Partition the grid into 1 by 1 squares and 1 by 2 dominoes in either orientation, marking dominoes with a line connecting the two adjacent squares, and 1 by 1 squares with an asterisk ($*$). No two 1 by 1 squares can share a side. A $border$ is a grid segment between two adjacent squares that contain dominoes of opposite orientations. All borders have been marked with thick lines in the grid. There is a unique solution, but you do not need to prove that your answer is the only one possible. You merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] unitsize(1cm); // makes asterisks larger (you can remove if you want) defaultpen(fontsize(30pt)); for(int i = 0; i < 10; ++i) { for(int j = 0; j < 10; ++j) { draw((i - 0.5, j - 0.5)--(i + 0.5, j - 0.5)--(i + 0.5, j + 0.5)--(i - 0.5, j + 0.5)--(i - 0.5, j - 0.5), gray(0.5)); } } draw((0 - 0.5, 2 - 0.5)--(1 - 0.5, 2 - 0.5), gray(0) + 3); draw((0 - 0.5, 6 - 0.5)--(0.5, 5.5), black+3); draw((-0.5, 6.5)--(0.5, 6.5)--(0.5, 7.5), black+3); draw((-0.5, 8.5)--(0.5, 8.5), black+3); draw((1.5, -0.5)--(1.5, 1.5), black+3); draw((1.5, 5.5)--(1.5, 6.5)--(2.5, 6.5)--(2.5, 7.5)--(1.5, 7.5), black+3); draw((1.5, 9.5)--(1.5, 8.5), black+3); draw((2.5, -0.5)--(2.5, 0.5)--(3.5, 0.5)--(3.5, 1.5), black+3); draw((4.5, 1.5)--(5.5, 1.5)--(5.5, 2.5), black+3); draw((4.5, 6.5)--(5.5, 6.5), black+3); draw((5.5, 8.5)--(6.5, 8.5)--(6.5, 7.5), black+3); draw((6.5, -0.5)--(6.5, 0.5), black+3); draw((6.5, 1.5)--(7.5, 1.5), black+3); draw((8.5, 5.5)--(8.5, 4.5), black+3); string[] grid = { "----------", "----------", "----------", "----------", "----------", "----------", "----------", "----------", "----------", "----------" }; /* L is the left side of a domino R is the right T is the top B is the bottom */ for(int j = 9; j >= 0; --j) { for(int i = 0; i < 10; ++i) { string identifier = substr(grid[9 - j], i, 1); if (identifier == "*") label("$*$", (i, j)); else if (identifier == "L") draw((i, j)--(i + 0.5, j)); else if (identifier == "R") draw((i, j)--(i - 0.5, j)); else if (identifier == "T") draw((i, j)--(i, j - 0.5)); else if (identifier == "B") draw((i, j)--(i, j + 0.5)); } } [/asy]

Given positive integers $a_1$, $a_2$, $...$, $a_m$ ($m \ge 1$). Consider the sequence $\{u_n\}_{n=1}^{\infty}$, with $$u_n = a_1^n + a_2^n + ... + a_m^n.$$ We know that this sequence has a finite number of prime divisors. Prove that $a_1 = a_2 = ...= a_m$.

The four congruent circles below touch one another and each has radius 1. [asy] unitsize(30); fill(box((-1,-1), (1, 1)), gray); filldraw(circle((1, 1), 1), white); filldraw(circle((1, -1), 1), white); filldraw(circle((-1, 1), 1), white); filldraw(circle((-1, -1), 1), white); [/asy] What is the area of the shaded region?

Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.

A valid lottery ticket is formed by choosing $5$ distinct numbers from $1, 2,3,..., 90$. What is the probability that the winning ticket contains at least two consecutive numbers?

Let be a real number $ a\in \left[ 2+\sqrt 2,4 \right] . $ Find $ \inf_{\stackrel{z\in\mathbb{C}}{|z|\le 1}} \left| z^2-az+a \right| . $

Seven boys and three girls are playing basketball. I how many different ways can they make two teams of five players so that both teams have at least one girl?

A random rectangle (not necessarily a square) with positive integer dimensions is selected from the $2\times4$ grid below. The probability that the selected rectangle contains only white squares can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$. [asy] fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,blue); fill((2,1)--(3,1)--(3,2)--(2,2)--cycle,blue); draw((0,0)--(4,0),black); draw((0,0)--(0,2),black); draw((4,0)--(4,2),black); draw((4,2)--(0,2),black); draw((0,1)--(4,1),black); draw((1,0)--(1,2),black); draw((2,0)--(2,2),black); draw((3,0)--(3,2),black); [/asy]

Let $R(x)$ be a function that takes a natural number as input and returns a rectangle. $R(1)$ is known to have integer side lengths. Let $p(x)$ be the perimeter of $R(x)$ and let $a(x)$ be the area of $R(x)$. Suppose that $p(x+5)=6 p(x)$ for all $x$ in the domain of $R$ and that $a(x+2)=12a(x)$ for all $x> 6$ in the domain of $R$. For $x \leq 6$, $a(x+1)=a(x)+2$. Suppose $p(16)=1296$, and let the side lengths of $R(11)$ be $a$ and $b$ with $a\leq b$. Find the ordered pair $(a,b)$. [i]Proposed by Matthew Weiss

Let $e_1,\ldots, e_n$ be semilines on the plane starting from a common point. Prove that if there is no $u\not\equiv 0$ harmonic function on the whole plane that vanishes on the set $e_1\cup \cdots \cup e_n$, then there exists a pair $i,j$ of indices such that no $u\not\equiv 0$ harmonic function on the whole plane exists that vanishes on $e_i\cup e_j$.

Rectangle $ABCD$ has perimeter $178$ and area $1848$. What is the length of the diagonal of the rectangle? [i]2016 CCA Math Bonanza Individual Round #2[/i]

Prove that $$\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{(n+1)^2} < n \cdot \left(1-\frac{1}{\sqrt[n]{2}}\right).$$

Circle $\omega_1$ with center $K$ of radius $4$ and circle $\omega_2$ of radius $6$ intersect at points $W$ and $U$. If the incenter of $\triangle KWU$ lies on circle $\omega_2$, find the length of $\overline{WU}$. (Note: The incenter of a triangle is the intersection of the angle bisectors of the angles of the triangle) [i]Proposed by Bradley Guo[/i]

Compute the least positive $x$ such that $25x - 6$ is divisible by $1001$.

Let $[n]$ denote the set of integers $\left\{ 1, 2, \ldots, n \right\}$. We randomly choose a function $f:[n] \to [n]$, out of the $n^n$ possible functions. We also choose an integer $a$ uniformly at random from $[n]$. Find the probability that there exist positive integers $b, c \geq 1$ such that $f^b(1) = a$ and $f^c(a) = 1$. ($f^k(x)$ denotes the result of applying $f$ to $x$ $k$ times.)

Let $ABCD$ be a cyclic quadrilateral. Let $M, N$ be the midpoints of the diagonals $AC$ and $BD$ and let $P$ be the midpoint of $MN$. Let $A',B',C',D'$ be the intersections of the rays $AP$, $BP$, $CP$ and $DP$ respectively with the circumcircle of the quadrilateral $ABCD$. Find, with proof, the value of the sum \[ \sigma = \frac{ AP}{PA'} + \frac{BP}{PB'} + \frac{CP}{PC'} + \frac{DP}{PD'} . \]

On the circumference of a circle, there are $3n$ colored points that divide the circle on $3n$ arches, $n$ of which have lenght $1$, $n$ of which have length $2$ and the rest of them have length $3$ . Prove that there are two colored points on the same diameter of the circle.

Real numbers $a$ and $b$ are chosen so that each of two quadratic trinomials $x^2+ax+b$ and $x^2+bx+a$ has two distinct real roots,and the product of these trinomials has exactly three distinct real roots.Determine all possible values of the sum of these three roots. [i](S.Berlov)[/i]

\[\bf{\sum_{n=1}^{\infty}3^n.sin^3(\frac{\pi}{3^n})=?}\]

Let $n$ be a natural number greater than 6. $X$ is a set such that $|X| = n$. $A_1, A_2, \ldots, A_m$ are distinct 5-element subsets of $X$. If $m > \frac{n(n - 1)(n - 2)(n - 3)(4n - 15)}{600}$, prove that there exists $A_{i_1}, A_{i_2}, \ldots, A_{i_6}$ $(1 \leq i_1 < i_2 < \cdots, i_6 \leq m)$, such that $\bigcup_{k = 1}^6 A_{i_k} = 6$.

If the expression $ \begin{pmatrix}a & c \\ d & b \end{pmatrix}$ has the value $ ab\minus{}cd$ for all values of $a, b, c$ and $d$, then the equation $ \begin{pmatrix}2x & 1 \\ x & x \end{pmatrix} = 3$: $\textbf{(A)}\ \text{Is satisfied for only 1 value of }x \qquad\\ \textbf{(B)}\ \text{Is satisified for only 2 values of }x \qquad\\ \textbf{(C)}\ \text{Is satisified for no values of }x \qquad\\ \textbf{(D)}\ \text{Is satisfied for an infinite number of values of }x \qquad\\ \textbf{(E)}\ \text{None of these.}$

The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$. [i]Proposed by Mariusz Skalba, Poland[/i]

The triangle $ABC$ is equilateral. Find the locus of the points $M$ such that the triangles $ABM$ and $ACM$ are both isosceles.

Let $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follows: the first term is $5$, and every consequent term is obtained by multiplying its precedent by $a$ and adding $b$. (For example, if $a = 2$ and $b = 4$, the first three terms are $5,14,32$.) What is the maximum possible number of primes that can occur before encoutering the first composite term?

Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.