In acute triangle $\triangle ABC$, $BC>AC$, $\Gamma$ is its circumcircle, $D$ is a point on segment $AC$ and $E$ is the intersection of the circle with diameter $CD$ and $\Gamma$. $M$ is the midpoint of $AB$ and $CM$ meets $\Gamma$ again at $Q$. The tangents to $\Gamma$ at $A,B$ meet at $P$, and $H$ is the foot of perpendicular from $P$ to $BQ$. $K$ is a point on line $HQ$ such that $Q$ lies between $H$ and $K$. Prove that $\angle HKP=\angle ACE$ if and only if $\frac{KQ}{QH}=\frac{CD}{DA}$.

Given two odd natural numbers $ a,b$ prove that for each $ n\in\mathbb{N}$ there exists $ m\in\mathbb{N}$ such that either $ a^mb^2-1$ or $ b^ma^2-1$ is multiple of $ 2^n.$

Let $\mathcal{C}$ be a family of subsets of $A=\{1,2,\dots,100\}$ satisfying the following two conditions: 1) Every $99$ element subset of $A$ is in $\mathcal{C}.$ 2) For any non empty subset $C\in\mathcal{C}$ there is $c\in C$ such that $C\setminus\{c\}\in \mathcal{C}.$ What is the least possible value of $|\mathcal{C}|$?

Given is a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $|f(x+y)-f(x)-f(y)|\leq 1$. Prove the existence of an additive function $g:\mathbb{R}\rightarrow \mathbb{R}$ (that is $g(x+y)=g(x)+g(y)$) such that $|f(x)-g(x)|\leq 1$ for any $x \in \mathbb{R}$

In triangle $\triangle ABC$, $BC>AC$, $I_B$ is the $B$-excenter, the line through $C$ parallel to $AB$ meets $BI_B$ at $F$. $M$ is the midpoint of $AI_B$ and the $A$-excircle touches side $AB$ at $D$. Point $E$ satisfies $\angle BAC=\angle BDE, DE=BC$, and lies on the same side as $C$ of $AB$. Let $EC$ intersect $AB,FM$ at $P,Q$ respectively. Prove that $P,A,M,Q$ are concyclic.