Let $ABC$ be an acute triangle with $AB < AC$. The points $A_1$ and $A_2$ are located on the line $BC$ so that $AA_1$ and $AA_2$ are the inner and outer angle bisectors at $A$ for the triangle $ABC$. Let $A_3$ be the mirror image $A_2$ with respect to $C$, and let $Q$ be a point on $AA_1$ such that $\angle A_1QA_3 = 90^o$. Show that $QC // AB$.

Circle $\omega$ is inscribed in acute triangle $ABC$. Let $I$ denote the center of $\omega$, and let $D,E,F$ be the points of tangency of $\omega$ with $BC, CA, AB$ respectively. Let $M$ be the midpoint of $BC$, and $P$ be the intersection of the line through $I$ perpendicular to $AM$ and line $EF$. Suppose that $AP=9$, $EC=2EA$, and $BD=3$. Find the sum of all possible perimeters of $\triangle ABC$. [i]Proposed by Harry Kim[/i]

Let $ABC$ be a right triangle with $AB=3$, $BC=4$, and $\angle B = 90^\circ$. Points $P$, $Q$, and $R$ are chosen on segments $AB$, $BC$, and $CA$, respectively, such that $PQR$ is an equilateral triangle, and $BP=BQ$. Given that $BP$ can be written as $\frac{\sqrt{a}-b}{c}$, where $a,b,c$ are positive integers and $\gcd(b,c)=1$, what is $a+b+c$? [i]2021 CCA Math Bonanza Individual Round #6[/i]

Let $m,n > 2$ be even integers. Consider a board of size $m \times n$ whose every cell is colored either black or white. The Guesser does not see the coloring of the board but may ask the Oracle some questions about it. In particular, she may inquire about two adjacent cells (sharing an edge) and the Oracle discloses whether the two adjacent cells have the same color or not. The Guesser eventually wants to find the parity of the number of adjacent cell-pairs whose colors are different. What is the minimum number of inquiries the Guesser needs to make so that she is guaranteed to find her answer?(Czech Republic)

Let $a,b$ be positive real numbers.Prove that $(1+a)^{8}+(1+b)^{8}\geq 128ab(a+b)^{2}$.

If $A$, $B$, $C$, $D$ are four distinct points such that every circle through $A$ and $B$ intersects (or coincides with) every circle through $C$ and $D$, prove that the four points are either collinear (all on one line) or concyclic (all on one circle).

For all positive integers $m,n$ define $f(m,n) = m^{3^{4n}+6} - m^{3^{4n}+4} - m^5 + m^3$. Find all numbers $n$ with the property that $f(m, n)$ is divisible by 1992 for every $m$. [i]Bulgaria[/i]

Isosceles triangle $\triangle{ABC}$ has $\angle{BAC}=\angle{ABC}=30^\circ$ and $AC=BC=2$. If the midpoints of $BC$ and $AC$ are $M$ and $N$, respectively, and the circumcircle of $\triangle{CMN}$ meets $AB$ at $D$ and $E$ with $D$ closer to $A$ than $E$ is, what is the area of $MNDE$? [i]2019 CCA Math Bonanza Individual Round #9[/i]

In a ten-mile race First beats Second by $2$ miles and First beats Third by $4$ miles. If the runners maintain constant speeds throughout the race, by how many miles does Second beat Third? $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 2\frac{1}{4}\qquad\textbf{(C)}\ 2\frac{1}{2}\qquad\textbf{(D)}\ 2\frac{3}{4}\qquad\textbf{(E)}\ 3 $

Two non-zero real numbers, $a$ and $b$, satisfy $ab=a-b$. Which of the following is a possible value of $\frac ab+\frac ba-ab$? $\text{(A)}\ -2\qquad\text{(B)}\ -\frac12\qquad\text{(C)}\ \frac13\qquad\text{(D)}\ \frac12\qquad\text{(E)}\ 2$

Given an integer $n\ge\ 3$, find the least positive integer $k$, such that there exists a set $A$ with $k$ elements, and $n$ distinct reals $x_{1},x_{2},\ldots,x_{n}$ such that $x_{1}+x_{2}, x_{2}+x_{3},\ldots, x_{n-1}+x_{n}, x_{n}+x_{1}$ all belong to $A$.

Let $ A,B,C,D $ be four $ 2\times 2 $ complex matrices such that $ A-D $ is invertible and such that $$ A^2+BA+C=0=D^2+BD+C. $$ Prove that $ \text{tr} (A+D) =-\text{tr} B $ and $ \det (AD) =\det C. $

There are $100$ random, distinct real numbers corresponding to $100$ points on a circle. Prove that you can always choose $4$ consecutive points in such a way that the sum of the two numbers corresponding to the points on the outside is always greater than the sum of the two numbers corresponding to the two points on the inside.

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of positive integers such that $a_1=1$. For each $n \geq 1$, $a_{n+1}$ is the smallest positive integer, distinct from $a_1,a_2,...,a_n$, such that $\gcd(a_{n+1}a_n+1,a_i)=1$ for each $i=1,2,...,n$. Prove that every positive integer appears in $\{a_n\}_{n=1}^{\infty}$.

If $A$ is the $3\times 3$ square matrix $\begin{bmatrix} 5 & 3 & 8\\ 2 & 2 & 5\\ 3 & 5 & 1 \end{bmatrix}$ and $B$ is the $4\times 4$ square matrix $\begin{bmatrix} 32 & 2 & 4 & 3 \\ 3 & 4 & 8 & 3 \\ 11 & 3 & 6 & 1 \\ 5 & 5 & 10 & 1 \end{bmatrix} $ find the sum of the determinants of $A$ and $B$.

For any $n$ consecutive integers $a_1, \cdots, a_n$, prove that $$(a_1+\cdots+a_n)\cdot\left(\frac{1}{a_1}+\cdots+\frac{1}{a_n}\right)\leqslant \frac{n(n+1)\ln(\text{e}n)}{2}.$$

It is known that a cells square can be cut into $n$ equal figures of $k$ cells. Prove that it is possible to cut it into $k$ equal figures of $n$ cells.

Let $x,y,z\in \mathbb{R}^+$ be such that $xy+yz+zx=x+y+z$. Prove the following inequality: $\frac{1}{x^2+y+1}+\frac{1}{y^2+z+1}+\frac{1}{z^2+x+1}\leq 1$.

Compute the number of ordered pairs of integers $(a, b)$, where $0 \le a < 17$ and $0 \le b < 17$, such that $y^2 \equiv x^3 +ax +b \pmod{17}$ has an even number of solutions $(x, y)$, where $0 \le x < 17$ and $0 \le y < 17$ are integers.

Let $P(x)$ be a polynomial taking integer values at integer inputs. Are there infinitely many natural numbers that are not representable in the form $P(k)-2^n$ where $n{}$ and $k{}$ are non-negative integers? [i]Proposed by F. Petrov[/i]

Let $\{g_i\}_{i=0}^{\infty}$ be a sequence of positive integers such that $g_0=g_1=1$ and the following recursions hold for every positive integer $n$: \begin{align*} g_{2n+1} &= g_{2n-1}^2+g_{2n-2}^2 \\ g_{2n} &= 2g_{2n-1}g_{2n-2}-g_{2n-2}^2 \end{align*} Compute the remainder when $g_{2011}$ is divided by $216$.

For any positive integer $n>1$, let $p(n)$ be the greatest prime divisor of $n$. Prove that there are infinitely many positive integers $n$ with \[p(n)<p(n+1)<p(n+2).\]

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

Let the vertex set \( V \) of a graph be partitioned into \( h \) parts \( (V = V_1 \cup V_2 \cup \cdots \cup V_h) \), with \(|V_1| = n_1, |V_2| = n_2, \ldots, |V_h| = n_h \). If there is an edge between any two vertices only when they belong to different parts, the graph is called a complete \( h \)-partite graph, denoted as \( k(n_1, n_2, \ldots, n_h) \). Let \( n \) and \( r \) be positive integers, \( n \geq 6 \), \( r \leq \frac{2}{3}n \). Consider the complete \( r + 1 \)-partite graph \( k\left(\underbrace{1, 1, \ldots, 1}_{r}, n - r\right) \). Answer the following questions: 1. Find the maximum number of disjoint circles (i.e., circles with no common vertices) in this complete \( r + 1 \)-partite graph. 2. Given \( n \), for all \( r \leq \frac{2}{3}n \), find the maximum number of edges in a complete \( r + 1 \)-partite graph \( k(1, 1, \ldots, 1, n - r) \) where no more than one circle is disjoint.

A toboggan sled is traveling at $\text{2.0 m/s}$ across the snow. The sled and its riders have a combined mass of $\text{120 kg}$. Another child ($m_{\text{child}} = \text{40 kg}$) headed in the opposite direction jumps on the sled from the front. She has a speed of $\text{5.0 m/s}$ immediately before she lands on the sled. What is the new speed of the sled? Neglect any effects of friction. (a) $\text{0.25 m/s}$ (b) $\text{0.33 m/s}$ (c) $\text{2.75 m/s}$ (d) $\text{3.04 m/s}$ (e) $\text{3.67 m/s}$