Solve in equation: $ x^2+y^2+z^2+w^2=3(x+y+z+w) $ where $ x,y,z,w $ are positive integers.

Find all rational terms of sequence defined by formula $ a_n=\sqrt{\frac{9n-2}{n+1}}, n \in N $

Determine strictly positive real numbers $ a_{1},a_{2},...,a_{n}$ if for any $ n\in N^*$ takes place equality: $ a_{1}^2+a_{2}^2+...+a_{n}^2=a_{1}+a_{2}+...+a_{n}+\frac{n(n^2+6n+11)}{3}$

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?

Define the sequence $(x_{n})$: $x_{1}=\frac{1}{3}$ and $x_{n+1}=x_{n}^{2}+x_{n}$. Find $\left[\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\dots+\frac{1}{x_{2007}+1}\right]$, wehere $[$ $]$ denotes the integer part.

Define $a_{n}$ as satisfying: $\left(1+\frac{1}{n}\right)^{n+a_{n}}=e$. Find $\lim_{n\rightarrow\infty}a_{n}$.

$ABCDA_{1}B_{1}C_{1}D_{1}$ is a cube with side length $4a$. Points $E$ and $F$ are taken on $(AA_{1})$ and $(BB_{1})$ such that $AE=B_{1}F=a$. $G$ and $H$ are midpoints of $(A_{1}B_{1})$ and $(C_{1}D_{1})$, respectively. Find the minimum value of the $CP+PQ$, where $P\in[GH]$ and $Q\in[EF]$.

The function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfies $f(\textrm{cot}x)=\sin2x+\cos2x$, for any $x\in(0,\pi)$. Find the minimum and maximum value of $g: [-1;1]\rightarrow\mathbb{R}$, $g(x)=f(x)\cdot f(1-x)$.

Real numbers $a_{1},a_{2},\dots,a_{n}$ satisfy $a_{i}\geq\frac{1}{i}$, for all $i=\overline{1,n}$. Prove the inequality: \[\left(a_{1}+1\right)\left(a_{2}+\frac{1}{2}\right)\cdot\dots\cdot\left(a_{n}+\frac{1}{n}\right)\geq\frac{2^{n}}{(n+1)!}(1+a_{1}+2a_{2}+\dots+na_{n}).\]

Define $(b_{n})$ to be: $b_{0}=12$, $b_{1}=\frac{\sqrt{3}}{2}$ adn $b_{n+1}+b_{n-1}=b_{n}\cdot\sqrt{3}$. Find $b_{0}+b_{1}+\dots+b_{2007}$. Note. Maybe this seems too easy, but I want to post all the problems...

Given a tetrahedron $VABC$ with edges $VA$, $VB$ and $VC$ perpendicular any two of them. The sum of the lengths of the tetrahedron's edges is $3p$. Find the maximal volume of $VABC$.

The continuous function and twice differentiable function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfies $2007^{2}\cdot f(x)+f''(x)=0$. Prove that there exist two such real numbers $k$ and $l$ such that $f(x)=l\cdot\sin(2007x)+k\cdot\cos(2007x)$.

For $a\in C^{*}$ find all $n\in N$ such that $X^{2}(X^{2}-aX+a^{2})^{2}$ divides $(X^{2}+a^{2})^{n}-X^{2n}-a^{2n}$

For $p\in (0;\infty)$ find the area of the region bounded by the curves $y^{2}=4px$ and $16py^{2}=5(x-p)^{3}$

For $a,b \in [1;\infty)$ show that \[ab\leq e^{a-1}+b\ln b\]

If the function $f\colon [1,2]\to R$ is such that $\int_{1}^{2}f(x) dx=\frac{73}{24}$, then show that there exists a $x_{0}\in (1;2)$ such that \[x_{0}^{2}<f(x_{0})<x_{0}^{3}\] [Edit: $f$ is continuous]

Find all polynomials $P\in \mathbb C[X]$ such that \[P(X^{2})=P(X)^{2}+2P(X)\]

Show that the distance between a point on the hyperbola $xy=5$ and a point on the ellipse $x^{2}+6y^{2}=6$ is at least $\frac{9}{7}$.

Find the limit \[\lim_{n\to \infty}\frac{\sqrt[n+1]{(2n+3)(2n+4)\ldots (3n+3)}}{n+1}\]

Find all continuous functions $f\colon [0;1] \to R$ such that \[\int_{0}^{1}f(x)dx = 2\int_{0}^{1}(f(x^{4}))^{2}dx+\frac{2}{7}\]