Given a triangle $XYZ$ with $\angle Y = 90^{\circ}, XY=1,$ and $XZ=2,$ mark a point $Q$ on $YZ$ such that $\frac{ZQ}{ZY}=\frac{1}{3}.$ A laser beam is shot from $Q$ perpendicular to $YZ,$ and it reflects off the sides of $XYZ$ indefinitely. How far has the laser traveled when it reaches its $2010$th bounce?

For a competition a school wants to nominate a team of $k$ students, where $k$ is a given positive integer. Each member of the team has to compete in the three disciplines juggling, singing and mental arithmetic. To qualify for the team, the $n \ge 2$ students of the school compete in qualifying competitions, determining a unique ranking in each of the three disciplines. The school now wants to nominate a team satisfying the following condition: $(*)$ [i]If a student $X$ is not nominated for the team, there is a student $Y$ on the team who defeated $X$ in at least two disciplines.[/i] Determine all positive integers $n \ge 2$ such that for any combination of rankings, a team can be chosen to satisfy the condition $(*)$, when a) $k=2$, b) $k=3$.

Gabriela found an encyclopedia with $2023$ pages, numbered from $1$ to $2023$. She noticed that the pages formed only by even digits have a blue mark, and that every three pages since page two have a red mark. How many pages of the encyclopedia have both colors?

Points $A$, $B$, and $C$ lie on a semicircle with diameter $\overline{PQ}$ such that $AB = 3$, $AC = 4$, $BC = 5$, and $A$ is on $\overline{PQ}$. Given $\angle PAB = \angle QAC$, compute the area of the semicircle.

Sequences $(a_n)_{n=0}^{\infty}$ and $(b_n)_{n=0}^{\infty}$ are defined with recurrent relations : $$a_0=0 , \;\;\; a_1=1, \;\;\;\; a_{n+1}=\frac{2018}{n} a_n+ a_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$ and $$b_0=0 , \;\;\; b_1=1, \;\;\;\; b_{n+1}=\frac{2020}{n} b_n+ b_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$ Prove that :$$\frac{a_{1010}}{1010}=\frac{b_{1009}}{1009}$$

In triangle ABC, the altitude, angle bisector and median from C divide the angle C into four equal angles. Find angle B.

Let $d \geq 3$ and let $A_1 \dots A_{d + 1}$ be a simplex in $\mathbb{R}^d$. (A simplex is the convex hull of $d + 1$ points not lying in a common hyperplane.) For every $i = 1, \dots , d + 1$ let $O_i$ be the circumcentre of the face $A_1 \dots A_{i - 1}A_{i+1}\dots A_{d+1}$, i.e. $O_i$ lies in the hyperplane $A_1 \dots A_{i - 1}A_{i+1}\dots A_{d+1}$ and it has the same distance from all points $A_1, \dots , A_{i-1}, A_{i+1}, \dots , A_{d+1}$. For each $i$ draw a line through $A_i$ perpendicular to the hyperplane $O_1 \dots O_{i-1}O_{i+1} \dots O_{d+1}$. Prove that either these lines are parallel or they have a common point.

Let $z_1$, $z_2$, $z_3$, and $z_4$ be the four distinct complex solutions of the equation \[ z^4 - 6z^2 + 8z + 1 = -4(z^3 - z + 2)i. \] Find the sum of the six pairwise distances between $z_1$, $z_2$, $z_3$, and $z_4$.

Let $ABC$ be a right triangle with $\angle ABC = 90^o$ . Points $D$ and $E$, located on the legs $(AC)$ and $(AB)$ respectively, are the legs of the inner bisectors taken from the vertices $B$ and $C$, respectively. Let $I$ be the center of the circle inscribed in the triangle $ABC$. If $BD \cdot CE = m^2 \sqrt2$ , find the area of the triangle $BIC$ (in terms of parameter $m$)

[b]p1.[/b] Given two irreducible fractions. The denominator of the first fraction is $4$, the denominator of the second fraction is $6$. What can the denominator of the product of these fractions be equal to if the product is represented as an irreducible fraction? [b]p2.[/b] Three horses compete in the race. The player can bet a certain amount of money on each horse. Bets on the first horse are accepted in the ratio $1: 4$. This means that if the first horse wins, then the player gets back the money bet on this horse, and four more times the same amount. Bets on the second horse are accepted in the ratio $1:3$, on the third -$ 1:1$. Money bet on a losing horse is not returned. Is it possible to bet in such a way as to win whatever the outcome of the race? [b]p3.[/b] A quadrilateral is inscribed in a circle, such that the center of the circle, point $O$, is lies inside it. Let $K$, $L$, $M$, $N$ be the midpoints of the sides of the quadrilateral, following in this order. Prove that the bisectors of angles $\angle KOM$ and $\angle LOC$ are perpendicular (Fig.). [img]https://cdn.artofproblemsolving.com/attachments/b/8/ea4380698eba7f4cc2639ce20e3057e0294a7c.png[/img] [b]p4.[/b] Prove that the number$$\underbrace{33...33}_{1999 \,\,\,3s}1$$ is not divisible by $7$. [b]p5.[/b] In triangle $ABC$, the median drawn from vertex $A$ to side $BC$ is four times smaller than side $AB$ and forms an angle of $60^o$ with it. Find the greatest angle of this triangle. [b]p6.[/b] Given a $7\times 8$ rectangle made up of 1x1 cells. Cut it into figures consisting of $1\times 1$ cells, so that each figure consists of no more than $5$ cells and the total length of the cuts is minimal (give an example and prove that this cannot be done with a smaller total length of the cuts). You can only cut along the boundaries of the cells. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Let $f_n = 2^{2^n}+ 1$, $n = 1,2,3,...$, be the Fermat’s numbers. Find the least real number $C$ such that $$\frac{1}{f_1}+\frac{2}{f_2}+\frac{2^2}{f_3}+...+\frac{2^{n-1}}{f_n} <C$$ for all positive integers $n$

Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 27$. Prove that $x + y + z \ge \sqrt{3xyz}$. When does equality hold?

Consider a sequence $\{a_n\}_{n\geq 0}$ such that $a_{n+1}=a_n-\lfloor{\sqrt{a_n}}\rfloor\ (n\geq 0),\ a_0\geq 0$. (1) If $a_0=24$, then find the smallest $n$ such that $a_n=0$. (2) If $a_0=m^2\ (m=2,\ 3,\ \cdots)$, then for $j$ with $1\leq j\leq m$, express $a_{2j-1},\ a_{2j}$ in terms of $j,\ m$. (3) Let $m\geq 2$ be integer and for integer $p$ with $1\leq p\leq m-1$, let $a\0=m^2-p$. Find $k$ such that $a_k=(m-p)^2$, then find the smallest $n$ such that $a_n=0$.

A dog has infinitely many pieces of meat, but on each piece of meat there is a fly. At each move, the dog does the following: [list=1] [*] He eats a piece of meat together with all flies lying on it; [*] He moves a fly from a piece of meat to another. [/list] The dog doesn't want to eat more than one milion flies. Prove that he cannot ensure that each piece of meat is eaten at some point. [i]Proposed by I. Mitrofanov[/i]

Let us say that a circle intersects a quadrilateral [i]properly[/i] if it intersects each of the quadrilateral’s sides at two distinct interior points. Is it true that for each convex quadrilateral there exists a circle which intersects it properly? [i]Alexandr Perepechko[/i]

Let f be a function such that $f(0) = 0, f(1) = 1$, and $f(n) = 2f(n-1)- f(n- 2) + (-1)^n(2n - 4)$ for all integers $n \ge 2$. Find f(n) in terms of $n$.

The altitueds $A {{A} _ {1}} $, $B {{B} _ {1}}$ and $C {C} _ 1$ are drawn in the acute triangle $ABC$. . The perpendicular $AK$ is drawn from the vertex $A$ on the line ${{A} _ {1}} {{B} _ {1}}$, and the perpendicular $BL$ is drawn from the vertex $B$ on the line ${{C} _ {1}} {{B} _ {1}}$. Prove that ${{A} _ {1}} K = {{B} _ {1}} L$. (Maria Rozhkova)

The difference of the squares of two odd numbers is always divisible by $ 8$. If $ a>b$, and $ 2a\plus{}1$ and $ 2b\plus{}1$ are the odd numbers, to prove the given statement we put the difference of the squares in the form: $ \textbf{(A)}\ (2a\plus{}1)^2\minus{}(2b\plus{}1)^2 \\ \textbf{(B)}\ 4a^2\minus{}4b^2\plus{}4a\minus{}4b \\ \textbf{(C)}\ 4[a(a\plus{}1)\minus{}b(b\plus{}1)] \\ \textbf{(D)}\ 4(a\minus{}b)(a\plus{}b\plus{}1) \\ \textbf{(E)}\ 4(a^2\plus{}a\minus{}b^2\minus{}b)$

Let $f(x)=\frac{P(x)}{Q(x)}$. Where $P(x), Q(x)$ are two non constant polynomials with no common zeros and $P(0)=P(1)=0$. Suppose $f(x)f(\frac{1}{x})=f(x)+f(\frac{1}{x})$ for all infinitely many values of $x$. a. Show that $deg(P) <deg(Q).$ b. Show that $P'(1)=2Q'(1)- deg(Q). Q(1)$ Here $P'(x)$ denotes the derivatives of $P(x)$ as usual

In one galaxy, there exist more than one million stars. Let $M$ be the set of the distances between any $2$ of them. Prove that, in every moment, $M$ has at least $79$ members. (Suppose each star as a point.)

Does there exist an integer $n \ge 3$ and an arithmetic sequence $a_0, a_1, ... , a_n$ such that the polynomial $a_nx^n +... + a_1x + a_0$ has $n$ roots which also form an arithmetic sequence?

Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A$, $B$, $C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $AOP$, $BOP$, and $COP$ by $O_A$, $O_B$, and $O_C$, respectively. The lines $\ell_A$, $\ell_B$, $\ell_C$ perpendicular to $BC$, $CA$, and $AB$ pass through $O_A$, $O_B$, and $O_C$, respectively. Prove that the circumcircle of triangle formed by $\ell_A$, $\ell_B$, and $\ell_C$ is tangent to the line $OP$.

Jacob has a piece of bread shaped like a figure $8$, marked into sections and all initially connected as one piece of bread. The central part of the “$8$” is a single section, and each of the two loops of “$8$” is divided into an additional $1010$ pieces. For each section, there is a $50$ percent chance that Jacob will decide to cut it out and give it to a friend, and this is done independently for each section. The remaining sections of bread form some number of connected pieces. If $E$ is the expected number of these pieces, and $k$ is the smallest positive integer so that $2^k(E - \lfloor E \rfloor ) \ge 1$, find $\lfloor E \rfloor +k$. (Here, we say that if Jacob donates all pieces, there are $0$ pieces left).

Let $ \mathcal{P}$ be a convex polygon with $ n$ sides, $ n\ge3$. Any set of $ n \minus{} 3$ diagonals of $ \mathcal{P}$ that do not intersect in the interior of the polygon determine a [i]triangulation[/i] of $ \mathcal{P}$ into $ n \minus{} 2$ triangles. If $ \mathcal{P}$ is regular and there is a triangulation of $ \mathcal{P}$ consisting of only isosceles triangles, find all the possible values of $ n$.

Consider the minimum positive real number $\lambda$ such that for any two squares $A,B$ satisfying $\text{Area}(A) + \text{Area}(B)=1$, there always exists some rectangle $C$ of area $\lambda$, such that $A,B$ can be put inside $C$ and satisfy the following two constraints: 1. $A,B$ are non-overlapping; 2. the sides of $A$ and $B$ are parallel to some side of $C$. $\lambda$ can be written as $\frac{\sqrt{m}+n}{p}$ for positive integers $m$, $n$, and $p$ where $n$ and $p$ are relatively prime. Find $m+n+p$.