A circle rolls along a side of an equilateral triangle. The radius of the circle is equal to the height of the triangle. Prove that the measure of the arc intercepted by the sides of the triangle on this circle is equal to $60^o$ at all times.

Let $O$ and $H$ be the circumcenter and orthocenter, respectively, of an acute scalene triangle $ABC$. The perpendicular bisector of $\overline{AH}$ intersects $\overline{AB}$ and $\overline{AC}$ at $X_A$ and $Y_A$ respectively. Let $K_A$ denote the intersection of the circumcircles of triangles $OX_AY_A$ and $BOC$ other than $O$. Define $K_B$ and $K_C$ analogously by repeating this construction two more times. Prove that $K_A$, $K_B$, $K_C$, and $O$ are concyclic. [i]Hongzhou Lin[/i]

Let $a$, $b$ and $c$ be rational numbers for which $a+bc$, $b+ac$ and $a+b$ are all non-zero and for which we have \[\frac{1}{a+bc}+\frac{1}{b+ac}=\frac{1}{a+b}.\] Prove that $\sqrt{(c-3)(c+1)}$ is rational.

Let $ A = \{ 1, 2, 3, \cdots , 12 \} $. Find the number of one-to-one function $ f :A \to A $ satisfying following condition: for all $ i \in A $, $ f(i)-i $ is not a multiple of $ 3 $.

Marko lives on the origin of the Cartesian plane. Every second, Marko moves $1$ unit up with probability $\tfrac{2}{9}$, $1$ unit right with probability $\tfrac{2}{9}$, $1$ unit up and $1$ unit right with probability $\tfrac{4}{9}$, and he doesn’t move with probability $\tfrac{1}{9}$. After $2019$ seconds, Marko ends up on the point $(A, B)$. What is the expected value of $A\cdot B$?

All the integers from $1$ to $100$ are arranged in a $10 \times 10$ table as shown below. Prove that if some ten numbers are removed from the table, the remaining $90$ numbers contain 10 numbers in Arithmetic Progression. $1 \,\,\,\,2\,\, \,\,3 \,\,\,\,... \,\,10$ $11 \,\,12 \,\,13 \,\,... \,\,20$ $\,\,.\,\,\,\,.\,\,\,.$ $\,\,.\,\,\,\,.\,\,\,\,.$ $91 \,\,92 \,\,93\,\, ... \,\,100$

Let $\mathcal{P}$ be the set of all primes, and let $M$ be a subset of $\mathcal{P}$, having at least three elements, and such that for any proper subset $A$ of $M$ all of the prime factors of the number $ -1+\prod_{p\in A}p$ are found in $M$. Prove that $M= \mathcal{P}$. [i]Valentin Vornicu[/i]

At this year's Olympiad, some of the students are friends (friendship is symmetric), however there are also students which are not friends. No matter how the students are partitioned in two contest halls, there are always two friends in different halls. Let $A$ be a fixed student. Show that there exist students $B$ and $C$ such that there are exactly two friendships in the group $\{ A,B,C \}$. [i]Authored by Mirko Petrushevski[/i]

Construct a triangle $ABC$ given its side $a = BC$, its circumradius $R \ (2R \geq a)$, and the difference $\frac{1}{k} = \frac{1}{c}-\frac{1}{b}$, where $c = AB$ and $ b = AC.$

(a) Let $a_0, a_1,a_2$ be real numbers and consider the polynomial $P(x) = a_0 + a_1x + a_2x^2$ . Assume that $P(-1), P(0)$ and $P(1)$ are integers. Prove that $P(n)$ is an integer for all integers $n$. (b) Let $a_0,a_1, a_2, a_3$ be real numbers and consider the polynomial $Q(x) = a0 + a_1x + a_2x^2 + a_3x^3 $. Assume that there exists an integer $i$ such that $Q(i),Q(i+1),Q(i+2)$ and $Q(i+3)$ are integers. Prove that $Q(n)$ is an integer for all integers $n$.

In the game of Colonel Blotto, you have 100 troops to distribute among 10 castles. Submit a 10-tuple $(x_1, x_2, \dots x_{10})$ of nonnegative integers such that $x_1 + x_2 + \dots + x_{10} = 100$, where each $x_i$ represent the number of troops you want to send to castle $i$. Your troop distribution will be matched up against each opponent's and you will win 10 points for each castle that you send more troops to (if you send the same number, you get 5 points, and if you send fewer, you get none). Your aim is to score the most points possible averaged over all opponents. For example, if team $A$ submits $(90,10,0,\dots,0)$, team B submits $(11,11,11,11,11,11,11,11,11,1)$, and team C submits $(10,10,10,\dots 10)$, then team A will win 10 points against team B and 15 points against team C, while team B wins 90 points against team C. Team A averages 12.5 points, team B averages 90 points, and team C averages 47.5 points. [i]2016 CCA Math Bonanza Lightning #5.4[/i]

Define a function $f$ on the unit interval $0 \le x \le 1$ by the rule \[ f(x) = \begin{cases} 1-3x & \text{if } 0 \le x < 1/3 \, ; \\ 3x-1 & \text{if } 1/3 \le x < 2/3 \, ; \\ 3-3x & \text{if } 2/3 \le x \le 1 \, . \end{cases} \] Determine $f^{(2018)}(1/730)$. Recall that $f^{(n)}$ denotes the $n$th iterate of $f$; for example, $f^{(3)}(1/730) = f(f(f(1/730)))$.

Let $n$ be a positive integer. Nancy is given a rectangular table in which each entry is a positive integer. She is permitted to make either of the following two moves: (1) select a row and multiply each entry in this row by $n$; (2) select a column and subtract $n$ from each entry in this column. Find all possible values of $n$ for which the following statement is true: Given any rectangular table, it is possible for Nancy to perform a finite sequence of moves to create a table in which each entry is $0$.

Let $\triangle ABC$ be an equilateral triangle with side length $1$ and $M$ be the midpoint of side $AC$. Let $N$ be the foot of the perpendicular from $M$ to side $AB$. The angle bisector of angle $\angle BAC$ intersects $MB$ and $MN$ at $X$ and $Y$, respectively. If the area of triangle $\triangle MXY$ is $\mathcal{A}$, and $\mathcal{A}^2$ can be expressed as a common fraction in the form $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m + n$.

On a competition called [i]"Mathematical duels"[/i] students were given $n$ problems and each student solved exactly 3 of them. For each two students there is at most one problem that is solved from both of them. Prove that, if $s\in \mathbb{N}$ is a number for which $s^2-s+1<2n$, then there are $s$ problems among the $n$, no three of which solved by one student.

Triangle $ABC$ has circumcircle $\omega$. Let $D$ be the foot of the altitude from $A$ to $BC$ and let $AD$ intersect $\omega$ at $E \neq A$. Let $M$ be the midpoint of $AD$. If $\angle{BMC} = 90^\circ$, $AB = 9$ and $AE = 10$, the area of $\triangle{ABC}$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a,b,c$ are positive integers and $b$ is square-free. Find $a+b+c$. [i]Proposed by Andy Xu[/i]

Let $ABC$ be a triangle such that $\angle A=30^\circ$ and $\angle B=80^\circ$. Let $D$ and $E$ be points on sides $AC$ and $BC$ respectively so that $\angle ABD=\angle DBC$ and $DE\parallel AB$. Determine the measure of $\angle EAC$.

If $ 4^x \minus{} 4^{x \minus{} 1} \equal{} 24$, then $ (2x)^x$ equals: $ \textbf{(A)}\ 5\sqrt{5}\qquad \textbf{(B)}\ \sqrt{5}\qquad \textbf{(C)}\ 25\sqrt{5}\qquad \textbf{(D)}\ 125\qquad \textbf{(E)}\ 25$

Let be $n\geq 3$ fixed positive integer.Let be real numbers $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ such that satisfied this conditions: [b]$i)$[/b] $ $ $a_n\geq a_{n-1}$ and $b_n\geq b_{n-1}$ [b]$ii)$[/b] $ $ $0<a_1\leq b_1\leq a_2\leq b_2\leq ... \leq a_{n-1}\leq b_{n-1}$ [b]$iii)$[/b] $ $ $a_1+a_2+...+a_n=b_1+b_2+...+b_n$ [b]$iv)$[/b] $ $ $a_{1}\cdot a_2\cdot ...\cdot a_n=b_1\cdot b_2\cdot ...\cdot b_n$ Show that $a_i=b_i$ for all $i=1,2,...,n$

A sequence of integers is defined as follows: $a_1=1,a_2=2,a_3=3$ and for $n>3$, $$a_n=\textsf{The smallest integer not occurring earlier, which is relatively prime to }a_{n-1}\textsf{ but not relatively prime to }a_{n-2}.$$Prove that every natural number occurs exactly once in this sequence. [i]M. Ivanov[/i]

Let $k$ be an integer and $k > 1$. Define a sequence $\{a_n\}$ as follows: $a_0 = 0$, $a_1 = 1$, and $a_{n+1} = ka_n + a_{n-1}$ for $n = 1,2,...$. Determine, with proof, all possible $k$ for which there exist non-negative integers $l,m (l \not= m)$ and positive integers $p,q$ such that $a_l + ka_p = a_m + ka_q$.

For real numbers $a,b,c$, if $$a^2-bc-8a+7=b^2+c^2+bc-6a-6=0,$$ then the range value of $a$ is $\text{(A)}(-\infty,+\infty)\qquad\text{(B)}(-\infty,1]\cup[9,+\infty)\qquad\text{(C)}(0,7)\qquad\text{(D)}[1,9]$

Show that, for any integer $n$, the number $n^3 - 9n + 27$ is not divisible by $81$.

Prove that if $ x \not\equal{} 0$ is a real number, then: $ x^8\minus{}x^5\minus{}\frac{1}{x}\plus{}\frac{1}{x^4} \ge 0$.

On the real number line, paint red all points that correspond to integers of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer point blue. Find a point $P$ on the line such that, for every integer point $T$, the reflection of $T$ with respect to $P$ is an integer point of a different colour than $T$.