The excircle of a triangle $ABC$ corresponding to $A$ touches the side $BC$ at $K$ and the rays $AB$ and $AC$ at $P$ and $Q$, respectively. The lines $OB$ and $OC$ intersect $PQ$ at $M$ and $N$, respectively. Prove that $$\frac{QN}{AB}=\frac{NM}{BC}=\frac{MP}{CA}.$$

Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$, there exists an element $z \in M$ such that $x < z < y.$

Find the maximum possible value of $k$ for which there exist distinct reals $x_1,x_2,\ldots ,x_k $ greater than $1$ such that for all $1 \leq i, j \leq k$, $$x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}.$$ [i]Proposed by Morteza Saghafian[/i]

An $n\times n$ square ($n$ is a positive integer) consists of $n^2$ unit squares.A $\emph{monotonous path}$ in this square is a path of length $2n$ beginning in the left lower corner of the square,ending in its right upper corner and going along the sides of unit squares. For each $k$, $0\leq k\leq 2n-1$, let $S_k$ be the set of all the monotonous paths such that the number of unit squares lying below the path leaves remainder $k$ upon division by $2n-1$.Prove that all $S_k$ contain equal number of elements.

Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden? $\textbf{(A)}\ 256\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 384\qquad\textbf{(D)}\ 448\qquad\textbf{(E)}\ 512$

We are given a unit square in the plane. A point $M$ in the plane is called [i]median [/i]if there exists points $P$ and $Q$ on the boundary of the square such that $PQ$ has length one and $M$ is the midpoint of $PQ$. Determine the geometric locus of all median points.

Let $P$ be a convex polygon. Prove that there exists a convex hexagon that is contained in $P$ and whose area is at least $\frac34$ of the area of the polygon $P$. [i]Alternative version.[/i] Let $P$ be a convex polygon with $n\geq 6$ vertices. Prove that there exists a convex hexagon with [b]a)[/b] vertices on the sides of the polygon (or) [b]b)[/b] vertices among the vertices of the polygon such that the area of the hexagon is at least $\frac{3}{4}$ of the area of the polygon. [i]Proposed by Ben Green and Edward Crane, United Kingdom[/i]

[b]a)[/b] Solve in $ \mathbb{R} $ the equation $ 2^x=x+1. $ [b]b)[/b] If a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ has the property that $$ (f\circ f)(x)=2^x-1,\quad\forall x\in\mathbb{R} , $$ then $ f(0)+f(1)=1. $

Quadrilateral $ABCD$ with perpendicular diagonals is inscribed in a circle with centre $O$. The tangents to this circle at $A$ and $C$ together with line $BD$ form the triangle $\Delta$. Prove that the circumcircles of $BOD$ and $\Delta$ are tangent. [hide=Additional information for Junior League]Show that this point lies belongs to $\omega$, the circumcircle of $OAC$[/hide] [i]Proposed by A. Kuznetsov[/i]

Find the number of positive integers $n$ not greater than 2017 such that $n$ divides $20^n + 17k$ for some positive integer $k$.

Define sequence $(a_{n})_{n=1}^{\infty}$ by $a_1=a_2=a_3=1$ and $a_{n+3}=a_{n+1}+a_{n}$ for all $n \geq 1$. Also, define sequence $(b_{n})_{n=1}^{\infty}$ by $b_1=b_2=b_3=b_4=b_5=1$ and $b_{n+5}=b_{n+4}+b_{n}$ for all $n \geq 1$. Prove that $\exists N \in \mathbb{N}$ such that $a_n = b_{n+1} + b_{n-8}$ for all $n \geq N$.

A container contains five red balls. On each turn, one of the balls is selected at random, painted blue, and returned to the container. The expected number of turns it will take before all five balls are colored blue is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Suppose $a, b$ are positive real numbers such that $a\sqrt{a} + b\sqrt{b} = 183, a\sqrt{b} + b\sqrt{a} = 182$. Find $\frac95 (a + b)$.

Let $T_1$ be a triangle having $a, b, c$ as lengths of its sides and let $T_2$ be another triangle having $u, v,w$ as lengths of its sides. If $P,Q$ are the areas of the two triangles, prove that \[16PQ \leq a^2(-u^2 + v^2 + w^2) + b^2(u^2 - v^2 + w^2) + c^2(u^2 + v^2 - w^2).\] When does equality hold?

Let $ABC$ be an acute angled triangle (with $AB<AC<BC$) inscribed in circle $c(O,R)$ (with center $O$ and radius $R$). Circle $c_1(A,AB)$ (with center $A$ and radius $AB$) intersects side $BC$ at point $D$ and the circumcircle $c(O,R)$ at point $E$. Prove that side $AC$ bisects angle $\angle DAE$.

Give an example of a continuous real-valued function $f$ form $[0,1]$ to $[0,1]$ which takes on every value in $[0,1]$ an infinite number of times.

The number 7 is written on a board. Alice and Bob in turn (Alice begins) write an additional digit in the number on the board: it is allowed to write the digit at the beginning (provided the digit is nonzero), between any two digits or at the end. If after someone’s turn the number on the board is a perfect square then this person wins. Is it possible for a player to guarantee the win? [i]Alexandr Gribalko[/i]

There are 60 empty boxes $B_1,\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game. In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps: (a) Bob chooses an integer $k$ with $1\leq k\leq 59$ and splits the boxes into the two groups $B_1,\ldots,B_k$ and $B_{k+1},\ldots,B_{60}$. (b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group. Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning. [i]Czech Republic[/i]

A scalene acute triangle has angles whose measures (in degrees) are whole numbers. What is the smallest possible measure of one of the angles, in degrees?

The number of real quadruples $(x,y,z,w)$ satisfying $x^3+2=3y, y^3+2=3z, z^3+2=3w, w^3+2=3x$ is $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \text{None}$

A triangle has side lengths of $x,75,100$ where $x<75$ and altitudes of lengths $y,28,60$ where $y<28$. What is the value of $x+y$? [i]2019 CCA Math Bonanza Team Round #2[/i]

A circle with center $O$ is drawn in the first quadrant of the 2D Cartesian plane (the quadrant with both positive $x$ and $y$ values) such that it lies tangent to the $x$ and $y$-axes. A line is drawn with slope $m>1$ and passing through the origin; the line intersects the circle at two points $A$ and $B$, with $A$ closer to the origin than $B$. Suppose that $ABO$ is an equilateral triangle. Compute $m$.

Two players play a game on four piles of pebbles labeled with the numbers $1,2,3,4$ respectively. The players take turns in an alternating fashion. On his or her turn, a player selects integers $m$ and $n$ with $1\leq m<n\leq 4$, removes $m$ pebbles from pile $n$, and places one pebble in each of the piles $n-1,n-2,\dots,n-m$. A player loses the game if he or she cannot make a legal move. For each starting position, determine the player with a winning strategy.

There are 25 ants on a number line; five at each of the coordinates $1$, $2$, $3$, $4$, and $5$. Each minute, one ant moves from its current position to a position one unit away. What is the minimum number of minutes that must pass before it is possible for no two ants to be on the same coordinate? [i]Ray Li[/i]

If $ |x| + x + y = 10$ and $x + |y| - y = 12$, find $x + y$. $ \textbf{(A)}\ -2\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{18}{5}\qquad\textbf{(D)}\ \frac{22}{3}\qquad\textbf{(E)}\ 22 $