We call a bar of width ${w}$ on the surface of the unit sphere ${\Bbb{S}^2}$, a spherical segment, centered at the origin, which has width ${w}$ and is symmetric with respect to the origin. Prove that there exists a constant ${c>0}$, such that for any positive integer ${n}$ the surface ${\Bbb{S}^2}$ can be covered with ${n}$ bars of the same width so that any point is contained in no more than ${c\sqrt{n}}$ bars.

In a country between each pair of cities there is at most one direct road. There is a connection (using one or more roads) between any two cities even after the elimination of any given city and all roads incident to this city. We say that the city $A$ can be[i] k -directionally[/i] connected to the city $B$, if : we can orient at most $k$ roads such that after[i] arbitrary[/i] orientation of remaining roads for any fixed road $l$ (directly connecting two cities) there is a path passing through roads in the direction of their orientation starting at $A$, passing through $l$ and ending at $B$ and visiting each city at most once. Suppose that in a country with $n$ cities, any two cities can be[i] k - directionally[/i] connected. What is the minimal value of $k$?

A circle is drawn through vertices $A$ and $B$ of triangle $ABC$, intersecting sides $AC$ and $BC$ at points $M$ and $P$. It is known that the segment $MP$ contains the center of the circle inscribed in $ABC$. Find $MP$ if $AB = c$, $BC = a$, $CA=b$.

a) In how many ways can we read the word SARAJEVO from the table below, if it is allowed to jump from cell to an adjacent cell (by vertex or a side) cell? b) After the letter in one cell was deleted, only $525$ ways to read the word SARAJEVO remained. Find all possible positions of that cell.

In $\triangle ABC$, $\angle ACB=3\angle BAC$, $BC=5$, $AB=11$. Find $AC$

For how many different primes $p$, there exists an integer $n$ such that $ p\mid n^3+3$ and $p\mid n^5+5$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{Infinitely many} $

The natural numbers $a$ and $b$ are such that $a^a$ is divisible by $b^b$. Can we say that then $a$ is divisible by $b$?

Prove that if a triangle has integral side lengths and its circumradius is a prime number then the triangle is right-angled.

There are $n$ non-coplanar points in space. Prove that there exists a circle exactly passes through three points of them.

Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$. Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$. Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$, and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$. Rays $PM$ and $QN$ meet at $X$. Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$. Show that $A,X,Y$ are collinear.

The circle inscribed in triangle $ABC$ touches $AC$ at point $F$. The perpendicular from point $F$ on $BC$ intersects the bisector of angle $C$ at point $N$. Prove that segment $FN$ is equal to the radius of the circle inscribed in triangle $ABC$. (Oleksii Karliuchenko)

Two players, the builder and the destroyer, plays the following game. Builder starts and players chooses alternatively different elements from the set $\{0,1,\ldots,10\}.$ Builder wins if some four integer of those six integer he chose forms an arithmetic sequence. Destroyer wins if he can prevent to form such an arithmetic four-tuple. Which one has a winning strategy?

Find all positive integers $n$ that have precisely $\sqrt{n+1}$ natural divisors.

For each positive integer $k$ greater than $1$, find the largest real number $t$ such that the following hold: Given $n$ distinct points $a^{(1)}=(a^{(1)}_1,\ldots, a^{(1)}_k)$, $\ldots$, $a^{(n)}=(a^{(n)}_1,\ldots, a^{(n)}_k)$ in $\mathbb{R}^k$, we define the score of the tuple $a^{(i)}$ as \[\prod_{j=1}^{k}\#\{1\leq i'\leq n\textup{ such that }\pi_j(a^{(i')})=\pi_j(a^{(i)})\}\] where $\#S$ is the number of elements in set $S$, and $\pi_j$ is the projection $\mathbb{R}^k\to \mathbb{R}^{k-1}$ omitting the $j$-th coordinate. Then the $t$-th power mean of the scores of all $a^{(i)}$'s is at most $n$. Note: The $t$-th power mean of positive real numbers $x_1,\ldots,x_n$ is defined as \[\left(\frac{x_1^t+\cdots+x_n^t}{n}\right)^{1/t}\] when $t\neq 0$, and it is $\sqrt[n]{x_1\cdots x_n}$ when $t=0$. [i]Proposed by Cheng-Ying Chang and usjl[/i]

Which of the followings is false for the sequence $9,99,999,\dots$? $\textbf{(A)}$ The primes which do not divide any term of the sequence are finite. $\textbf{(B)}$ Infinitely many primes divide infinitely many terms of the sequence. $\textbf{(C)}$ For every positive integer $n$, there is a term which is divisible by at least $n$ distinct prime numbers. $\textbf{(D)}$ There is an inteter $n$ such that every prime number greater than $n$ divides infinitely many terms of the sequence. $\textbf{(E)}$ None of above

Let $ABC$ be a triangle with circumcircle $\Gamma$. If the internal angle bisector of $\angle A$ meets $BC$ and $\Gamma$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $A$ and $D$ tangent to $BC$, let the external angle bisector of $\angle A$ meet $\Gamma$ at $F$, and let $FO_1$ meet $\Gamma$ at some point $P \neq F$. Show that the circumcircle of $DEP$ is tangent to $BC$.

Given an arbitrary set of $2k+1$ integers $\{a_1,a_2,...,a_{2k+1}\}$. We make a new set $$ \{(a_1+a_2)/2, (a_2+a_3)/2, (a_{2k}+a_{2k+1})/2, (a_{2k+1}+a_1)/2\}$$ and a new one, according to the same rule, and so on... Prove that if we obtain integers only, the initial set consisted of equal integers only.

Find all solutions in positive integers to: \begin{eqnarray*} a + b + c = xyz \\ x + y + z = abc \end{eqnarray*}

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

[i]The Marathon.[/i] Let $\omega$ denote the incircle of triangle $ABC$. The segments $BC$, $CA$, and $AB$ are tangent to $\omega$ at $D$, $E$ and $F$, respectively. Point $P$ lies on $EF$ such that segment $PD$ is perpendicular to $BC$. The line $AP$ intersects $BC$ at $Q$. The circles $\omega_1$ and $\omega_2$ pass through $B$ and $C$, respectively, and are tangent to $AQ$ at $Q$; the former meets $AB$ again at $X$, and the latter meets $AC$ again at $Y$. The line $XY$ intersects $BC$ at $Z$. Given that $AB=15$, $BC=14$, and $CA=13$, find $\lfloor XZ\cdot YZ\rfloor$.

Let $x_0,x_1,x_2,\dots$ be a infinite sequence of real numbers, such that the following three equalities are true: I- $x_{2k}=(4x_{2k-1}-x_{2k-2})^2$, for $k\geq 1$ II- $x_{2k+1}=|\frac{x_{2k}}{4}-k^2|$, for $k\geq 0$ III- $x_0=1$ a) Determine the value of $x_{2022}$ b) Prove that there are infinite many positive integers $k$, such that $2021|x_{2k+1}$

The numbers from $1$ to $2017$ are written on a board. Deka and Farid play the following game : each of them, on his turn, erases one of the numbers. Anyone who erases a multiple of $2, 3$ or $5$ loses and the game is over. Is there a winning strategy for Deka ?

Positive integers $a,b,z$ satisfy the equation $ab=z^2+1$. Prove that there exist positive integers $x,y$ such that $$\frac{a}{b}=\frac{x^2+1}{y^2+1}$$

Let $I \subset \mathbb{R}$ be a nondegenerate interval and $f:I \to \mathbb{R}$ a differentiable function. We denote $J= \left\{ \frac{f(b)-f(a)}{b-a} : a,b \in I, a<b \right\}.$ Prove that: $a)$ $J$ is an interval; $b)$ $J \subset f'(I),$ and the set $f'(I) \setminus J$ contains at most two elements; $c)$ Using parts $a)$ and $b),$ deduce that $f'$ has the intermediate value property.

Suppose $ABC$ is a scalene right triangle, and $P$ is the point on hypotenuse $\overline{AC}$ such that $\angle ABP=45^\circ$. Given that $AP=1$ and $CP=2$, compute the area of $ABC$.