Find the value of $(1+2)(1+2^2)(1+2^4)(1+2^8)...(1+2^{2048})$.

If $a>1,b>1$ are the legths of the catheti of an right triangle and $c$ the length of its hypotenuse, prove that $a+b\leq c\sqrt 2$

The integers $a_1,a_2,...,a_{2012}$ are given. Exactly $29$ of them are divisible by $3$. Prove that the sum $a_1^2+a_2^2+...+a_{2012}^2$ is divisible by $3$.

Inside of the square $ABCD$ the point $P$ is given such that $|PA|:|PB|:|PC|=1:2:3$. Find $\angle APB$.

The following square table is given with seven raws and seven columns: $a_{11},a_{12},a_{13},a_{14},a_{15},a_{16},a_{17}$ $a_{21},a_{22},a_{23},a_{24},a_{25},a_{26},a_{27}$ $a_{31},a_{32},a_{33},a_{34},a_{35},a_{36},a_{37}$ $a_{41},a_{42},a_{43},a_{44},a_{45},a_{46},a_{47}$ $a_{51},a_{52},a_{53},a_{54},a_{55},a_{56},a_{57}$ $a_{61},a_{62},a_{63},a_{64},a_{65},a_{66},a_{67}$ $a_{71},a_{72},a_{73},a_{74},a_{75},a_{76},a_{77}$ Suppose $a_{ij}\in\{0,1\},\forall i,j\in\{1,...,7\}$. Prove that there exists at least one combination of the numbers $1$ and $0$ so that the following conditions hold: $(i)$ Each raw and each column has exactly three $1$'s. $(ii)$$\sum_{j=1}^7a_{lj}a_{ij}=1,\forall l,i\in\{1,...,7\}$ and $l\neq i$.(so for any two distinct raws there is exactly one $r$ so that the both raws have $1$ in the $r$-th place). $(iii)$$\sum_{i=1}^7a_{ij}a_{ik}=1,\forall j,k\in\{1,...,7\}$ and $j\neq k$.(so for any two distinct columns there is exactly one $s$ so that the both columns have $1$ in the $s$-th place).

Find the two last digits of $2012^{2012}$.

Let $a,b,c$ be the lengths of the sides of a triangle. Prove that, $\left|\frac {a}{b}+\frac {b}{c}+\frac {c}{a}-\frac {b}{a}-\frac {c}{b}-\frac {a}{c}\right|<1$

Let $n\not\equiv 2\pmod{3}$. Is $\sqrt{\lfloor n+\tfrac {2n}{3}\rfloor+7},\forall n \in \mathbb {N}$, a natural number?

The right triangle $ABC$ with a right angle at $C$. From all the rectangles $CA_1MB_1$, where $A_1\in BC, M\in AB$ and $B_1\in AC$ which one has the biggest area?

If $(x^2-x-1)^n=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0$, where $a_i,i\in\{0,1,2,..,2n\}$, find $a_1+a_3+...+a_{2n-1}$ and $a_0+a_2+a_4+...+a_{2n}$.

The bisector of acute angle $\alpha$ of the right triangle $ABC$ splits the side $a$ in two segments with lengths $m$ and $n$. Find the acute angle $\beta$ and the lenths of the other two sides by knowing $m$ and $n$.

Prove that for any integer $n\geq 2$ it holds that $\dbinom {2n}{n}>\frac {4^n}{2n}$.

Find the set of solutions to the equation $\log_{\lfloor x\rfloor}(x^2-1)=2$

Prove that for all $n\in\mathbb{N}$ we have $\sum_{k=0}^n\dbinom {n}{k}^2=\dbinom {2n}{n}$.

In a sphere $S_0$ we radius $r$ a cube $K_0$ has been inscribed. Then in the cube $K_0$ another sphere $S_1$ has been inscribed and so on to infinity. Calculate the volume of all spheres created in this way.

Solve the recurrence $R_0=1, R_n=nR_{n-1}+2^n\cdot n!$.

Let $x,y$ be positive real numbers such that $x+y+xy=3$. Prove that $x+y\geq 2$. For what values of $x$ and $y$ do we have $x+y=2$?