In a room are several people, some of which always lie and all others always tell the truth. Their ages are pairwise distinct. Each person says one of the following phrases: "In this room, there is an equal number of truth-sayers older than me and of liars younger than me" or "In this room, there is an equal number of truth-sayers younger than me and of liars older than me" What is the maximum possible number of truth-sayers in the room? Find an example in which this maximum is achieved and prove a higher number is impossible.

We consider sports tournaments with $n \ge 4$ participating teams and where every pair of teams plays against one another at most one time. We call such a tournament [i]balanced [/i] if any four participating teams play exactly three matches between themselves. So, not all teams play against one another. Determine the largest value of $n$ for which a balanced tournament with $n$ teams exists.

All the angles of the hexagon $ABCDEF$ are equal. Prove that \[AB-DE=EF-BC=CD-FA \]

The sequence $(x_n)$ is defined as follows: $$x_1=2,\, x_{n+1}=\sqrt{x_n+8}-\sqrt{x_n+3}$$ for all $n\geq 1$. a. Prove that $(x_n)$ has a finite limit and find that limit. b. For every $n\geq 1$, prove that $$n\leq x_1+x_2+\dots +x_n\leq n+1.$$

For the positive integers $a, b$, $lcm(a, b) = \gcd(a, b) = p^2q^4$, where $p$ and $q$ are prime numbers. Find $lcm(ap, bq)$. Here lcm and gcd represent the least common multiple and the greatest common divisor respectively.

Let $\Delta ABC$ be a right triangle with $\angle ACB=90^\circ$. The points $P$ and $Q$ on the side $BC$ and $R$ and $S$ on the side $CA$ are such that $\angle BAP=\angle PAQ=\angle QAC$ and $\angle ABS=\angle SBR=\angle RBC$. If $AP\cap BS=T$, prove that $120^\circ<\angle RTB<150^\circ$.

Let a positive integer $n$ be $\textit{nice}$ if there exists a positive integer $m$ such that \[ n^3 < 5mn < n^3 +100. \] Find the number of [i]nice[/i] positive integers. [i]Proposed by Akshaj[/i]

Let $I=3\sqrt2\int^x_0\frac{\sqrt{1+\cos t}}{17-8\cos t}dt$. If $0<x<\pi$ and $\tan I=\frac2{\sqrt3}$, what is $x$?

We call a natural number $n$ good if each of the numbers $n$, $ n+1$, $n+2$ and $n+3$ are divided by the sum of their digits. (For example, $n = 60398$ is good.) Does the penultimate digit of a good number ending in eight have to be nine?

Let $ a, b, c, m, n$ be positive integers. Consider the trinomial $ f (x) = ax^{2}+bx+c$. Show that there exist $ n$ consecutive natural numbers $ a_{1}, a_{2}, . . . , a_{n}$ such that each of the numbers $ f (a_{1}), f (a_{2}), . . . , f (a_{n})$ has at least $ m$ different prime factors.

Let $ABC$ be a triangle with incenter $I$, and $A$-excenter $\Gamma$. Let $A_1,B_1,C_1$ be the points of tangency of $\Gamma$ with $BC,AC$ and $AB$, respectively. Suppose $IA_1, IB_1$ and $IC_1$ intersect $\Gamma$ for the second time at points $A_2,B_2,C_2$, respectively. $M$ is the midpoint of segment $AA_1$. If the intersection of $A_1B_1$ and $A_2B_2$ is $X$, and the intersection of $A_1C_1$ and $A_2C_2$ is $Y$, prove that $MX=MY$.

The number $2021$ can be written as the sum of $2021$ consecutive integers. What is the largest term in the sequence of $2021$ consecutive integers? [i]Proposed by Taiki Aiba[/i]

Let $m, n, p$ be fixed positive real numbers which satisfy $mnp = 8$. Depending on these constants, find the minimum of $$x^2+y^2+z^2+ mxy + nxz + pyz,$$ where $x, y, z$ are arbitrary positive real numbers satisfying $xyz = 8$. When is the equality attained? Solve the problem for: [list=a][*]$m = n = p = 2,$ [*] arbitrary (but fixed) positive real numbers $m, n, p.$[/list] [i]Stijn Cambie[/i]

In a mathematical version of baseball, the umpire chooses a positive integer $m$, $m \leq n$, and you guess positive integers to obtain information about $m$. If your guess is smaller than the umpire's $m$, he calls it a "ball"; if it is greater than or equal to $m$, he calls it a "strike." To "hit" it you must state the the correct value of $m$ after the 3rd strike or the 6th guess, whichever comes first. What is the largest $n$ so that there exists a strategy that will allow you to bat 1.000, i.e. always state $m$ correctly? Describe your strategy in detail.

$ OA, OB, OC, OD$ are 4 rays in space such that the angle between any two is the same. Show that for a variable ray $ OX,$ the sum of the cosines of the angles $ XOA, XOB, XOC, XOD$ is constant and the sum of the squares of the cosines is also constant.

Two players $A$ and $B$ participate in the following game. Initially we have a pile of 2003 stones. $A$ plays first, and he picks a divisor of 2003 and removes that number of stones from the pile. Then $B$ picks a divisor of the number of remaining stones, and removes that number of stones from the pile, and so forth. The players who removes the last stone loses. Prove that one of the players has a winning strategy and describe it.

Find all the integers $x,y$ satisfying equation $x^2+x=y^4+y^3+y^2+y$.

A beam of light strikes $\overline{BC}$ at point $C$ with angle of incidence $\alpha=19.94^\circ$ and reflects with an equal angle of reflection as shown. The light beam continues its path, reflecting off line segments $\overline{AB}$ and $\overline{BC}$ according to the rule: angle of incidence equals angle of reflection. Given that $\beta=\alpha/10=1.994^\circ$ and $AB=AC,$ determine the number of times the light beam will bounce off the two line segments. Include the first reflection at $C$ in your count. [asy] size(250);defaultpen(linewidth(0.7)); real alpha=24, beta=32; pair B=origin, C=(1,0), A=dir(beta), D=C+0.5*dir(alpha); pair EE=2*dir(180-alpha), E=intersectionpoint(C--EE, A--B); pair EEE=reflect(B,A)*EE, EEEE=reflect(C,B)*EEE, F=intersectionpoint(E--EEE, B--C), G=intersectionpoint(F--EEEE, A--B); draw((1.4,0)--B--1.4*dir(beta)); draw(D--C, linetype("4 4"),EndArrow(5)); draw(C--E, linetype("4 4"),EndArrow(5)); draw(E--F, linetype("4 4"),EndArrow(5)); draw(F--G, linetype("4 4"),EndArrow(5)); markscalefactor=0.01; draw(anglemark(C,B,A)); draw(anglemark((1.4,0), C,D)); label("$\beta$", 0.07*dir(beta/2), dir(beta/2), fontsize(10)); label("$\alpha$", C+0.07*dir(alpha/2), dir(alpha/2), fontsize(10)); label("$A$", A, dir(90)*dir(A)); label("$B$", B, dir(beta/2+180)); label("$C$", C, S);[/asy]

Angle $A$ in triangle $ABC$ is equal to $a$. A circle passing through $A$ and $B$ and tangent to $BC$ intersects the median to side $BC$ (or its extension) at a point $M$ different from $A$. Find the angle $\angle BMC$.

Is $$\prod_{k=0}^\infty \left(1-\frac{1}{2022^{k!}}\right)$$ rational?

[b]a)[/b] Show that , if $ a,b>1 $ are two distinct real numbers, then $ \log_a\log_a b >\log_b\log_a b. $ [b]b)[/b] Show that if $ a_1>a_2>\cdots >a_n>1 $ are $ n\ge 2 $ real numbers, then $$ \log_{a_1}\log_{a_1} a_2 +\log_{a_2}\log_{a_2} a_3 +\cdots +\log_{a_{n-1}}\log_{a_{n-1}} a_n +\log_{a_n}\log_{a_n} a_1 >0. $$

Let $n$ be an odd positive integer and consider an $n \times n$ chessboard of $n^2$ unit squares. In some of the cells of the chessboard, we place a knight. A knight in a cell $c$ is said to [i]attack [/i] a cell $c'$ if the distance between the centers of $c$ and $c'$ is exactly $\sqrt{5}$ (in particular, a knight does not attack the cell which it occupies). Suppose each cell of the board is attacked by an even number of knights (possibly zero). Show that the configuration of knights is symmetric with respect to all four axes of symmetry of the board (i.e. the configuration of knights is both horizontally and vertically symmetric, and also unchanged by reflection along either diagonal of the chessboard). [i]NIkolai Beluhov[/i]

$\sqrt{\frac{8^{10}+4^{10}}{8^4+4^{11}}}=$ $ \textbf{(A)}\ \sqrt{2} \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 32 \qquad\textbf{(D)}\ 12^{\frac{2}{3}} \qquad\textbf{(E)}\ 512.5 $

There are $2003$ coins distributed in several bags. The bags are then distributed in several pockets. It is known that the total number of bags is greater than the number of coins in each of the pockets. Is it true that the total number of pockets is greater than the number of coins in some of the bags?

Let $f(n)=\sum_{k=0}^{2010}n^k$. Show that for any integer $m$ satisfying $2\leqslant m\leqslant 2010$, there exists no natural number $n$ such that $f(n)$ is divisible by $m$. [i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 1)[/i]