Let n be a set of integers. $S(n)$ is defined as the sum of the elements of n. $T=\{1,2,3,4,5,6,7,8,9\}$ and A and B are subsets of T such that A $\cup$ $B=T$ and A $\cap$ $B=\varnothing$. The probability that $S(A)\geq4S(B)$ can be expressed as $\frac{p}{q}$. Compute $p+q$. [i]2022 CCA Math Bonanza Team Round #8[/i]

A paper triangle with the angles $20^o$, $20^o$ and $140^o$ is cut into two triangles by the bisector of one of its angles. Then one of these triangles is cut into two by its bisector, and so on. Prove that it is impossible to get a triangle similar to the initial one. (AI Galochkin)

Determine all pairs $\{a, b\}$ of positive integers with the property that, in whatever manner you color the positive integers with two colors $A$ and $B$, there always exist two positive integers of color $A$ having their difference equal to $a$ [b]or[/b] of color $B$ having their difference equal to $b$.

Let {$a_1, a_2,\cdots,a_n$} be a set of $n$ real numbers whos sym equals S. It is known that each number in the set is less than $\frac{S}{n-1}$. Prove that for any three numbers $a_i$, $a_j$ and $a_k$ in the set, $a_i+a_j>a_k$.

Consider the sequence defined recursively by $(x_1,y_1)=(0,0)$, $(x_{n+1},y_{n+1})=\left(\left(1-\frac{2}{n}\right)x_n-\frac{1}{n}y_n+\frac{4}{n},\left(1-\frac{1}{n}\right)y_n-\frac{1}{n}x_n+\frac{3}{n}\right)$. Find $\lim_{n\to \infty}(x_n,y_n)$.

On a checkerboard composed of 64 unit squares, what is the probability that a randomly chosen unit square does [b] not [/b] touch the outer edge of the board? [asy] unitsize(10); draw((0,0)--(8,0)--(8,8)--(0,8)--cycle); draw((1,8)--(1,0)); draw((7,8)--(7,0)); draw((6,8)--(6,0)); draw((5,8)--(5,0)); draw((4,8)--(4,0)); draw((3,8)--(3,0)); draw((2,8)--(2,0)); draw((0,1)--(8,1)); draw((0,2)--(8,2)); draw((0,3)--(8,3)); draw((0,4)--(8,4)); draw((0,5)--(8,5)); draw((0,6)--(8,6)); draw((0,7)--(8,7)); fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,black); fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,black); fill((4,0)--(5,0)--(5,1)--(4,1)--cycle,black); fill((6,0)--(7,0)--(7,1)--(6,1)--cycle,black); fill((0,2)--(1,2)--(1,3)--(0,3)--cycle,black); fill((2,2)--(3,2)--(3,3)--(2,3)--cycle,black); fill((4,2)--(5,2)--(5,3)--(4,3)--cycle,black); fill((6,2)--(7,2)--(7,3)--(6,3)--cycle,black); fill((0,4)--(1,4)--(1,5)--(0,5)--cycle,black); fill((2,4)--(3,4)--(3,5)--(2,5)--cycle,black); fill((4,4)--(5,4)--(5,5)--(4,5)--cycle,black); fill((6,4)--(7,4)--(7,5)--(6,5)--cycle,black); fill((0,6)--(1,6)--(1,7)--(0,7)--cycle,black); fill((2,6)--(3,6)--(3,7)--(2,7)--cycle,black); fill((4,6)--(5,6)--(5,7)--(4,7)--cycle,black); fill((6,6)--(7,6)--(7,7)--(6,7)--cycle,black); fill((1,1)--(2,1)--(2,2)--(1,2)--cycle,black); fill((3,1)--(4,1)--(4,2)--(3,2)--cycle,black); fill((5,1)--(6,1)--(6,2)--(5,2)--cycle,black); fill((7,1)--(8,1)--(8,2)--(7,2)--cycle,black); fill((1,3)--(2,3)--(2,4)--(1,4)--cycle,black); fill((3,3)--(4,3)--(4,4)--(3,4)--cycle,black); fill((5,3)--(6,3)--(6,4)--(5,4)--cycle,black); fill((7,3)--(8,3)--(8,4)--(7,4)--cycle,black); fill((1,5)--(2,5)--(2,6)--(1,6)--cycle,black); fill((3,5)--(4,5)--(4,6)--(3,6)--cycle,black); fill((5,5)--(6,5)--(6,6)--(5,6)--cycle,black); fill((7,5)--(8,5)--(8,6)--(7,6)--cycle,black); fill((1,7)--(2,7)--(2,8)--(1,8)--cycle,black); fill((3,7)--(4,7)--(4,8)--(3,8)--cycle,black); fill((5,7)--(6,7)--(6,8)--(5,8)--cycle,black); fill((7,7)--(8,7)--(8,8)--(7,8)--cycle,black);[/asy] $ \textbf{(A)}\frac{1}{16}\qquad\textbf{(B)}\frac{7}{16}\qquad\textbf{(C)}\frac12\qquad\textbf{(D)}\frac{9}{16}\qquad\textbf{(E)}\frac{49}{64} $

Let $C$ be a circle of radius $1$ and $O$ its center. Let $\overline{AB}$ be a chord of the circle and $D$ a point on $\overline{AB}$ such that $OD =\frac{\sqrt2}{2}$ such that $D$ is closer to $ A$ than it is to $ B$, and if the perpendicular line at $D$ with respect to $\overline{AB}$ intersects the circle at $E $and $F$, $AD = DE$. The area of the region of the circle enclosed by $\overline{AD}$, $\overline{DE}$, and the minor arc $AE$ may be expressed as $\frac{a + b\sqrt{c} + d\pi}{e}$ where $a, b, c, d, e$ are integers, gcd $(a, b, d, e) = 1$, and $c$ is squarefree. Find $a + b + c + d + e$

a) Solve the system of equations $\begin{cases} 1 - x_1x_2 = 0 \\ 1 - x_2x_3 = 0 \\ ...\\ 1 - x_{14}x_{15} = 0 \\ 1 - x_{15}x_1 = 0 \end{cases}$ b) Solve the system of equations $\begin{cases} 1 - x_1x_2 = 0 \\ 1 - x_2x_3 = 0 \\ ...\\ 1 - x_{n-1}x_{n} = 0 \\ 1 - x_{n}x_1 = 0 \end{cases}$ How does the solution vary for distinct values of $n$?

Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as \[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \] where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.

Let $x,y,z$ be real numbers with $|x|,|y|,|z| > 2$. What is the smallest possible value of $|xyz+2(x+y+z)|$ ?

Ryan uses $91$ puzzle pieces to make a rectangle. Each of them is identical to one of the tiles shown. Given that pieces can be flipped or rotated, find the number of pieces that are red in the puzzle. (He is not allowed to join two ``flat sides'' together.)

Let $\Gamma$ is a circle with diameter $AB$. Let $\ell$ be the tangent of $\Gamma$ at $A$, and $m$ be the tangent of $\Gamma$ through $B$. Let $C$ be a point on $\ell$, $C \ne A$, and let $q_1$ and $q_2$ be two lines that passes through $C$. If $q_i$ cuts $\Gamma$ at $D_i$ and $E_i$ ($D_i$ is located between $C$ and $E_i$) for $i = 1, 2$. The lines $AD_1, AD_2, AE_1, AE_2$ intersects $m$ at $M_1, M_2, N_1, N_2$ respectively. Prove that $M_1M_2 = N_1N_2$.

Let $ E$ be the set of all triangles whose only points with integer coordinates (in the Cartesian coordinate system in space), in its interior or on its sides, are its three vertices, and let $ f$ be the function of area of a triangle. Determine the set of values $ f(E)$ of $ f.$

Let $ u_1, u_2, \ldots, u_m$ be $ m$ vectors in the plane, each of length $ \leq 1,$ with zero sum. Show that one can arrange $ u_1, u_2, \ldots, u_m$ as a sequence $ v_1, v_2, \ldots, v_m$ such that each partial sum $ v_1, v_1 \plus{} v_2, v_1 \plus{} v_2 \plus{} v_3, \ldots, v_1, v_2, \ldots, v_m$ has length less than or equal to $ \sqrt {5}.$

Say that an integer $A$ is [i]yummy[/i] if there exist several consecutive integers (including $A$) that add up to 2014. What is the smallest yummy integer?

Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.

Let $ABCD$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $AC$ is tangent to the circle $ADQ$, and the line $BD$ is tangent to the circle $BCP$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AD$, respectively. Prove that the following three lines are concurrent: line $CD$, the tangent of circle $ANQ$ at point $A$, and the tangent to circle $BMP$ at point $B$.

There are $n$ natural numbers written on a blackboard, where $n\in\mathbb{N},\ n\geq 2$. During each step two chosen numbers $a,b$, having the property that none of them divides the other, are replaced by their greatest common divisor and least common multiple. Prove that after a number of steps, all the numbers on the blackboard cease modifying. Prove that the respective number of steps is at most $(n-1)!$.

Oleksii chose $11$ pairwise distinct positive integer numbers not exceeding $2025$. Prove that among them, it is possible to choose two numbers \(a < b\) such that the number \(b\) gives an even remainder when divided by the number \(a\). [i]Proposed by Anton Trygub[/i]

Let $ p(x) \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c$, where $ a$, $ b$, and $ c$ are complex numbers. Suppose that \[ p(2009 \plus{} 9002\pi i) \equal{} p(2009) \equal{} p(9002) \equal{} 0 \]What is the number of nonreal zeros of $ x^{12} \plus{} ax^8 \plus{} bx^4 \plus{} c$? $ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 12$

Let $b_1 = 1$ and $ b_{n+1} = 1 + \frac{1}{n(n+1)b_1b_2...b_n}$ for $n \ge 1$. Find $b_12$.

Let $ P$ be the product of all non-zero digits of the positive integer $ n$. For example, $ P(4) \equal{} 4$, $ P(50) \equal{} 5$, $ P(123) \equal{} 6$, $ P(2009) \equal{} 18$. Find the value of the sum: P(1) + P(2) + ... + P(2008) + P(2009).

Triangle $ABC$ satisfies $AB=28$, $BC=32$, and $CA=36$, and $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively. Let point $P$ be the unique point in the plane $ABC$ such that $\triangle PBM\sim\triangle PNC$. What is $AP$?

Let $ABC$ be a given triangle. Consider any painting of points of the plane in red and green. Show that there exist either two red points on the distance $1$, or three green points forming a triangle congruent to triangle $ABC$.

A knight is modified so that it moves $p$ fields horizontally or vertically and $q$ fields in the perpendicular direction. It is placed on an infinite chessboard. If the knight returns to the initial field after $n$ moves, show that $n$ must be even.