An infinite sequence of integers $a_1,a_2,a_3, ...$ is given with $a_1 = 0$ and further holds for every natural number $n$ that $a_{n+1} = a_n - n$ if $a_n \ge n$ and $a_{n+1} = a_n + n$ if $a_n < n$ . (a) Prove that there are infinitely many numbers in the sequence equal to $0$. (b) Express in terms of $k$ the ordinal number of the $k^e$ number from the sequence, which is equal to $0$.

In the arrangement of $pn$ real numbers below, the difference between the greatest and smallest numbers in each row is at most $d$, $d > 0$. \[ \begin{array}{l} a_{11} \,\, a_{12} \,\, ... \,\, a_{1n}\\ a_{21} \,\, a_{22} \,\, ... \,\, a_{2n}\\ \,\, . \,\, \,\, \,\, \,\, . \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, .\\ \,\, . \,\, \,\, \,\, \,\, . \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, .\\ \,\, . \,\, \,\, \,\, \,\, . \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, .\\ a_{n1} \,\, a_{n2} \,\, ... \,\, a_{nn}\\ \end{array} \] Prove that, when the numbers in each column are rearranged in decreasing order, the difference between the greatest and smallest numbers in each row will still be at most d.

Each vertex $v$ and each edge $e$ of a graph $G$ are assigned numbers $f(v)\in\{1,2\}$ and $f(e)\in\{1,2,3\}$, respectively. Let $S(v)$ be the sum of numbers assigned to the edges incident to $v$ plus the number $f(v)$. We say that an assignment $f$ is [i]cool [/i]if $S(u) \ne S(v)$ for every pair $(u,v)$ of adjacent (i.e. connected by an edge) vertices in $G$. Prove that for every graph there exists a cool assignment.

Let $f : \mathbb{N} \longrightarrow \mathbb{N}$ be a function such that (a) $ f(m) < f(n)$ whenever $m < n$. (b) $f(2n) = f(n) + n$ for all $n \in \mathbb{N}$. (c) $n$ is prime whenever $f(n)$ is prime. Find $$\sum_{n=1}^{2022} f(n).$$

Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$ and $a+b$ are all distinct squares. What is the smallest possible value of $b$ ?

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that has limits at any point and has no local extrema. Show that: $a)$ $f$ is continuous; $b)$ $f$ is strictly monotone.

A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures 8 inches high and 10 inches wide. What is the area of the border, in square inches? $\textbf{(A)}\hspace{.05in}36 \qquad \textbf{(B)}\hspace{.05in}40 \qquad \textbf{(C)}\hspace{.05in}64 \qquad \textbf{(D)}\hspace{.05in}72 \qquad \textbf{(E)}\hspace{.05in}88 $

For which real values of $x,y$ the expression$\frac{2-\left(\dfrac{x+y}{3}-1\right)^2}{\left(\dfrac{x-3}{2}+\dfrac{2y-x}{3}\right)^2+4}$ becomes maximum? Which is that maximum value?

Let $T$ be a triangle with inscribed circle $C.$ A square with sides of length $a$ is circumscribed about the same circle $C.$ Show that the total length of the parts of the edge of the square interior to the triangle $T$ is at least $2 \cdot a.$

Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be points on segments $AB$, $BC$, $CD$, $DA$, respectively, and let $P$ be the intersection of $EG$ and $FH$. Given that quadrilaterals $HAEP$, $EBFP$, $FCGP$, $GDHP$ all have inscribed circles, prove that $ABCD$ also has an inscribed circle. [i]Evan O'Dorney.[/i]

What is the minimum $n$ so that grid $nxn$ can be covered with equal number of 2x2 squares and angle triminoes (2x2 without one square)

Let $ f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} $ be a function such that for all $n > 1$ there is a prime divisor $p$ of $n$ such that \[ f(n)=f\left(\frac{n}{p}\right)-f(p). \] Furthermore, it is given that $ f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 $. Determine $ f(2014^2)+f(2015^3)+f(2016^5) $.

Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$.

Given two odd integers $a$ and $b$; prove that $a^3 -b^3$ is divisible by $2^n$ if and only if $a-b$ is divisible by $2^n$.

In the scalene triangle $ABC$, the bisector of angle A cuts side $BC$ at point $D$. The tangent lines to the circumscribed circunferences of triangles $ABD$ and $ACD$ on point D, cut lines $AC$ and $AB$ on points $E$ and $F$ respectively. Let $G$ be the intersection point of lines $BE$ and $CF$. Prove that angles $EDG$ and $ADF$ are equal.

Inside the triangle $ABC$ choose the point $M$, and on the side $BC$ - the point $K$ in such a way that $MK || AB$. The circle passing through the points $M, \, \, K, \, \, C,$ crosses the side $AC$ for the second time at the point $N$, a circle passing through the points $M, \, \, N, \, \, A, $ crosses the side $AB$ for the second time at the point $Q$. Prove that $BM = KQ$. (Nagel Igor)

The board contains $20$ non-constant linear functions, not necessarily distinct. For each pair $(f, g)$ of these functions ($190$ pairs in total), Victor writes on the board a quadratic function $f(x)\cdot g(x) - 2$, and Solomiya writes on the board a quadratic function $f(x)g(x)-1$. Victor calculated that exactly $V$ of his quadratic functions have a root, and Solomiya calculated that exactly $S$ of her quadratic functions have a root. Find the largest possible value of $S-V$. [i]Remarks.[/i] A linear function $y = kx+b$ is called non-constant if $k\neq 0$. [i]Proposed by Oleksiy Masalitin[/i]

Maya lists all the positive divisors of $ 2010^2$. She then randomly selects two distinct divisors from this list. Let $ p$ be the probability that exactly one of the selected divisors is a perfect square. The probability $ p$ can be expressed in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.

Compute the smallest integer $n\geq 4$ such that $\textstyle\binom n4$ ends in $4$ or more zeroes (i.e. the rightmost four digits of $\textstyle\binom n4$ are $0000$).

The first two terms of a sequence are $2$ and $3$. Each next term thereafter is the sum of the nearestly previous two terms if their sum is not greather than $10, 0$ otherwise. The $2014$th term is: (A): $0$, (B): $8$, (C): $6$, (D): $4$, (E) None of the above.

Mr. Jones teaches algebra. He has a whiteboard with a pre-drawn coordinate grid that runs from $-10$ to $10$ in both the $x$ and $y$ coordinates. Consequently, when he illustrates the graph of a quadratic, he likes to use a quadratic with the following properties: I. The quadratic sends integers to integers. II. The quadratic has distinct integer roots both between $-10$ and $10$, inclusive. III. The vertex of the quadratic has integer $x$ and $y$ coordinates both between $-10$ and $10$, inclusive. How many quadratics are there with these properties?

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

Let $ABCD$ be a rectangle of center $O$, such that $\angle DAC = 60^o$. The angle bisector of $\angle DAC$ meets $DC$ at $S$. Lines $OS$ and $AD$ meet at $L$ and lines $BL$ and $AC$ meet at $M$. Prove that lines $SM$ and $CL$ are parallel.

In each cell of a $2019\times 2019$ board is written the number $1$ or the number $-1$. Prove that for some positive integer $k$ it is possible to select $k$ rows and $k$ columns so that the absolute value of the sum of the $k^2$ numbers in the cells at the intersection of the selected rows and columns is more than $1000$. [i]Folklore[/i]

Let $s(n)$ be the final value obtained after repeatedly summing the digits of $n$ until a single-digit number is reached. (For example: $s(187) = 7$, because the digit sum of $187$ is $16$ and the digit sum of $16$ is $7$). Evaluate the sum: $$ s(1^2) + s(2^2) + s(3^2) + \ldots + s(2025^2)$$ [i]Proposed by Lia Chitishvili, Georgia [/i]