Let be a natural number $ n\ge 2 $ and an expression of $ n $ variables $$ E\left( x_1,x_2,...,x_n\right) =x_1^2+x_2^2+\cdots +x_n^2-x_1x_2-x_2x_3-\cdots -x_{n-1}x_n -x_nx_1. $$ Determine $ \sup_{x_1,...,x_n\in [0,1]} E\left( x_1,x_2,...,x_n\right) $ and the specific values at which this supremum is attained.

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?