Let $ABC$ be an acute triangle with incenter $I$ and circumcircle $\omega$. Suppose a circle $\omega_B$ is tangent to $BA,BC$, and internally tangent to $\omega$ at $B_1$, while a circle $\omega_C$ is tangent to $CA, CB$, and internally tangent to $\omega$ at $C_1$. If $B_2, C_2$ are the points opposite to $B,C$ on $\omega$, respectively, and $X$ denotes the intersection of $B_1C_2, B_2C_1$, prove that $XA=XI$. [i]Proposed by Vincent Huang and Nathan Weckwerth

Let $P$ be a regular $2006$-gon. A diagonal is called [i]good[/i] if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called [i]good[/i]. Suppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a common point in the interior of $P$. Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.

A circle with diameter $20$ has points $A, B, C, D, E,$ and $F$ equally spaced along its circumference. A second circle is tangent to the lines $AB$ and $AF$ and internally tangent to the circle. If the second circle has diameter $\sqrt{m}+n$ for integers $m$ and $n$, find $m + n.$ [asy] import geometry; size(180); draw(circle((0,0),5)); pair[] p; string[] l={"A","B","C","D","E","F"}; for (int i=0; i<6; ++i){ p.append(new pair[]{dir(i*60+180)*5}); dot(p[i]); label(l[i],p[i],p[i]/3); } draw(p[0]--p[1]^^p[0]--p[5]); p.append(new pair[]{intersectionpoint(p[0]--p[0]+dir(-60)*90,p[3]--p[3]+(0,-100))}); p.append(new pair[]{intersectionpoint(p[0]--p[0]+dir(+60)*90,p[3]--p[3]+(0,+100))}); draw(incircle(p[0],p[6],p[7]));[/asy]

[i]Biribol[/i] is a game played between two teams of 4 people each (teams are not fixed). Find all the possible values of $ n$ for which it is possible to arrange a tournament with $ n$ players in such a way that every couple of people plays a match in opposite teams exactly once.

Let a chain denote a row of positive integers which continue infinitely in both directions, such that for each number $n$, the $n$ numbers directly to the left of $n$ yield $n$ distinct remainders upon division by $n$. (a) If a chain has a maximum integer, what are the possible values of that integer? (b) Does there exist a chain which does not have a maximum integer?

Eight rooks are placed on a chessboard so that no two rooks attack each other. Prove that one can always move all rooks, each by a move of a knight so that in the final position no two rooks attack each other as well. (In intermediate positions several rooks can share the same square).

Let $k$ be a constant such that exactly three real values of $x$ satisfy $$x-|x^2-4x+3| = k$$ The sum of all possible values of $k$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Andy Xu[/i]

Fix integers $n\ge k\ge 2$. We call a collection of integral valued coins $n-diverse$ if no value occurs in it more than $n$ times. Given such a collection, a number $S$ is $n-reachable$ if that collection contains $n$ coins whose sum of values equals $S$. Find the least positive integer $D$ such that for any $n$-diverse collection of $D$ coins there are at least $k$ numbers that are $n$-reachable. [I]Proposed by Alexandar Ivanov, Bulgaria.[/i]

Show that any triangle can be subdivided into isosceles triangles.

Prove that for no integer $ n$ is $ n^7 \plus{} 7$ a perfect square.

In the morning, Esther biked from home to school at an average speed of $x$ miles per hour. In the afternoon, having lent her bike to a friend, Esther walked back home along the same route at an average speed of 3 miles per hour. Her average speed for the round trip was 5 miles per hour. What is the value of $x$?

Equilateral octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is constructed such that $A_1A_3A_5A_7$ is a square of side length $\sqrt{2}$ and $A_2A_4A_6A_8$ is a square of side length 4/3. For each vertex $A_i$ of the octagon, let $B_i$ be the intersection of lines $A_{i+1}A_{i+2}$ and $A_{i-1}A_{i-2}$, where $A_{i-8} = A_i = A_{i+8}$. Compute $[B_1B_2B_3B_4B_5B_6B_7B_8]^2$. [i]2022 CCA Math Bonanza Team Round #9[/i]

[b]Q15.[/b] Determine the greatest value of the sum $M=xy+yz+zx$, where $x,y,z$ are real numbers satisfying the following condition $x^2+2y^2+5z^2=22.$

$A$ and $B$ play a game, given an integer $N$, $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$, but his number cannot be bigger than $N$. The player who writes $N$ wins. For which values of $N$ does $B$ win? [i]Proposed by A. Slinko & S. Marshall, New Zealand[/i]

The interior angles of a convex polygon form an arithmetic progression with a common difference of $4^\circ$. Determine the number of sides of the polygon if its largest interior angle is $172^\circ.$

Let $\left(a_n\right)_{n\geq1}$ be a sequence of nonnegative real numbers which converges to $a \in \mathbb{R}$. [list=1] [*]Calculate$$\lim \limits_{n\to \infty}\sqrt[n]{\int \limits_0^1 \left(1+a_nx^n \right)^ndx}$$ [*]Calculate$$\lim \limits_{n\to \infty}\sqrt[n]{\int \limits_0^1 \left(1+\frac{a_1x+a_3x^3+\cdots+a_{2n-1}x^{2n-1}}{n} \right)^ndx}$$ [/list]

Prove that for every $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ in the interval $(0,\pi/2)$ \[\left({1\over \sin \alpha_{1}}+{1\over \sin \alpha_{2}}+\ldots+{1\over \sin \alpha_{n}}\right) \left({1\over \cos \alpha_{1}}+{1\over \cos \alpha_{2}}+\ldots+{1\over \cos \alpha_{n}}\right) \leq\] \[\leq 2 \left({1\over \sin 2\alpha_{1}}+{1\over \sin 2\alpha_{2}}+\ldots+{1\over \sin 2\alpha_{n}}\right)^{2}.\] [i]Proposed by A. Khrabrov[/i]

Let $T$ be the set of all 2-digit numbers whose digits are in $\{1,2,3,4,5,6\}$ and the tens digit is strictly smaller than the units digit. Suppose $S$ is a subset of $T$ such that it contains all six digits and no three numbers in $S$ use all six digits. If the cardinality of $S$ is $n$, find all possible values of $n$.

$ p >3 $ is a prime number such that $ p | 2^{p-1} -1 $ and $ p \not | 2^x - 1 $ for $ x = 1, 2, \cdots , p-2 $. Let $ p = 2k+3 $. Now we define sequence $ \{ a_n \} $ as \[ a_i = a_{i+k}= 2^i ( 1 \le i \le k ) , \ a_{j+2k} = a_j a_{j+k} \ ( j \ge 1 ) \] Prove that there exist $2k$ consecutive terms of sequence $ a_{x+1} , a_{x+2} , \cdots , a_{x+2k} $ such that for all $ 1 \le i < j \le 2k $, $ a_{x+i} \not \equiv a_{x+j} \ (mod \ p) $.

Let $N$ be the product of all odd primes less than $2^4$. What remainder does $N$ leave when divided by $2^4$? $\text{(A) }5\qquad\text{(B) }7\qquad\text{(C) }9\qquad\text{(D) }11\qquad\text{(E) }13$

How many $8$-digit numbers are $*2*0*1*7$, where four unknown numbers are replaced by stars, which are divisible by $7$?

Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]

Let $ N\equal{}100^2\plus{}99^2\minus{}98^2\minus{}97^2\plus{}96^2\plus{}\cdots\plus{}4^2\plus{}3^2\minus{}2^2\minus{}1^2$, where the additions and subtractions alternate in pairs. Find the remainder when $ N$ is divided by $ 1000$.

[b]2.[/b] Show that threre exist a compact set $K \subset \mathbb{R}$ and a set $A \subset \mathbb{R}$ of type $F_{\sigma}$ such that the set $\{ x\in \mathbb{R} : K+x \subset A\}$ is not Borel-measurable (here $K+x = \{y+x : y \in K\}$). ([b]M.16[/b]) [M. Laczkovich]

Is it possible to draw $10$ bus routes with stops such that for any $8$ routes there is a stop that does not belong to any of the routes, but any $9$ routes pass through all the stops?