Let $O$ be the circumcenter of triangle $ABC$. Suppose the perpendicular bisectors of $\overline{OB}$ and $\overline{OC}$ intersect lines $AB$ and $AC$ at $D\neq A$ and $E\neq A$, respectively. Determine the maximum possible number of distinct intersection points between line $BC$ and the circumcircle of $\triangle ADE$. [i]Andrew Wen[/i]

Find all polygons $P$ that can be covered completely by three (possibly overlapping) smaller dilated versions of itself. [i]Proposed by Evan Chang (squareman), USA[/i]

Given a regular 103-sided polygon. 79 vertices are colored red and the remaining vertices are colored blue. Let $A$ be the number of pairs of adjacent red vertices and $B$ be the number of pairs of adjacent blue vertices. a) Find all possible values of pair $(A,B).$ b) Determine the number of pairwise non-similar colorings of the polygon satisfying $B=14.$ 2 colorings are called similar if they can be obtained from each other by rotating the circumcircle of the polygon.

A convex quadrilateral $ABCD$ inscribed in a circle $k$ has the property that the rays $DA$ and $CB$ meet at a point $E$ for which $CD^2=AD\cdot
ED.$ The perpendicular to $ED$ at $A$ intersects $k$ again at point $F.$ Prove that the segments $AD$ and $CF$ are congruent if and only if the circumcenter of $\triangle ABE$ lies on $ED.$

Is there a convex polygon with each side equal to some diagonal, and each diagonal equal to some side?

Find all function $f:\mathbb R^+ \rightarrow \mathbb R^+$ such that: \[f\left(\frac{f(x)}{x}+y\right)=1+f(y), \quad \forall x,y \in \mathbb R^+.\]

Let $a_{0,1}, a_{0,2}, . . . , a_{0, 2016}$ be positive real numbers. For $n\geq 0$ and $1 \leq k < 2016$ set $$a_{n+1,k} = a_{n,k} +\frac{1}{2a_{n,k+1}} \ \ \text{and} \ \ a_{n+1,2016} = a_{n,2016} +\frac{1}{2a_{n,1}}.$$ Show that $\max_{1\leq k \leq 2016} a_{2016,k} > 44.$

Let $S$ be a set of size $n$, and $k$ be a positive integer. For each $1 \le i \le kn$, there is a subset $S_i \subset S$ such that $|S_i| = 2$. Furthermore, for each $e \in S$, there are exactly $2k$ values of $i$ such that $e \in S_i$. Show that it is possible to choose one element from $S_i$ for each $1 \le i \le kn$ such that every element of $S$ is chosen exactly $k$ times.

Given a triangle $ABC$, let $D$ be the intersection of the line through $A$ perpendicular to $AB$, and the line through $B$ perpendicular to $BC$. Let $P$ be a point inside the triangle. Show that $DAPB$ is cyclic if and only if $\angle BAP = \angle CBP$.

The points $M$ and $N$ are chosen on the angle bisector $AL$ of a triangle $ABC$ such that $\angle ABM=\angle ACN=23^{\circ}$. $X$ is a point inside the triangle such that $BX=CX$ and $\angle BXC=2\angle BML$. Find $\angle MXN$.

Let \(a\), \(b\), and \(n\) be positive integers. A lemonade stand owns \(n\) cups, all of which are initially empty. The lemonade stand has a [i]filling machine[/i] and an [i]emptying machine[/i], which operate according to the following rules: [list] [*]If at any moment, \(a\) completely empty cups are available, the filling machine spends the next \(a\) minutes filling those \(a\) cups simultaneously and doing nothing else. [*]If at any moment, \(b\) completely full cups are available, the emptying machine spends the next \(b\) minutes emptying those \(b\) cups simultaneously and doing nothing else. [/list] Suppose that after a sufficiently long time has passed, both the filling machine and emptying machine work without pausing. Find, in terms of \(a\) and \(b\), the least possible value of \(n\). [i]Proposed by Raymond Feng[/i]

High school graduate Igor Petrov, who dreamed of becoming a diplomat, took the entrance exam in mathematics to Moscow University. Igor remembered all the problems offered during the exam, but forgot some numerical data in one. This is the task: “When multiplying two natural numbers, the difference of which is $10$, an error was made: the hundreds digit in the product was increased by $2$. When dividing the resulting (incorrect) product by the smaller of the factors, the result was quotient $k$ and remainder $r$.. Find the numbers that needed to be multiplied.” . The values of $k$ and $r$ were given in the condition, but Igor forgot them. However, he remembered that the problem had two answers. What could the numbers $ k$ and $r$ be equal to (they are both integers and positive)? [i]Note. The problem in question was proposed at one of the humanities faculties of Moscow State University in 1991. [/i]

a) Prove that if the sum of the non-zero digits $a_1, a_2, ... , a_n$ is a multiple of $27$, then it is possible to permute these digits in order to obtain an $n$-digit number that is a multiple of $27$. b) Prove that if the non-zero digits $a_1, a_2, ... , a_n$ have the property that every ndigit number obtained by permuting these digits is a multiple of $27$, then the sum of these digits is a multiple of $27$

Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that\[f\left(\Big \lceil \frac{f(m)}{n} \Big \rceil\right)=\Big \lceil \frac{m}{f(n)} \Big \rceil\]for all $m,n \in \mathbb{N}$. [i]Proposed by Md. Ashraful Islam Fahim[/i]

Some of the people attending a meeting greet each other. Let $n$ be the number of people who greet an odd number of people. Prove that $n$ is even.

Let $x$ and $y$ be two-digit positive integers with mean 60. What is the maximum value of the ratio $\frac{x}{y}$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10} $

If $T$ and $T_1$ are two triangles with angles $x, y, z$ and $x_1, y_1, z_1$, respectively, prove the inequality \[\frac{\cos x_1}{\sin x}+\frac{\cos y_1}{\sin y}+\frac{\cos z_1}{\sin z} \leq \cot x+\cot y+\cot z.\]

Find all positive numbers $x$ such that:$$\frac{1}{[x]}-\frac{1}{[2x]}=\frac{1}{6\{x\}}$$ where $[x]$ represents the integer part of $x$ and $\{x\}=x-[x]$.

Let $ABC$ be an acute, non-isosceles triangle with altitude $AD$ ($D \in BC$), $M$ is the midpoint of $AD$ and $O$ is the circumcenter. Line $AO$ meets $BC$ at $K$ and circle of center $K$, radius $KA$ cuts $AB,AC$ at $E, F$ respectively. Prove that $AO$ bisects $EF$.

Let $ABCD$ be a parallelogram of center $O$. Points $M$ and $N$ are the midpoints of $BO$ and $CD$, respectively. Prove that if the triangles $ABC$ and $AMN$ are similar, then $ABCD$ is a square.

Prove that with $ n\ge 1 $ distinct numbers we can form an arithmetic progression if and only if there are exactly $ n-1 $ distinct elements in the set of positive differences between any two of these numbers.

Let $n$ be a positive integer divisible by $39$. What is the smallest possible sum of digits that $n$ can have (in base $10$)?

Sterling draws 6 circles on the plane, which divide the plane into regions (including the unbounded region). What is the maximum number of resulting regions?

Prove that the following inequality holds: \[ \left( \log_{24}48 \right)^2+ \left( \log_{12}54 \right)^2>4\]

The sets $A_1,A_2,...,A_n$ are finite. With $d$ we denote the number of elements in $\bigcup_{i=1}^n A_i$ which are in odd number of the sets $A_i$. Prove that the number: $D(k)=d-\sum_{i=1}^n|A_i|+2\sum_{i<j}|A_i\cap A_j |+...+(-1)^k2^{k-1}\sum_{i_1<i_2<...<i_k}|A_{i_1}\cap A_{i_2}\cap ...\cap A_{i_k}|$ is divisible by $2^k$.