Let $ABC$ be an acute triangle inscribed in circle $(O)$, with orthocenter $H$. Median $AM$ of triangle $ABC$ intersects circle $(O)$ at $A$ and $N$. $AH$ intersects $(O)$ at $A$ and $K$. Three lines $KN$, $BC$ and the line through $H$ and perpendicular to $AN$ intersect each other and form triangle $XYZ$. Prove that the circumcircle of triangle $XYZ$ is tangent to $(O)$.
A convex polygon is divided into some triangles. Let $V$ and $E$ be respectively the set of vertices and the set of edges of all triangles (each vertex in $V$ may be some vertex of the polygon or some point inside the polygon). The polygon is said to be good if the following conditions hold:
i. There are no 3 vertices in $V$ which are collinear;
ii. Each vertex in $V$ belongs to an even number of edges in $E$.
Here are $2017$ points on the plane, no three of them are collinear. Some pairs of the points are connected by $n$ segments. Find the smallest value of $n$ so that there always exists two disjoint segments in any case.
Let $ABC$ be a triangle inscribed in circle $(O)$, with its altitudes $BH_b$, $CH_c$ intersecting at orthocenter $H$ ($H_b \in AC$, $H_c \in AB$). $H_b H_c$ meets $BC$ at $P$. Let $N$ be the midpoint of $AH$, $L$ be the orthogonal projection of $O$ on the symmedian with respect to angle $A$ of triangle $ABC$. Prove that $\angle NLP = 90^\circ$.
For each pair of positive integers $x$, $y$ a nonnegative integer $x \Delta y$ is defined. It's known that for all positive integers $a$ and $b$ the following equalities hold:
i. $(a+b) \Delta b = a \Delta b + 1$.
ii. $(a \Delta b) \cdot (b \Delta a) = 0$.
Find values of the expressions $2016 \Delta 121$ and $2016 \Delta 144$.
Let $ABC$ be a triangle inscribed in circle $(O)$ such that two points $B$, $C$ are fixed, $A$ moves on major arc $BC$ of $(O)$. Two tangents through $B$, $C$ of $(O)$ intersect at $P$. Circle with diameter $OP$ intersects $AC$, $AB$ at $D$, $E$, respectively. Prove that $DE$ is tangent to a fixed circle whose radius is half the radius of $(O)$.
Let $S = \{-17, -16, \ldots, 16, 17\}$. We call a subset $T$ of $S$ a good set if $-x \in T$ for any $x \in T$ and if $x, y, z \in T$ ($x, y, z$ may be equal) then $x + y + z \neq 0$. Find the largest number of elements in a good set.
Let $a$, $b$, $c$ be positive real numbers such that $a + b + c = 6$. Prove that
$$
\frac{1}{a^2 b + 16} + \frac{1}{b^2 c + 16} + \frac{1}{c^2 a + 16} \geq \frac{1}{8}
$$
Let $ABC$ be a triangle inscribed in circle $(O)$, with orthocenter $H$. Let $d$ be an arbitrary line which passes through $H$ and intersects $(O)$ at two points $P$ and $Q$. Draw diameter $AA'$ of circle $(O)$. $A'P$, $A'Q$ meet $BC$ at $K$, $L$, respectively. Prove that $O$, $K$, $L$, $A'$ are concyclic.
A chessboard has $64$ cells painted black and white in the usual way. A bishop path is a sequence of distinct cells such that two consecutive cells have exactly one common point. At least how many bishop paths can the set of all white cells be divided into?
Let $a$ be a positive integer such that for every positive integer $n$ the number $n^{2} a-1$ has at least one divisor greater than 1 which leaves remainder 1 when divided by $n$. Prove that $a$ is a perfect square.
Determine the largest number of queens that can be placed on a $2017 \times 2017$ board so that each queen attacks at most one of the remaining queens.
(Bojan Bašić and the problem committee)
Let $k$ be the circle circumscribed about $\triangle A B C$, and let $k_{a}$ be the excircle opposite vertex $A$. The two common tangents of the circles $k$ and $k_{a}$ intersect the line $B C$ at points $P$ and $Q$. Prove that $\varangle P A B=\varangle Q A C$.
Let $a, b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove:
$$
a \sqrt{2 b+1}+b \sqrt{2 c+1}+c \sqrt{2 a+1} \leqslant \sqrt{2-\left(a^{2}+b^{2}+c^{2}\right)}
$$
There are $2n-1$ light bulbs in a row. Initially the middle ($n$-th) bulb is on and all the others are off. In one step it is allowed to choose two non-adjacent bulbs that are off, such that all bulbs between them are on, and to change the state of these two bulbs and of all bulbs between them (for example, from the configuration $\bullet\circ\circ\circ\bullet$ one obtains $\circ\bullet\bullet\bullet\circ$, where $\circ$ denotes a bulb that is on and $\bullet$ a bulb that is off). What is the largest number of steps that can be performed?
(Dušan Đukić)
Let $Q$ be a square of size $12 \times 12$ given on an infinite cell grid. Student wants to paint some (not necessary all!) cells with seven colors (each cell with one color), such that no two three-cell rectangles (out of 288) with centers in $Q$ are colored identically. Is this possible?
(Two three-cell rectangles are colored identically, if it is possible to shift and/or rotate so that each cell of a rectangle matches the same color cell of the other rectangle.)
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$. Its diagonals $AC$ and $BD$ intersect at the point $O$. Let $E$ and $F$ be points on the segments $AO$ and $DO$ respectively. The line $EF$ intersects $\omega$ at the points $E_1$ and $F_1$. Circumcircles of triangles $ADE$ and $BCF$ intersect the line $EF$ at the points $E_2$, $F_2$ respectively. Prove that $E_1E_2 = F_1F_2$.
Prove that among every 42 numbers from the interval $[1, 10^6]$ it is possible to choose four numbers in such a way that for any permutation $(a, b, c, d)$ of these numbers, the following inequality holds:
$$
25(ab + cd)(ad + bc) \geq 16(ac + bd)^2.
$$