The $C$-excircle of a triangle $ABC$ touches $AB, AC, BC$ at $M, N, K$. The points $P, Q$ lie on $NK$ so that $AN=AP, BK=BQ$. Prove that the circumradius of $\triangle MPQ$ is equal to the inradius of $\triangle ABC$.

$a,b,c$ are positive real numbers such that $a+b+c\ge 3$ and $a^2+b^2+c^2=2abc+1$. Prove that $$a+b+c\le 2\sqrt{abc}+1$$

Given $x,y>0$ such that $x^2y^2+2x^3y=1$. Find the minimum value of sum $x+y$

Solve the given equation in prime numbers $$p^3+q^3+r^3=p^2qr$$

The altitudes of an acute triangle $ABC$ intersect at $H$. The tangent line at $H$ to the circumcircle of triangle $BHC$ intersects the lines $AB$ and $AC$ at points $Q$ and $P$ respectively. The circumcircles of triangles $ABC$ and $APQ$ intersect at point $K$ ($K\neq A$). The tangent lines at the points $A$ and $K$ to the circumcircle of triangle $APQ$ intersect at $T$. Prove that $TH$ passes through the midpoint of segment $BC$.

A triangle $ABC$ with obtuse angle $C$ and $AC>BC$ has center $O$ of its circumcircle $\omega$. The tangent at $C$ to $\omega$ meets $AB$ at $D$. Let $\Omega$ be the circumcircle of $AOB$. Let $OD, AC$ meet $\Omega$ at $E, F$ and let $OF \cap CE=T$, $OD \cap BC=K$. Prove that $OTBK$ is cyclic.

Let $n>100$ be an integer. The numbers $1,2 \ldots, 4n$ are split into $n$ groups of $4$. Prove that there are at least $\frac{(n-6)^2}{2}$ quadruples $(a, b, c, d)$ such that they are all in different groups, $a<b<c<d$ and $c-b \leq |ad-bc|\leq d-a$.

$a,b,c$ are positive real numbers such that $\max\{\frac{a(b+c)}{a^2+bc},\frac{b(c+a)}{b^2+ca},\frac{c(a+b)}{c^2+ab}\}\le \frac{5}{2}$. Prove inequality $$\frac{a(b+c)}{a^2+bc}+\frac{b(c+a)}{b^2+ca}+\frac{c(a+b)}{c^2+ab}\le 3$$

Given are positive integers $a, b, m, k$ with $k \geq 2$. Prove that there exist infinitely many $n$, such that $\gcd (\varphi_m(n), \lfloor \sqrt[k] {an+b} \rfloor)=1$, where $\varphi_m(n)$ is the $m$-th iteration of $\varphi(n)$.

Inside an equilateral triangle with side $3$ there are two rhombuses with sides $1,061$ and acute angles $60^\circ$. Prove that these two rhombuses intersect each other. (The vertices of the rhombus are strictly inside the triangle.)