Given quadrilateral $ABCD$ inscribed into a circle with diagonal $AC$ as diameter. Let $E$ be a point on segment $BC$ s.t. $\sphericalangle DAC=\sphericalangle EAB$. Point $M$ is midpoint of $CE$. Prove that $BM=DM$.

The diagram shows two equilateral triangles with side length $4$ mounted on two adjacent sides of a square also with side length $4$. The distance between the two vertices marked $A$ and $B$ can be written as $\sqrt{m}+\sqrt{n}$ for two positive integers $m$ and $n$. Find $m + n$. [asy] size(120); defaultpen(linewidth(0.7)+fontsize(11pt)); draw(unitsquare); draw((0,1)--(1/2,1+sqrt(3)/2)--(1,1)--(1+sqrt(3)/2,1/2)--(1,0)); label("$A$",(1/2,1+sqrt(3)/2),N); label("$B$",(1+sqrt(3)/2,1/2),E); [/asy]

Let $a < b < c < d < e$ be real numbers. We calculate all possible sums in pairs of these 5 numbers. Of these 10 sums, the three smaller ones are 32, 36, 37, while the two larger ones are 48 and 51. Determine all possible values ​​that $e$ can take.

Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$. [i]Proposed by Lewis Chen[/i]

An ice cream cone has radius $1$ and height $4$ inches. What is the number of inches in the radius of a sphere of ice cream which has the same volume of the cone? $\text{(A) }\frac{1}{2}\qquad\text{(B) }1\qquad\text{(C) }\frac{3}{2}\qquad\text{(D) }2\qquad\text{(E) }\frac{5}{2}$

For all positive integer $ n\geq 2$, prove that product of all prime numbers less or equal than $ n$ is smaller than $ 4^{n}$.

Determine all pairs $(x, y)$ of natural numbers satisfying the equation $5^x=y^4+4y+1$.

21) A particle of mass $m$ moving at speed $v_0$ collides with a particle of mass $M$ which is originally at rest. The fractional momentum transfer $f$ is the absolute value of the final momentum of $M$ divided by the initial momentum of $m$. If the collision is perfectly $elastic$, what is the maximum possible fractional momentum transfer, $f_{max}$? A) $0 < f_{max} < \frac{1}{2}$ B) $f_{max} = \frac{1}{2}$ C) $\frac{1}{2} < f_{max} < \frac{3}{2}$ D) $f_{max} = 2$ E) $f_{max} \ge 3$

In square $ABCD$ with side length $2$, let $M$ be the midpoint of $AB$. Let $N$ be a point on $AD$ such that $AN = 2ND$. Let point $P$ be the intersection of segment $MN$ and diagonal $AC$. Find the area of triangle $BPM$. [i]Proposed by Jacob Xu[/i]

Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$. [i]Proposed by Netherlands[/i]

Let $i$ be a root of the equation $x^2+1=0$ and let $\omega$ be a root of the equation $x^2+x+1=0$ . Construct a polynomial $$f(x)=a_0+a_1x+\cdots+a_nx^n$$ where $a_0,a_1,\cdots,a_n$ are all integers such that $f(i+\omega)=0$ .

Find all nonnegative integers $x, y, z$ satisfying the equation $$2^x+31^y=z^2.$$

A satellite is considered accessible from the point $A{}$ of the planet's surface if it is located relative to the tangent plane drawn at point $A{}$, strictly on the other side than the planet. What is the smallest number of satellites that need to be launched over a spherical planet so that at some point the signals of at least two satellites are available from each point on the planet's surface? [i]Proposed by S. Volchenkov[/i]

Let $O$ be the center of circle $\Gamma$, and $A$, $B$ be two points on $\Gamma$ so that $O, A$ and $B$ are not collinear. Let $M$ be the midpoint of $AB$. Let $P$ and $Q$ be points on $OA$ and $OB$, respectively, so that $P \neq A$ and $P, M, Q$ are collinear. Let $X$ be the intersection of the line passing through $P$ and parallel to $AB$ and the line passing through $Q$ and parallel to $OM$. Let $Y$ be the intersection of the line passing through $X$ and parallel to $OA$ and the line passing through $B$ and orthogonal to $OX$. Prove that: if $X$ is on $\Gamma$, then $Y$ is also on $\Gamma$. [i] Proposed by usjl[/i]

Let three circles $ \Gamma_1, \Gamma_2, \Gamma_3$, which are non-overlapping and mutually external, be given in the plane. For each point $ P$ in the plane, outside the three circles, construct six points $ A_1, B_1, A_2, B_2, A_3, B_3$ as follows: For each $ i \equal{} 1, 2, 3$, $ A_i, B_i$ are distinct points on the circle $ \Gamma_i$ such that the lines $ PA_i$ and $ PB_i$ are both tangents to $ \Gamma_i$. Call the point $ P$ exceptional if, from the construction, three lines $ A_1B_1, A_2 B_2, A_3 B_3$ are concurrent. Show that every exceptional point of the plane, if exists, lies on the same circle.

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let the internal angle bisector of $\angle BAC$ intersect $BC$ and $\Gamma$ at $E$ and $N$, respectively. Let $A'$ be the antipode of $A$ on $\Gamma$ and let $V$ be the point where $AA'$ intersects $BC$. Given that $EV=6$, $VA'=7$, and $A'N=9$, compute the radius of $\Gamma$. [i]Proposed by James Lin.[/i]

Let $x$ and $y$ be positive real numbers. 1. Prove: if $x^3 - y^3 \ge 4x$, then $x^2 > 2y$. 2. Prove: if $x^5 - y^3 \ge 2x$, then $x^3 \ge 2y$.

Let $O, H$ be the orthocenter and circumcenter of of an acute-angled triangke $ABC$ with $AB<AC$.Let $K$ be the midpoint of $AH$.The line through $K$ perpendicular to $OK$ meet $AB$ and the tangent to the circumcircle at $A$ at $X$ and $Y$ respectively. Prove that $\angle XOY=\angle AOB$

A positive integer $n$ is defined as a $\textit{stepstool number}$ if $n$ has one less positive divisor than $n + 1$. For example, $3$ is a stepstool number, as $3$ has $2$ divisors and $4$ has $2 + 1 = 3$ divisors. Find the sum of all stepstool numbers less than $300$. [i]Proposed by [b]Th3Numb3rThr33[/b][/i]

Akash's birthday cake is in the form of a $4 \times 4 \times 4$ inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into $64$ smaller cubes, each measuring $1 \times 1 \times 1$ inch, as shown below. How many of the small pieces will have icing on exactly two sides? [asy] /* Created by SirCalcsALot and sonone Code modfied from https://artofproblemsolving.com/community/c3114h2152994_the_old__aops_logo_with_asymptote */ import three; currentprojection=orthographic(1.75,7,2); //++++ edit colors, names are self-explainatory ++++ //pen top=rgb(27/255, 135/255, 212/255); //pen right=rgb(254/255,245/255,182/255); //pen left=rgb(153/255,200/255,99/255); pen top = rgb(170/255, 170/255, 170/255); pen left = rgb(81/255, 81/255, 81/255); pen right = rgb(165/255, 165/255, 165/255); pen edges=black; int max_side = 4; //+++++++++++++++++++++++++++++++++++++++ path3 leftface=(1,0,0)--(1,1,0)--(1,1,1)--(1,0,1)--cycle; path3 rightface=(0,1,0)--(1,1,0)--(1,1,1)--(0,1,1)--cycle; path3 topface=(0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle; for(int i=0; i<max_side; ++i){ for(int j=0; j<max_side; ++j){ draw(shift(i,j,-1)*surface(topface),top); draw(shift(i,j,-1)*topface,edges); draw(shift(i,-1,j)*surface(rightface),right); draw(shift(i,-1,j)*rightface,edges); draw(shift(-1,j,i)*surface(leftface),left); draw(shift(-1,j,i)*leftface,edges); } } picture CUBE; draw(CUBE,surface(leftface),left,nolight); draw(CUBE,surface(rightface),right,nolight); draw(CUBE,surface(topface),top,nolight); draw(CUBE,topface,edges); draw(CUBE,leftface,edges); draw(CUBE,rightface,edges); // begin made by SirCalcsALot int[][] heights = {{4,4,4,4},{4,4,4,4},{4,4,4,4},{4,4,4,4}}; for (int i = 0; i < max_side; ++i) { for (int j = 0; j < max_side; ++j) { for (int k = 0; k < min(heights[i][j], max_side); ++k) { add(shift(i,j,k)*CUBE); } } } [/asy] $\textbf{(A)}\ 12\qquad~~\textbf{(B)}\ 16\qquad~~\textbf{(C)}\ 18\qquad~~\textbf{(D)}\ 20\qquad~~\textbf{(E)}\ 24$

Let $\mathbb{Z}$ be the set of integers. Determine, with proof, all primes $p$ for which there exists a function $f:\mathbb{Z}\to\mathbb{Z}$ such that for any integer $x,$ $\quad \bullet \ f(x+p)=f(x)\text{ and}$ $\quad \bullet \ p \text{ divides } f(x+f(x))-x.$

There's a convex $3n$-polygon on the plane with a robot on each of it's vertices. Each robot fires a laser beam toward another robot. On each of your move,you select a robot to rotate counter clockwise until it's laser point a new robot. Three robots $A$, $B$ and $C$ form a triangle if $A$'s laser points at $B$, $B$'s laser points at $C$, and $C$'s laser points at $A$. Find the minimum number of moves that can guarantee $n$ triangles on the plane.

Find the remainder when \[ \sum_{i=2}^{63} \frac{i^{2011}-i}{i^2-1}. \] is divided by 2016. [i]Author: Alex Zhu[/i]

Isosceles $\triangle{ABC}$ has interior point $O$ such that $AO = \sqrt{52}$, $BO = 3$, and $CO = 5$. Given that $\angle{ABC}=120^{\circ}$, find the length $AB$. [i]Proposed by Powell Zhang[/i]

Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. [i]Proposed by Gurgen Asatryan, Armenia[/i]