Find the number of solutions to the equation$$x_1^4+x_2^4+\ldots+x_{10}^4=2011$$in the set of positive integers.

Given a segment $AB$ of length $2003$ in a coordinate plane, determine the maximal number of unit squares with vertices in the lattice points whose intersection with the given segment is non-empty.

Let $a,b$ and $c$ be the lengths of the edges of a triangle whose angles are $\alpha=40^\circ,\beta=60^\circ$ and $\gamma=80^\circ$. Prove that $$a(a+b+c)=b(b+c).$$

An acute angle with the vertex $O$ and the rays $Op_1$ and $Op_2$ is given in a plane. Let $k_1$ be a circle with the center on $Op_1$ which is tangent to $Op_2$. Let $k_2$ be the circle that is tangent to both rays $Op_1$ and $Op_2$ and to the circle $k_1$ from outside. Find the locus of tangency points of $k_1$ and $k_2$ when center of $k_1$ moves along the ray $Op_1$.

Given a $\triangle ABC$ with the edges $a,b$ and $c$ and the area $S$: (a) Prove that there exists $\triangle A_1B_1C_1$ with the sides $\sqrt a,\sqrt b$ and $\sqrt c$. (b) If $S_1$ is the area of $\triangle A_1B_1C_1$, prove that $S_1^2\ge\frac{S\sqrt3}4$.

Let ABCD be a square inscribed in a circle k and P be an arbitrary point of that circle. Prove that at least one of the numbers PA, PB, PC and PD is not rational.

Let $ABCD$ be a rectangle. Determine the set of all points $P$ from the region between the parallel lines $AB$ and $CD$ such that $\angle APB=\angle CPD$.

Let $S$ be the subset of $N$($N$ is the set of all natural numbers) satisfying: i)Among each $2003$ consecutive natural numbers there exist at least one contained in $S$; ii)If $n \in S$ and $n>1$ then $[\frac{n}{2}] \in S$ Prove that:$S=N$ I hope it hasn't posted before.

Prove that the number $\left\lfloor\left(5+\sqrt{35}\right)^{2n-1}\right\rfloor$ is divisible by $10^n$ for each $n\in\mathbb N$.

Let $ f : [0, 1] \to\ R $ be a function such that :- $1.)$ $f(x) \ge 0$ for all $x$ in $[0,1]$ . $2.)$ $f(1) = 1$ . $3.)$ If $x_1 , x_2$ are in $[0,1]$ such that $x_1 + x_2 \le 1$ , then $f(x_1) + f(x_2) \le f(x_1 + x_2)$ . Show that $f(x) \le 2x $ for all $x$ in $ [0,1] $.

Given a circle $k$ and the point $P$ outside it, an arbitrary line $s$ passing through $P$ intersects $k$ at the points $A$ and $B$ . Let $M$ and $N$ be the midpoints of the arcs determined by the points $A$ and $B$ and let $C$ be the point on $AB$ such that $PC^2=PA\cdot PB$ . Prove that $\angle MCN$ doesn't depend on the choice of $s$. [Moderator edit: This problem has also been discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=56295 .]

Let $ n$ be an even number, and $ S$ be the set of all arrays of length $ n$ whose elements are from the set $ \left\{0,1\right\}$. Prove that $ S$ can be partitioned into disjoint three-element subsets such that for each three arrays $ \left(a_i\right)_{i = 1}^n$, $ \left(b_i\right)_{i = 1}^n$, $ \left(c_i\right)_{i = 1}^n$ which belong to the same subset and for each $ i\in\left\{1,2,...,n\right\}$, the number $ a_i + b_i + c_i$ is divisible by $ 2$.