles $\Gamma_1$ and $\Gamma_2$ with the midpoints $O_1$ resp. $O_2$ intersect in the two distinct points $A$ and $B$. A line through $A$ meets $\Gamma_1$ in $C \neq A$ and $\Gamma_2$ in $D \neq A$. The lines $CO_1$ and $DO_2$ intersect in $X$. Prove that the four points $O_1,O_2,B$ and $X$ are concyclic.

Find all functions $f:\mathbb{R}\to \mathbb{R},g:\mathbb{R}\to \mathbb{R}$ such that $$f(x-2f(y))= xf(y)-yf(x)+g(x)$$ for all real $x,y$

Points $H$ and $L$ are, respectively, the feet of the altitude and the angle bisector drawn from the vertex $A$ of the triangle $ABC$, $K$ is the touchpoint of the circle inscribed in the triangle $ABC$ with the side $BC$. Under what conditions will $AK$ be the bisector of the angle $\angle LAH$? (Hryhorii Filippovskyi)

Let $\triangle ABC$ be an acute angled triangle and let $k$ be its circumscribed circle. A point $O$ is given in the interior of the triangle, such that $CE=CF$, where $E$ and $F$ are on $k$ and $E$ lies on $AO$ while $F$ lies on $BO$. Prove that $O$ is on the angle bisector of $\angle ACB$ if and only if $AC=BC$.

A circle with center $O$ has radius $5,$ and has two points $A,B$ on the circle such that $\angle AOB = 90^{\circ}.$ Rays $OA$ and $OB$ are extended to points $C$ and $D,$ respectively, such that $AB$ is parallel to $CD,$ and the length of $CD$ is $200\%$ more than the radius of circle $O.$ Determine the length of $AC.$

In every vertex of a regular $n$ -gon exactly one chip is placed. At each $step$ one can exchange any two neighbouring chips. Find the least number of steps necessary to reach the arrangement where every chip is moved by $[\frac{n}{2}]$ positions clockwise from its initial position.

Let $ a_{0},a_{1},\ldots,a_{n}$ be integers, one of which is nonzero, and all of the numbers are not less than $ \minus{} 1$. Prove that if \[ a_{0} \plus{} 2a_{1} \plus{} 2^{2}a_{2} \plus{} \cdots \plus{} 2^{n}a_{n} \equal{} 0,\] then $ a_{0} \plus{} a_{1} \plus{} \cdots \plus{} a_{n} > 0$.

The pentagons $P_1P_2P_3P_4P_5$ and$I_1I_2I_3I_4I_5$ are cyclic, where $I_i$ is the incentre of the triangle $P_{i-1}P_iP_{i+1}$ (reckoned cyclically, that is $P_0=P_5$ and $P_6=P_1$). Show that the lines $P_1I_1, P_2I_2, P_3I_3, P_4I_4$ and $P_5I_5$ meet in a single point.

Consider the sequence $(3^{2^n} + 1)_{n\geq 1}$. i) Prove there exist infinitely many primes, none dividing any term of the sequence. ii) Prove there exist infinitely many primes, each dividing some term of the sequence. [i](Dan Schwarz)[/i]

The sum of $n$ terms of an arithmetic progression is $153$, and the common difference is $2$. If the first interm is an integer, and $n>1$, then the number of possible values for $n$ is: $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6 $

Chim Tu has a large rectangular table. On it, there are finitely many pieces of paper with nonoverlapping interiors, each one in the shape of a convex polygon. At each step, Chim Tu is allowed to slide one piece of paper in a straight line such that its interior does not touch any other piece of paper during the slide. Can Chim Tu always slide all the pieces of paper off the table in finitely many steps?

Find all real quadruples $(a,b,c,d)$ satisfying the system of equations $$ \left\{ \begin{array}{ll} ab+cd = 6 \\ ac + bd = 3 \\ ad + bc = 2 \\ a + b + c + d = 6. \end{array} \right. $$

For how many values of $ a$ is it true that the line $ y \equal{} x \plus{} a$ passes through the vertex of the parabola $ y \equal{} x^2 \plus{} a^2$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ \text{infinitely many}$

In triangle $ABC$ points $P$ and $Q$ lies on the external bisector of $\angle A$ such that $B$ and $P$ lies on the same side of $AC$. Perpendicular from $P$ to $AB$ and $Q$ to $AC$ intersect at $X$. Points $P'$ and $Q'$ lies on $PB$ and $QC$ such that $PX=P'X$ and $QX=Q'X$. Point $T$ is the midpoint of arc $BC$ (does not contain $A$) of the circumcircle of $ABC$. Prove that $P',Q'$ and $T$ are collinear if and only if $\angle PBA+\angle QCA=90^{\circ}$.