All vertices of triangles $ABC$ and $A_1B_1C_1$ lie on the hyperbola $y=1/x$. It is known that $AB \parallel A_1B_1$ and $BC \parallel B_1C_1$. Prove that $AC_1 \parallel A_1C$. (I. Gorodnin)

Given a triangle $ABC$. Let $S$ be the circle passing through $C$, centered at $A$. Let $X$ be a variable point on $S$ and let $K$ be the midpoint of the segment $CX$ . Find the locus of the midpoints of $BK$, when $X$ moves along $S$. (I. Gorodnin)

$N$ cells are marked on an $n\times n$ table so that at least one marked cel is among any four cells of the table which form the figure (tbe figure may be rotated). Find the smallest possible value of $N$. (E. Barabanov)

Find all integers $a$ and $b$ satisfying the equality $3^a - 5^b = 2$. (I. Gorodnin)

Find all functions$ f : R_+ \to R_+$ such that $f(f(x)+y)=x+f(y)$ , for all $x, y \in R_+$ (Folklore) PSUsing search terms + ''f(x+f(y))'' + ''f(x)+y'' I found the same problem in Q, continuous in R, strictly monotone in R , without extra conditions in R

Given positive real numbers $a,b,c$ with $ab+bc+ca\ge a+b+c$ , prove that $$(a + b + c)(ab + bc+ca) + 3abc \ge 4(ab + bc + ca).$$ (I. Gorodnin)

Point $L$ is marked on the side $AB$ of a triangle $ABC$. The incircle of the triangle $ABC$ meets the segment $CL$ at points $P$ and $Q$ .Is it possible that the equalities $CP = PQ = QL$ hold if $CL$ is a) the median? b) the bisector? c) the altitude? d) the segment joining vertex $C$ with the point $L$ of tangency of the excircle of the triangie $ABC$ with $AB$ ? (I. Gorodnin)

Thirty rays with the origin at the same point are constructed on a plane. Consider all angles between any two of these rays. Let $N$ be the number of acute angles among these angles. Find the smallest possible value of $N$. (E. Barabanov)

Circles $\Gamma_1$ and $\Gamma_2$ meet at points $X$ and $Y$. A circle $S_1$ touches internally $\Gamma_1$ at $A$ and $\Gamma_2$ externally at $B$. A circle $S_2$ touches $\Gamma_2$ internally at $C$ and $\Gamma_1$ externally at $D$. Prove that the points $A, B, C, D$ are either collinear or concyclic. (A. Voidelevich)

Let $x,y,z$ be pairwise distinct real numbers such that $x^2-1/y = y^2 -1/z = z^2 -1/x$. Given $z^2 -1/x = a$, prove that $(x + y + z)xyz= -a^2$. (I. Voronovich)

$n$ points are marked on a plane. Each pair of these points is connected with a segment. Each segment is painted one of four different colors. Find the largest possible value of $n$ such that one can paint the segments so that for any four points there are four segments (connecting these four points) of four different colors. (E. Barabanov)

Given triangle $ABC$ with $\angle A = a$. Let $AL$ be the bisector of the triangle $ABC$. Let the incircle of $\vartriangle ABC$ touch the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $X$ be the intersection point of the lines $AQ$ and $LP$. Prove that the lines $BX$ and $AL$ are perpendicular. (V. Karamzin)

Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that $$\frac{a^2}{(b+c)^3}+\frac{b^2}{(c+a)^3}+\frac{c^2}{(a+b)^3}\geq \frac98$$

Do there exist functions $f$ and $g$, $f : R \to R$, $g : R \to R$ such that $f(x + f(y)) = y^2 + g(x)$ for all real $x$ and $y$ ? (I. Gorodnin)

Let $O$ be the circumcenter of an acute-angled triangle $ABC$. Let $AH$ be the altitude of this triangle, $M,N,P,Q$ be the midpoints of the segments $AB, AC, BH, CH$, respectively. Let $\omega_1$ and $\omega_2$ be the circumferences of the triangles $AMN$ and $POQ$. Prove that one of the intersection points of $\omega_1$ and $\omega_2$ belongs to the altitude $AH$. (A. Voidelevich)

Find all sequences $(a_n)$ of positive integers satisfying the equality $a_n=a_{a_{n-1}}+a_{a_{n+1}}$ a) for all $n\ge 2$ b) for all $n \ge 3$ (I. Gorodnin)

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

Let $I$ be the incenter of a triangle $ABC$. The circle passing through $I$ and centered at $A$ meets the circumference of the triangle $ABC$ at points $M$ and $N$. Prove that the line $MN$ touches the incircle of the triangle $ABC$. (I. Kachan)

Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

Given $a,b,c$ ,$(a, b,c \in (0,2)$), with $a + b + c = ab+bc+ca$, prove that $$\frac{a^2}{a^2-a+1}+\frac{b^2}{b^2-b+1}+\frac{c^2}{c^2-c+1} \le 3$$(D. Pirshtuk)

Let $\Gamma_B$ and $\Gamma_C$ be excircles of an acute-angled triangle $ABC$ opposite to its vertices $B$ and $C$, respectively. Let $C_1$ and $L$ be the tangent points of $\Gamma_C$ and the side $AB$ and the line $BC$ respectively. Let $B_1$ and $M$ be the tangent points of $\Gamma_B$ and the side $AC$ and the line $BC$, respectively. Let $X$ be the point of intersection of the lines $LC_1$ and $MB_1$. Prove that $AX$ is equal to the inradius of the triangle $ABC$. (A. Voidelevich)

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

Find the maximum possible number of edges of a simple graph with $8$ vertices and without any quadrilateral. (a simple graph is an undirected graph that has no loops (edges connected at both ends to the same vertex) and no more than one edge between any two different vertices.)

Let $AA_1, BB_1$ be the altitudes of an acute non-isosceles triangle $ABC$. Circumference of the triangles $ABC$ meets that of the triangle $A_1B_1C$ at point $N$ (different from $C$). Let $M$ be the midpoint of $AB$ and $K$ be the intersection point of $CN$ and $AB$. Prove that the line of centers the circumferences of the triangles $ABC$ and $KMC$ is parallel to the line $AB$. (I. Kachan)

Prove that for all even positive integers $n$ the following inequality holds a) $\{n\sqrt6\} > \frac{1}{n}$ b)$ \{n\sqrt6\}> \frac{1}{n-1/(5n)} $ (I. Voronovich)

Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.