Solve the equation $|x-m|+|x+m|=x$ depending on the value of the parameter $m$.

Let $\triangle ABC$ have $\angle A=20^{\circ}$ and $\angle C=40^{\circ}$. We've constructed the angle bisector $AL$ ($L\in BC$) and the external angle bisector $CN$ ($N\in AB$). Find $\angle CLN$.

Prove that there exists a prime number $p$, such that the sum of digits of $p$ is a composite odd integer. Find the smallest such $p$.

Let $M$ be a set of $99$ different rays with a common end point in a plane. It's known that two of those rays form an obtuse angle, which has no other rays of $M$ inside in. What is the maximum number of obtuse angles formed by two rays in $M$?

Solve the system $\begin{cases} x^2y^2+|xy|=\frac{4}{9}\\ xy+1=x+y^2\\ \end{cases}$

Given a $\triangle ABC$ and the altitude $CH$ ($H$ lies on the segment $AB$) and let $M$ be the midpoint of $AC$. Prove that if the circumcircle of $\triangle ABC$, $k$ and the circumcircle of $\triangle MHC$, $k_{1}$ touch, then the radius of $k$ is twice the radius of $k_{1}$.

Let $n$ be a natural number. Prove that if $n^5+n^4+1$ has $6$ divisors then $n^3-n+1$ is a square of an integer.

Stoyan and Nikolai have two $100\times 100$ chess boards. Both of them number each cell with the numbers $1$ to $10000$ in some way. Is it possible that for every two numbers $a$ and $b$, which share a common side in Nikolai's board, these two numbers are at a knight's move distance in Stoyan's board (that is, a knight can move from one of the cells to the other one with a move)? Nikolai Beluhov

For which values of the parameter $a$ does the equation \[(2x-a)\sqrt{ax^2-(a^2+a+2)x+2(a+1)}=0\]has three different real roots.

Let $\triangle ABC$ have $M$ as the midpoint of $BC$ and let $P$ and $Q$ be the feet of the altitudes from $M$ to $AB$ and $AC$ respectively. Find $\angle BAC$ if $[MPQ]=\frac{1}{4}[ABC]$ and $P$ and $Q$ lie on the segments $AB$ and $AC$.

Find all natural numbers $x,y,z$, such that $7^{x}+13^{y}=2^{z}$.

There are $3\leq n\leq 25$ passengers in a bus some of which are friends. Every passenger has exactly $k$ friends among the passengers, no two friends have a common friend and every two people, who are not friends have a common friend. Find $n$.

Let $a_{1},a_{2},\ldots$ be an infinite arithmetic progression. It's known that there exist positive integers $p,q,t$ such that $a_{p}+tp=a_{q}+tq$. If $a_{t}=t$ and the sum of the first $t$ numbers in the sequence is $18$, determine $a_{2008}$.

On the sides $AB$ and $AC$ of the right $\triangle ABC$ ($\angle A=90^{\circ}$) are chosen points $C_{1}$ and $B_{1}$ respectively. Prove that if $M=CC_{1}\cap BB_{1}$ and $AC_{1}=AB_{1}=AM$, then $[AB_{1}MC_{1}]+[AB_{1}C_{1}]=[BMC]$.

In a convex $2008$-gon some of the diagonals are coloured red and the rest blue, so that every vertex is an endpoint of a red diagonal and no three red diagonals concur at a point. It's known that every blue diagonal is intersected by a red diagonal in an interior point. Find the minimal number of intersections of red diagonals.

a) Prove that $\lfloor x\rfloor$ is odd iff $\Big\lfloor 2\{\frac{x}{2}\}\Big\rfloor=1$ ($\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$ and $\{x\}=x-\lfloor x\rfloor$). b) Let $n$ be a natural number. Find the number of square free numbers $a$, such that $\Big\lfloor\frac{n}{\sqrt{a}}\Big\rfloor$ is odd. (A natural number is square free if it's not divisible by any square of a prime number).

Determine the values of the real parameter $a$, such that the solutions of the system of inequalities $\begin{cases} \log_{\frac{1}{3}}{(3^{x}-6a)}+\frac{2}{\log_{a}{3}}<x-3\\ \log_{\frac{1}{3}}{(3^{x}-18)}>x-5\\ \end{cases}$ form an interval of length $\frac{1}{3}$.

Let $ABC$ be a triangle, such that the midpoint of $AB$, the incenter and the touchpoint of the excircle opposite $A$ with $\overline{AC}$ are collinear. Find $AB$ and $BC$ if $AC=3$ and $\angle ABC=60^{\circ}$.

Find all continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that \[(f(x)f(y)-1)f(x+y)=2f(x)f(y)-f(x)-f(y)\quad \forall x,y\in \mathbb{R}\]

Veni writes down finitely many real numbers (possibly one), squares them, and then subtracts 1 from each of them and gets the same set of numbers as in the beginning. What were the starting numbers?