$\boxed{A1}$ Find all ordered triplets of $(x,y,z)$ real numbers that satisfy the following system of equation $x^3=\frac{z}{y}-\frac {2y}{z}$ $y^3=\frac{x}{z}-\frac{2z}{x}$ $z^3=\frac{y}{x}-\frac{2x}{y}$

Let $ S$ be the smallest subset of the integers with the property that $ 0\in S$ and for any $ x\in S$, we have $ 3x\in S$ and $ 3x \plus{} 1\in S$. Determine the number of non-negative integers in $ S$ less than $ 2008$.

$1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}$ is equal to $\textbf{(A)}\ \dfrac{15}{8} \qquad \textbf{(B)}\ 1\dfrac{3}{14} \qquad \textbf{(C)}\ \dfrac{11}{8} \qquad \textbf{(D)}\ 1\dfrac{3}{4} \qquad \textbf{(E)}\ \dfrac{7}{8}$

Divide the grid shown to the right into more than one region so that the following rules are satisfied. 1. Each unit square lies entirely within exactly 1 region. 2. Each region is a single piece connected by the edges of its unit squares. 3. Each region contains the same number of whole unit squares. 4. Each region contains the same sum of numbers. You do not need to prove that your configuration is the only one possible; you merely need to find a configuration that works. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] size(6cm); for (int i=0; i<=8; ++i) draw((i,0)--(i,7), linewidth(.8)); for (int j=0; j<=7; ++j) draw((0,j)--(8,j), linewidth(.8)); void draw_num(pair ll_corner, int num) { label(string(num), ll_corner + (0.5, 0.5), p = fontsize(18pt)); } draw_num((0, 0), 1); draw_num((1, 0), 1); draw_num((2, 0), 1); draw_num((0, 5), 4); draw_num((1, 1), 4); draw_num((1, 4), 3); draw_num((2, 2), 4); draw_num((3, 4), 3); draw_num((3, 5), 2); draw_num((4, 1), 4); draw_num((4, 3), 4); draw_num((5, 4), 4); draw_num((5, 6), 6); draw_num((6, 2), 3); draw_num((6, 5), 4); draw_num((6, 6), 5); draw_num((7, 1), 4); draw_num((7, 6), 6);[/asy]

Given a sheet of paper and the use of a rule, compass and pencil, show how to draw a straight line that passes through two given points, if the length of the ruler and the maximum opening of the compass are both less than half the distance between the two points. You may not fold the paper.

[b]a) [/b]Is the number $ 1111\cdots11$ (with $ 2010$ ones) a prime number? [b]b)[/b] Prove that every prime factor of $ 1111\cdots11$ (with $ 2011$ ones) is of the form $ 4022j\plus{}1$ where $ j$ is a natural number.

There is a queue of $n{}$ girls on one side of a tennis table, and a queue of $n{}$ boys on the other side. Both the girls and the boys are numbered from $1{}$ to $n{}$ in the order they stand. The first game is played by the girl and the boy with the number $1{}$ and then, after each game, the loser goes to the end of their queue, and the winner remains at the table. After a while, it turned out that each girl played exactly one game with each boy. Prove that if $n{}$ is odd, then a girl and a boy with odd numbers played in the last game. [i]Proposed by A. Gribalko[/i]

Given any $60$ points on a circle of radius $1$, prove that there is a point on the circle the sum of whose distances to these $60$ points is at most $80$.

Let the curve $ C: y\equal{}x\sqrt{9\minus{}x^2}\ (x\geq 0)$. (1) Find the maximum value of $ y$. (2) Find the area of the figure bounded by the curve $ C$ and the $ x$ axis. (3) Find the volume of the solid generated by rotation of the figure about the $ y$ axis.

In an infinite arithmetic progression of positive integers there are two integers with the same sum of digits. Will there necessarily be one more integer in the progression with the same sum of digits? [i]Proposed by A. Shapovalov[/i]

Let $n$ be a positive integer, and let $S_1,S_2,…,S_n$ be a collection of finite non-empty sets such that $$\sum_{1\leq i<j\leq n}{\frac{|S_i \cap S_j|}{|S_i||S_j|}} <1.$$ Prove that there exist pairwise distinct elements $x_1,x_2,…,x_n$ such that $x_i$ is a member of $S_i$ for each index $i$.

Suppose $ABCD$ is a cyclic quadrilateral with $\angle ABC = \angle ADC = 90^{\circ}$. Let $E$ and $F$ be the feet of perpendiculars from $A$ and $C$ to $BD$ respectively. Prove that $BE = DF$.

Jay is given a permutation $\{p_1, p_2,\ldots, p_8\}$ of $\{1, 2,\ldots, 8\}$. He may take two dividers and split the permutation into three non-empty sets, and he concatenates each set into a single integer. In other words, if Jay chooses $a,b$ with $1\le a< b< 8$, he will get the three integers $\overline{p_1p_2\ldots p_a}$, $\overline{p_{a+1}p_{a+2}\ldots p_{b}}$, and $\overline{p_{b+1}p_{b+2}\ldots p_8}$. Jay then sums the three integers into a sum $N=\overline{p_1p_2\ldots p_a}+\overline{p_{a+1}p_{a+2}\ldots p_b}+\overline{p_{b+1}p_{b+2}\ldots p_8}$. Find the smallest positive integer $M$ such that no matter what permutation Jay is given, he may choose two dividers such that $N\le M$. [i]Proposed by James Lin[/i]

Let $ m\geq n\geq 4$ be two integers. We call a $ m\times n$ board filled with 0's or 1's [i]good[/i] if 1) not all the numbers on the board are 0 or 1; 2) the sum of all the numbers in $ 3\times 3$ sub-boards is the same; 3) the sum of all the numbers in $ 4\times 4$ sub-boards is the same. Find all $ m,n$ such that there exists a good $ m\times n$ board.

Show that there is no integer $ a$ such that $ a^2 \minus{} 3a \minus{} 19$ is divisible by $ 289$.

We are given 4 similar dices. Denote $x_i (1\le x_i \le 6)$ be the number of dots on a face appearing on the $i$-th dice $1\le i \le 4$ a) Find the numbers of $(x_1,x_2,x_3,x_4)$ b) Find the probability that there is a number $x_j$ such that $x_j$ is equal to the sum of the other 3 numbers c) Find the probability that we can divide $x_1,x_2,x_3,x_4$ into 2 groups has the same sum

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

[b](a)[/b] Sketch the diagram of the function $f$ if \[f(x)=4x(1-|x|) , \quad |x| \leq 1.\] [b](b)[/b] Does there exist derivative of $f$ in the point $x=0 \ ?$ [b](c)[/b] Let $g$ be a function such that \[g(x)=\left\{\begin{array}{cc}\frac{f(x)}{x} \quad : x \neq 0\\ \text{ } \\ 4 \ \ \ \ \quad : x=0\end{array}\right.\] Is the function $g$ continuous in the point $x=0 \ ?$ [b](d)[/b] Sketch the diagram of $g.$

Find all pairs of real numbers $(x, y) $ such that: a) $x\ge y\ge1$ b) $2x^2-xy-5x +y + 4 = 0 $

Determine the sum of all positive integers $n<100$ satisfying the following expression. \[\sum_{k=0}^{\lfloor{\log_{10} n}\rfloor}\frac{1}{10^k}\left(n \; (\bmod \;{10^{k+1})}-n \;(\bmod \;{10^k)}\right)=\prod_{k=0}^{\lfloor{\log_{10} n}\rfloor}\frac{1}{10^k}\left(n \; (\bmod\; 10^{k+1})-n \;(\bmod\; 10^k)\right).\] Here, $\textstyle\sum$ and $\textstyle\prod$ represent sum and product, respectively. [i]2022 CCA Math Bonanza Lightning Round 3.3[/i]

Let $n$ be a positive integer, and Megavan has a $(3n+1)\times (3n+1)$ board. All squares, except one, are tiled by non-overlapping $1\times 3$ triominoes. In each step, he can choose a triomino that is untouched in the step right before it, and then shift this triomino horizontally or vertically by one square, as long as the triominoes remain non-overlapping after this move. Show that there exist some $k$, such that after $k$ moves Megavan can no longer make any valid moves irregardless of the initial configuration, and find the smallest possible $k$ for each $n$. [i](Note: While he cannot undo a move immediately before the current step, he may still choose to move a triomino that has already been moved at least two steps before.)[/i] [i]Proposed by Ivan Chan Kai Chin[/i]

Let $x_1,x_2,\dots,x_{100}$ be real numbers such that $|x_1|=63$ and $|x_{n+1}|=|x_n+1|$ for $n=1,2\dots,99$. Find the largest possible value of $(-x_1-x_2-\cdots-x_{100})$.

There are three bags. One bag contains three green candies and one red candy. One bag contains two green candies and two red candies. One bag contains one green candy and three red candies. A child randomly selects one of the bags, randomly chooses a first candy from that bag, and eats the candy. If the first candy had been green, the child randomly chooses one of the other two bags and randomly selects a second candy from that bag. If the first candy had been red, the child randomly selects a second candy from the same bag as the first candy. If the probability that the second candy is green is given by the fraction $m/n$ in lowest terms, find $m + n$.

What is the product of all the solutions to the equation $$\log_{7x}2023 \cdot \log_{289x} 2023 = \log_{2023x} 2023?$$ $\textbf{(A) }(\log_{2023}7 \cdot \log_{2023}289)^2 \qquad\textbf{(B) }\log_{2023}7 \cdot \log_{2023}289\qquad\textbf{(C) }1\qquad\textbf{(D) }\log_{7}2023 \cdot \log_{289}2023\qquad\textbf{(E) }(\log_{7}2023 \cdot \log_{289}2023)^2$

In the sequence $ (a_n)$ with general term $ a_n \equal{} n^3 \minus{} (2n \plus{} 1)^2$, does there exist a term that is divisible by 2006?