On an infinite chessboard (whose squares are labeled by $ (x, y)$, where $ x$ and $ y$ range over all integers), a king is placed at $ (0, 0)$. On each turn, it has probability of $ 0.1$ of moving to each of the four edge-neighboring squares, and a probability of $ 0.05$ of moving to each of the four diagonally-neighboring squares, and a probability of $ 0.4$ of not moving. After $ 2008$ turns, determine the probability that the king is on a square with both coordinates even. An exact answer is required.

Let $S=\{(a,b)|a=1,2,\dots,n,b=1,2,3\}$. A [i]rook tour[/i] of $S$ is a polygonal path made up of line segments connecting points $p_1,p_2,\dots,p_{3n}$ is sequence such that (i) $p_i\in S,$ (ii) $p_i$ and $p_{i+1}$ are a unit distance apart, for $1\le i<3n,$ (iii) for each $p\in S$ there is a unique $i$ such that $p_i=p.$ How many rook tours are there that begin at $(1,1)$ and end at $(n,1)?$ (The official statement includes a picture depicting an example of a rook tour for $n=5.$ This example consists of line segments with vertices at which there is a change of direction at the following points, in order: $(1,1),(2,1),(2,2),(1,2), (1,3),(3,3),(3,1),(4,1), (4,3),(5,3),(5,1).$)

Find all pairs of integers $(x,y)$ such that $x \geq 0$ and \[ (6^x-y)^2 = 6^{x+1}-y. \]

In an equilateral trapezoid, the point $O$ is the midpoint of the base $AD$. A circle with a center at a point $O$ and a radius $BO$ is tangent to a straight line $AB$. Let the segment $AC$ intersect this circle at point $K(K \ne C)$, and let $M$ is a point such that $ABCM$ is a parallelogram. The circumscribed circle of a triangle $CMD$ intersects the segment $AC$ at a point $L(L\ne C)$. Prove that $AK=CL$.

Given that positive integers $a,b$ satisfy \[\frac{1}{a+\sqrt{b}}=\sum_{i=0}^\infty \frac{\sin^2\left(\frac{10^\circ}{3^i}\right)}{\cos\left(\frac{30^\circ}{3^i}\right)},\] where all angles are in degrees, compute $a+b$. [i]2021 CCA Math Bonanza Team Round #10[/i]

In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.

Clara and Nick each randomly and independently pick an integer between $0$ and $2017$, inclusive. What is the probability that the two integers they pick sum to an even number?

A black pawn and a white pawn are placed on the first square and the last square of a $ 1\times n$ chessboard, respectively. Wiwit and Siti move alternatingly. Wiwit has the white pawn, and Siti has the black pawn. The white pawn moves first. In every move, the player moves her pawn one or two squares to the right or to the left, without passing the opponent's pawn. The player who cannot move anymore loses the game. Which player has the winning strategy? Explain the strategy.

In a tournament with $2015$ teams, each team plays every other team exactly once and no ties occur. Such a tournament is [i]imbalanced [/i]if for every group of $6$ teams, there exists either a team that wins against the other $5$ or a team that loses to the other $5$. If the teams are indistinguishable, what is the number of distinct imbalanced tournaments that can occur?

Suppose that positive integers $a_1,a_2,\ldots,a_{2014}$ (not necessarily distinct) satisfy the condition that: $\frac{a_1}{a_2},\frac{a_2}{a_3},\ldots,\frac{a_{2013}}{a_{2014}}$ are pairwise distinct. What is the minimal possible number of distinct numbers in $\{a_1,a_2,\ldots,a_{2014}\}$?

Michael and James are playing a game where they alternate throwing darts at a simplified dartboard. Each dart throw is worth either 25 points or 50 points. They track the sequence of scores per throw (which is shared between them), and on the first time the three most recent scores sum to 125, the person who threw the last dart wins. On each throw, a given player has a $2/3$ chance of getting the score they aim for, and a $1/3$ chance of getting the other score. Suppose Michael goes first, and the first two throws are both 25. If both players use an optimal strategy, what is the probability Michael wins? [i]Proposed by Michael Duncan[/i]

Determine all non negative integers $x$ and $y$ such that $6^x$ + $2^y$ + 2 is a perfect square.

Define the sequence $\{x_{n}\}$ by $x_{0}=a\in (0,1)$ and $x_{n+1}=\frac{4}{\pi^{2}}(\cos^{-1}x_{n}+\frac{\pi}{2})\sin^{-1}x_{n}(n=0,1,2,...)$. Show that the sequence converges and find its limit.

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

Let $A_0A_1 \dots A_{11}$ be a regular $12$-gon inscribed in a circle with diameter $1$. For how many subsets $S \subseteq \{1,\dots,11\}$ is the product \[ \prod_{s \in S} A_0A_s \] equal to a rational number? (The empty product is declared to be $1$.) [i]Proposed by Evan Chen[/i]

Let \[f(n) = \sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k\frac{1}{n-k}\binom{n-k}k,\] for each positive integer $n$. If $|f(2007) + f(2008)| = a/b$ for relatively prime positive integers $a$ and $b$, find the remainder when $a$ is divded by $1000$.

Given a prime $p$, let $d(a,b)$ be the number of integers $c$ such that $1 \leq c < p$, and the remainders when $ac$ and $bc$ are divided by $p$ are both at most $\frac{p}{3}$. Determine the maximum value of \[\sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)(x_a + 1)(x_b + 1)} - \sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)x_ax_b}\] over all $(p-1)$-tuples $(x_1,x_2,\ldots,x_{p-1})$ of real numbers. [i]Brian Hamrick.[/i]

In each cell of a table with $m$ rows and $n$ columns, where $m<n$, we put a non-negative real number such that each column contains at least one positive number. Show that there is a cell with a positive number such that the sum of the numbers in its row is larger than the sum of the numbers in its column.

Let $n,k$ be positive integers such that $n$ is not divisible by $3$ and $k\ge n$. Prove that there is an integer $m$ divisible by $n$ whose sum of digits in base $10$ equals $k$.

Players $1,2,\ldots,10$ are playing a game on Christmas. Santa visits each player's house according to a set of rules: -Santa first visits player $1$. After visiting player $i$, Santa visits player $i+1$, where player $11$ is the same as player $1$. -Every time Santa visits someone, he gives them either a present or a piece of coal (but not both). -The absolute difference between the number of presents and pieces of coal that Santa has given out is at most $3$ at every point in time. -If Santa has a choice between giving out a present and a piece of coal, he chooses with equal probability. Let $p$ be the probability that player $1$ gets a present before player $2$ does. If $p=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $100m+n$. [i]Proposed by Tristan Shin

Outside of the plane of the triangle $ABC$ is given point $D$. (a) prove that if the segment $DA$ is perpendicular to the plane $ABC$ then orthogonal projection of the orthocenter of the triangle $ABC$ on the plane $BCD$ coincides with the orthocenter of the triangle $BCD$. (b) for all tetrahedrons $ABCD$ with base, the triangle $ABC$ with smallest of the four heights that from the vertex $D$, find the locus of the foot of that height.

Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

Wiles county contains eight townships as shown on the map. If there are four colors available, in how many ways can the the map be colored so that each township is colored with one color and no two townships that share a border are colored with the same color? [asy] path[] P= { (0,0)--(13,0)--(13,11)--(0,11)--cycle, (5,0)--(13,6)--(13,0)--cycle, (13,7)--(13,11)--(7,11)--cycle, (0,0)--(7,0)--(7,11)--(0,11)--cycle, (0,5)--(0,11)--(11,11)--cycle, circle((4,7),2.5), (0,0)--(5,0)--(2,11)--(0,11)--cycle, (0,5)--(0,11)--(5,11)--cycle, }; for(int k=0;k<P.length;++k) { unfill(P[k]); draw(P[k]); }[/asy]

Compute the number of squares of positive area whose vertices all are points on the grid shown below. [asy] unitsize(1cm); dot((0,0)); dot((1,0)); dot((2,0)); dot((3,0)); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1)); dot((0,2)); dot((1,2)); dot((2,2)); dot((3,2)); dot((0,3)); dot((1,3)); dot((2,3)); dot((3,3)); [/asy]

Let $a^2_1+ a^2_2+ a^2_3+ a^2_4+ a^2_5= b^2$, where $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, and $b$ are integers. Prove that not all of these numbers can be odd.