Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

Prove the inequality for all positive integer $n$ : \[ \left(\frac{2n-1}{e}\right)^{\frac{2n-1}{2}}<1\cdot 3\cdot 5\cdots (2n-1)<\left(\frac{2n+1}{e}\right)^{\frac{2n+1}{2}} \]

Anna writes a sequence of integers starting with the number 12. Each subsequent integer she writes is chosen randomly with equal chance from among the positive divisors of the previous integer (including the possibility of the integer itself). She keeps writing integers until she writes the integer 1 for the first time, and then she stops. One such sequence is \[ 12, 6, 6, 3, 3, 3, 1. \] What is the expected value of the number of terms in Anna’s sequence?

Let $ABC$ be an acute angled triangle with orthocenter ${H}$ and circumcenter ${O}$. Assume the circumcenter ${X}$ of ${BHC}$ lies on the circumcircle of ${ABC}$. Reflect $O$ across ${X}$ to obtain ${O'}$, and let the lines ${XH}$and ${O'A}$ meet at ${K}$. Let $L,M$ and $N$ be the midpoints of $\left[ XB \right],\left[ XC \right]$ and $\left[ BC \right]$, respectively. Prove that the points $K,L,M$ and ${N}$ are concyclic.

Let $A=2^7(7^{14}+1)+2^6\cdot 7^{11}\cdot 10^2+2^6\cdot 7^7\cdot 10^{4}+2^4\cdot 7^3\cdot 10^6$. Prove that the number $A$ ends with $14$ zeros. [i](I. Gorodnin)[/i]

A sequence $\{a_n\}$ is defined recursively by $a_1=\frac{1}{2}, $ and for $n\ge 2,$ $0<a_n\leq a_{n-1}$ and \[a_n^2(a_{n-1}+1)+a_{n-1}^2(a_n+1)-2a_na_{n-1}(a_na_{n-1}+a_n+1)=0.\] $(1)$ Determine the general formula of the sequence $\{a_n\};$ $(2)$ Let $S_n=a_1+\cdots+a_n.$ Prove that for $n\ge 1,$ $\ln\left(\frac{n}{2}+1\right)<S_n<\ln(n+1).$

Tnag is repeating the phrase "sigma sigma on the wall'' an infinite number times. Between each word, there is exactly one second of pause. Adam has heard the phrase so many times that he has come up with a game using two numbers $x$ and $y$: Start with a score of 0. [list] [*] At a random time, Adam will hear the word $a$ (each of the 5 words are equally likely to be heard). [*] Then [list] [*] if $a$ is "sigma'', Adam will multiply his score by $x$, and [*] if $a$ is any of the other words, Adam will add $y$ to his score. [/list] [/list] Let $f(x,y)$ be Adam's expected score after infinitely many steps. Find \[ \sum_{n=2}^{\infty}f\left(\frac{1}{n}, \frac{1}{n^2}\right). \]

Jacob likes to watchMickeyMouse Clubhouse! One day, he decides to create his own MickeyMouse head shown below, with two circles $\omega_1$ and $\omega_2$ and a circle $\omega$, and centers $O_1$, $O_2$, and $O$, respectively. Let $\omega_1$ and $\omega$ meet at points $P_1$ and $Q_1$, and let $\omega_2$ and $\omega$ meet at points $P_2$ and $Q_2$. Point $P_1$ is closer to $O_2$ than $Q_1$, and point $P_2$ is closer to $O_1$ than $Q_2$. Given that $P_1$ and $P_2$ lie on $O_1O_2$ such that $O_1P_1 = P_1P_2 = P_2O_2 = 2$, and $Q_1O_1 \parallel Q_2O_2$, the area of $\omega$ can be written as $n \pi$. Find $n$. [img]https://cdn.artofproblemsolving.com/attachments/6/d/d98a05ee2218e80fd84d299d47201669736d99.png[/img]

Let $D$ be a point inside of equilateral $\triangle ABC$, and $E$ be a point outside of equilateral $\triangle ABC$ such that $m(\widehat{BAD})=m(\widehat{ABD})=m(\widehat{CAE})=m(\widehat{ACE})=5^\circ$. What is $m(\widehat{EDC})$ ? $ \textbf{(A)}\ 45^\circ \qquad\textbf{(B)}\ 40^\circ \qquad\textbf{(C)}\ 35^\circ \qquad\textbf{(D)}\ 30^\circ \qquad\textbf{(E)}\ 25^\circ $

We say that a doubly infinite sequence $. . . , s_{−2}, s_{−1}, s_{0}, s_1, s_2, . . .$ is subaveraging if $s_n = (s_{n−1} + s_{n+1})/4$ for all integers n. (a) Find a subaveraging sequence in which all entries are different from each other. Prove that all entries are indeed distinct. (b) Show that if $(s_n)$ is a subaveraging sequence such that there exist distinct integers m, n such that $s_m = s_n$, then there are infinitely many pairs of distinct integers i, j with $s_i = s_j$ .

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?