There are $n+1$ containers arranged in a circle. One container has $n$ stones, the others are empty. A move is to choose two containers $A$ and $B$, take a stone from $A$ and put it in one of the containers adjacent to $B$, and to take a stone from $B$ and put it in one of the containers adjacent to $A$. We can take $A = B$. For which $n$ is it possible by series of moves to end up with one stone in each container except that which originally held $n$ stones.

$2^{n-1}$ subsets are choosen from a set with $n$ elements, such that every three of these subsets have an element in common. Show that all subsets have an element in common.

Let $A'\in(BC),$ $B'\in(CA),C'\in(AB)$ be the points of tangency of the excribed circles of triangle $\triangle ABC$ with the sides of $\triangle ABC.$ Let $R'$ be the circumradius of triangle $\triangle A'B'C'.$ Show that \[ R'=\frac{1}{2r}\sqrt{2R\left(2R-h_{a}\right)\left(2R-h_{b}\right)\left(2R-h_{c}\right)}\] where as usual, $R$ is the circumradius of $\triangle ABC,$ r is the inradius of $\triangle ABC,$ and $h_{a},h_{b},h_{c}$ are the lengths of altitudes of $\triangle ABC.$

A snail begins a journey starting at the origin of a coordinate plane. The snail moves along line segments of length $\sqrt{10}$ and in any direction such that the horizontal and vertical displacements are both integers. As the snail moves, it leaves a trail tracing out its entire journey. After a while, this trail can form various polygons. What is the smallest possible area of a polygon that could be created by the snail's trail?

A [i]squidward[/i] is a piece that moves on a board in the following way: it advances three squares in one direction and then two squares in a perpendicular direction. For example, in the figure below, by making one move, the squidward can move to any of the $8$ squares indicated with arrows. Initially, there is one squidward on each of the $35$ squares of a $5 \times 7$ board. At the same time, each squidward makes exactly one move. What is the smallest possible number of empty squares after these moves? [center][img]https://i.imgur.com/rqgG95C.png[/img][/center]

Let $S \subset \{ 1, \dots, n \}$ be a nonempty set, where $n$ is a positive integer. We denote by $s$ the greatest common divisor of the elements of the set $S$. We assume that $s \not= 1$ and let $d$ be its smallest divisor greater than $1$. Let $T \subset \{ 1, \dots, n \}$ be a set such that $S \subset T$ and $|T| \ge 1 + \left[ \frac{n}{d} \right]$. Prove that the greatest common divisor of the elements in $T$ is $1$. ----------- [Second Version] Let $n(n \ge 1)$ be a positive integer and $U = \{ 1, \dots, n \}$. Let $S$ be a nonempty subset of $U$ and let $d (d \not= 1)$ be the smallest common divisor of all elements of the set $S$. Find the smallest positive integer $k$ such that for any subset $T$ of $U$, consisting of $k$ elements, with $S \subset T$, the greatest common divisor of all elements of $T$ is equal to $1$.

The rectangle $ABCD$ below has dimensions $AB = 12 \sqrt{3}$ and $BC = 13 \sqrt{3}$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$. If triangle $ABP$ is cut out and removed, edges $\overline{AP}$ and $\overline{BP}$ are joined, and the figure is then creased along segments $\overline{CP}$ and $\overline{DP}$, we obtain a triangular pyramid, all four of whose faces are isosceles triangles. Find the volume of this pyramid. [asy] pair D=origin, A=(13,0), B=(13,12), C=(0,12), P=(6.5, 6); draw(B--C--P--D--C^^D--A); filldraw(A--P--B--cycle, gray, black); label("$A$", A, SE); label("$B$", B, NE); label("$C$", C, NW); label("$D$", D, SW); label("$P$", P, N); label("$13\sqrt{3}$", A--D, S); label("$12\sqrt{3}$", A--B, E);[/asy]

Let $a_1 = 1$, $a_2 = 1$, and for $n \ge 2$, let $$a_{n+1} =\frac{1}{n} a_n + a_{n-1}.$$ What is $a_{12}$?

Al, Betty, and Clare split $ \$1000$ among them to be invested in different ways. Each begins with a different amount. At the end of one year they have a total of $ \$1500$. Betty and Clare have both doubled their money, whereas Al has managed to lose $ \$100$. What was Al’s original portion? $ \textbf{(A)}\ \$ 250 \qquad \textbf{(B)}\ \$ 350 \qquad \textbf{(C)}\ \$ 400 \qquad \textbf{(D)}\ \$ 450 \qquad \textbf{(E)}\ \$ 500$

There are $30$ cities in the empire of Euleria. Every week, Martingale City runs a very well-known lottery. $900$ visitors decide to take a trip around the empire, visiting a different city each week in some random order. $3$ of these cities are inhabited by mathematicians, who will talk to all visitors about the laws of statistics. A visitor with this knowledge has probability $0$ of buying a lottery ticket, else they have probability $0.5$ of buying one. What is the expected number of visitors who will play the Martingale Lottery?

A $2010\times 2010$ board is divided into corner-shaped figures of three cells. Prove that it is possible to mark one cell in each figure such that each row and each column will have the same number of marked cells. [i]I. Bogdanov & O. Podlipsky[/i]

On the side $AC$ of triangle $ABC$ point $D$ is chosen. The perpendicular bisector of segment $BD$ intersects the circumcircle $\Omega$ of triangle $ABC$ at $P$, $Q$. Point $E$ lies on the arc $AC$ of circle $\Omega$, that doesn't contain point $B$, such that $\angle ABD=\angle CBE$. Prove that the orthocenter of the triangle $PQE$ lies on the line $AC$ [i]M. Zorka[/i]

Prove the inequality \[ \sqrt {a^{1 \minus{} a}b^{1 \minus{} b}c^{1 \minus{} c}} \le \frac {1}{3} \] holds for all positive real numbers $ a$, $ b$ and $ c$ with $ a \plus{} b \plus{} c \equal{} 1$.

Let $(a_n)_{n\geq 1}$ be a positive real sequence given by $a_n=\sum \limits_{k=1}^n \frac{1}{k}$. Compute $$\lim \limits_{n \to \infty}e^{-2a_n} \sum \limits_{k=1}^n \left \lfloor \left(\sqrt[2k]{k!}+\sqrt[2(k+1)]{(k+1)!}\right)^2 \right \rfloor$$where we denote by $\lfloor x\rfloor$ the integer part of $x$.

Are there three pairwise distinct non-zero integers whose sum is zero and whose sum of thirteenth powers is the square of some natural number?

Inside acute $\angle{XOY},$ points $M$ and $N$ are taken so that $\angle{XON}=\angle{YOM}$. Point $Q$ is taken on segment $OX$ such that $\angle{NQO}=\angle{MQX}.$ Point $P$ is taken such that $\angle{NPO}=\angle{MPY}.$ Prove the lengths of the broken lines $MPN$ and $MQN$ are equal.

Suppose that $p> 2$ is a prime number. For each integer $k = 1, 2,..., p-1$, denote $r_k$ as the remainder of the division $k^p$ by $p^2$. Prove that $r_1+r_2+r_3+...+r_{p-1}=\frac{p^2(p-1)}{2}$

There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \dots < a_k$ such that \[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}.\] What is $k?$ $\textbf{(A) } 117 \qquad \textbf{(B) } 136 \qquad \textbf{(C) } 137 \qquad \textbf{(D) } 273 \qquad \textbf{(E) } 306$

How many integers between 123 and 321 inclusive have exactly two digits that are 2? [i] Proposed by Yannick Yao [/i]

In the eight-term sequence $A, B, C, D, E, F, G, H$, the value of $C$ is $5$ and the sum of any three consecutive terms is $30$. What is $A + H$? $ \textbf{(A)}\ 17 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 26 \qquad \textbf{(E)}\ 43 $

What is the sum of the digits of the square of $ 111,111,111$? $ \textbf{(A)}\ 18 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 63 \qquad \textbf{(E)}\ 81$

Given a triangle $ ABC $, $ AD $ is its angle bisector. Let $ E, F $ be the centers of the circles inscribed in the triangles $ ADC $ and $ ADB $, respectively. Denote by $ \omega $, the circle circumscribed around the triangle $ DEF $, and by $ Q $, the intersection point of $ BE $ and $ CF $, and $ H, J, K, M $ , respectively the second intersection point of the lines $ CE, CF, BE, BF $ with circle $ \omega $. Let $\omega_1, \omega_2 $ the circles be circumscribed around the triangles $ HQJ $ and $ KQM $ Prove that the intersection point of the circles $\omega_1, \omega_2 $ different from $ Q $ lies on the line $ AD $. (Kivva Bogdan)

Compute the perimeter of the triangle that has area $3-\sqrt{3}$ and angles $45^\circ$, $60^\circ$, and $75^\circ$.

An alien race has three genders: male, female and emale. A married triple consists of three persons, one from each gender who all like each other. Any person is allowed to belong to at most one married triple. The feelings are always mutual ( if $x$ likes $y$ then $y$ likes $x$). The race wants to colonize a planet and sends $n$ males, $n$ females and $n$ emales. Every expedition member likes at least $k$ persons of each of the two other genders. The problem is to create as many married triples so that the colony could grow. a) Prove that if $n$ is even and $k\geq 1/2$ then there might be no married triple. b) Prove that if $k \geq 3n/4$ then there can be formed $n$ married triple ( i.e. everybody is in a triple).

On a board there are $2022$ numbers: $1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots,\frac{1}{2022}$. During a $move$ two numbers are chosen, $a$ and $b$, they are erased and $a+b+ab$ is written in their place. The moves take place until only one number is left on the board. What are the possible values of this number?