Is it possible to divide $\mathbb{N}$ into six disjoint sets $A_1, A_2, A_3, A_4, A_5, A_6$, such that $x,y,z$ are not in the same set if $x+2y=5z$?

If $x,y,z$ are positive integers and $z(xz+1)^2=(5z+2y)(2z+y)$, prove that $z$ is an odd perfect square.

Starting from 37, adding 5 before each previous term, forms the following sequence: \[37,537,5537,55537,555537,...\] How many prime numbers are there in this sequence?

Prove that for positive reals $a,b,c$, \[\frac{8a^2+2ab}{(b+\sqrt{6ac}+3c)^2}+\frac{2b^2+3bc}{(3c+\sqrt{2ab}+2a)^2}+\frac{18c^2+6ac}{(2a+\sqrt{3bc}+b})^2\geq 1\]

Let $\Delta ABC$ be a triangle with $AB=AC$ and $\angle A = \alpha$, and let $O,H$ be its circumcenter and orthocenter, respectively. If $P,Q$ are points on $AB$ and $AC$, respectively, such that $APHQ$ forms a rhombus, determine $\angle POQ$ in terms of $\alpha$.

A V-tromino is a diagram formed by three unit squares.(As attachment.) (a)Is it possible to cover a $3\times 2013$ table by $3\times 671$ V-trominoes? (b)Is it possible to cover a $5\times 2013$ table by $5\times 671$ V-trominoes?

Let P be a point in an acute triangle $ABC$, and $d_A, d_B, d_C$ be the distance from P to vertices of the triangle respectively. If the distance from P to the three edges are $d_1, d_2, d_3$ respectively, prove that \[d_A+d_B+d_C\geq 2(d_1+d_2+d_3)\]

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

Find all $g:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in R$, \[(4x+g(x)^2)g(y)=4g(\frac{y}{2}g(x))+4xyg(x)\]

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers. a) Prove that there are at least $500$ friendly integers in the set $\{ 1,2,\ldots ,2012\}$. b) Decide whether $a=2$ is friendly.

Let $ABCD$ be a convex quadrilateral with non-parallel sides $BC$ and $AD$. Assume that there is a point $E$ on the side $BC$ such that the quadrilaterals $ABED$ and $AECD$ are circumscribed. Prove that there is a point $F$ on the side $AD$ such that the quadrilaterals $ABCF$ and $BCDF$ are circumscribed if and only if $AB$ is parallel to $CD$.