Find all positive integers which have exactly 16 positive divisors $1 = d_1 < d_2 < \ldots < d_{16} =n$ such that the divisor $d_k$, where $k = d_5$, equals $(d_2 + d_4) d_6$.

Let the equilateral triangles $ABC$ and $DEF$ be congruent with the centers $O_1$, respectively $O_2$, so that segment $AB$ intersects segments $DE$ and $DF$ at $M, N$, and the segment $AC$ intersects the segments $DF$ and $EF$ at $P$ and $Q$, respectively. We denote by $I$ the intersection point of the bisectors of the angles $EMN$ and $DPQ$ and by $J$ the intersection of the bisectors of the angles $FNM$ and $EQP$. Prove that $IJ$ is the perpendicular bisector of the segment $O_1O_2$.

A particular country has seven distinct cities, conveniently named $C_1,C_2,\dots,C_7.$ Between each pair of cities, a direction is chosen, and a one-way road is constructed in that direction connecting the two cities. After the construction is complete, it is found that any city is reachable from any other city, that is, for distinct $1 \leq i, j \leq 7,$ there is a path of one-way roads leading from $C_i$ to $C_j.$ Compute the number of ways the roads could have been configured. Pictured on the following page are the possible configurations possible in a country with three cities, if every city is reachable from every other city. [Insert Diagram] [i]Proposed by Ezra Erives[/i]

Elmo and Elmo's clone are playing a game. Initially, $n\geq 3$ points are given on a circle. On a player's turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo's clone goes first and players alternate turns, who wins? (Your answer may be in terms of $n$.) [i]Proposed by Milan Haiman[/i]

A triangle $ABC$ is given in the plane with $AB = \sqrt{7},$ $BC = \sqrt{13}$ and $CA = \sqrt{19},$ circles are drawn with centers at $A,B$ and $C$ and radii $\frac{1}{3},$ $\frac{2}{3}$ and $1,$ respectively. Prove that there are points $A',B',C'$ on these three circles respectively such that triangle $ABC$ is congruent to triangle $A'B'C'.$

A square of side length $ 1$ and a circle of radius $ \sqrt3/3$ share the same center. What is the area inside the circle, but outside the square? $ \textbf{(A)}\ \frac{\pi}3 \minus{} 1 \qquad\textbf{(B)}\ \frac{2\pi}{9} \minus{} \frac{\sqrt3}3 \qquad\textbf{(C)}\ \frac{\pi}{18} \qquad\textbf{(D)}\ \frac14 \qquad\textbf{(E)}\ 2\pi/9$

Let $ABC$ be a scalene triangle and let $I$ be its incenter. The projections of $I$ on $BC, CA$, and $AB$ are $D, E$ and $F$ respectively. Let $K$ be the reflection of $D$ over the line $AI$, and let $L$ be the second point of intersection of the circumcircles of the triangles $BFK$ and $CEK$. If $\frac{1}{3} BC = AC - AB$, prove that $DE = 2KL$.

What is the minimum number of movements that a horse must carry out on chess, on an $8\times 8$ board, to reach the upper right square starting at the lower left? Remember that the horse moves in the usual $L$-shaped manner.

In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$, $\beta$, and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$. If $\cos \alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?

Is it true that if a subgroup $G \leq \text{Sym}(\mathbb{N})$ is $n$-transitive for every positive integer $n$, then every group automorphism of $G$ extends to a group automorphism of $\text{Sym}(\mathbb{N})$?

Alice and Bob play a game. Alice writes an equation of the form $ax^2 + bx + c =0$, choosing $a$, $b$, $c$ to be real numbers (possibly zero). Bob can choose to add (or subtract) any real number to each of $a$, $b$, $c$, resulting in a new equation. Bob wins if the resulting equation is quadratic and has two distinct real roots; Alice wins otherwise. For which choices of $a$, $b$, $c$ does Alice win, no matter what Bob does?

A polyomino is region with connected interior that is a union of a finite number of squares from a grid of unit squares. Do there exist a positive integer $n>4$ and a polyomino $P$ contained entirely within and $n$-by-$n$ grid such that $P$ contains exactly $3$ unit squares in every row and every column of the grid? Proposed by [i]Nikolai Beluhov[/i]

Let $p=4k+3$ be a prime number. Find the number of different residues mod p of $(x^{2}+y^{2})^{2}$ where $(x,p)=(y,p)=1.$

Tanya and Serezha have a heap of $2016$ candies. They make moves in turn, Tanya moves first. At each move a player can eat either one candy or (if the number of candies is even at the moment) exactly half of all candies. The player that cannot move loses. Which of the players has a winning strategy?

Find all infinite sequences $a_1, a_2, \ldots$ of positive integers satisfying the following properties: (a) $a_1 < a_2 < a_3 < \cdots$, (b) there are no positive integers $i$, $j$, $k$, not necessarily distinct, such that $a_i+a_j=a_k$, (c) there are infinitely many $k$ such that $a_k = 2k-1$.

Find all functions \( f:\mathbb{Z} \to \mathbb{Z} \) such that \[ f(x + f(y) - 2y) + f(f(y)) = f(x) \] for all integers \( x \) and \( y \).

Let $ ABCDE$ be a convex pentagon such that $ AB\plus{}CD\equal{}BC\plus{}DE$ and $ k$ a circle with center on side $ AE$ that touches the sides $ AB$, $ BC$, $ CD$ and $ DE$ at points $ P$, $ Q$, $ R$ and $ S$ (different from vertices of the pentagon) respectively. Prove that lines $ PS$ and $ AE$ are parallel.

Define a sequence $\{a_n\}$ by\[S_1=1,\ S_{n+1}=\frac{(2+S_n)^2}{ 4+S_n} (n=1,\ 2,\ 3,\ \cdots).\] Where $S_n$ the sum of first $n$ terms of sequence $\{a_n\}$. For any positive integer $n$ ,prove that\[a_{n}\ge \frac{4}{\sqrt{9n+7}}.\]

Jorge chooses $6$ different positive integers and writes one on each face of a cube. He threw his bucket three times. The first time his cube showed the number $5$ facing up and also the sum of the numbers on the faces sides was $20$. The second time his cube showed the number $7$ facing up and also the sum of the numbers on the faces sides was $17$. The third time his cube showed the number $4$ up, plus all the numbers on the side faces. They turned out to be primes. What are the numbers that Jorge chose and how did he distribute them on the faces of the cube? Analyze all odds. Remember that $1$ is not prime.

Let $\vartriangle ABC$ be a triangle with $AB = 4$ and $AC = \frac72$ . Let $\omega$ denote the $A$-excircle of $\vartriangle ABC$. Let $\omega$ touch lines $AB$, $AC$ at the points $D$, $E$, respectively. Let $\Omega$ denote the circumcircle of $\vartriangle ADE$. Consider the line $\ell$ parallel to $BC$ such that $\ell$ is tangent to $\omega$ at a point $F$ and such that $\ell$ does not intersect $\Omega$. Let $\ell$ intersect lines $AB$, $AC$ at the points $X$, $Y$ , respectively, with $XY = 18$ and $AX = 16$. Let the perpendicular bisector of $XY$ meet the circumcircle of $\vartriangle AXY$ at $P$, $Q$, where the distance from $P$ to $F$ is smaller than the distance from $Q$ to$ F$. Let ray $\overrightarrow {PF}$ meet $\Omega$ for the first time at the point $Z$. If $PZ^2 = \frac{m}{n}$ for relatively prime positive integers $m$, $n$, find $m + n$.

In the triangle $ABC$, $\angle{B} = 120^{\circ}$, point $M$ is the midpoint of side $AC$. On the sides $AB$ and $BC$, the points $K$ and $L$ are chosen such that $KL \parallel AC$. Prove that $MK + ML \geq MA$.

Find all pairs of nonnegative integers $ (x, y)$ satisfying $ (14y)^x \plus{} y^{x\plus{}y} \equal{} 2007$.

Andy and Harry are trying to make an O for the MOAA logo. Andy starts with a circular piece of leather with radius 3 feet and cuts out a circle with radius 2 feet from the middle. Harry starts with a square piece of leather with side length 3 feet and cuts out a square with side length 2 feet from the middle. In square feet, what is the positive difference in area between Andy and Harry's final product to the nearest integer? [i]Proposed by Andy Xu[/i]

$p$ is a prime. Find the all $(m,n,p)$ positive integer triples satisfy $m^3+7p^2=2^n$.

Each diagonal of a quadrangle divides it into two isosceles triangles. Is it true that the quadrangle is a diamond?