Let $k,n,r$ be positive integers and $r<n$. Quirin owns $kn+r$ black and $kn+r$ white socks. He want to clean his cloths closet such there does not exist $2n$ consecutive socks $n$ of which black and the other $n$ white. Prove that he can clean his closet in the desired manner if and only if $r\geq k$ and $n>k+r$.

The game of rock-scissors is played just like rock-paper-scissors, except that neither player is allowed to play paper. You play against a poorly-designed computer program that plays rock with $50\%$ probability and scissors with $50\%$ probability. If you play optimally against the computer, find the probability that after $8$ games you have won at least $4$. [i]In the game of rock-paper-scissors, two players each choose one of rock, paper, or scissors to play. Rock beats scissors, scissors beats paper, and paper beats rock. If the players play the same thing, the match is considered a draw.[/i]

Let $ \Gamma$ be the circumcircle of a triangle $ ABC$. A circle passing through points $ A$ and $ C$ meets the sides $ BC$ and $ BA$ at $ D$ and $ E$, respectively. The lines $ AD$ and $ CE$ meet $ \Gamma$ again at $ G$ and $ H$, respectively. The tangent lines of $ \Gamma$ at $ A$ and $ C$ meet the line $ DE$ at $ L$ and $ M$, respectively. Prove that the lines $ LH$ and $ MG$ meet at $ \Gamma$.

Let $\Phi$ denote the Euler totient function. Prove that for infinitely many $k$ we have $\Phi (2^k+1) < 2^{k-1}$ and that for infinitely many $m$ one has $\Phi (2^m+1) > 2^{m-1}$

Find all real $x$ that satisfy the equation $$\frac{1}{x+1}+\frac{1}{x+2}=\frac{1}{x}$$ [i]2015 CCA Math Bonanza Lightning Round #2.2[/i]

Find all primes $p,q,r$ such that $qr-1$ is divisible by $p$, $pr-1$ is divisible by $q$, $pq-1$ is divisible by $r$.

Compute the smallest positive integer $n$ such that there do not exist integers $x$ and $y$ satisfying $n=x^3+3y^3$. [i]Proposed by Luke Robitaille[/i]

Compute the number of integer ordered pairs $(a, b)$ such that $10!$ is a multiple of $a^2 + b^2.$

Let $ABC$ be an acute triangle with an altitude $AD$. Let $H$ be the orthocenter of triangle $ABC$. The circle $\Omega$ passes through the points $A$ and $B$, and touches the line $AC$. Let $BE$ be the diameter of $\Omega$. The lines~$BH$ and $AH$ intersect $\Omega$ for the second time at points $K$ and $L$, respectively. The lines $EK$ and $AB$ intersect at the point~$T$. Prove that $\angle BDK=\angle BLT$.

Show that for some $c > 0$, we have $\left|\sqrt[3]{2} - \frac{m}{n}\right | > \frac{c}{n^3}$ for all integers $m, n$ with $n \ge 1$.

Given is a square board measuring $1 995 \times 1 995$. These cells are painted with black and white paints in a checkerboard pattern, so that the corner cells are black. A spider sitting on one of the black cells can crawl to the cell on the same side as the one it occupies in one step. Prove that a spider can always reach a fly sitting motionless in another black cell by visiting all the cells of the board once.

Find all functions $f : R^+ \to R^+$ such that for all $x, y > 0$ $$f(yf(x))(x + y) = x^2(f(x) + f(y)).$$

For which positive $x$ does the expression $x^{1000}+x^{900}+x^{90}+x^6+\frac{1996}{x}$ attain the smallest value?

Given two fractions $a/b$ and $c/d$ we define their [i]pirate sum[/i] as: $\frac{a}{b} \star \frac{c}{d} = \frac{a+c}{b+d}$ where the two initial fractions are simplified the most possible, like the result. For example, the pirate sum of $2/7$ and $4/5$ is $1/2$. Given an integer $n \ge 3$, initially on a blackboard there are the fractions: $\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, ..., \frac{1}{n}$. At each step we choose two fractions written on the blackboard, we delete them and write at their place their pirate sum. Continue doing the same thing until on the blackboard there is only one fraction. Determine, in function of $n$, the maximum and the minimum possible value for the last fraction.

Find all positive integers $N$ such that there exists a triangle which can be dissected into $N$ similar quadrilaterals. [i]Proposed by Nikolai Beluhov (Bulgaria) and Morteza Saghafian[/i]

Let $A$ be the set of all triples $(x, y, z)$ of positive integers satisfying $2x^2 + 3y^3 = 4z^4$ . a) Show that if $(x, y, z) \in A$ then $6$ divides all of $x, y, z$. b) Show that $A$ is an infinite set.

In the $xy$-coordinate plane, the $x$-axis and the line $y=x$ are mirrors. If you shoot a laser beam from the point $(126, 21)$ toward a point on the positive $x$-axis, there are $3$ places you can aim at where the beam will bounce off the mirrors and eventually return to $(126, 21)$. They are $(126, 0)$, $(105, 0)$, and a third point $(d, 0)$. What is $d$? (Recall that when light bounces off a mirror, the angle of incidence has the same measure as the angle of reflection.)

Determine all integers $a$ such that $a^k + 1$ is divisible by $12321$ for some $k$

There are $n{}$ passengers in the queue to board a $n{}$-seat plane. The first one in the queue is an absent-minded old lady who, after boarding the plane, sits down at a randomly selected place. Each subsequent passenger sits in his seat if it is free, and in a random seat otherwise. How many passengers will be out of their seats on average? [i]Proposed by A. Zaslavsky[/i]

Let $x+y=a$ and $xy=b$. Calculate exression $ x^4+y^4$ in terms of $a$ and $b$.

Find all positive integers $n$, such that $\sigma(n) =\tau(n) \lceil {\sqrt{n}} \rceil$.

Let a $9$-digit number be balanced if it has all numerals $1$ to $9$. Let $S$ be the sequence of the numerals which is constructed by writing all balanced numbers in increasing order consecutively. Find the least possible value of $k$ such that any two subsequences of $S$ which has consecutive $k$ numerals are different from each other.

Let $O$ and $H$ be the circumcenter and the orthocenter, respectively, of an acute triangle $ABC$. Points $D$ and $E$ are chosen from sides $AB$ and $AC$, respectively, such that $A$, $D$, $O$, $E$ are concyclic. Let $P$ be a point on the circumcircle of triangle $ABC$. The line passing $P$ and parallel to $OD$ intersects $AB$ at point $X$, while the line passing $P$ and parallel to $OE$ intersects $AC$ at $Y$. Suppose that the perpendicular bisector of $\overline{HP}$ does not coincide with $XY$, but intersect $XY$ at $Q$, and that points $A$, $Q$ lies on the different sides of $DE$. Prove that $\angle EQD = \angle BAC$. [i]Proposed by Shuang-Yen Lee[/i]

Let $(x_n)_{n\ge 1}$ a sequence defined by $x_1=2,\ x_{n+1}=\sqrt{x_n+\frac{1}{n}},\ (\forall)n\in \mathbb{N}^*$. Prove that $\lim_{n\to \infty} x_n=1$ and evaluate $\lim_{n\to \infty} x_n^n$.

Let $\mathcal{R}(f(x))$ denote the number of distinct real roots of $f(x)$. Compute $$\sum_{a=1}^{1009}\sum_{b=1010}^{2018}\mathcal{R}(x^{2018}-ax^{2016}+b).$$