Solve the equation $|x+1|-|x-1|=2$.

Someones age is equal to the sum of the digits of his year of birth. How old is he and when was he born, if it is known that he is older than $11$. P.s. the current year in the problem is $2010$.

Prove that in any polygon, there exist two sides whose radio is less than $2$.(Essentialy if $a_1\geq a_2\geq...\geq a_n$ are the sides of a polygon prove that there exist $i,j\in\{1,2,..,n\}$ so that $i<j$ and $\frac {a_i}{a_j}<2$).

Prove that $\sqrt 3$ is irrational.

Let $x,y$ be positive real numbers such that $x+y=1$. Prove that $\left(1+\frac {1}{x}\right)\left(1+\frac {1}{y}\right)\geq 9$.

Find the graph of the function $y=|2^{|x|}-1|$.

The equation is given $x^2-(m+3)x+m+2=0$. If $x_1$ and $x_2$ are its solutions find all $m$ such that $\frac{x_1}{x_1+1}+\frac{x_2}{x_2+1}=\frac{13}{10}$.

Let $a,b,c$ be non negative integers. Suppose that $c$ is even and $a^5+4b^5=c^5$. Prove that $b=0$.

Solve the inequation $\sqrt {3-x}-\sqrt {x+1}>\frac {1}{2}$.

Let $a_1,a_2,...,a_n$ be an arithmetic progression of positive real numbers. Prove that $\tfrac {1}{\sqrt a_1+\sqrt a_2}+\tfrac {1}{\sqrt a_2+\sqrt a_3}+...+\tfrac {1}{\sqrt a_{n-1}+\sqrt a_n}=\tfrac{n-1}{\sqrt {a_1}+\sqrt{a_n}}$.

Arrange the numbers $\cos 2, \cos 4, \cos 6$ and $\cos 8$ from the biggest to the smallest where $2,4,6,8$ are given as radians.

Prove that $\sqrt[3]{5}$ is irrational.

If the real function $f(x)=\cos x+\sum_{i=1}^{n}\cos(a_ix)$ is periodic, prove that $a_i,i\in\{1,2,...,n\}$, are rational numbers.

The set $S\subseteq \mathbb{R}$ is given with the properties: $(a) \mathbb{Z}\subset S$, $(b) (\sqrt 2 +\sqrt 3)\in S$, $(c)$ If $x,y\in S$ then $x+y\in S$, and $(d)$ If $x,y\in S$ then $x\cdot y\in S$. Prove that $(\sqrt 2+\sqrt 3)^{-1}\in S$.

Let $n\in \mathbb{N}$. Prove that the polynom $p(x)=x^{2n}-2x^{2n-1}+3x^{2n-2}-...-2nx+2n+1$ doesn't have real roots.

Let $(p_1,p_2,..., p_n)$ be a random permutation of the set $\{1,2,...,n)$. If $n$ is odd, prove that the product $(p_1-1)\cdot (p_2-2)\cdot ...\cdot (p_n-n)$ is an even number. @below fixed.