Let $ABC$ be a right-angled triangle with $\angle BAC = 90^{\circ}$. Let the height from $A$ cut its side $BC$ at $D$. Let $I, I_B, I_C$ be the incenters of triangles $ABC, ABD, ACD$ respectively. Let also $EB, EC$ be the excenters of $ABC$ with respect to vertices $B$ and $C$ respectively. If $K$ is the point of intersection of the circumcircles of $E_CIB_I$ and $E_BIC_I$, show that $KI$ passes through the midpoint $M$ of side $BC$.

Let $f:\mathbb{Z}\rightarrow\{ 1,2,\ldots ,n\}$ be a function such that $f(x)\not= f(y)$, for all $x,y\in\mathbb{Z}$ such that $|x-y|\in\{2,3,5\}$. Prove that $n\ge 4$. [i]Ioan Tomescu[/i]

Determine the largest two-digit number $d$ with the following property: for any six-digit number $\overline{aabbcc}$ number $d$ is a divisor of the number $\overline{aabbcc}$ if and only if the number $d$ is a divisor of the corresponding three-digit number $\overline{abc}$. Note The numbers $a \ne 0, b$ and $c$ need not be different.

Find all positive integers m, n such that $m^3 - n^3 = 999$.

Let $a,b,c,d,e,f$ be real numbers such that $a^2+b^2+c^2=14, d^2+e^2+f^2=77,$ and $ad+be+cf=32.$ Find $(bf-ce)^2+(cd-af)^2+(ae-bd)^2.$

Let $n$ be an integer larger than $1$ and let $S$ be the set of $n$-element subsets of the set $\{1,2,\ldots,2n\}$. Determine \[\max_{A\in S}\left (\min_{x,y\in A, x \neq y} [x,y]\right )\] where $[x,y]$ is the least common multiple of the integers $x$, $y$.

The base of a regular pyramid is a regular $2n$-gon $A_1A_2...A_{2n}$. A sphere passing through the top vertex $S$ of the pyramid cuts the edge $SA_i$ at $B_i$ (for $i = 1, 2, ... , 2n$). Show that $\sum\limits_{i=1}^n SB_{2i-1} = \sum\limits_{i=1}^n SB_{2i}$.

Determine all integers $a, b, c$ satisfying identities: $a + b + c = 15$ $(a - 3)^3 + (b - 5)^3 + (c -7)^3 = 540$

The perpendicular line issued from the center of the circumcircle to the bisector of angle $C$ in a triangle $ABC$ divides the segment of the bisector inside $ABC$ into two segments with ratio of lengths $\lambda$. Given $b = AC$ and $a = BC$, find the length of side $c.$

What is the smallest positive integer $k$ such that the number $\textstyle\binom{2k}k$ ends in two zeros? $\textbf{(A) }3\hspace{14em}\textbf{(B) }4\hspace{14em}\textbf{(C) }5$ $\textbf{(D) }6\hspace{14em}\textbf{(E) }7\hspace{14em}\textbf{(F) }8$ $\textbf{(G) }9\hspace{14em}\textbf{(H) }10\hspace{13.3em}\textbf{(I) }11$ $\textbf{(J) }12\hspace{13.8em}\textbf{(K) }13\hspace{13.3em}\textbf{(L) }14$ $\textbf{(M) }2007$

We have a stick of length $2n{}$ and a machine which cuts sticks of length $k\in\mathbb{N}$ with $k>1$ into two sticks with arbitrary positive integer lengths. What is the smallest number of cuts after which we can always find some sticks whose lengths sum up to $n{}$?

A polynomial $p(x)$ has remainder three when divided by $x-1$ and remainder five when divided by $x-3$. The remainder when $p(x)$ is divided by $(x-1)(x-3)$ is $\textbf{(A) }x-2\qquad\textbf{(B) }x+2\qquad\textbf{(C) }2\qquad\textbf{(D) }8\qquad \textbf{(E) }15$

Let $n \ge 2$ be a positive integer, and let $\sigma(n)$ denote the sum of the positive divisors of $n$. Prove that the $n^{\text{th}}$ smallest positive integer relatively prime to $n$ is at least $\sigma(n)$, and determine for which $n$ equality holds. [i]Proposed by Ashwin Sah[/i]

Let $ABC$ be a triangle of perimeter $100$ and $I$ be the point of intersection of its bisectors. Let $M$ be the midpoint of side $BC$. The line parallel to $AB$ drawn by$ I$ cuts the median $AM$ at point $P$ so that $\frac{AP}{PM} =\frac73$. Find the length of side $AB$.

In the triangle $ABC$, $M$ is the midpoint of $AB$ and $B'$ is the foot of $B$-altitude. $CB'M$ intersects the line $BC$ for the second time at $D$. Circumcircles of $CB'M$ and $ABD$ intersect each other again at $K$. The parallel to $AB$ through $C$ intersects the $CB'M$ circle again at $L$. Prove that $KL$ cuts $CM$ in half.

Let $G$ be the point of intersection of the medians in the triangle $ABC$. Let us denote $A_1, B_1, C_1$ the second points of intersection of lines $AG, BG, CG$ with the circle circumscribed around the triangle. Prove that $AG + BG + CG \le A_1C + B_1C + C_1C$. (Yasinsky V.A.)

The number of distinct points common to the graphs of $x^2+y^2=9$ and $y^2=9$ is: $ \textbf{(A) }\text{infinitely many} \qquad\textbf{(B) } \text{four}\qquad\textbf{(C) }\text{two}\qquad\textbf{(D) }\text{one}\qquad\textbf{(E) }\text{none} $

Solve the equation $$[2 \sin x] =2\cos \left(3x+\frac{\pi}{4} \right)$$ ($[x]$ is the integer part of $x$, $[x]$ is equal to the largest integer not exceeding $x$. For example, $[3,33] = 3$, $[2] = 2$, $[- 3.01] = -4$).

Find all 3-digit positive integers $\overline{abc}$ such that \[ \overline{abc} = abc(a+b+c) , \] where $\overline{abc}$ is the decimal representation of the number.

For a sequence $x_1,x_2,\ldots,x_n$ of real numbers, we define its $\textit{price}$ as \[\max_{1\le i\le n}|x_1+\cdots +x_i|.\] Given $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price $D$. Greedy George, on the other hand, chooses $x_1$ such that $|x_1 |$ is as small as possible; among the remaining numbers, he chooses $x_2$ such that $|x_1 + x_2 |$ is as small as possible, and so on. Thus, in the $i$-th step he chooses $x_i$ among the remaining numbers so as to minimise the value of $|x_1 + x_2 + \cdots x_i |$. In each step, if several numbers provide the same value, George chooses one at random. Finally he gets a sequence with price $G$. Find the least possible constant $c$ such that for every positive integer $n$, for every collection of $n$ real numbers, and for every possible sequence that George might obtain, the resulting values satisfy the inequality $G\le cD$. [i]Proposed by Georgia[/i]

Each subset of $97$ out of $1997$ given real numbers has positive sum. Show that the sum of all the $1997$ numbers is positive.

Let $n\ge 2$ be an integer. Ariane and Bérénice play a game on the number of the residue classes modulo $n$. At the beginning there is the residue class $1$ on each piece of paper. It is the turn of the player whose turn it is to replace the current residue class $x$ with either $x + 1$ or by $2x$. The two players take turns, with Ariane starting. Ariane wins if the residue class $0$ is reached during the game. Bérénice wins if she can prevent that permanently. Depending on $n$, determine which of the two has a winning strategy.

Does there exist such a power of ${2}$, that when written in the decimal system its digits are all different than zero and it is possible to reorder the other digits to form another power of ${2}$? Justify your answer.

Given that $n$ and $r$ are positive integers. Suppose that \[ 1 + 2 + \dots + (n - 1) = (n + 1) + (n + 2) + \dots + (n + r) \] Prove that $n$ is a composite number.

Consider an $ABCD$ parallelogram with $\overline{AD}$ $=$ $\overline{BD}$. Point E lies in segment $\overline{BD}$ in such a way that $\overline{AE}$ $=$ $\overline{DE}$. The extension of line $\overline{AE}$ cuts segment $\overline{BC}$ and $F$. if line $\overline{DF}$ is the bisector of the $\angle CED$. Find the value of the $\angle ABD$ $\textbf{4.1.}$ Point $E$ lies in segment $\overline{BD}$ means that exits a point $E$ in the segment $\overline{BD}$ in other words lies refers to the same thing found [i]Proposed by Alicia Smith, Francisco Morazan[/i]