Consider the sequence $a_1, a_2, a_3, \dots$ defined by $a_1 = 9$ and
$$
a_{n+1} = \frac{(n+5)a_n + 22}{n+3}
$$
for $n \ge 1$. Find all positive integers $n$ for which $a_n$ is a perfect square.
In the popular game of Minesweeper, some fields of an $a \times b$ board are marked with a mine and on all the remaining fields the number of adjacent fields that contain a mine is recorded. Two fields are considered adjacent if they share a common vertex. For which $k \in \{0, 1, 2, 3, 4, 5, 6, 7, 8\}$ is it possible for some $a$ and $b$, $ab \gt 2021$, to create a board whose fields are covered in mines, except for 2021 fields who are all marked with $k$?
Let $F$ be the set of all sequences $(a_1, a_2, \dots, a_{2020})$ with $a_i \in \{-1, 1\}$ for all $i = 1, 2, \dots, 2020$. Prove that there exists a set $S$, such that $S \subset F$, $|S| = 2020$ and for any $(a_1, a_2, \dots, a_{2020}) \in F$ there exists $(b_1, b_2, \dots, b_{2020}) \in S$, such that $\sum_{i=1}^{2020} a_i b_i = 0$.
Let $(a_n)_{n \ge 1}$ be a sequence given by $a_1 = 45$ and
$$
a_n = a_{n-1}^2 + 15a_{n-1}
$$
for $n \gt 1$. Prove that the sequence contains no perfect squares.
Let $ABC$ be a non-isosceles triangle with incenter $I$ and let the circumcircle of the triangle $ABC$ has radius $R$. Let $AL$ be the external angle bisector of $\angle BAC$ with $L \in BC$. Let $K$ be the point on the perpendicular bisector of $BC$ such that $IL \perp IK$. Prove that $OK = 3R$.
Let $ABC$ be an acute triangle with $AB \lt AC$ and inscribed in the circle $(O)$. Denote $I$ as the incenter of $ABC$ and $D, E$ as the intersections of $AI$ with $BC, (O)$ respectively. Take a point $K$ on $BC$ such that $\angle AIK = 90^\circ$ and $KA, KE$ meet $(O)$ again at $M, N$ respectively. The rays $ND, NI$ meet the circle $(O)$ at $Q, P$.
**Part 1.** Prove that the quadrilateral $MPQE$ is a kite.
**Part 2.** Take $J$ on $IO$ such that $AK \perp AJ$. The line through $I$ and perpendicular to $OI$ cuts $BC$ at $R$, cuts $EK$ at $S$. Prove that $OR \parallel JS$.
We have $n \gt 2$ nonzero integers such that every one of them is divisible by the sum of the other $n-1$ numbers. Show that the sum of the $n$ numbers is precisely $0$.
A set of $n$ points in space is given, no three of which are collinear and no four of which are co-planar (on a single plane), and each pair of points is connected by a line segment. Initially, all the line segments are colorless. A positive integer $b$ is given and Alice and Bob play the following game. In each turn Alice colors one segment red and then Bob colors up to $b$ segments blue. This is repeated until there are no more colorless segments left. If Alice colors a red triangle, Alice wins. If there are no more colorless segments and Alice hasn't succeeded in coloring a red triangle, Bob wins. Neither player is allowed to color over an already colored line segment.
**Part 1.** Prove that if $b \lt \sqrt{2n-2} - \frac{3}{2}$, then Alice has a winning strategy.
**Part 2.** Prove that if $b \ge 2\sqrt{n}$, then Bob has a winning strategy.
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 $\{1, 2, \ldots, n\}$ can be partitioned into nice subsets.
**Part a.** Prove that every perfect square is amazing
**Part b.** Show that there are infinitely many positive integers which are not amazing.
Given a positive integer $k$, show that there exists a prime $p$ such that one can choose distinct integers $a_1, a_2, \dots, a_{k+3} \in \{1, 2, \dots, p-1\}$ such that $p$ divides $a_i a_{i+1} a_{i+2} a_{i+3} - i$ for all $i = 1, 2, 3, \dots, k$.
Suppose that $a, b, c, d$ are positive real numbers satisfying $(a+c)(b+d) = ac+bd$. Find the smallest possible value of
$$
\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}
$$
Let $p$ be an odd prime and let $N = \frac{1}{4}(p^3 - p) - 1$. The numbers $1, 2, \dots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \le N$, denote by $r(n)$ the fraction of integers in $\{1, 2, \dots, n\}$ that are red (number of red numbers divided by $n$). Prove that there exists a positive integer $a \in \{1, 2, \dots, p-1\}$ such that $r(n) \ne \frac{a}{p}$ for all $n = 1, 2, \dots, N$
Let $P(x)$ be a nonconstant integer polynomial and positive integer $n$. The sequence $a_0, a_1, \ldots$ is defined by $a_0 = n$ and $a_k = P(a_{k-1})$ for $k \ge 1$. Given that for each positive integer $b$, the sequence contains a $b$-th power of some positive integer greater than 1. Prove that $\deg P = 1$.
Let $PAB$ and $PBC$ be two similar right-angled triangles (in the same plane) with $\angle PAB = \angle PBC = 90^\circ$ such that $A$ and $C$ lie on opposite sides of the line $PB$. If $PC = AC$, calculate the ratio $\frac{PA}{AB}$.
Let $ABC$ be a triangle with $\angle ABC \ne 90^\circ$ and $AB$ its shortest side. Denote by $H$ the intersection of the altitudes of triangle $ABC$. Let $K$ be the circle through $A$ with centre $B$. Let $D$ be the other intersection of $K$ and $AC$. Let $K$ intersect the circumcircle of $BCD$ again at $E$. If $F$ is the intersection of $DE$ and $BH$, show that $BD$ is tangent to the circle through $D, F$ and $H$.
Jacob and Laban take turns playing a game. Each of them starts with a list of square numbers $1, 4, 9, \dots, 2021^2$, and there is a whiteboard in front of them with the number $0$ on it. Jacob chooses a number $x^2$ from his list, removes it from his list, and replaces the number $W$ on the whiteboard with $W + x^2$. Laban then does the same with a number from his list, and they repeat back and forth until both of them have no more numbers in their list. Now every time that the number on the whiteboard is divisible by $4$ after a player has taken their turn, Jacob gets a sheep. Jacob wants to have as many sheep as possible by the end of the game, whereas Laban wants Jacob to have as few sheep as possible. What is the greatest number $K$ such that Jacob can guarantee to get at least $K$ sheep by the end of the game, no matter how Laban plays?
A sequence $s$ consisting of zeroes and ones is given. For each positive integer $k$ we define $v_k$ as the maximum number of ways to find consecutive digits forming the sequence $s$ in a sequence of $k$ digits. (For example, if $s = 0110$, then $v_7 = v_8 = 2$, since in the sequences 0110110 and 01101100 consecutive digits 0110 are found in two places, and three pairs of ones surrounded by zeroes can not occur in a sequence of 7 or 8 digits.) It is known that $v_n \lt v_{n+1} \lt v_{n+2}$ for some positive integer $n$. Prove that all the digits in the sequence $s$ are identical.