Show that if a triangle has two excircles of the same size, then the triangle is isosceles. (Note: The excircle $ABC$ to the side $ a$ touches the extensions of the sides $AB$ and $AC$ and the side $BC$.)

There is no sequence $x_n$ strictly increasing with terms natural numbers such that : $$ x_n+x_{k}=x_{nk}, \ \ for \, any \,\,\, n, k \in \mathbb{N}^*$$

A quadrilateral $ABCD$ is such that the sides $AB$ and $DC$ are parallel, and $|BC| =|AB| + |CD|$. Prove that the angle bisectors of the angles $\angle ABC$ and $\angle BCD$ intersect at right angles on the side $AD$.

Let $n>1$ be an integer and $S_n$ the set of all permutations $\pi : \{1,2,\cdots,n \} \to \{1,2,\cdots,n \}$ where $\pi$ is bijective function. For every permutation $\pi \in S_n$ we define: \[ F(\pi)= \sum_{k=1}^n |k-\pi(k)| \ \ \text{and} \ \ M_{n}=\frac{1}{n!}\sum_{\pi \in S_n} F(\pi) \] where $M_n$ is taken with all permutations $\pi \in S_n$. Calculate the sum $M_n$.

Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, how many of the four quantities listed below change? $\mathrm{(A)}\ \text{the length of the segment} MN$ $\mathrm{(B)}\ \text{the perimeter of }\triangle PAB$ $\mathrm{(C)}\ \text{ the area of }\triangle PAB$ $\mathrm{(D)}\ \text{ the area of trapezoid} ABNM$ [asy] draw((2,0)--(8,0)--(6,4)--cycle); draw((4,2)--(7,2)); draw((1,4)--(9,4),Arrows); label("$A$",(2,0),SW); label("$B$",(8,0),SE); label("$M$",(4,2),W); label("$N$",(7,2),E); label("$P$",(6,4),N);[/asy] $\mathrm{(A)}\ 0 \qquad\mathrm{(B)}\ 1 \qquad\mathrm{(C)}\ 2 \qquad\mathrm{(D)}\ 3 \qquad\mathrm{(E)}\ 4$

For every complex number $z$ different from 0 and 1 we define the following function \[ f(z) := \sum \frac 1 { \log^4 z } \] where the sum is over all branches of the complex logarithm. a) Prove that there are two polynomials $P$ and $Q$ such that $f(z) = \displaystyle \frac {P(z)}{Q(z)} $ for all $z\in\mathbb{C}-\{0,1\}$. b) Prove that for all $z\in \mathbb{C}-\{0,1\}$ we have \[ f(z) = \frac { z^3+4z^2+z}{6(z-1)^4}. \]

The Blablabla set contains all the different seven-digit numbers that can be formed with the digits $2, 3, 4, 5, 6, 7$ and $8$. Prove that there are not two Blablabla numbers such that one of them is divisible by the other.

Let $a_1, a_2, \cdots$, be a sequence with the following properties. I. $a_1 = 1$, and II. $a_{2n}=n\cdot a_n$ for any positive integer $n$. What is the value of $a_{2^{100}}$? $ \textbf{(A)}\; 1\qquad \textbf{(B)}\; 2^{99}\qquad \textbf{(C)}\; 2^{100}\qquad \textbf{(D)}\; 2^{4950}\qquad \textbf{(E)}\; 2^{9999} $

Given a cube, a cubic box, that exactly suits for the cube, and six colours. First man paints each side of the cube with its (side's) unique colour. Another man does the same with the box. Prove that the third man can put the cube in the box in such a way, that every cube side will touch the box side of different colour.

On a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. The speed of sound is 1088 feet per second and one mile is 5280 feet. Estimate, to the nearest half-mile, how far Snoopy was from the flash of lightning. $ \text{(A)}\ 1\qquad\text{(B)}\ 1\frac{1}{2}\qquad\text{(C)}\ 2\qquad\text{(D)}\ 2\frac{1}{2}\qquad\text{(E)}\ 3 $

According to the figure, three equilateral triangles with side lengths $a,b,c$ have one common vertex and do not have any other common point. The lengths $x, y$, and $z$ are defined as in the figure. Prove that $3(x+y+z)>2(a+b+c)$. [i]Proposed by Mahdi Etesamifard[/i]

The limit of $ \frac {x^2\minus{}1}{x\minus{}1}$ as $x$ approaches $1$ as a limit is: $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ \text{Indeterminate} \qquad \textbf{(C)}\ x-1 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 1$

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 $a_n=0$.