Let $\omega$ be a circle centered at $O$ with radius $R=2018$. For any $0 < r < 1009$, let $\gamma$ be a circle of radius $r$ centered at a point $I$ satisfying $OI =\sqrt{R(R-2r)}$. Choose any $A,B,C\in \omega$ with $AC, AB$ tangent to $\gamma$ at $E,F$, respectively. Suppose a circle of radius $r_A$ is tangent to $AB,AC$, and internally tangent to $\omega$ at a point $D$ with $r_A=5r$. Let line $EF$ meet $\omega$ at $P_1,Q_1$. Suppose $P_2,P_3,Q_2,Q_3$ lie on $\omega$ such that $P_1P_2,P_1P_3,Q_1Q_2,Q_1Q_3$ are tangent to $\gamma$. Let $P_2P_3,Q_2Q_3$ meet at $K$, and suppose $KI$ meets $AD$ at a point $X$. Then as $r$ varies from $0$ to $1009$, the maximum possible value of $OX$ can be expressed in the form $\frac{a\sqrt{b}}{c}$, where $a,b,c$ are positive integers such that $b$ is not divisible by the square of any prime and $\gcd (a,c)=1$. Compute $10a+b+c$. [i]Proposed by Vincent Huang

A sequence $x_0, x_1, x_2, . . .$ is given by $x_0 = 8$ and $x_{n+1} =\frac{1 + x_n}{1- x_n}$ for $n = 0, 1, 2, . . . .$ Determine the number $x_{2013}$.

Let $f: [0,1]\to\mathbb{R}$ be a continuous function such that \[ \int_{0}^{1}f(x)dx=0. \] Prove that there is $c\in (0,1)$ such that \[ \int_{0}^{c}xf(x)dx=0. \] [i]Cezar Lupu, Tudorel Lupu[/i]

From a regular octahedron with edge $1$, cut off a pyramid about each vertex. The base of each pyramid is a square with edge $\frac 13$. Can copies of the polyhedron so obtained, whose faces are either regular hexagons or squares, be used to tile space?

Suppose that $N$ is a three digit number divisible by $7$ such that upon removing its middle digit, the remaining two digit number is also divisible by $7$. What is the minimum possible value of $N$? [i]2019 CCA Math Bonanza Lightning Round #3.1[/i]

Consider $h+1$ chess boards. Number the squares of each board from 1 to 64 in such a way that when the perimeters of any two boards of the collection are brought into coincidence in any possible manner, no two squares in the same position have the same number. What is the maximum value of $h?$

Let $n$ and $m$ be positive integers. The daycare nanny uses $n \times m$ square floor mats to construct an $n \times m$ rectangular area, with a baby on each of the mats. Each baby initially faces toward one side of the rectangle. When the nanny claps, all babies crawl one mat forward in the direction it is facing at, and then turn 90 degrees clockwise. If a baby crawls outside of the rectangle, it cries. If two babies simultaneously crawl onto the same mat, they bump into each other and cry. Suppose that it is possible for the nanny to arrange the initial direction of each baby so that, no matter how many times she claps, no baby would cry. Find all possible values of $n$ and $m$. [i]Proposed by Chu-Lan Kao[/i]

The real numbers $a,b,c,x,y,$ and $z$ are such that $a>b>c>0$ and $x>y>z>0$. Prove that \[\frac {a^2x^2}{(by+cz)(bz+cy)}+\frac{b^2y^2}{(cz+ax)(cx+az)}+\frac{c^2z^2}{(ax+by)(ay+bx)}\ge \frac{3}{4}\]

A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A?$ $\textbf{(A)}\ 385 \qquad \textbf{(B)}\ 665 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 1140 \qquad \textbf{(E)}\ 1330$

Consider the trinomial $f(x) = x^2 + 2bx + c$ with integer coefficients $b$ and $c$. Prove that if $f(n) \ge 0$ for all integers $n$, then $f(x) \ge 0$ even for all rational numbers $x$.

On the side $BC$ of the rectangle $ABCD$, a point $P{}$ is marked so that $\angle APD = 90^\circ$. On the straight line $AD$, points $Q{}$ and $R{}$ are selected outside the segment $AD$ such that $AQ = BP$ and $CP = DR$. The circle $\omega$ passes through the points $Q, D$ and the circumcenter of the triangle $PDQ$. The circle $\gamma$ passes through the points $A, R$ and the circumcenter of the triangle $APR$. Prove that the radius of one of the circles touching the line $AD$ and the circles $\omega$ and $\gamma$ is $2AB$.

$\textbf{(Caos)}$ A cao [sic] has 6 legs, 3 on each side. A walking pattern for the cao is defined as an ordered sequence of raising and lowering each of the legs exactly once (altogether 12 actions), starting and ending with all legs on the ground. The pattern is safe if at any point, he has at least 3 legs on the ground and not all three legs are on the same side. Estimate $N$, the number of safe patterns. An estimate of $E > 0$ earns $\left\lfloor 20\min(N/E, E/N)^4 \right\rfloor$ points.

Prove for positive numbers: $$(a^2+b+\frac{3}{4})(b^2+a+\frac{3}{4}) \geq (2a+\frac{1}{2})(2b+\frac{1}{2})$$

By dividing $2023$ by a natural number $m$, the remainder is $23$. How many numbers $m$ are there with this property?

The number of proper subsets of the set $\left\{x|-1\leq\log_{\frac{1}{x}}10<-\frac{1}{2},x\in\mathbb{Z}_{\geq0}\right\}$ is________.

Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime? $ \textbf{(A)}\ \left(\frac{1}{12}\right)^{12}\qquad \textbf{(B)}\ \left(\frac{1}{6}\right)^{12}\qquad \textbf{(C)}\ 2\left(\frac{1}{6}\right)^{11}\qquad \textbf{(D)}\ \frac{5}{2}\left(\frac{1}{6}\right)^{11}\qquad \textbf{(E)}\ \left(\frac{1}{6}\right)^{10}$

Can a triangle be a development of a quadrangular pyramid?

Given the progression $10^{\dfrac{1}{11}}, 10^{\dfrac{2}{11}}, 10^{\dfrac{3}{11}}, 10^{\dfrac{4}{11}},\dots , 10^{\dfrac{n}{11}}$. The least positive integer $n$ such that the product of the first $n$ terms of the progression exceeds $100,000$ is $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad \textbf{(E) }11$

$n$ students take a test with $m$ questions, where $m,n\ge 2$ are integers. The score given to every question is as such: for a certain question, if $x$ students fails to answer it correctly, then those who answer it correctly scores $x$ points, while those who answer it wrongly scores $0$. The score of a student is the sum of his scores for the $m$ questions. Arrange the scores in descending order $p_1\ge p_2\ge \ldots \ge p_n$. Find the maximum value of $p_1+p_n$.

Prove that there exist infinitely many integers $n$ such that $n, n+1, n+2$ are each the sum of the squares of two integers.

Suppose that we have $ n \ge 3$ distinct colours. Let $ f(n)$ be the greatest integer with the property that every side and every diagonal of a convex polygon with $ f(n)$ vertices can be coloured with one of $ n$ colours in the following way: (i) At least two colours are used, (ii) any three vertices of the polygon determine either three segments of the same colour or of three different colours. Show that $ f(n) \le (n\minus{}1)^2$ with equality for infintely many values of $ n$.

Let $ABC$ be an acute triangle such that $CA \neq CB$ with circumcircle $\omega$ and circumcentre $O$. Let $t_A$ and $t_B$ be the tangents to $\omega$ at $A$ and $B$ respectively, which meet at $X$. Let $Y$ be the foot of the perpendicular from $O$ onto the line segment $CX$. The line through $C$ parallel to line $AB$ meets $t_A$ at $Z$. Prove that the line $YZ$ passes through the midpoint of the line segment $AC$. [i]Proposed by Dominic Yeo, United Kingdom[/i]

\[ \begin{array}{c} {x^{2} \plus{} y^{2} \plus{} z^{2} \equal{} 21} \\ {x \plus{} y \plus{} z \plus{} xyz \equal{} \minus{} 3} \\ {x^{2} yz \plus{} y^{2} xz \plus{} z^{2} xy \equal{} \minus{} 40} \end{array} \] The number of real triples $ \left(x,y,z\right)$ satisfying above system is $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ \text{None}$

A circle is divided into $ n $ equal parts. Marceline sets out to assign whole numbers from $ 1 $ to $ n $ to each of these pieces so that the distance between two consecutive numbers is always the same. The numbers $ 887 $, $ 217 $ and $ 1556 $ occupy consecutive positions. How many parts was the circumference divided into?

Find the sum of all positive integers $n$ such that $7<n < 100$ and $1573_{n}$ has $6$ factors when written in base $10$. [i]Proposed by Aidan Duncan[/i]