A ticket for the tram costs 1 leva. On the queue in front of the ticket seller are standing $n$ people with a banknote of 1 leva and $m$ people with a banknote of 2 leva. The ticket seller has no money in his cash deck so he can only sell a ticket to a buyer with a banknote of 2 leva, if he has at least 1 banknote of 1 leva. Determine the probability that the ticket seller could sell tickets to all of the people standing in the queue.

In $\Delta ABC$ with $AC=10$ and $BC=15$ the points $G$ and $I$ are its centroid and the center of its inscribed circle respectively. Find $AB$, if $\angle GIC=90^\circ$.

Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

Prove that if $x$, $y$, and $z$ are non-negative numbers and $x^2+y^2+z^2=1$, then the following inequality is true: $\frac{x}{1-x^2}+\frac{y}{1-y^2}+\frac{z}{1-z^2 }\geq \frac{3\sqrt{3}}{2}$

Let $p$ be some odd prime number and let $k=\frac{p+1}{2}$. The natural numbers $a_1,a_2…a_k$ are such that $a_i\neq a_j$ and $a_i<p$ for $\forall i,j=1,2…k$. Prove that for each natural number $r<p$ there exist not necessarily different $a_i$ and $a_j$, for which $a_i a_j\equiv r\, (mod\, p)$.

If $a$, $b$, and $c$ are positive numbers, determine the least possible value of the following expression: $\frac{1}{\frac{a}{b}+\frac{b}{c}+\frac{c}{a}}-\frac{2}{\frac{a}{c}+\frac{c}{b}+\frac{b}{a}}$.

$\Delta ABC$ is such that $AC+BC=2$ and the sum of its altitude through $C$ and its base $AB$ is $CD+AB=\sqrt{5}$. Find the sides of the triangle.

On a chess tournament two teams $A$ and $B$ are playing between each other and each consists of $n$ participants. It was noticed that however they arranged them in pairs, there was at least one pair that already played a match. Prove that there can be chosen $a$ chess players from $A$ and $b$ chess players from $B$ so that $a+b>n$ and each from the first chosen group has played a match earlier with each from the second group.

Find the area of a triangle with angles $\frac{1}{7} \pi$, $\frac{2}{7} \pi$, and $\frac{4}{7} \pi $, and radius of its circumscribed circle $R=1$.

Find all natural numbers, which cannot be expressed in the form $\frac{a}{b}+\frac{a+1}{b+1}$ where $a,b\in \mathbb{N}$.

Let $A$ be a set of natural numbers, for which for $\forall n\in \mathbb{N}$ exactly one of the numbers $n$, $2n$, and $3n$ is an element of $A$. If $2\in A$, show whether $13824\in A$.

The numbers $x_i,i=1,2…6\in \mathbb{R}^+$ are such that $x_1+x_2+...+x_6=1$ and $x_1 x_3 x_5+x_2 x_4 x_6\geq \frac{1}{540}$. Let $S=x_1 x_2 x_3+x_2 x_3 x_4+...+x_6 x_1 x_2$. If $max\, S=\frac{p}{q}$ , where $gcd(p,q)=1$, find $p+q$.

We are given the following sequence: $a_1=8,a_2=20,a_{n+2}=a_{n+1}^2+12a_n a_{n+1}+11a_n$. Prove that none of the members of the sequence can be presented as a sum of three seventh powers of natural numbers.

Determine all functions $f:\Bbb{R}\to\Bbb{R}$ such that \[ f(x^2 + f(y)) = (f(x) + y^2)^ 2 \] , for all $x,y\in \Bbb{R}.$

The quadrilateral $ABCD$ is such that $AB=AD=1$ and $\angle A=90^\circ$. If $CB=c$, $CA=b$, and $CD=a$, then prove that $(2-a^2-c^2 )^2+(2b^2-a^2-c^2 )^2=4a^2 c^2$ and $(a-c)^2\leq 2b^2\leq (a+c)^2$.

The lengths of the sides of a convex decagon are no greater than 1. Prove that for each inner point $M$ of the decagon there is at least one vertex $A$, for which $MA\leq \frac{\sqrt{5}+1}{2}$.

Let $n\in \mathbb{N}$ be a number multiple of 4. We take all permutations $(a_1,a_2...a_n)$ of the numbers $(1,2...n)$, for which $\forall j$, $a_i+j=n+1$ where $i=a_j$. Prove that there exist $\frac{(\frac{1}{2}n)!}{(\frac{1}{4}n)!}$ such permutations.

There are 20 towns on the bay of a circular island. Each town has 20 teams for a mathematical duel. No two of these teams are of equal strength. When two teams meet in a duel, the stronger one wins. For a given number $n\in \mathbb{N}$ one town $A$ can be called “n-stronger” than $B$, if there exist $n$ different duels between a team from $A$ and team from $B$, for which the team from $A$ wins. Find the maximum value of $n$, for which it is possible for each town to be n-stronger by its neighboring one clockwise.

In a circle with radius 1 a regular n-gon $A_1 A_2...A_n$ is inscribed. Calculate the product: $A_1 A_2.A_1 A_3 \dots A_1 A_{n-1} .A_1 A_n$.

Let $n$ be a natural number. Find the number of real roots of the following equation: $1+\frac{x}{1}+\frac{x^2}{2}+...+\frac{x^n}{n}=0$.

Let $c_0,c_1>0$. And suppose the sequence $\{c_n\}_{n\ge 0}$ satisfies \[ c_{n+1}=\sqrt{c_n}+\sqrt{c_{n-1}}\quad \text{for} \;n\ge 1 \] Prove that $\lim_{n\to \infty}c_n$ exists and find its value. Proposed by Sadovnichy-Grigorian-Konyagin

Find all triples $(x,y,z)$ of real numbers satisfying the system of equations $\left\{\begin{matrix} 3(x+\frac{1}{x})=4(y+\frac{1}{y})=5(z+\frac{1}{z}),\\ xy+yz+zx=1.\end{matrix}\right.$

Let $\Delta ABC$ be a triangle with orthocenter $H$ and midpoints $M_a,M_b$, and $M_c$ of $BC$, $AC$, and $AB$ respectively. A circle with center $H$ intersects the lines $M_bM_a$, $M_bM_c$, and $M_cM_a$ in points $U_1,U_2,V_1,V_2,W_1,W_2$ respectively. Prove that $CU_1=CU_2=AV_1=AV_2=BW_1=BW_2$.

An equilateral triangle $ABC$ is inscribed in a square with side 1 (each vertex of the triangle is on a side of the square and no two are on the same side). Determine the greatest and smallest value of the side of $\Delta ABC$.

For a natural number $x$ we define $f(x)$ to be the sum of all natural numbers less than $x$ and coprime with it. Let $m$ and $n$ be some natural numbers where $n$ is odd. Prove that there exist $x$, which is a multiple of $m$ and for which $f(x)$ is a perfect n-th power.

The sequence $\{x_n\}_{n=0}^\infty$ is defined by the following equations: $x_n=\sqrt{x_{n-1} x_{n-2}+\frac{n}{2}}$ ,$\forall$ $n\geq 2$, $x_0=x_1=1$. Prove that there exist a real number $a$, such that $an<x_n<an+1$ for each natural number $n$.

Prove the following inequality: $tan \, 1>\frac{3}{2}$.

In the right-angled $\Delta ABC$, with area $S$, a circle with area $S_1$ is inscribed and a circle with area $S_2$ is circumscribed. Prove the following inequality: $\pi \frac{S-S_1}{S_2} <\frac{1}{\pi-1}$.

Let $\sum_{i=1}^n a_i x_i =0$, $a_i,x_i\in \mathbb{Z}$. It is known that however we color $\mathbb{Z}$ with finite number of colors, then the given equation has a monochromatic (of one color) solution. Prove that there is some non-empty sum of its coefficients equal to 0.

Let $A_1 B_1 C_1$ and $A_2 B_2 C_2$ be two oppositely oriented concentric equilateral triangles. Prove that the lines $A_1 A_2$ , $B_1 B_2$ , and $C_1 C_2$ intersect in one point.

A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Let $A_1 B_1 C_1 D_1$ be the image of $ABCD$ after rotation with center $O$ and angle $\alpha \in (0,90^\circ)$. The points $P,Q,R$ and $S$ are intersections of $AB$ and $A_1 B_1$, $BC$ and $B_1 C_1$, $CD$ and $C_1 D_1$, and $DA$ and $D_1 A_1$. Prove that $PQRS$ is a parallelogram.

Let $n$ be a natural number and $\alpha ,\beta ,\gamma$ be the angles of an acute triangle. Determine the least possible value of the sum: $T=tan^n \alpha+tan^n \beta+tan^n \gamma$.

Let $A_n$ be the set of all sequences with length $n$ and members of the set $\{1,2…q\}$. We denote with $B_n$ a subset of $A_n$ with a minimal number of elements with the following property: For each sequence $a_1,a_2,...,a_n$ from $A_n$ there exist a sequence $b_1,b_2,...,b_n$ from $B_n$ such that $a_i\neq b_i$ for each $i=1,2,....,n$. Prove that, if $q>n$, then $|B_n |=n+1$.

Let $p$ and $q=4p+1$ be prime numbers. Determine the least power $i$ of 2 for which $2^i\equiv 1\,(mod\, q)$.

The polynomial $p(x)$ is of degree $9$ and $p(x)-1$ is exactly divisible by $(x-1)^{5}$. Given that $p(x) + 1$ is exactly divisible by $(x+1)^{5}$, find $p(x)$.

Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

We denote with $p_n(k)$ the number of permutations of the numbers $1,2,...,n$ that have exactly $k$ fixed points. a) Prove that $\sum_{k=0}^n kp_n (k)=n!$. b) If $s$ is an arbitrary natural number, then: $\sum_{k=0}^n k^s p_n (k)=n!\sum_{i=1}^m R(s,i)$, where with $R(s,i)$ we denote the number of partitions of the set $\{1,2,...,s\}$ into $i$ non-empty non-intersecting subsets and $m=min(s,n)$.

Calculate the sum $1+\frac{\binom{2}{1}}{8}+\frac{\binom{4}{2}}{8^2}+\frac{\binom{6}{3}}{8^3}+...+\frac{\binom{2n}{n}}{8^n}+...$

Let $M=\{1,2,...,n\}$. Prove that the number of pairs $(A,a)$, where $A\subset M$ and $a$ is a permutation of $M$, for which $a(A)\cap A=\emptyset $, is equal to $n!.F_{n+1}$, where $F_{n+1}$ is the $n+1$ member of the Fibonacci sequence.

In a non-isosceles $\Delta ABC$ with angle bisectors $AL_a$, $BL_b$, and $CL_c$ we have that $L_aL_c=L_bL_c$. Prove that $\angle C$ is smaller than $120^\circ$.