What is $20 - 2^1$? $\textbf{(A) }1\qquad\textbf{(B) }18\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21$

Does there exist a natural number $a$ so that: a) $\Big ((a^2-3)^3+1\Big) ^a-1$ is a perfect square? b) $\Big ((a^2-3)^3+1\Big) ^{a+1}-1$ is a perfect square?

Show for all integers $ n \ge 2005$ the following chaine of inequalities: $ (n\plus{}830)^{2005}<n(n\plus{}1)\dots(n\plus{}2004)<(n\plus{}1002)^{2005}$

Find the smallest prime $q$ such that $$q = a_1^2 + b_1^2 = a_2^2 + 2b_2^2 = a_3^2 + 3b_3^2 = ... = a_{10}^ 2 + 10b_{10}^2$$ where $a_i, b_i(i = 1, 2, ...,10)$ are positive integers

The first $510$ positive integers are written on a blackboard: $1, 2, 3, ..., 510$. An [i]operation [/i] consists of of erasing two numbers whose sum is a prime number. What is the maximum number of operations in a row what can be done? Show how it is accomplished and explain why it can be done in no more operations.

Let $F$ be the set of all polynomials $\Gamma$ such that all the coefficients of $\Gamma (x)$ are integers and $\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\Gamma \in F$ for which $\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.

A knight is on an infinite chessboard. After exactly $100$ legal moves, how many different possible squares can it end on? A knight can move to any of the $8$ closest squares not on the same row, column, or diagonal, as illustrated in the figure below. [center][img]https://cdn.artofproblemsolving.com/attachments/0/7/144226144fb3ead533e7b517f5f65d8a70da5a.png[/img] [/center]

Let $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ be two finite sequences consisting of $2n$ real different numbers. Rearranging each of the sequences in increasing order we obtain $a_1',a_2',\ldots,a_n'$ and $b_1',b_2',\ldots,b_n'$. Prove that \[\max_{1\le i\le n}|a_i-b_i|\ge\max_{1\le i\le n}|a_i'-b_i'|.\]

Suppose the side lengths of triangle $ABC$ are the roots of polynomial $x^3 - 27x^2 + 222x - 540$. What is the product of its inradius and circumradius?

The quadrilateral $ABCD$ is inscribed in a circle. The point $P$ lies in the interior of $ABCD$, and $\angle P AB = \angle P BC = \angle P CD = \angle P DA$. The lines $AD$ and $BC$ meet at $Q$, and the lines $AB$ and $CD$ meet at $R$. Prove that the lines $P Q$ and $P R$ form the same angle as the diagonals of $ABCD$.

Let $ABC$ be a triangle with $\angle BAC = 60^{\circ}$. Let $AP$ bisect $\angle BAC$ and let $BQ$ bisect $\angle ABC$, with $P$ on $BC$ and $Q$ on $AC$. If $AB + BP = AQ + QB$, what are the angles of the triangle?

You shoot an arrow in the air. It falls to earth, you know not where. But you do know that the arrow’s height in feet after ${t}$ seconds is $-16t^2 + 80t + 96.$ After how many seconds does the arrow hit the ground? (the ground has height 0) $$ \mathrm a. ~ 2\qquad \mathrm b.~3\qquad \mathrm c. ~4 \qquad \mathrm d. ~5 \qquad \mathrm e. ~6 $$

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