In the triangle $ ABC$ let $ D$ and $ E$ be the boundary points of the incircle with the sides $ BC$ and $ AC$. Show that if $ AD\equal{}BE$ holds, then the triangle is isoceles.

Kevin the Koala eats $1$ leaf on the first day of its life, $3$ leaves on the second, $5$ on the third, and in general eats $2n-1$ leaves on the $n$th day. What is the smallest positive integer $n>1$ such that the total number of leaves Kevin has eaten his entire $n$-day life is a perfect sixth power? [i]2015 CCA Math Bonanza Individual Round #4[/i]

Triangle $\triangle ABC$ has circumcenter $O$ and incircle $\gamma$. Suppose that $\angle BAC =60^\circ$ and $O$ lies on $\gamma$. If \[ \tan B \tan C = a + \sqrt{b} \] for positive integers $a$ and $b$, compute $100a+b$. [i]Proposed by Kaan Dokmeci[/i]

Find all functions $f:R \to R$, such that $f(x)+f(y)=f(x+y)$, and there exists non-constant polynomials $P(x)$, $Q(x)$ such that $P(x)f(Q(x))=f(P(x)Q(x))$

Let $f(x)$ be a polynomial with real coefficients of degree 2. Suppose that for some pairwise distinct real numbers , $a,b,c$ we have:\\ \[f(a)=bc , f(b)=ac, f(c)=ab\] Dertermine $f(a+b+c)$ in terms of $a,b,c$.

Consider a regular $n$-gon with $n$ odd. Given two adjacent vertices $A_{1}$ and $A_{2},$ define the sequence $(A_{k})$ of vertices of the $n$-gon as follows: For $k\ge 3,\, A_{k}$ is the vertex lying on the perpendicular bisector of $A_{k-2}A_{k-1}.$ Find all $n$ for which each vertex of the $n$-gon occurs in this sequence.

Let $a, b, c$ be positive real numbers such that $a\le b \le c \le 2a$. Find the maximum possible value of $$\frac{b}{a} +\frac{c}{b} +\frac{a}{c}.$$

Five distinct numbers are drawn successively and at random from the set $\{1, \cdots , n\}$. Show that the probability of a draw in which the first three numbers as well as all five numbers can be arranged to form an arithmetic progression is greater than $\frac{6}{(n-2)^3}$

Prove that it is possible to find $2^{n+1}$ of $2^n$ digit numbers containing only "$1$" and "$2$" as digits, such that every two of them distinguish at least in $2^{n-1}$ digits.

Let $p$ be a real constant. The parabola $y^2=-4px$ rolls without slipping around the parabola $y^2=4px$. Find the equation of the locus of the vertex of the rolling parabola.

Do there exist positive integers $a$ and $b$ such that $a^n+n^b$ and $b^n+n^a$ are relatively prime for all natural $n$?

Richard starts with the string $HHMMMMTT$. A move consists of replacing an instance of $HM$ with $MH$, replacing an instance of $MT$ with $TM$, or replacing an instance of $TH$ with $HT$. Compute the number of possible strings he can end up with after performing zero or more moves.

Let $x_1, x_2, x_3, x_4$ be the distances from an arbitrary point inside the tetrahedron to the planes of its faces, and let $h_1, h_2, h_3, h_4$ be the corresponding heights of the tetrahedron. Prove that $$\sqrt{h_1+h_2+h_3+h_4} \ge \sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}$$ (Dmitry Nomirovsky)

Let $ X$ be a set of $ k$ vertexes on a plane such that no three of them are collinear. Let $ P$ be the family of all $ {k \choose 2}$ segments that connect each pair of points. Determine $ \tau(P)$.

Let $ ABC$ be an equilateral triangle with side length equal to $ N \in \mathbb{N}.$ Consider the set $ S$ of all points $ M$ inside the triangle $ ABC$ satisfying \[ \overrightarrow{AM} \equal{} \frac{1}{N} \cdot \left(n \cdot \overrightarrow{AB} \plus{} m \cdot \overrightarrow{AC} \right)\] with $ m, n$ integers, $ 0 \leq n \leq N,$ $ 0 \leq m \leq N$ and $ n \plus{} m \leq N.$ Every point of S is colored in one of the three colors blue, white, red such that [b](i) [/b]no point of $ S \cap [AB]$ is coloured blue [b](ii)[/b] no point of $ S \cap [AC]$ is coloured white [b](iii)[/b] no point of $ S \cap [BC]$ is coloured red Prove that there exists an equilateral triangle the following properties: [b](1)[/b] the three vertices of the triangle are points of $ S$ and coloured blue, white and red, respectively. [b](2)[/b] the length of the sides of the triangle is equal to 1. [i]Variant:[/i] Same problem but with a regular tetrahedron and four different colors used.

Let $a,b,c$ are non-negative numbers such that $$2(a^2+b^2+c^2)+3(ab+bc+ca)=5(a+b+c)$$ then prove that $4(a^2+b^2+c^2)+2(ab+bc+ca)+7abc\le 25$

16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems.

In the game of Tristack Solitaire, you start with three stacks of cards, each with a different positive integer number of cards. At any time, you can double the number of cards in any one stack of cards by moving cards from exactly one other, larger, stack of cards to the stack you double. You win the game when any two of the three stacks have the same number of cards. For example, if you start with stacks of $3$, $5$, and $7$ cards, then you have three possible legal moves: [list] [*]You may move $3$ cards from the $5$-card stack to the $3$-card stack, leaving stacks of $6$, $2$, and $7$ cards. [*]You may move $3$ cards from the $7$-card stack to the $3$-card stack, leaving stacks of $6,$ $5$, and $4$ cards. [*]You may move $5$ cards from the $7$-card stack to the $5$-card stack, leaving stacks of $3$, $10$, and $2$ cards.[/list] Can you win Tristack Solitaire from any starting position? If so, then give a strategy for winning. If not, then explain why.

Find the remainder when $26!^{26} + 27!^{27}$ is divided by $29$.

Given a strictly increasing infinite sequence $\{a_n\}$ of positive real numbers such that for any $n\in N$: $$a_{n+2}=(a_{n+1}-a_{n})^{\sqrt{n}}+n^{-\sqrt{n}}$$ Prove that for any $C>0$ there exist a positive integer $m(C)$ (depended on $C$) such that $a_{m(C)}>C$.

The $64$ cells of an $8 \times 8$ chessboard have $64$ different colours. A Knight stays in one cell. In each move, the Knight jumps from one cell to another cell (the $2$ cells on the diagonal of an $2 \times 3$ board) also the colours of the $2$ cells interchange. In the end, the Knight goes to a cell having common side with the cell it stays at first. Can it happen that: there are exactly $3$ cells having the colours different from the original colours?

Suppose $f$ and $g$ are differentiable functions such that \[xg(f(x))f^\prime(g(x))g^\prime(x)=f(g(x))g^\prime(f(x))f^\prime(x)\] for all real $x$. Moreover, $f$ is nonnegative and $g$ is positive. Furthermore, \[\int_0^a f(g(x))dx=1-\dfrac{e^{-2a}}{2}\] for all reals $a$. Given that $g(f(0))=1$, compute the value of $g(f(4))$.

Is there an infinite sequence $ a_0, a_1, a_2, \cdots $ of nonzero real numbers such that for $ n = 1, 2, 3, \cdots $ the polynomial \[ p_n(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \] has exactly $n$ distinct real roots?

Prove that $n^4 - n^2$ is divisible by $12$ for all integers $n > 1$.

Suppose that $2^{2n+1}+ 2^{n}+1=x^{k}$, where $k\geq2$ and $n$ are positive integers. Find all possible values of $n$.