$9.$ What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length $12 ?$

$3.$ Congruent circles $k_{1}$ and $k_{2}$ intersect in the points $A$ and $B$. Let $P$ be a variable point of arc $AB$ of circle $k_{2}$ which is inside $k_{1}$ and let $AP$ intersect $k_{1}$ once more in point $C$, and the ray $CB$ intersects $k_{2}$ once more in $D$. Let the angle bisector of $\angle CAD$ intersect $k_{1}$ in $E$, and the circle $k_{2}$ in $F$. Ray $FB$ intersects $k_{1}$ in $Q$. If $X$ is one of the intersection points of circumscribed circles of triangles $CDP$ and $EQF$, prove that the triangle $CFX$ is equilateral.

[b]4.[/b] Consider $n$ circles of radius $1$ in the planea. Prove that at least one of the circles contains an are of length greater than $\frac{2\pi}{n}$ not intersected by any other of these circles. [b](G. 4)[/b]

A two-digit integer is divided by the sum of its digits. Find the largest remainder that can occur.

In a chess tournament , each of two players have only one game played. After 2 rounds 5 players left the tournament. At the final of tournament was found that the number of total games played is 100. How many players were at the start of the tournament?

Given two directly congruent triangles $ABC$ and $A'B'C'$ in a plane, assume that the circles with centers $C$ and $C'$ and radii $CA$ and $C'A'$ intersect. Denote by $\mathcal M$ the transformation that maps $\triangle ABC$ to $\triangle A'B'C'$. Prove that $\mathcal M$ can be expressed as a composition of at most three rotations in the following way: The first rotation has the center in one of $A,B,C$ and maps $\triangle ABC$ to $\triangle A_1B_1C_1$; The second rotation has the center in one of $A_1,B_1,C_1$, and maps $\triangle A_1B_1C_1$ to $\triangle A_2B_2C_2$; The third rotation has the center in one of $A_2,B_2,C_2$ and maps $\triangle A_2B_2C_2$ to $\triangle A'B'C'$.

Let $ n$ and $ k$ be positive integers such as either $ n$ is odd or both $ n$ and $ k$ are even. Prove that exists integers $ a$ and $ b$ such as $ GCD(a,n) \equal{} GCD(b,n) \equal{} 1$ and $ k \equal{} a \plus{} b$

Consider four real numbers $x$, $y$, $a$, and $b$, satisfying $x + y = a + b$ and $x^2 + y^2 = a^2 + b^2$. Prove that $x^n + y^n = a^n + b^n$, for all $n \in \mathbb{N}$.

Let $ABCDEF$ be a convex hexagon with $\angle A = \angle C=\angle E$ and $\angle B = \angle D=\angle F$. [list=a] [*] Prove that there is a unique point $P$ which is equidistant from sides $AB,CD$ and $EF$. [*] If $G_1$ and $G_2$ are the centers of mass of $\triangle ACE$ and $\triangle BDF$, show that $\angle G_1PG_2=60^{\circ}$.

Let $ABC$ be a triangle and let $I$ be the incenter. Let $N$, $M$ be the midpoints of the sides $AB$ and $CA$ respectively. The lines $BI$ and $CI$ meet $MN$ at $K$ and $L$ respectively. Prove that $AI+BI+CI>BC+KL$. [i]Greece[/i]

Let $ a $ be a positive real number and $ \left( x_n\right)_{n\ge 1} $ be a sequence of pairwise distinct real numbers satisfying the properties: $ \text{(i) } x_n\in (0,a) , $ for any natural numbers $ n $ $ \text{(ii) } \left| x_n-x_m \right|\geqslant\frac{m+n}{amn} , $ for all pairs $ (m,n) $ of distinct natural numbers Show that $ a\geqslant 2. $

In $3$ dimensional space we are given a parallelogram $ABCD$ and plane $M$. The distances from vertices $A, B$ and $C$ to plane $M$ are $a, b$ and $c$ respectively. Find the distance $d$ from vertex $D$ to the plane $M$ .

There are ten coins a line, which are indistinguishable. It is known that two of them are false and have consecutive positions on the line. For each set of positions, you may ask how many false coins it contains. Is it possible to identify the false coins by making only two of those questions, without knowing the answer to the first question before making the second?

One side of a convex polygon is equal to $a$, the sum of exterior angles at the vertices not adjacent to this side are equal to $120^o$. Among such polygons, find the polygon of the largest area.

Define the unique $9$-digit integer $M$ that has the following properties: (1) its digits are all distinct and nonzero; and (2) for every positive integer $m=2,3,4,...,9$, the integer formed by the leftmost $m$ digits of $M$ is divisible by $m$.

A line intersects a semicircle with diameter $AB$ and center $O$ at $C$ and $D$, and the line $AB$ at $M$, where $MB < MA$ and $MD < MC.$ If the circumcircles of the triangles $AOC$ and $DOB$ meet again at $K,$ prove that $\angle MKO$ is right.

Four right triangles, each with the sides $1$ and $2$, are assembled to a figure as shown. How large a fraction does the area of the small circle make up of that of the big one? [img]https://1.bp.blogspot.com/-XODK1XKCS0Q/XzXDtcA-xAI/AAAAAAAAMWA/zSLPpf3IcX0rgaRtOxm_F2begnVdUargACLcBGAsYHQ/s0/2010%2BMohr%2Bp1.png[/img]

If $ x$, $ y$, $ z$ are nonnegative real numbers with the sum $ 1$, find the maximum value of $ S \equal{} x^2(y \plus{} z) \plus{} y^2(z \plus{} x) \plus{} z^2(x \plus{} y)$ and $ C \equal{} x^2y \plus{} y^2z \plus{} z^2x$.

Find all functions $ f :\mathbb{Z}\mapsto\mathbb{Z} $ such that following conditions holds: $a)$ $f(n) \cdot f(-n)=f(n^2)$ for all $n\in\mathbb{Z}$ $b)$ $f(m+n)=f(m)+f(n)+2mn$ for all $m,n\in\mathbb{Z}$

Given is the acute triangle $ABC$. Let us denote $k$ a circle with diameter $AB$. Another circle, tangent to $AB$ at point $A$ and passing through point $C$ intersects the circle $k$ at point $P, P \ne A$. Another circle which touches AB at point $B$ and passes point $C$, intersects the circle $k$ at point $Q, Q \ne B$. Prove that the intersection of the line $AQ$ and $BP$ lies on one of the sides of angle $ACB$. (Peter Novotný)

Let $p,\ q$ be positive integers. Consider integrs $a,\ b,\ c$ satisfying conditions: \[-q\leq b\leq 0\leq a\leq p,\ \ b\leq c\leq a\] Call such $a,\ b,\ c$ in arranging in the form of $[a,\ b\ ;\ c]$ as $(p,\ q)$ pattern. For each $(p,\ q)$ pattern $[a,\ b\ ;\ c]$, let $w([a,\ b\ ;\ c])=p-q-(a+b)$. (1) Find the number of $(p,\ q)$ pattern such that $w([a,\ b\ ;\ c])=-q$, then find the number of $(p,\ q)$ pattern such that $w([a,\ b\ ;\ c])=p$. From now on, we consider the case of $p=q$. (2) Let $s$ be integer. Find the number of $(p,\ p)$ pattern such that $w([a,\ b\ ;\ c])=-p+s$. (3) Find the total number of $(p,\ p)$ pattern. [i]2011 Tokyo University entrance exam/Science, Problem 5[/i]

On the faces of a cube several positive integer numbers are written. On every edge the sum of the numbers of it's two faces is written, and in every vertex the sum of numbers on the three faces that have this vertex. It turned out that all the written numbers are different. Find the smallest possible amount of the sum of all written numbers.

Prove that there exist infinitely many even positive integers $k$ such that for every prime $p$ the number $p^2+k$ is composite.

We will call a pair of positive integers $(n, k)$ with $k > 1$ a $lovely$ $couple$ if there exists a table $nxn$ consisting of ones and zeros with following properties: • In every row there are exactly $k$ ones. • For each two rows there is exactly one column such that on both intersections of that column with the mentioned rows, number one is written. Solve the following subproblems: a) Let $d \neq 1$ be a divisor of $n$. Determine all remainders that $d$ can give when divided by $6$. b) Prove that there exist infinitely many lovely couples. Proposed by Miroslav Marinov, Daniel Atanasov

In a triangle $ABC$, $D$ is a point on $BC$ such that $AD$ is the internal bisector of $\angle A$. Suppose $\angle B = 2 \angle C$ and $CD =AB$. prove that $\angle A = 72^{\circ}$.