The cells of an $n\times n$ board with $n\ge 5$ are coloured black or white so that no three adjacent squares in a row, column or diagonal are the same colour. Show that for any $3\times 3$ square within the board, two of its corner squares are coloured black and two are coloured white.

Assume that $\triangle ABC$ is acute. Let $a=BC, b=CA, c=AB$. a) Denote $H$ by the orthocenter of $\triangle ABC$. Prove that:$$a.\frac{{\overrightarrow {HA} }}{{HA}} + b.\frac{{\overrightarrow {HB} }}{{HB}} + c.\frac{{\overrightarrow {HC} }}{{HC}} = \overrightarrow 0 .$$ b) Consider a point $P$ lying on the plane. Prove that the sum:$$aPa+bPB+cPC$$ get its minimum value iff $P\equiv H$.

Prove that every positive $a$ and $b$ satisfy inequality $$\frac{(a+b)^2}{2} + \frac{a+b}{4} \ge a\sqrt b + b\sqrt a$$

Let us choose arbitrarily $n$ vertices of a regular $2n$-gon and color them red. The remaining vertices are colored blue. We arrange all red-red distances into a non-decreasing sequence and do the same with the blue-blue distances. Prove that the sequences are equal.

Let $S$ be a nonempty set of primes satisfying the property that for each proper subset $P$ of $S$, all the prime factors of the number $\left(\prod_{p\in P}p\right)-1$ are also in $S$. Determine all possible such sets $S$.

The line that is tangent to the circle $x^2+y^2=25$ at the point $\left(3,4\right)$ intersects the $x$-axis at $\left(k,0\right)$. What is $k$? $\text{(A) }\frac{25}{4}\qquad\text{(B) }\frac{19}{3}\qquad\text{(C) }25\qquad\text{(D) }\frac{25}{3}\qquad\text{(E) }-\frac{7}{3}$

In cyclic quadrilateral $ABCD$, $AB= AD$. If $AC = 6$ and $\frac{AB}{BD} =\frac35$ , find the maximum possible area of $ABCD$.

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

In a game, two players control an adventurer in a dungeon. The adventurer starts with $H{}$ hit points, an integer greater than one. The dungeon consists of several chambers. There are some passageways in the dungeon, each leading from a chamber to a chamber. These passageways are one-way, and a passageway may return to its starting chamber. Every chamber can be exited through at least one passageway. There are 5 types of chambers: [list] [*]Entrance: the adventurer starts here, no passageway comes in here; [*]Hollow: nothing happens; [*]Spike: the adventurer loses a hit point; [*]Trap: the adventurer gets shot by an arrow; [*]Catacomb: the adventurer loses hit points equal to the total number of times they have been hit by an arrow. [/list] The two players control the adventurer alternatively. At a turn, a player can move him through one passageway. A player loses if the adventurer’s hit points fall below zero due to their action (at 0 hit points, the character is alive). Construct a dungeon map, which has at most 20 chambers in total and exactly one entrance, with the following condition: the first player has a winning strategy if $H{}$ is a prime, and the second player has a winning strategy if $H{}$ is composite. [i]Note: If the game doesn’t end after a finite number of moves, neither player wins.[/i]

A $9\times 1$ rectangle is divided into unit squares. A broken line, from the lower left to the upper right corner, goes through all $20$ vertices of the unit squares and consists of $19$ line segments. How many such lines are there?

Find all real values $x$ that satisfy the following equation: $$\frac{\sin 3x cos \left(\frac{\pi}{3}-4x \right)+ 1}{\sin \left(\frac{\pi}{3}-7x \right) - cos\left(\frac{\pi}{6}+x \right)+m}= 0$$ where $m$ is a given real number.

Parallelogram $ABCD$ has $AB=CD=6$ and $BC=AD=10$, where $\angle ABC$ is obtuse. The circumcircle of $\triangle ABD$ intersects $BC$ at $E$ such that $CE=4$. Compute $BD$.

Find all real-valued functions $f$ defined on pairs of real numbers, having the following property: for all real numbers $a, b, c$, the median of $f(a,b), f(b,c), f(c,a)$ equals the median of $a, b, c$. (The [i]median[/i] of three real numbers, not necessarily distinct, is the number that is in the middle when the three numbers are arranged in nondecreasing order.)

Let $ ABC$ be a triangle where $ \angle ACB\equal{}90^{\circ}$. Let $ D$ be the midpoint of $ BC$ and let $ E$, and $ F$ be points on $ AC$ such that $ CF\equal{}FE\equal{}EA$. The altitude from $ C$ to the hypotenuse $ AB$ is $ CG$, and the circumcentre of triangle $ AEG$ is $ H$. Prove that the triangles $ ABC$ and $ HDF$ are similar.

The quadrilateral $ ABCD$ inscribed in a circle wich has diameter $ BD$. Let $ A',B'$ are symmetric to $ A,B$ with respect to the line $ BD$ and $ AC$ respectively. If $ A'C \cap BD \equal{} P$ and $ AC\cap B'D \equal{} Q$ then prove that $ PQ \perp AC$

Tom has a collection of $13$ snakes, $4$ of which are purple and $5$ of which are happy. He observes that $\bullet$ all of his happy snakes can add $\bullet$ none of his purple snakes can subtract, and $\bullet$ all of his snakes that can’t subtract also can’t add Which of these conclusions can be drawn about Tom’s snakes? $\textbf{(A)}$ Purple snakes can add. $\textbf{(B)}$ Purple snakes are happy. $\textbf{(C)}$ Snakes that can add are purple. $\textbf{(D)}$ Happy snakes are not purple. $\textbf{(E)}$ Happy snakes can't subtract.

Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

Somewhere in the universe, $n$ students are taking a $10$-question math competition. Their collective performance is called [i]laughable[/i] if, for some pair of questions, there exist $57$ students such that either all of them answered both questions correctly or none of them answered both questions correctly. Compute the smallest $n$ such that the performance is necessarily laughable.

The circle $C_1$ and the circle $C_2$ passing through the center of $C_1$ intersect each other at $A$ and $B$. The line tangent to $C_2$ at $B$ meets $C_1$ at $B$ and $D$. If the radius of $C_1$ is $\sqrt 3$ and the radius of $C_2$ is $2$, find $\dfrac{|AB|}{|BD|}$. $ \textbf{(A)}\ \dfrac 12 \qquad\textbf{(B)}\ \dfrac {\sqrt 3}2 \qquad\textbf{(C)}\ \dfrac {2\sqrt 3}2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \dfrac {\sqrt 5}2 $

Given triangle $ABC$. Find the locus of points $M$ such that there is a rotation with center at $M$ that takes $C$ to a certain point on side $AB$.

From a square board $11$ squares long and $11$ squares wide, the central square is removed. Prove that the remaining $120$ squares cannot be covered by $15$ strips each $8$ units long and one unit wide.

Prove that there exist subfields of $\mathbb R$ that are a) non-measurable and b) of measure zero and continuum cardinality. (translated by Miklós Maróti)

Given that $x$, $y$ are positive integers with $x(x+1)|y(y+1)$, but neither $x$ nor $x+1$ divides either of $y$ or $y+1$, and $x^2 + y^2$ as small as possible, find $x^2 + y^2$.

Among the $n$ inhabitants of an island, every two are either friends or enemies. Some day, the chief of the island orders that each inhabitant (including himself) makes and wears a necklace consisting of marbles, in such a way that the necklaces of two friends have at least one marble of the same type and that the necklaces of two enemies differ at all marbles. (A necklace may also be marbleless). Show that the chief’s order can be achieved by using $\left\lfloor\frac{n^2}4\right\rfloor$ different types of stones, but not necessarily by using fewer types.

Given positive integers $x$ and $y$ such that $xy^2$ is a perfect cube, prove that $x^2y$ is also a perfect cube.