Let $ABC$ be a triangle with $\angle ABC \neq 90^\circ$, and $AB$ its shortest side. Let $H$ be the orthocenter of $ABC$. Let $\Gamma$ be the circle with center $B$ and radius $BA$. Let $D$ be the second point where the line $CA$ meets $\Gamma$. Let $E$ be the second point where $\Gamma$ meets the circumcircle of the triangle $BCD$. Let $F$ be the intersection point of the lines $DE$ and $BH$. Prove that the line $BD$ is tangent to the circumcircle of the triangle $DFH$.

Find all $3$-tuples $(a, b, c)$ of positive integers, with $a \geq b \geq c$, such that $a^2 + 3b$, $b^2 + 3c$, and $c^2 + 3a$ are all squares.

Let $n$ be a positive integer, and $a_1, a_2, \dots, a_{2n}$ be a sequence of positive real numbers whose product is equal to $2$. For $k = 1, 2, \dots, 2n$, set $a_{2n + k} = a_k$, and define $$ A_k = \frac{1 + a_k + a_k a_{k + 1} + \dots + a_k a_{k + 1} \cdots a_{k + n - 2}}{1 + a_k + a_k a_{k + 1} + \dots + a_k a_{k + 1} \cdots a_{k + 2n - 2}}. $$ Suppose that $A_1, A_2, \dots, A_{2n}$ are pairwise distinct; show that exactly half of them are less than $\sqrt{2} - 1$.

Find all functions $f$ and $g$ defined from $\mathbb{R}_{>0}$ to $\mathbb{R}_{>0}$ such that for all $x, y > 0$ the two equations hold $$ (f(x) + y - 1)(g(y) + x - 1) = {(x + y)}^2 $$$$ (-f(x) + y)(g(y) + x) = (x + y + 1)(y - x - 1) $$ Note: $\mathbb{R}_{>0}$ denotes the set of positive real numbers.

Let $r$ be a positive integer. Find the smallest positive integer $m$ satisfying the condition: For all sets $A_1, A_2, \dots, A_r$ with $A_i \cap A_j = \emptyset$, for all $i \neq j$, and $\bigcup_{k = 1}^{r} A_k = \{ 1, 2, \dots, m \}$, there exists $a, b \in A_k$ for some $k$ such that $1 \leq \frac{b}{a} \leq 1 + \frac{1}{2022}$.

Does there exist positive integers $n_1, n_2, \dots, n_{2022}$ such that the number $$ \left( n_1^{2020} + n_2^{2019} \right)\left( n_2^{2020} + n_3^{2019} \right) \cdots \left( n_{2021}^{2020} + n_{2022}^{2019} \right)\left( n_{2022}^{2020} + n_1^{2019} \right) $$is a power of $11$?