Let $ABC$ be a triangle with symmedian point $K$. Select a point $A_1$ on line $BC$ such that the lines $AB$, $AC$, $A_1K$ and $BC$ are the sides of a cyclic quadrilateral. Define $B_1$ and $C_1$ similarly. Prove that $A_1$, $B_1$, and $C_1$ are collinear. [i]Proposed by Sammy Luo[/i]

Let $a$, $b$, $c$ be non-negative real numbers such that $ab + bc + ca = 3$. Suppose that \[a^3 b + b^3 c + c^3 a + 2abc(a + b + c) = \frac{9}{2}.\] What is the maximum possible value of $ab^3 + bc^3 + ca^3$?

How many integers from $1$ to $2008$ have the sum of their digits divisible by $5$ ?

Consider the set $S$ of lattice points $(x,y)$ with $0\le x,y\le 8$. Call a function $f:S\to \{1,2,\dots, 9\}$ a [i]Sudoku function[/i] if: [list] [*] $\{ f(x,0), f(x,1), \dots, f(x,8)\} = \{1,2,\dots, 9\}$ for each $0\le x\le 8$ and $\{ f(0,y), f(1,y), \dots, f(8,y) \} = \{1,2,\dots, 9\}$ for each $0\le y\le 8$. [*] For any integers $0\le m,n\le 2$ and any $0\le i_1,j_1,i_2,j_2\le 2$, $f(3m+i_1, 3n+j_1)\neq f(3m+i_2, 3n+j_2)$ unless $i_1=i_2$ and $j_1=j_2$. [/list] Over all Sudoku functions $f$, compute the maximum possible value of $\sum_{0\le i\le 8} f(i,i) + \sum_{0\le i\le 7} f(i, i+1)$. [i]Proposed by Brandon Wang[/i]

If $a,b,c$ are positive real numbers prove that: $\frac{a}{b}+\sqrt{\frac{b}{c}}+\sqrt[3]{\frac{c}{a}}>2.$

Two players, $A$ and $B,$ alternatively take stones from a pile of $n \geq 2$ stones. $A$ plays first and in his first move he must take at least one stone and at most $n-1$ stones. Then each player must take at least one stone and at most as many stones as his opponent took in the previous move. The player who takes the last stone wins. Which player has a winning strategy?

Let $a$, $b$, $c$, and $d$ be positive integers such that $77^a \cdot 637^b = 143^c \cdot 49^d$. Compute the minimal value of $a+b+c+d$. [i]2022 CCA Math Bonanza Team Round #1[/i]

Draw a tangent with positive slope to a parabola $y=x^2+1$. Find the $x$-coordinate such that the area of the figure bounded by the parabola, the tangent and the coordinate axisis is $\frac{11}{3}.$

Let $a_1<a_2<a_3<a_4<\cdots$ be an infinite sequence of real numbers in the interval $(0,1)$. Show that there exists a number that occurs exactly once in the sequence \[ \frac{a_1}{1},\frac{a_2}{2},\frac{a_3}{3},\frac{a_4}{4},\ldots.\] [i]Merlijn Staps[/i]

Prove that no number of the form $a^3+3a^2+a$, for a positive integer $a$, is a perfect square.

A $ 6\times 6$ square is dissected in to 9 rectangles by lines parallel to its sides such that all these rectangles have integer sides. Prove that there are always [b]two[/b] congruent rectangles.

Pairwise distinct real numbers $a, b, c$ satisfies the equality \[a +\frac 1b =b + \frac 1c =c+\frac 1a.\] Find all possible values of $abc .$

On blackboard is written $3$ digit number so all three digits are distinct than zero. Out of it, we made three $2$ digit numbers by crossing out first digit of original number, crossing out second digit of original number and crossing out third digit of original number. Sum of those three numbers is $293$. Which number is written on blackboard?

Two players $A$ and $B$ take stones one after the other from a heap with $n \ge 2$ stones. $A$ begins the game and takes at least one stone, but no more than $n -1$ stones. Thereafter, a player on turn takes at least one, but no more than the other player has taken before him. The player who takes the last stone wins. Who of the players has a winning strategy?

Given a positive integer $ n> 2 $. Prove that there exists a natural $ K $ such that for all integers $ k \ge K $ on the open interval $ ({{k} ^{n}}, \ {{(k + 1)} ^{n}}) $ there are $n$ different integers, the product of which is the $n$-th power of an integer.

Let \(ABC\) be a triangle with \(AB < AC\) and let \(\omega\) be its circumcircle. Let \(M\) denote the midpoint of side \(BC\) and \(N\) the midpoint of arc \(BC\) of \(\omega\) that contains \(A\). The circumcircle of triangle \(AMN\) intersects sides \(AB\) and \(AC\) at points \(P\) and \(Q\), respectively. Prove that \(BP = CQ\).

Prove that the midpoints of the altitudes of a triangle are collinear if and only if the triangle is right. [i]Dorin Popovici[/i]

Which of the following is equal to the product \[ \frac {8}{4}\cdot\frac {12}{8}\cdot\frac {16}{12}\cdots\frac {4n \plus{} 4}{4n}\cdots\frac {2008}{2004}? \]$ \textbf{(A)}\ 251 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 2008 \qquad \textbf{(E)}\ 4016$

Let $\mathfrak F$ be the family of all $k$-element subsets of the set $\{1, 2, \ldots, 2k + 1\}$. Prove that there exists a bijective function $f :\mathfrak F \to \mathfrak F$ such that for every $A \in \mathfrak F$, the sets $A$ and $f(A)$ are disjoint.

$n, k$ are positive integers. $A_0$ is the set $\{1, 2, ... , n\}$. $A_i$ is a randomly chosen subset of $A_{i-1}$ (with each subset having equal probability). Show that the expected number of elements of $A_k$ is $\dfrac{n}{2^k}$

A fair 6-sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number? $ \textbf{(A)}\dfrac16\qquad\textbf{(B)}\dfrac5{12}\qquad\textbf{(C)}\dfrac12\qquad\textbf{(D)}\dfrac7{12}\qquad\textbf{(E)}\dfrac56 $

Let $ABC$ be a scalene triangle. Let $I_0=A$ and, for every positive integer $t$, let $I_t$ be the incenter of triangle $I_{t-1}BC$. Suppose that the points $I_0,I_1,I_2,\ldots$ all lie on some hyperbola $\mathcal{H}$ whose asymptotes are lines $\ell_1$ and $\ell_2$. Let the line through $A$ perpendicular to line $BC$ intersect $\ell_1$ and $\ell_2$ at points $P$ and $Q$ respectively. Suppose that $AC^2=\frac{12}{7}AB^2+1$. Then the smallest possible value of the area of quadrilateral $BPCQ$ is $\frac{j\sqrt{k}+l\sqrt{m}}{n}$ for positive integers $j$, $k$, $l$, $m$, and $n$ such that $\gcd(j,l,n)=1$, both $k$ and $m$ are squarefree, and $j>l$. Compute $10000j+1000k+100l+10m+n$. [i]Proposed by Gopal Goel, Luke Robitaille, Ashwin Sah, & Eric Shen[/i]

Given a polynomial \[P(x) = ab(a - c)x^3 + (a^3 - a^2c + 2ab^2 - b^2c + abc)x^2 +(2a^2b + b^2c + a^2c + b^3 - abc)x + ab(b + c),\] where $a, b, c \neq 0$, prove that $P(x)$ is divisible by \[Q(x) = abx^2 + (a^2 + b^2)x + ab\] and conclude that $P(x_0)$ is divisible by $(a + b)^3$ for $x_0 = (a + b + 1)^n, n \in \mathbb N$.

Let $r = 1-\sqrt[5]{2}+ \sqrt[5]{4}-\sqrt[5]{8}+ \sqrt[5]{16}$. There exists a unique fifth-degree polynomial $P$ such that its leading coefficient is positive, all of its coefficients are integers whose greatest common factor (among all of them) is $1$, and $P(r) = 0$. Evaluate $P(10)$.

Find all pairs of positive integers $(a,b)$ such that $\sqrt{a-1}+\sqrt{b-1}=\sqrt{ab-1}.$