There are $5$ people who start with $1, 2, 3, 4,$ and $5$ cookies, respectively. Every minute, two different people are chosen uniformly at random. If they have $a$ and $b$ cookies and $a\neq b,$ the person with more cookies eats $|a-b|$ of their own cookies. If $a = b,$ the minute still passes with nothing happening. Compute the expected number of minutes until all $5$ people have an equal number of cookies.

For how many integers $n$ between $1$ and $50$, inclusive, is \[ \frac{(n^2-1)!}{(n!)^n} \]an integer? (Recall that $0! = 1$.) $\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35$

Let $k$ be the incircle of the triangle $ABC$ with the center of the incircle $I$. The circle $k$ touches the sides $BC, CA$ and $AB$ in points $D, E$ and $F$. Let $G$ be the intersection of the straight line $AI$ and the circle $k$, which lies between $A$ and $I$. Assume $BE$ and $FG$ are parallel. Show that $BD = EF$.

Let $$P(x) = \sum^n_{k=1}(x^{3^k}+ x^{-3^k}- 1), Q(x) = \sum^n_{k=1}(x^{3^k}+ x^{-3^k}+ 1).$$ Given that $$P(x)Q(x) =\sum^{2\cdot 3^n}_{k=-2\cdot 3^n} a_kx^k,$$ Compute $\sum^{3^n}_{k=0}a_k$ in terms of $n$.

Bisectors of a right triangle $\triangle ABC$ with right angle $B$ meet at point $I.$ The perpendicular to $IC$ drawn from $B$ meets the line $IA$ at $D;$ the perpendicular to $IA$ drawn from $B$ meets the line $IC$ at $E.$ Prove that the circumcenter of the triangle $\triangle IDE$ lies on the line $AC.$ [i](A. Kuznetsov )[/i]

A grid of cells is tiled with dominoes such that every cell is covered by exactly one domino. A subset $S$ of dominoes is chosen. Is it true that at least one of the following 2 statements is false? (1) There are $2022$ more horizontal dominoes than vertical dominoes in $S$. (2) The cells covered by the dominoes in $S$ can be tiled completely and exactly by $L$-shaped tetrominoes.

Let $\alpha=1+a_1x+a_2x^2+\ldots$ be a formal power series with coefficients in the field of two elements. Let $$a_n=\begin{cases}1&\text{if every block of zeroes in the binary expansion of }n\text{ has an even number of zeroes}\\0&\text{otherwise}\end{cases}$$(For example, $a_{36}=1$ since $36=100100_2$) Prove that $\alpha^3+x\alpha+1=0$.

Let $ABC$ be a right-angled triangle where ∠$ACB$=90°.Let $CD$ be an altitude of that triangle and points $M$ and $N$ be the midpoints of $CD$ and $BC$, respectively.If $S$ is the circumcenter of the triangle $AMN$, prove that $AS$ and $BC$ are paralel.

Let $ABCD$ be a convex cyclic quadrilateral with circumcircle $\omega$. Let $BA$ produced beyond $A$ meet $CD$ produced beyond $D$, at $L$. Let $\ell$ be a line through $L$ meeting $AD$ and $BC$ at $M$ and $N$ respectively, so that $M,D$ (respectively $N,C$) are on opposite sides of $A$ (resp. $B$). Suppose $K$ and $J$ are points on the arc $AB$ of $\omega$ not containing $C,D$ so that $MK, NJ$ are tangent to $\omega$. Prove that $K,J,L$ are collinear. [i]Proposed by Rijul Saini[/i]

Let $ f$ be an integrable real-valued function on the closed interval $ [0, 1]$ such that $$\int_{0}^{1} x^{m}f(x) dx=\begin{cases} 0 \;\; \text{for}\; m=0,1,\ldots,n-1;\\ 1\;\; \text{for}\; m=n. \end{cases} $$ Show that $|f(x)|\geq2^{n}(n+1)$ on a set of positive measure.

Let $G$ be a simple graph on $n$ vertices with at least one edge, and let us consider those $S:V(G)\to\mathbb R^{\ge 0}$ weighings of the vertices of the graph for which $\sum_{v\in V(G)} S(v)=1$. Furthermore define \[f(G)=\max_S\min_{(v,w)\in E(G)}S(v)S(w),\] where $S$ runs through all possible weighings. Prove that $f(G)=\frac1{n^2}$ if and only if the vertices of $G$ can be covered with a disjoint union of edges and odd cycles. ($V(G)$ denotes the vertices of graph $G$, $E(G)$ denotes the edges of graph $G$.)

Find all functions $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$ with the following properties: 1) For every natural number $n\geq 3$, $\gcd(f(n),n)\neq 1$. 2) For every natural number $n\geq 3$, there exists $i_n\in\mathbb{Z}_{>0}$, $1\leq i_n\leq n-1$, such that $f(n)=f(i_n)+f(n-i_n)$. [i]Proposed by Pavel Ciurea[/i]

How many of the first $1000$ positive integers can be written as the sum of finitely many distinct numbers from the sequence $3^0$, $3^1$, $3^2$ ,$...$?

The square $ABCD$ and the equilateral triangle $MKL$ are located as shown in the figure. Find the angle $\angle PQD$. [img]https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQKgjvzy1WhwkMJbcV_C0iveelYmm75FpaGlWgZ-Ap_uQUiegaKYafelo-J_3rMgKMgpMp5soYc1LVYLI8H4riC6R-f8eq2DiWTGGII08xQkwu7t2KVD4pKX4_IN-gC7DVRhdVZSjbaj2S/s1600/oral+moscow+geometry+2015+8.9+p2.png[/img]

Find all positive integers $n$, such that there exist positive integers $a,b$, such that $a+2^b=n^{2022}$ and $a^2+4^b=n^{2023}$.

Given triangle $ABC$. The midperpendicular of side $AB$ meets one of the remaining sides at point $C'$. Points $A'$ and $B'$ are defined similarly. Find all triangles $ABC$ such that triangle $A'B'C'$ is regular.

From a point $P$ outside a circle, two tangents are drawn touching the circle at points $A$ and $C$. Let $B$ be a point on segment $AC$, and let segment $PB$ intersect the circle at point $Q$. The angle bisector of $\angle AQC$ intersects segment $AC$ at $R$. Show that $$\frac{AB}{BC} =\left(\frac{ AR}{RC}\right)^2$$

In the diagram, if points $ A$, $ B$ and $ C$ are points of tangency, then $ x$ equals: [asy]unitsize(5cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3; pair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16); pair O=(0,3/8); draw((-2/3,9/16)--(2/3,9/16)); draw((-2/3,1/2)--(-sqrt(3)/6,1/2)--(0,0)--(sqrt(3)/6,1/2)--(2/3,1/2)); draw(Circle(O,3/16)); draw((-2/3,0)--(2/3,0)); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$\frac{3}{8}$",O); draw(O+.07*dir(60)--O+3/16*dir(60),EndArrow(3)); draw(O+.07*dir(240)--O+3/16*dir(240),EndArrow(3)); label("$\frac{1}{2}$",(.5,.25)); draw((.5,.33)--(.5,.5),EndArrow(3)); draw((.5,.17)--(.5,0),EndArrow(3)); label("$x$",midpoint((.5,.5)--(.5,9/16))); draw((.5,5/8)--(.5,9/16),EndArrow(3)); label("$60^{\circ}$",(0.01,0.12)); dot(A); dot(B); dot(C);[/asy]$ \textbf{(A)}\ \frac {3}{16}" \qquad \textbf{(B)}\ \frac {1}{8}" \qquad \textbf{(C)}\ \frac {1}{32}" \qquad \textbf{(D)}\ \frac {3}{32}" \qquad \textbf{(E)}\ \frac {1}{16}"$

Prove that \[ (x^4+y)(y^4+z)(z^4+x) \geq (x+y^2)(y+z^2)(z+x^2) \] for all positive real numbers $x, y, z$ satisfying $xyz \geq 1$.

If $q_1 ( x )$ and $r_ 1$ are the quotient and remainder, respectively, when the polynomial $x^ 8$ is divided by $x + \tfrac{1}{2}$ , and if $q_ 2 ( x )$ and $r_2$ are the quotient and remainder, respectively, when $q_ 1 ( x )$ is divided by $x + \tfrac{1}{2}$, then $r_2$ equals $\textbf{(A) }\frac{1}{256}\qquad\textbf{(B) }-\frac{1}{16}\qquad\textbf{(C) }1\qquad\textbf{(D) }-16\qquad\textbf{(E) }256$

Let $ABC$ be a triangle with $\angle A=90^{\circ}$ and $AB=AC$. Let $D$ and $E$ be points on the segment $BC$ such that $BD:DE:EC = 1:2:\sqrt{3}$. Prove that $\angle DAE= 45^{\circ}$

A fair die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it may be expressed in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

A contractor has two teams of workers: team $A$ and team $B$. Team $A$ can complete a job in $12$ days and team $B$ can do the same job in $36$ days. Team $A$ starts working on the job and team $B$ joins team $A$ after four days. The team $A$ withdraws after two more days. For how many more days should team $B$ work to complete the job?

Given an $m \times n$ table consisting of $mn$ unit cells. Alice and Bob play the following game: Alice goes first and the one who moves colors one of the empty cells with one of the given three colors. Alice wins if there is a figure, such as the ones below, having three different colors. Otherwise Bob is the winner. Determine the winner for all cases of $m$ and $n$ where $m, n \ge 3$. Proposed by [i]Toghrul Abbasov, Azerbaijan[/i]

We know $\mathbb Z_{210} \cong \mathbb Z_2 \times \mathbb Z_3 \times \mathbb Z_5 \times \mathbb Z_7$. Moreover,\begin{align*} 53 & \equiv 1 \pmod{2} \\ 53 & \equiv 2 \pmod{3} \\ 53 & \equiv 3 \pmod{5} \\ 53 & \equiv 4 \pmod{7}. \end{align*} Let \[ M = \left( \begin{array}{ccc} 53 & 158 & 53 \\ 23 & 93 & 53 \\ 50 & 170 & 53 \end{array} \right). \] Based on the above, find $\overline{(M \mod{2})(M \mod{3})(M \mod{5})(M \mod{7})}$.