$ABCD$ is a trapezoid with $BC\parallel AD$ and $BC = 2AD$. Point $M$ is chosen on the side $CD$ such that $AB = AM$. Prove that $BM \perp CD$. [i]Proposed by Bogdan Rublov[/i]

A hemisphere is inscribed in a cone so that its base lies on the base of the cone. The ratio of the area of the entire surface of the cone to the area of the hemisphere (without the base) is $\frac{18}5$. Compute the angle at the vertex of the cone.

$\sqrt{3+2\sqrt{2}}-\sqrt{3-2\sqrt{2}}$ is equal to $\textbf{(A) }2\qquad\textbf{(B) }2\sqrt{3}\qquad\textbf{(C) }4\sqrt{2}\qquad\textbf{(D) }\sqrt{6}\qquad \textbf{(E) }2\sqrt{2}$

Let \( \varphi \) be the Euler's totient function. 1. For any given integer \( a > 1 \), does there exist \( l \in \mathbb{N}_+ \) such that for any \( k \in \mathbb{N}_+ \), \( l \mid k \) and \( a^2 \nmid l \), \( \frac{\varphi(k)}{\varphi(l)} \) is a non-negative power of \( a \)? 2. For integer \( x > a \), are there integers \( k_1 \) and \( k_2 \) satisfying: \[ \varphi(k_i) \in \left ( \frac{x}{a} ,x \right ], i = 1,2; \quad \varphi(k_1) \neq \varphi(k_2). \] And these two different \( k_i \) correspond to the same \( l_1 \) and \( l_2 \) as described in (1), yet \( \varphi(l_1) = \varphi(l_2) \). 3. Define \( \#E \) as the number of elements in set \( E \). For integer \( x > a \), let \( V(x) = \#\{v \in \mathbb{N}_+ \mid v = \varphi(k) \leq x\} \) and \( W(x) = \#\{w \in \mathbb{N}_+ \mid w = \varphi(l) \leq x, a^2 \mid l\} \). Compare \( V\left( \frac{x}{a} \right) \) with \( W(x) \).

The Cincinnati Reals are playing the Houston Alphas in the last game of the Swirled Series. The Alphas are leading by $1$ run in the bottom of the $9\text{th}$ (last) inning, and the Reals are at bat. Each batter has a $\dfrac{1}{3}$ chance of hitting a single and a $\dfrac{2}{3}$ chance of making an out. If the Reals hit $5$ or more singles before they make $3$ outs, they will win. If the Reals hit exactly $4$ singles before they make $3$ outs, they will tie the game and send it into extra innings, and they will have a $\dfrac{3}{5}$ chance of eventually winning the game (since they have the added momentum of coming from behind). If the Reals hit fewer than $4$ singles, they will LOSE! What is the probability that the Alphas hold off the Reals and win, sending the packed Alphadome into a frenzy? Express the answer as a fraction.

The quadratic polynomial $f(x)$ has the expansion $2x^2 - 3x + r$. What is the largest real value of $r$ for which the ranges of the functions $f(x)$ and $f(f(x))$ are the same set?

Three distinct points with integer coordinates lie in the plane on a circle of radius $r>0$. Show that two of these points are separated by a distance of at least $r^{1/3}$.

When $1999{}^2{}^0{}^0{}^0$ is divided by $5$ , the remainder is $ \text{(A)}\ 4\qquad\text{(B)}\ 1\qquad\text{(C)}\ 2\qquad\text{(D)}\ 3\qquad\text{(E)}\ 0 $

Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let $p$ be the probability that the object reaches $(2,2)$ in six or fewer steps. Given that $p$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

A convex quadrilateral $ABCD$ satisfies $AB\cdot CD = BC\cdot DA$. Point $X$ lies inside $ABCD$ so that \[\angle{XAB} = \angle{XCD}\quad\,\,\text{and}\quad\,\,\angle{XBC} = \angle{XDA}.\] Prove that $\angle{BXA} + \angle{DXC} = 180^\circ$. [i]Proposed by Tomasz Ciesla, Poland[/i]

Which of the following triangles cannot exist? $\textbf{(A)}\ \text{An acute isosceles triangle}$ $\textbf{(B)}\ \text{An isosceles right triangle}$ $\textbf{(C)}\ \text{An obtuse right triangle}$ $\textbf{(D)}\ \text{A scalene right triangle}$ $\textbf{(E)}\ \text{A scalene obtuse triangle}$

According to the figure, three equilateral triangles with side lengths $a,b,c$ have one common vertex and do not have any other common point. The lengths $x, y$, and $z$ are defined as in the figure. Prove that $3(x+y+z)>2(a+b+c)$. [i]Proposed by Mahdi Etesamifard[/i]

Show that a set $A$ consisting of $16$ consecutive non-negative integers can be partitioned in two disjoint sets $X$ and $Y$ each containing $8$ elements so that \(\sum\limits_{x\in X}x^k=\sum\limits_{y\in Y} y^k,\) for $k=1,2,3.$

Let $a$, $b$, $c$, $d$ be complex numbers satisfying \begin{align*} 5 &= a+b+c+d \\ 125 &= (5-a)^4 + (5-b)^4 + (5-c)^4 + (5-d)^4 \\ 1205 &= (a+b)^4 + (b+c)^4 + (c+d)^4 + (d+a)^4 + (a+c)^4 + (b+d)^4 \\ 25 &= a^4+b^4+c^4+d^4 \end{align*} Compute $abcd$. [i]Proposed by Evan Chen[/i]

Find all quadruples $(a,b,c,d)$ of positive real numbers such that $abcd=1,a^{2012}+2012b=2012c+d^{2012}$ and $2012a+b^{2012}=c^{2012}+2012d$.

The non-isosceles triangle $ABC$ is inscribed in the circle ω. The tangent to this circle at the point $C$ intersects the line $AB$ at the point $D$. Let the bisector of the angle $CDB$ intersect the segments $AC$ and $BC$ at the points $K$ and $L$, respectively. On the side $AB$, the point $M$ is taken such that $AK / BL = AM / BM$. Let the perpendiculars from the point $M$ to the lines $KL$ and $DC$ intersect the lines $AC$ and $DC$ at the points $P$ and $Q$, respectively. Prove that the angle $CQP$ is half of the angle $ACB$.

Given a concyclic quadrilateral $ABCD$ with circumcenter $O$. Let $E$ be the intersection of $AD$ and $BC$, while $F$ be the intersection of $AC$ and $BD$. A circle $w$ are tangent to $BD$ and $AC$ such that $F$ is the orthocenter of $\triangle QEP$ where $PQ$ is a diameter of $w$. Prove that $EO$ passes through the center of $w$.

Prove that if a set $X\subset S^n$ takes up more than half a Riemannian volume of a unit sphere $S^n$, then the set of all possible geodesic segments length less than $\pi$ with endpoints in the set $X$ covers the entire sphere. Geodetic on sphere $S^n$ is a curve lying on some circle of intersection of the sphere $S^n\subset R^{n+1}$ two-dimensional linear subspace $L \subset R^{n+1}$

The plane $\alpha$ intersects the edges $AB$, $BC$, $CD$ and $DA$ of the tetrahedron $ABCD$ at points $X, Y, Z$ and $T$, respectively. It turned out, that points $Y$ and $T$ lie on a circle $\omega$ constructed with segment $XZ$ as the diameter. Point $P$ is marked in the plane $\alpha$ so that the lines $P Y$ and $P T$ are tangent to the circle $\omega$.Prove that the midpoints of the edges are $AB$, $BC$, $CD,$ $DA$ and the point $P$ lie in the same plane.

Let $ABC$ be a triangle with $AB = 4024$, $AC = 4024$, and $BC=2012$. The reflection of line $AC$ over line $AB$ meets the circumcircle of $\triangle{ABC}$ at a point $D\ne A$. Find the length of segment $CD$. [i]Ray Li.[/i]

Prove that for any positive integer $n$, the number of odd integers among the binomial coefficients $\binom nh \ ( 0 \leq h \leq n)$ is a power of 2.

Find the maximum of \[P=\frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3}+\frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3}+\frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3}\] where $x,y,z$ are positive real numbers.

A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, $A_0,$ and the hunter's starting point, $B_0$ are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}.$ In the $n^{\text{th}}$ round of the game, three things occur in order: [list=i] [*]The rabbit moves invisibly to a point $A_n$ such that the distance between $A_{n-1}$ and $A_n$ is exactly $1.$ [*]A tracking device reports a point $P_n$ to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between $P_n$ and $A_n$ is at most $1.$ [*]The hunter moves visibly to a point $B_n$ such that the distance between $B_{n-1}$ and $B_n$ is exactly $1.$ [/list] Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after $10^9$ rounds, she can ensure that the distance between her and the rabbit is at most $100?$ [i]Proposed by Gerhard Woeginger, Austria[/i]

Find the polynomial $f$ with the following properties: $\bullet$ its leading coefficient is $1$, $\bullet$ its coefficients are nonnegative integers, $\bullet$ $72|f(x)$ if $x$ is an integer, $\bullet$ if $g$ is another polynomial with the same properties, then $g - f$ has a nonnegative leading coecient.

In a convex quadrilateral $ ABCD $ with area $ S $, each side was divided into 3 equal parts and segments were drawn connecting the appropriate points of division of the opposite sides in such a way that the quadrilateral was divided into 9 quadrilaterals. Prove that the sum of the areas of the following three quadrilaterals resulting from the division: the one containing the vertex $ A $, the middle one and the one containing the vertex $ C $ is equal to $ \frac{S}{3} $.