Find all positive integers $a$, $b$, $c$, and $p$, where $p$ is a prime number, such that $73p^2 + 6 = 9a^2 + 17b^2 + 17c^2$.

In a circle $O$, there are six points, $ A$, $ B$, $C$, $D$, $E$, $F$ in a counterclockwise order such that $BD \perp CF$ , and $CF$, $BE$, $AD$ are concurrent. Let the perpendicular from $B$ to $AC$ be $M$, and the perpendicular from $D$ to $CE$ be $N$. Prove that $AE \parallel MN$.

Consider the sequence $a_1, a_2, a_3, ...$ defined by $a_1 = 9$ and $a_{n + 1} = \frac{(n + 5)a_n + 22}{n + 3}$ for $n \ge 1$. Find all natural numbers $n$ for which $a_n$ is a perfect square of an integer.

Let $F$ is the set of all sequences $\{(a_1, a_2, . . . , a_{2020})\}$ with $a_i \in \{-1, 1\}$ for all $i = 1,2,...,2020$. Prove that there exists a set $S$, such that $S \subset F$, $|S| = 2020$ and for any $(a_1,a_2,...,a_{2020}) \in F$ there exists $(b_1,b_2,...,b_{2020}) \in S$, such that $\sum_{i=1}^{2020} a_ib_i = 0$.

Let $(a_n)_{n\ge 1}$ be a sequence given by $a_1 = 45$ and $$a_n = a^2_{n-1} + 15a_{n-1}$$for $n > 1$. Prove that the sequence contains no perfect squares.

In a triangle $ABC$, let $K$ be a point on the median $BM$ such that $CM = CK$. It turned out that $\angle CBM = 2\angle ABM$. Show that $BC = KM$.

We have $n > 2$ nonzero integers such that everyone of them is divisible by the sum of the other $n - 1$ numbers, Show that the sum of the $n$ numbers is precisely $0$.

Let us call a set of positive integers nice if the number of its elements equals to the average of its numbers. Call a positive integer $n$ an amazing number if the set $\{1, 2 , . . . , n\}$ can be partitioned into nice subsets. a) Prove that every perfect square is amazing. b) Show that there are infinitely many positive integers which are not amazing.