Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.

In trapezoid $ABCD$ with $AB || CD$, $\omega_1$ and $\omega_2$ are two circles with diameters $AD$ and $BC$, respectively. Let $X$ and $Y$ be two arbitrary points on $\omega_1$ and $\omega_2$, respectively. Show that the length of segment $XY$ is not more than half the perimeter of $ABCD$. [i]Proposed by Mahdi Etesami Fard[/i]

A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.

Let $P$ be a point inside a quadrilateral $ABCD$ such that $\angle APB+\angle CPD=180^{\circ}$. Points $P_{a}$, $P_{b}$, $P_{c},$ $P_{d}$ are isogonally conjugated to $P$ with respect to the triangles $BCD$, $CDA$, $DAB$, $ABC$ respectively. Prove that the diagonals of the quadrilaterals $ABCD$ and $P_{a}P_{b}P_{c}P_{d}$ concur. Proposed by: G.Galyapin

Each face of a cube has been divided into four equal quarters and each quarter is painted with one of three available colours. Quarters with common sides are painted with different colours . Prove that each of the available colours was used in painting $8$ quarters.

Prove that in a plane, arbitrary $ n$ points can be overlapped by discs that the sum of all the diameters is less than $ n$, and the distances between arbitrary two are greater than $ 1$. (where the distances between two discs that have no common points are defined as that the distances between its centers subtract the sum of its radii; the distances between two discs that have common points are zero)

Let $a_1, a_2, a_3, \dots, a_{20}$ be a permutation of the numbers $1, 2, \dots, 20$. How many different values can the expression $a_1-a_2+a_3-\dots - a_{20}$ have?

Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$. [i]Proposed by Morteza Saghafian, Iran[/i]

The game of circulate is played with a deck of $kn$ cards each with a number in $1,2,\ldots,n$ such that there are $k$ cards with each number. First, $n$ piles numbered $1,2,\ldots,n$ of $k$ cards each are dealt out face down. The player then flips over a card from pile $1$, places that card face up at the bottom of the pile, then next flips over a card from the pile whose number matches the number on the card just flipped. The player repeats this until he reaches a pile in which every card has already been flipped and wins if at that point every card has been flipped. Hamster has grown tired of losing every time, so he decides to cheat. He looks at the piles beforehand and rearranges the $k$ cards in each pile as he pleases. When can Hamster perform this procedure such that he will win the game? [i]Brian Hamrick.[/i]

Prove that no convex $13$-gon can be cut into parallelograms.

Solve in the real numbers the equation $ 3^{x+1}=(x-1)(x-3). $

Let $ABCD$ be a trapezoid of bases $AD$ and $BC$ , with $AD> BC$, whose diagonals are cut at point $E$. Let $P$ and $Q$ be the feet of the perpendicular drawn from $E$ on the sides $AD$ and $BC$, respectively, with $P$ and $Q$ in segments $AD$ and $BC,$ respectively. Let $I$ be the center of the triangle $AED$ and let $K$ be the point of intersection of the lines $AI$ and $CD$. If $AP + AE = BQ + BE$, show that $AI = IK$.

Polynomial $ P(t)$ is such that for all real $ x$, \[ P(\sin x) \plus{} P(\cos x) \equal{} 1. \] What can be the degree of this polynomial?

Every day, Heesu talks to Sally with some probability $p$. One day, after not talking to Sally the previous day, Heesu resolves to ask Sally out on a date. From now on, each day, if Heesu has talked to Sally each of the past four days, then Heesu will ask Sally out on a date. Heesu’s friend remarked that at this rate, it would take Heesu an expected $2800$ days to finally ask Sally out. Suppose $p=\tfrac{m}{n}$, where $\gcd(m, n) = 1$ and $m, n > 0$. What is $m + n$?

$2022$ points are arranged in a circle, one of which is colored in black, and others in white. In one operation, The Hedgehog can do one of the following actions: 1) Choose two adjacent points of the same color and flip the color of both of them (white becomes black, black becomes white) 2) Choose two points of opposite colors with exactly one point in between them, and flip the color of both of them Is it possible to achieve a configuration where each point has a color opposite to its initial color with these operations? [i](Proposed by Oleksii Masalitin)[/i]

Points $ B_1$ and $ B_2$ lie on ray $ AM$, and points $ C_1$ and $ C_2$ lie on ray $ AK$. The circle with center $ O$ is inscribed into triangles $ AB_1C_1$ and $ AB_2C_2$. Prove that the angles $ B_1OB_2$ and $ C_1OC_2$ are equal.

Let $n$ be a positive integer. Find all $n$- rows $( a_1 , a_2 ,..., a_n )$ of different positive integers such that $$ \frac{(a_1 + d ) (a_2 + d ) \cdot\cdot\cdot ( a_n + d )}{a_1a_2\cdot \cdot \cdot a_n }$$ is integer for every integer $d\ge 0$

$ABC$ triangle with $|AB|<|BC|<|CA|$ has the incenter $I$. The orthocenters of triangles $IBC, IAC$ and $IAB$ are $H_A, H_A$ and $H_A$. $H_BH_C$ intersect $BC$ at $K_A$ and perpendicular line from $I$ to $H_BH_B$ intersect $BC$ at $L_A$. $K_B, L_B, K_C, L_C$ are defined similarly. Prove that $$|K_AL_A|=|K_BL_B|+|K_CL_C|$$

How many primes less than $100$ have $7$ as the ones digit? (Assume the usual base ten representation) $\text{(A)} \ 4 \qquad \text{(B)} \ 5 \qquad \text{(C)} \ 6 \qquad \text{(D)} \ 7 \qquad \text{(E)} \ 8$

We say that a table with three rows or infinite columns is [i]cool [/i] if it was filled with natural numbers, and also whenever the same number m appears in two or more different places in the table, the numbers that appear in the cells immediately below said places (when they exist) are equal. For example, the following table is cool: [img]https://cdn.artofproblemsolving.com/attachments/5/7/16583a6a9434fd2792a4df48a733226cf2f560.png[/img] For each of the following two tables, decide whether it is possible to fill in the empty cells before the resulting tables are cool, explaining how to do this, or why it is not possible to do this. In both tables from the fifth column, the number in the third line is two units greater than the number in the first line. [img]https://cdn.artofproblemsolving.com/attachments/8/a/56d2f05ea09555c39da88f09eb5901a57567f0.png[/img]

In the convex hexagon $ABCDEF$, the angles at the vertices $B$, $C$, $E$, and $F$ are equal. Moreover, the equality \[AB+DE=AF+CD\] holds. Prove that the line $AD$ and the bisectors of the segments $BC$ and $EF$ have a common point.

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that any real numbers $x{}$ and $y{}$ satisfy \[f(xf(x)+f(y))=f(f(x^2))+y.\]

Let $ABCD$ be a parallelogram. Let $M$ be the midpoint of the arc $AC$ containing $B$ of the circumcircle of $ABC$ . Let $E$ be a point on segment $AD$ and $F$ a point on segment $CD$ such that $ME=MD=MF$. Show that $BMEF$ is cyclic. [i]Proposed by A. Tereshin[/i]

(a) Prove that for every natural number $k$, there are positive integers $a_1<a_2<\ldots <a_k$ such that $a_i-a_j$ divides $a_i$ for all $1\leq i, j\leq k, i\neq j$. (b) Show that there is an absolute constant $C>0$ such that $a_1>k^{Ck}$ for every sequence $a_1,\ldots, a_k$ of numbers that satisfy the above divisibility condition. [A. Balogh, I. Z. Ruzsa]

Let $p$ and $q$ be different prime numbers. Solve the following system in integers: \[\frac{z+ p}x+\frac{z-p}y= q,\\ \frac{z+ p}y -\frac{z-p}x= q.\]