Two vertices of the rectangle are located on side $BC$ of triangle $ABC$, and the other two are on sides $AB$ and $AC$. It is known that the midpoint of the altitude of this triangle, drawn on the side $BC$, lies on one of the diagonals of the rectangle, and the side of the rectangle located on $BC$ is three times less than $BC$. In what ratio does the altitude of the triangle divide the side $BC$ ?

Let $k$ be a natural number. Bijection $f:\mathbb{Z} \rightarrow \mathbb{Z}$ has the following property: for any integers $i$ and $j$, $|i-j|\leq k$ implies $|f(i) - f(j)|\leq k$. Prove that for every $i,j\in \mathbb{Z}$ it stands: \[|f(i)-f(j)|= |i-j|.\]

An infinite array of rational numbers $G(d, n)$ is defined for integers $d$ and $ n$ with $1\leq d \leq n$ as follows: $$G(1, n)= \frac{1}{n}, \;\;\; G(d,n)= \frac{d}{n} \sum_{i=d}^{n} G(d-1, i-1) \; \text{for} \; d>1.$$ For $1 < d < p$ and $p$ prime, prove that $G(d, p)$ is expressible as a quotient $s\slash t$ of integers $s$ and $t$ with $t$ not divisible by $p.$

$ABC$ is a triangle such that $AC<AB<BC$ and $D$ is a point on side $AB$ satisfying $AC=AD$. The circumcircle of $ABC$ meets with the bisector of angle $A$ again at $E$ and meets with $CD$ again at $F$. $K$ is an intersection point of $BC$ and $DE$. Prove that $CK=AC$ is a necessary and sufficient condition for $DK \cdot EF = AC \cdot DF$.

The function $g : [0, \infty) \to [0, \infty)$ satisfies the functional equation: $g(g(x)) = \frac{3x}{x + 3}$, for all $x \ge 0$. You are also told that for $2 \le x \le 3$: $g(x) = \frac{x + 1}{2}$. (a) Find $g(2021)$. (b) Find $g(1/2021)$.

Let $P_1, P_2, \ldots, P_6$ be points in the complex plane, which are also roots of the equation $x^6+6x^3-216=0$. Given that $P_1P_2P_3P_4P_5P_6$ is a convex hexagon, determine the area of this hexagon.

Let $ABCDE$ be a convex pentagon such that $BC=DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB=TD,TC=TE$ and $\angle ABT = \angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P,B,A,Q$ occur on their line in that order. Let line $AE$ intersect $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R,E,A,S$ occur on their line in that order. Prove that the points $P,S,Q,R$ lie on a circle.

Consider a triangle $ABC$ with $AB=AC$, and $D$ the foot of the altitude from the vertex $A$. The point $E$ lies on the side $AB$ such that $\angle ACE= \angle ECB=18^{\circ}$. If $AD=3$, find the length of the segment $CE$.

Let $\{a_n\}$ be the sequance with $a_0=0$, $a_n=\frac{1}{a_{n-1}-2}$ ($n\in N_+$). Select an arbitrary term $a_k$ in the sequence $\{a_n\}$ and construct the sequence $\{b_n\}$: $b_0=a_k$, $b_n=\frac{2b_{n-1}+1} {b_{n-1}}$ ($n\in N_+$) . Determine whether the sequence $\{b_n\}$ is a finite sequence or an infinite sequence and give proof.

Assume that every root of polynomial $P(x) = x^d - a_1x^{d-1} + ... + (-1)^{d-k}a_d$ is in $[0,1]$. Show that for every $k = 1,2,...,d$ the following inequality holds: $ a_k - a_{k+1} + ... + (-1)^{d-k}a_d \geq 0 $

What is the shortest distance between the plane $Ax + By + Cz + 1 = 0$ and the ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1.$ You may find it convenient to use the notation $h = (A^2 + B^2 + C^2)^{\frac{-1}{2}}, m = (a^2A^2 + b^2B^2 + c^2C^2)^{\frac{1}{2}}.$ What is the algebraic condition for the plane not to intersect the ellipsoid?

Compute all real values of $b$ such that, for $f(x) = x^2+bx-17, f(4)=f'(4)$.

Viktor and Natalia play a colouring game with a 3-dimensional cube taking turns alternatingly. Viktor goes first, and on each of his turns, he selects an unpainted edge, and paints it violet. On each of Natalia's turns, she selects an unpainted edge, or at most once during the game a face diagonal, and paints it neon green. If the player on turn cannot make a legal move, then the turn switches to the other player. The game ends when nobody can make any more legal moves. Natalia wins if at the end of the game every vertex of the cube can be reached from every other vertex by traveling only along neon green segments (edges or diagonal), otherwise Viktor wins. Who has a winning strategy? (Prove your answer.) [i]Authored by Viktor Simjanoski[/i]

From a point $C$, tangents $CA$ and $CB$ are drawn to a circle $O$. From an arbitrary point $N$ on the circle, perpendiculars $ND, NE, NF$ are drawn on $AB, CA$ and $CB$, respectively. Prove that the length of $ND$ is the mean proportional of the lengths of $NE$ and $NF$.

$6k+2$ people play in odd or even championship. In each odd or even match they participate exactly two people. Six rounds have been arranged so that in each round there are $3k + 1$ simultaneous matches, and no player participates in two games of the same round. It is known that two people do not play with each other more than one turn. Prove that there are $k + 1$ people where any two of them have not played each other. [hide=original wording] 6k+2 pessoas jogam em campeonato de par ou impar. Em cada partida de par ou impar participam exatamente duas pessoas. Seis rodadas foram organizadas, de modo que, em cada rodada, ha 3k + 1 partidas simultaneas, e nenhum jogador participa de dois jogos da mesma rodada. Sabe-se que duas pessoas nao jogam entre si mais de uma vez. Prove que existem k + 1 pessoas em que quaisquer duas delas nao jogaram entre si. [/quote]

Let $ABC$ be an equilateral triangle. From the vertex $A$ we draw a ray towards the interior of the triangle such that the ray reaches one of the sides of the triangle. When the ray reaches a side, it then bounces off following the law of reflection, that is, if it arrives with a directed angle $\alpha$, it leaves with a directed angle $180^{\circ}-\alpha$. After $n$ bounces, the ray returns to $A$ without ever landing on any of the other two vertices. Find all possible values of $n$.

Let $a,b$ be positive integers satisfying $$\sqrt{\dfrac{ab}{2b^2-a}}=\dfrac{a+2b}{4b}$$. Find $|10(a-5)(b-15)|+8$.

At each vertex of a $13$-sided polygon we write one of the numbers $1,2,3,…, 12,13$, without repeating. Then, on each side of the polygon we write the difference of the numbers of the vertices of its ends (the largest minus the smallest). For example, if two consecutive vertices of the polygon have the numbers $2$ and $11$, the number $9$ is written on the side they determine. a) Is it possible to number the vertices of the polygon so that only the numbers $3, 4$ and $5$ are written on the sides? b) Is it possible to number the vertices of the polygon so that only the numbers $3, 4$ and $6$ are written on the sides?

Find the greatest integer $n < 1000$ for which $4n^3 - 3n$ is the product of two consecutive odd integers.

We say that the real numbers $a$ and $b$ have property $P$ if: $a^2+b \in Q$ and $b^2 + a \in Q$.Prove that: a) The numbers $a= \frac{1+\sqrt2}{2}$ and $b= \frac{1-\sqrt2}{2}$ are irrational and have property $P$ b) If $a, b$ have property $P$ and $a+b \in Q -\{1\}$, then $a$ and $b$ are rational numbers c) If $a, b$ have property $P$ and $\frac{a}{b} \in Q$, then $a$ and $b$ are rational numbers.

Incircle of $ABC$ tangent $AB,AC,BC$ in $C_1,B_1,A_1$. $AA_1$ intersect incircle in $E$. $N$ is midpoint $B_1A_1$. $M$ is symmetric to $N$ relatively $AA_1$. Prove that $\angle EMC= 90$

Let $ k$ and $ n$ be integers with $ 0\le k\le n \minus{} 2$. Consider a set $ L$ of $ n$ lines in the plane such that no two of them are parallel and no three have a common point. Denote by $ I$ the set of intersections of lines in $ L$. Let $ O$ be a point in the plane not lying on any line of $ L$. A point $ X\in I$ is colored red if the open line segment $ OX$ intersects at most $ k$ lines in $ L$. Prove that $ I$ contains at least $ \dfrac{1}{2}(k \plus{} 1)(k \plus{} 2)$ red points. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

In an acute $ABC$ triangle which has a circumcircle center called $O$, there is a line that perpendiculars to $AO$ line cuts $[AB]$ and $[AC]$ respectively on $D$ and $E$ points. There is a point called $K$ that is different from $AO$ and $BC$'s junction point on $[BC]$. $AK$ line cuts the circumcircle of $ADE$ on $L$ that is different from $A$. $M$ is the symmetry point of $A$ according to $DE$ line. Prove that $K$,$L$,$M$,$O$ are circular.

There are $32$ boxers in a tournament. Each boxer can fight no more often than once per day. It is known that the boxers are of different strength, and the stronger man always wins. Prove that a $15$ day tournament can be organised so as to determine their classification (put them in the order of strength). The schedule of fights for each day is fixed on the evening before and cannot be changed during the day. (A. Andjans, Riga)

Let $ X$ be a set containing $ 2k$ elements, $ F$ is a set of subsets of $ X$ consisting of certain $ k$ elements such that any one subset of $ X$ consisting of $ k \minus{} 1$ elements is exactly contained in an element of $ F.$ Show that $ k \plus{} 1$ is a prime number.