A square is cut into red and blue rectangles. The sum of areas of red triangles is equal to the sum of areas of the blue ones. For each blue rectangle, we write the ratio of the length of its vertical side to the length of its horizontal one and for each red rectangle, the ratio of the length of its horizontal side to the length of its vertical side. Find the smallest possible value of the sum of all the written numbers.

Betty has a $4$-by-$4$ square box of chocolates. Every time Betty eats a chocolate, she picks one from a row with the greatest number of remaining chocolates. In how many ways can Betty eat $5$ chocolates from her box, where order matters?

A directed graph has each vertex with outdegree 2. Prove that it is possible to split the vertices into 3 sets so that for each vertex $v$, $v$ is not simultaneously in the same set with both of the vertices that it points to. [i]David Yang.[/i] [hide="Stronger Version"]See [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=492100]here[/url].[/hide]

Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. Let $l$ be the line tangent to $\Gamma$ at $A$. Let $D$ and $E$ be the intersections of the circumference with center $B$ and radius $AB$ with lines $l$ and $AC$, respectively. Prove the orthocenter of $ABC$ lies on line $DE$.

Juan did not like the criticism of his classmates published in his school newspaper. He found nothing better than to start ripping up the diary. First he tore it into $4$ parts and then he continued to break it in a very methodical way: namely, each piece of newspaper he found he would tear it back into $4$ or $10$ pieces randomly. Breaking this way, was he able to get exactly $2003$ pieces of the diary?

Let $K$ and $M$ be points on the side $AB$ of a triangle $\triangle{ABC}$, and let $L$ and $N$ be points on the side $AC$. The point $K$ is between $M$ and $B$, and the point $L$ is between $N$ and $C$. If $\frac{BK}{KM}=\frac{CL}{LN}$, then prove that the orthocentres of the triangles $\triangle{ABC}$, $\triangle{AKL}$ and $\triangle{AMN}$ lie on one line.

If $k$ raisins are distributed evenly to eleven children, four raisins would be left over. How many raisins would be left over if $3k$ raisins were distributed instead? $\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }7$

Let $A$ and $B$ be $2\times 2$ matrices with integer entries, such that $AB=BA$ and $\det B=1$. Prove tht if $\det(A^3+B^3)=1$, then $A^2=O$.

Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that: [b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color, and [b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$. [i]Author: Gerhard Wöginger, Netherlands[/i]

Let $A, B, C, A', B', C'$ be six pairs of different points. Prove that the Circles $BCA'$, $CAB'$ and $ABC'$ have a common point, then the Circles $B'C'A, C'A'B$ and $A'B'C$ also share a common point. Note: For three pairs of different points $X, Y$ and $Z$ we define the [i]Circle [/i] $XYZ$ as the circumcircle of the triangle $XYZ$, or - in the case when the points $X, Y$ and $Z$ lie on a straight line - this straight line.

If an angle of a triangle remains unchanged but each of its two including sides is doubled, then the area is multiplied by: $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ \text{more than }6$

Find how many integer values $3\le n \le 99$ satisfy that the polynomial $x^2 + x + 1$ divides $x^{2^n} + x + 1$.

Leticia has a $9\times 9$ board. She says that two squares are [i]friends[/i] is they share a side, if they are at opposite ends of the same row or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of three colors: green, blue or red. In each square a number will be written based on the following rules: - If the square is green, write the number of red friends plus twice the number of blue friends. - If the square is red, write the number of blue friends plus twice the number of green friends. - If the square is blue, write the number of green friends plus twice the number of red friends. Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares.

Ben has a square of side length $2.$ He wants to put a circle and an equilateral triangle inside the square such that the circle and equilateral triangle do not overlap. The maximum possible sum of the areas of the circle and triangle is $\tfrac{a\pi+b\sqrt{c}+d\sqrt{e}}{f},$ where $a,c,e,f$ are positive integers, $b$ and $d$ are integers, $c$ and $e$ are square-free, and $\gcd(a,b,d,f)=1.$ Find $a+b+c+d+e+f.$

At 12 o'clock in the afternoon, "Zaporozhets" and "Moskvich" were at a distance of 90 km and began to move towards each other at a constant speed. Two hours later they were again at a distance of 90 km. Dunno claims that ''Zaporozhets'' before meeting with ''Moskvich'' and ''Moskvich'' after the meeting with ''Zaporozhets'' , have drove a total of 60 km. Prove that he is wrong. [hide=original wording]В 12 часов дня ''Запорожец'' и ''Москвич'' находилисьна расстоянии 90 км и начали двигаться навстречу друг другу с постоянной скоростью. Через два часа они снова оказались на расстоянии 90 км. Незнайка утверждает, что ''Запорожец'' до встречи с ''Москвичом'' и ''Москвич'' после встречи с ''Запорожцем'' проехали в сумме 60 км. Докажите, что он не прав. [/hide]

Let $S = {1, 2, \cdots, 100}.$ $X$ is a subset of $S$ such that no two distinct elements in $X$ multiply to an element in $X.$ Find the maximum number of elements of $X$. [i]2022 CCA Math Bonanza Individual Round #3[/i]