Sam Wang decides to evaluate an expression of the form $x +2 \cdot 2+ y$. However, he unfortunately reads each ’plus’ as a ’times’ and reads each ’times’ as a ’plus’. Surprisingly, he still gets the problem correct. Find $x + y$. [i]Proposed by Edwin Zhao[/i] [hide=Solution] [i]Solution.[/i] $\boxed{4}$ We have $x+2*2+y=x \cdot 2+2 \cdot y$. When simplifying, we have $x+y+4=2x+2y$, and $x+y=4$. [/hide]

In the Bank of Shower, a bored customer lays $n$ coins in a row. Then, each second, the customer performs ``The Process." In The Process, all coins with exactly one neighboring coin heads-up before The Process are placed heads-up (in its initial location), and all other coins are placed tails-up. The customer stops once all coins are tails-up. Define the function $f$ as follows: If there exists some initial arrangement of the coins so that the customer never stops, then $f(n) = 0$. Otherwise, $f(n)$ is the average number of seconds until the customer stops over all initial configurations. It is given that whenever $n = 2^k-1$ for some positive integer $k$, $f(n) > 0$. Let $N$ be the smallest positive integer so that \[ M = 2^N \cdot \left(f(2^2-1) + f(2^3-1) + f(2^4-1) + \cdots + f(2^{10}-1)\right) \]is a positive integer. If $M = \overline{b_kb_{k-1}\cdots b_0}$ in base two, compute $N + b_0 + b_1 + \cdots + b_k$. [i]Proposed by Edward Wan and Brandon Wang[/i]

Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$? [asy] unitsize(8); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((9,0)--(10,0)--(10,3)--(9,3)--cycle); draw((8,1)--(11,1)--(11,2)--(8,2)--cycle); draw((19,0)--(20,0)--(20,5)--(19,5)--cycle); draw((18,1)--(21,1)--(21,4)--(18,4)--cycle); draw((17,2)--(22,2)--(22,3)--(17,3)--cycle); draw((32,0)--(33,0)--(33,7)--(32,7)--cycle); draw((29,3)--(36,3)--(36,4)--(29,4)--cycle); draw((31,1)--(34,1)--(34,6)--(31,6)--cycle); draw((30,2)--(35,2)--(35,5)--(30,5)--cycle); label("Figure",(0.5,-1),S); label("$0$",(0.5,-2.5),S); label("Figure",(9.5,-1),S); label("$1$",(9.5,-2.5),S); label("Figure",(19.5,-1),S); label("$2$",(19.5,-2.5),S); label("Figure",(32.5,-1),S); label("$3$",(32.5,-2.5),S);[/asy]$ \textbf{(A)}\ 10401 \qquad \textbf{(B)}\ 19801 \qquad \textbf{(C)}\ 20201 \qquad \textbf{(D)}\ 39801 \qquad \textbf{(E)}\ 40801$

In the $5\times6$ grid shown, fill in all of the grid cells with the digits $0\textendash9$ so that the following conditions are satisfied: [list=1][*] Each digit gets used exactly $3$ times. [*] No digit is greater than the digit directly above it. [*] In any four cells that form a $2\times2$ subgrid, the sum of the four digits must be a multiple of $3$.[/list] 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.) \[\begin{Large}\begin{array}{|c|c|c|c|c|c|}\hline&&&&\,7\,&\\ \hline&\,8\,&&&&\,6\,\\\hline&&\,2\,&\,4\,&&\\ \hline\,5\,&&&&1&\\ \hline&3&&&&\\ \hline\end{array}\end{Large}\]

There is a board with $2010$ rows and $2001$ columns, on it there is a token located in the upper left box that can perform one of the following operations: (A) Walk 3 steps horizontally or vertically. (B) Walk 2 steps to the right and 3 steps down. (C) Walk 2 steps to the left and 2 steps up. With the condition that immediately after carrying out an operation on (B) or (C) it is mandatory to take a step to the right before perform the following operation. It is possible to exit the board, so count the number of steps necessary, entering through the other end of the row or column from which it exits, as if the board outside circular (example: from the beginning you can walk to the square located in row $1$ and column $1999$). Will it be possible that after $2011$ operations allowed the checker to land exactly on the bottom square right?

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N} $ satisfying \[ f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) ) \] for all positive integer $m,n$.

The circles $\gamma$ and $\delta$ are internally tangent to the circle $\omega$ at $A$ and $B$. From $A$, draw two tangent lines $\ell_1, \ell_2$ to $\delta$, . From $B$ draw two tangent lines $t_1, t_2$ to $\gamma$ . Let $\ell_1$ intersect $t_1$ at $X$ and $\ell_2$ intersect $t_2$ at $Y$ . Prove that the quadrilateral $AX BY$ is cyclic. Nguyen Van Linh, High School of Natural Sciences, Hanoi National University

Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying (for all $x,y \in \mathbb{R}$): $f(x+y)^2 - f(2x^2) = f(y-x)f(y+x) + 2x\cdot f(y)$.

We are given a $\Delta ABC$ with $\angle BAC=39^\circ$ and $\angle ABC=77^\circ$. Points $M$ and $N$ are chosen on $BC$ and $CA$ respectively, so that $\angle MAB=34^\circ$ and $\angle NBA=26^\circ$. Find $\angle BNM$.

Let $n$ be a positive integer. Determine all positive real numbers $x$ satisfying $nx^2 +\frac{2^2}{x + 1}+\frac{3^2}{x + 2}+...+\frac{(n + 1)^2}{x + n}= nx + \frac{n(n + 3)}{2}$

For integer $n\geq2$, let $x_1, x_2, \ldots, x_n$ be real numbers satisfying \[x_1+x_2+\ldots+x_n=0, \qquad \text{and}\qquad x_1^2+x_2^2+\ldots+x_n^2=1.\]For each subset $A\subseteq\{1, 2, \ldots, n\}$, define\[S_A=\sum_{i\in A}x_i.\](If $A$ is the empty set, then $S_A=0$.) Prove that for any positive number $\lambda$, the number of sets $A$ satisfying $S_A\geq\lambda$ is at most $2^{n-3}/\lambda^2$. For which choices of $x_1, x_2, \ldots, x_n, \lambda$ does equality hold?

Box is thinking of a number, whose digits are all “$1$”. When he squares the number, the sum of its digit is $85$. How many digits is Box’s number?

An equilateral triangle with side length 2023 has area $A$ and a regular hexagon with side length 289 has area $B$. If $\frac{A}{B}$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime, find $m+n$. [i]Proposed by Andy Xu[/i]

Let $a, b$, and $c$ denote the three sides of a billiard table in the shape of an equilateral triangle. A ball is placed at the midpoint of side $a$ and then propelled toward side $b$ with direction defined by the angle $\theta$. For what values of $\theta$ will the ball strike the sides $b, c, a$ in that order?

Compute the number of ways to fill each cell in a $8 \times 8$ square grid with one of the letters $H, M,$ or $T$ such that every $2 \times 2$ square in the grid contains the letters $H, M, M, T$ in some order.

Prove that the inequality $ (x^{k}\minus{}y^{k})^{n}<(x^{n}\minus{}y^{n})^{k}$ holds forall reals $ x>y>0$ and positive integers $ n>k$.

In acute triangle $\triangle ABC$, point $R$ lies on the perpendicular bisector of $AC$ such that $\overline{CA}$ bisects $\angle BAR$. Let $Q$ be the intersection of lines $AC$ and $BR$. The circumcircle of $\triangle ARC$ intersects segment $\overline{AB}$ at $P\neq A$, with $AP=1$, $PB=5$, and $AQ=2$. Compute $AR$.

Let $x_1,x_2,\ldots,x_n$ be real numbers. Prove that \[ \sum_{i,j=1}^n |x_i+x_j|\geq n\sum_{i=1}^n |x_i| \]

Prove that for any positive integer $n$ there exists a matrix of the form $$A=\begin{pmatrix}1&a&b&c\\0&1&a&b\\0&0&1&a\\0&0&0&1\end{pmatrix},$$ (a) with nonzero entries, (b) with positive entries, such that the entries of $A^n$ are all perfect squares.

The fifth and eighth terms of a geometric sequence of real numbers are $ 7!$ and $ 8!$ respectively. What is the first term? $ \textbf{(A)}\ 60\qquad \textbf{(B)}\ 75\qquad \textbf{(C)}\ 120\qquad \textbf{(D)}\ 225\qquad \textbf{(E)}\ 315$

There are a total of $n$ boys and girls sitting in a big circle. Now, Dave wants to walk around the circle. For a start point, if at any time, one of the following two conditions holds: 1. he doesn't see any girl 2. the number of boys he saw $\ge$ the number of girls he saw $+k$ Then we say this point is [i]good[/i]. What is the maximum of $r$ with the property that there is at least one good point whenever the number of girls is $r$?

Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards. Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells. [*]Prove that every minimal blocking set containing at most $3m^2$ cells.

There are $5$ accents in French, each applicable to only specific letters as follows: [list] [*] The cédille: ç [*] The accent aigu: é [*] The accent circonflexe: â, ê, î, ô, û [*] The accent grave: à, è, ù [*] The accent tréma: ë, ö, ü [/list] Cédric needs to write down a phrase in French. He knows that there are $3$ words in the phrase and that the letters appear in the order: \[cesontoiseaux.\] He does not remember what the words are and which letters have what accents in the phrase. If $n$ is the number of possible phrases that he could write down, then determine the number of distinct primes in the prime factorization of $n$.

How many natural numbers $n$ are there so that $n^2 + 2014$ is a perfect square? (A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.

Given a triangle $ABC$ with $M$ the midpoint of minor arc $BC$. Let $H$ be the feet of altitude from $A$ to $BC$. Let $S$ and $T$ be the reflections of $B$ and $C$ with respect to line $AM$. Suppose the circle $(HST)$ meets $BC$ again at a point $P$. Prove that $\angle AMP = 90^\circ$. [i](Proposed by Tan Rui Xuen)[/i]