Find all the pairs of positive integers $ (a,b)$ such that $ a^2 \plus{} b \minus{} 1$ is a power of prime number $ ; a^2 \plus{} b \plus{} 1$ can divide $ b^2 \minus{} a^3 \minus{} 1,$ but it can't divide $ (a \plus{} b \minus{} 1)^2.$

In the regular hexagon $ABCDEF$ on the line $AF$, the point $X$ is taken so that the angle $XCD$ is $45^o$. Find the angle $\angle FXE$. (Kiev Olympiad)

Young Guy likes to make friends with numbers, so he calls a number “friendly” if the sum of its digits is equal to the product of its digits. How many $3 \text{digit friendly numbers}$ are there?

In an arcade game, the "monster" is the shaded sector of a circle of radius $ 1$ cm, as shown in the figure. The missing piece (the mouth) has central angle $ 60^{\circ}$. What is the perimeter of the monster in cm? [asy]size(100); defaultpen(linewidth(0.7)); filldraw(Arc(origin,1,30,330)--dir(330)--origin--dir(30)--cycle, yellow, black); label("1", (sqrt(3)/4, 1/4), NW); label("$60^\circ$", (1,0)); [/asy] $ \textbf{(A)}\ \pi \plus{} 2 \qquad \textbf{(B)}\ 2\pi \qquad \textbf{(C)}\ \frac53 \pi \qquad \textbf{(D)}\ \frac56 \pi \plus{} 2 \qquad \textbf{(E)}\ \frac53 \pi \plus{} 2$

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

Some squares of the infinite sheet of cross-lined paper are red. Each $2\times 3$ rectangle (of $6$ squares) contains exactly two red squares. How many red squares can be in the $9\times 11$ rectangle of $99$ squares?

Let $ f : \mathbb{R}\to \mathbb{R}$ be a continuous function. Suppose that for any $ c > 0$, the graph of $ f$ can be moved to the graph of $ cf$ using only a translation or a rotation. Does this imply that $ f(x) = ax+b$ for some real numbers $ a$ and $ b$?

Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

Given a triangle $ABC$, in which the angle $B$ is three times the angle $C$. On the side $AC$, point $D$ is chosen such that the angle $BDC$ is twice the angle $C$. Prove that $BD + BA = AC$.

Let $A$ be an infinite subset of $\mathbb{N}$, and $n$ a fixed integer. For any prime $p$ not dividing $n$, There are infinitely many elements of $A$ not divisible by $p$. Show that for any integer $m >1, (m,n) =1$, There exist finitely many elements of $A$, such that their sum is congruent to 1 modulo $m$ and congruent to 0 modulo $n$.

Given a tetrahedron $ABCD.$ The sphere $\omega$ inscribed in it touches the face $ABC$ at point $T$. Sphere $\omega' $ touches face $ABC$ at point $T'$ and extensions of faces $ABD$, $BCD$, $CAD$. Prove that the lines $AT$ and $AT'$ are symmetric wrt bisector of angle $\angle BAC$

Find all integers $m>1$ such that $m^3$ is a sum of $m$ squares of consecutive integers.

Determine all positive integers $x$, $y$ and $z$ such that $x^5 + 4^y = 2013^z$. ([i]Serbia[/i])

Let $m$ and $n$ be positive integers. Some squares of an $m \times n$ board are coloured red. A sequence $a_1, a_2, \ldots , a_{2r}$ of $2r \ge 4$ pairwise distinct red squares is called a [i]bishop circuit[/i] if for every $k \in \{1, \ldots , 2r \}$, the squares $a_k$ and $a_{k+1}$ lie on a diagonal, but the squares $a_k$ and $a_{k+2}$ do not lie on a diagonal (here $a_{2r+1}=a_1$ and $a_{2r+2}=a_2$). In terms of $m$ and $n$, determine the maximum possible number of red squares on an $m \times n$ board without a bishop circuit. ([i]Remark.[/i] Two squares lie on a diagonal if the line passing through their centres intersects the sides of the board at an angle of $45^\circ$.)

$N$ is a $50$-digit number (in decimal representation). All digits except the $26$th digit (from the left) are $1$. If $N$ is divisible by $13$, find its $26$-th digit.

Find all continious $f: \mathbb R\longrightarrow\mathbb R$ that for any $x,y$ \[f(x)+f(y)+f(xy)=f(x+y+xy)\]

Determine all the pairs $(a,b)$ of positive integers, such that all the following three conditions are satisfied: 1- $b>a$ and $b-a$ is a prime number 2- The last digit of the number $a+b$ is $3$ 3- The number $ab$ is a square of an integer.

Find all positive integers $p$, $q$, $r$ such that $p$ and $q$ are prime numbers and $\frac{1}{p+1}+\frac{1}{q+1}-\frac{1}{(p+1)(q+1)} = \frac{1}{r}.$

Some blue and red circular disks of identical size are packed together to form a triangle. The top level has one disk and each level has 1 more disk than the level above it. Each disk not at the bottom level touches two disks below it and its colour is blue if these two disks are of the same colour. Otherwise its colour is red. Suppose the bottom level has 2048 disks of which 2014 are red. What is the colour of the disk at the top?

Let $N$ be an odd number, $N\geq 3$. $N$ tennis players take part in a championship. Before starting the championship, a commission puts the players in a row depending on how good they think the players are. During the championship, every player plays with every other player exactly once, and each match has a winner. A match is called [i]suprising[/i] if the winner was rated lower by the commission. At the end of the tournament, players are arranged in a line based on the number of victories they have achieved. In the event of a tie, the commission's initial order is used to decide which player will be higher. It turns out that the final order is exactly the same as the commission's initial order. What is the maximal number of suprising matches that could have happened.

The sports plane flew along a diamond-shaped route in windy weather. He flew through the first three sides of the rhombus in $a $, $b$ and $c$ hours, respectively. How long did it take him to cover the fourth side of the diamond? (The speed of an aircraft is a vector equal to the sum of two vectors: the aircraft’s own speed and the wind speed. Wind speed is a constant vector. The aircraft’s own speed is a vector of constant length).

Compute the number of ordered pairs of non-negative integers $(x, y)$ which satisfy $x^2 + y^2 = 32045.$

Find all functions $f : \mathbb R \to \mathbb R$ for which \[f(a - b) f(c - d) + f(a - d) f(b - c) \leq (a - c) f(b - d),\] for all real numbers $a, b, c$ and $d$. Note that there is only one occurrence of $f$ on the right hand side!

Assume that all angles of a triangle $ABC$ are acute. Let $D$ and $E$ be points on the sides $AC$ and $BC$ of the triangle such that $A, B, D,$ and $E$ lie on the same circle. Further suppose the circle through $D,E,$ and $C$ intersects the side $AB$ in two points $X$ and $Y$. Show that the midpoint of $XY$ is the foot of the altitude from $C$ to $AB$.

Let $ABCD$ be a convex quadrilateral. Construct equilateral triangles $AQB$, $BRC$, $CSD$ and $DPA$ externally on the sides $AB$, $BC$, $CD$ and $DA$ respectively. Let $K, L, M, N$ be the mid-points of $P Q, QR, RS, SP$. Find the maximum value of $$\frac{KM + LN}{AC + BD}$$ .