Let $\triangle{ABC}$ be an equilateral triangle. Let $E_{AB}$ be the ellipse with foci $A, B$ passing through $C,$ and in the parallel manner define $E_{BC}, E_{AC}.$ Let $\triangle{GHI}$ be a (nondegenerate) triangle with vertices where two ellipses intersect such that the edges of $\triangle{GHI}$ do not intersect those of $\triangle{ABC}.$ Compute the ratio of the largest sides of $\triangle{GHI}$ and $\triangle{ABC}.$

If the integers $1,2,\dots,n$ can be divided into two sets such that each of the two sets does not contain the arithmetic mean of its any two elements, what is the largest possible value of $n$? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ \text{None of the preceding} $

In acute triangle $ABC$ bisector of angle $BAC$ intersects side $BC$ in point $D$. Bisector of line segment $AD$ intersects circumcircle of triangle $ABC$ in points $E$ and $F$. Show that circumcircle of triangle $DEF$ is tangent to line $BC$.

Let $ f:\mathbb{D}\longrightarrow\mathbb{C} $ be a continuous function, where $ \mathbb{D} $ is the closed unit disk. Suppose that $ f $ is holomorphic on the open unit disk and that $ e^{i\theta } $ are roots, for any $ \theta\in\left[ 0,\pi /4 \right] . $ Show that $ f=0_{\mathbb{D}} . $

Let $n\geq 2$ be some fixed positive integer and suppose that $a_1, a_2,\dots,a_n$ are positive real numbers satisfying $a_1+a_2+\cdots+a_n=2^n-1$. Find the minimum possible value of $$\frac{a_1}{1}+\frac{a_2}{1+a_1}+\frac{a_3}{1+a_1+a_2}+\cdots+\frac{a_n}{1+a_1+a_2+\cdots+a_{n-1}}$$

There is a $50\times 50$ grid board.. Carlos is going to write a number in each box with the following procedure. He first chooses $100$ distinct numbers that we denote $f_1,f_2,f_3,…,f_{50},c_1,c_2,c_3,…,c_{50}$ among which there are exactly $50$ that they are rational. Then he writes in each box ($i,j)$ the number $f_i \cdot c_j$ (the multiplication of $f_i$ by $c_j$). Determine the maximum number of rational numbers that the squares on the board can contain.

Prove that for every natural number $k$ ($k \geq 2$) there exists an irrational number $r$ such that for every natural number $m$, \[[r^m] \equiv -1 \pmod k .\] [i]Remark.[/i] An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients. [i]Proposed by Yugoslavia.[/i]

Consider $n$ segments in the plane which no two intersect and between their $2n$ endpoints no three are collinear. Is the following statement true? Statement: There exists a simple $2n$-gon such that it's vertices are the $2n$ endpoints of the segments and each segment is either completely inside the polygon or an edge of the polygon.

Let $p$ be the least prime number for which there exists a positive integer $n$ such that $n^{4}+1$ is divisible by $p^{2}$. Find the least positive integer $m$ such that $m^{4}+1$ is divisible by $p^{2}$.

Rectangle $ABCD$ has side lengths $AB = 6\sqrt3$ and $BC = 8\sqrt3$. The probability that a randomly chosen point inside the rectangle is closer to the diagonal $\overline{AC}$ than to the outside of the rectangle is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Alice’s age in years is twice Eve’s age in years. In $10$ years, Eve will be as old as Alice is now. Compute Alice’s age in years now.

Find all integers$ n$ such that $n^2 + 24n + 35$ is a square.

A continuous function $f : [a,b] \to [a,b]$ is called piecewise monotone if $[a, b]$ can be subdivided into finitely many subintervals $$I_1 = [c_0,c_1], I_2 = [c_1,c_2], \dots , I_\ell = [ c_{\ell - 1},c_\ell ]$$ such that $f$ restricted to each interval $I_j$ is strictly monotone, either increasing or decreasing. Here we are assuming that $a = c_0 < c_1 < \cdots < c_{\ell - 1} < c_\ell = b$. We are also assuming that each $I_j$ is a maximal interval on which $f$ is strictly monotone. Such a maximal interval is called a lap of the function $f$, and the number $\ell = \ell (f)$ of distinct laps is called the lap number of $f$. If $f : [a,b] \to [a,b]$ is a continuous piecewise-monotone function, show that the sequence $( \sqrt[n]{\ell (f^n )})$ converges; here $f^n$ means $f$ composed with itself $n$-times, so $f^2 (x) = f(f(x))$ etc.

An integer sequence is defined by \[{ a_n = 2 a_{n-1} + a_{n-2}}, \quad (n > 1), \quad a_0 = 0, a_1 = 1.\] Prove that $2^k$ divides $a_n$ if and only if $2^k$ divides $n$.

The circles $ S_1 $ and $ S_2 $ intersect at points $ A $ and $ B $. Circle $ S_3 $ externally touches $ S_1 $ and $ S_2 $ at points $ C $ and $ D $ respectively. Let $ K $ be the midpoint of the chord cut by the line $ AB $ on circles $ S_3 $. Prove that $ \angle CKA = \angle DKA $.

In a game two players alternately choose larger natural numbers. At each turn the difference between the new and the old number must be greater than zero but smaller than the old number. The original number is 2. The winner is considered to be the player who chooses the number $1987$. In a perfect game, which player wins?