There are two boxes containing balls. One of them contains $m$ balls, and the other contains $n$ balls, where $m, n > 0$. Two actions are permitted: (i) Remove an equal number of balls from both boxes. (ii) Increase the number of balls in one of the boxes by a factor $k$. Is it possible to remove all of the balls from both boxes with just these two actions, 1. if $k = 2$? 2. if $k = 3$?

Let $S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$. Then $S$ equals: ${{ \textbf{(A)}\ (x-2)^4 \qquad\textbf{(B)}\ (x-1)^4 \qquad\textbf{(C)}\ x^4 \qquad\textbf{(D)}\ (x+1)^4 }\qquad\textbf{(E)}\ x^4+1} $

Given: \[ S = 1 + \frac{1}{1 + \frac{1}{3}} + \frac{1}{1 + \frac{1}{3} + \frac{1} {6}} + \cdots + \frac{1}{1 + \frac{1}{3} + \frac{1}{6} + \cdots + \frac{1} {1993006}} \] where the denominators contain partial sums of the sequence of reciprocals of triangular numbers (i.e. $k=\frac{n(n+1)}{2}$ for $n = 1$, $2$, $\ldots$,$1996$). Prove that $S>1001$.

Maisy is at the origin of the coordinate plane. On her first step, she moves $1$ unit up. On her second step, she moves $ 1$ unit to the right. On her third step, she moves $2$ units up. On her fourth step, she moves $2$ units to the right. She repeats this pattern with each odd-numbered step being $ 1$ unit more than the previous step. Given that the point that Maisy lands on after her $21$st step can be written in the form $(x, y)$, find the value of $x + y$. Proposed by Audrey Chun

Consider in the plane a circle $\Gamma$ with centre O and a line l not intersecting the circle. Prove that there is a unique point Q on the perpendicular drawn from O to line l, such that for any point P on the line l, PQ represents the length of the tangent from P to the given circle.

Find all integer solutions $(a,b)$ of the equation \[ (a+b+3)^2 + 2ab = 3ab(a+2)(b+2)\]

Determine all natural $ n$ for which there exists natural $ m$ divisible by all natural numbers from 1 to $ n$ but not divisible by any of the numbers $ n \plus{} 1$, $ n \plus{} 2$, $ n \plus{} 3$.

Prove the inequality: $$\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{1974}-\frac{1}{1975}<\frac{2}{5}$$

There are two distinct pairs of positive integers $a_1 < b_1$ and $a_2 < b_2$ such that both $(a_1 + ib_1)(b_1 - ia_1) $ and $(a_2 + ib_2)(b_2 - ia_2)$ equal $2020$, where $i =\sqrt{-1}$. Find $a_1 + b_1 + a_2 + b_2$.

The grid to the right consists of 74 unit squares, marked by gridlines. Partition the grid into five regions along the gridlines so that the areas of the regions are 1, 13, 19, 20, and 21. A square with a number should be contained in the region with that area. [asy]unitsize(20); path p = (5,0)--(3,0)--(3,1)--(1,1)--(1,2)--(0,2)--(0,7)--(1,7)--(1,8)--(3,8)--(3,9)--(5,9); draw(p^^reflect((5,0),(5,3.14))*p); int[] v = {0,1,1,0,0,0,0,0,1,1}; int[] h = {0,1,0,0,0,0,0,1,1}; for(int i=1; i<10; ++i) { draw((i,v[i])--(i,9-v[i]),dotted); } for(int i=1; i<9; ++i) { draw((h[i],i)--(10-h[i],i),dotted); } int[][] dord = {{1,4,21},{2,4,19},{3,4,1},{4,4,13},{5,4,20},{6,4,19},{7,4,20},{8,4,19},{2,2,21},{2,6,19},{4,7,19},{5,1,21},{7,6,13},{7,2,13}}; for(int i=0; i<14; ++i){ label(string(dord[i][2]),(dord[i][0]+.5,.5+dord[i][1])); } [/asy] 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.)

A moving point $P(x,y)$ rotate anticlockwise around unit circle, who seangular speed is $\omega$. Then how does $Q(-2xy,y^2-x^2)$ moves? $\text{(A)}$ Rotate clockwise around unit circle, who seangular speed is $\omega$. $\text{(B)}$ Rotate anticlockwise around unit circle, who seangular speed is $\omega$. $\text{(C)}$ Rotate clockwise around unit circle, who seangular speed is $2\omega$. $\text{(D)}$ Rotate anticlockwise around unit circle, who seangular speed is $2\omega$.

Let $A_1,A_2,...$ be a sequence of sets such that for any positive integer $i$, there are only finitely many values of $j$ such that $A_j\subseteq A_i$. Prove that there is a sequence of positive integers $a_1,a_2,...$ such that for any pair $(i,j)$ to have $a_i\mid a_j\iff A_i\subseteq A_j$.

Consider $\triangle XPQ$ and $\triangle YPQ$ such that $X$ and $Y$ are on the opposite sides of $PQ$ and the circumradius of $\triangle XPQ$ and the circumradius of $\triangle YPQ$ are the same. $I$ and $J$ are the incenters of $\triangle XPQ$ and $\triangle YPQ$ respectively. Let $M$ be the midpoint of $PQ$. Suppose $I, M, J$ are collinear. Prove that $XPYQ$ is a parallelogram.

In CMI, each person has atmost $3$ friends. A disease has infected exactly $2023$ peoplein CMI . Each day, a person gets infected if and only if atleast two of their friends were infected on the previous day. Once someone is infected, they can neither die nor be cured. Given that everyone in CMI eventually got infected, what is the maximum possible number of people in CMI?

Indicate whether it is possible to write the integers $1, 2, 3, 4, 5, 6, 7, 8$ at the vertices of an regular octagon such that the sum of the numbers of any $3$ consecutive vertices is greater than: a) $11$. b) $13$.

Let $a,b,c,d$ be positive integers, and let $p=a+b+c+d$. Prove that if $p$ is a prime, then $p$ is not a divisor of $ab-cd$. [i]Marian Andronache[/i]

Let $A_1,A_2,\ldots ,A_{n+1}$ be positive integers such that $(A_i,A_{n+1})=1$ for every $i=1,2,\ldots ,n$. Show that the equation \[x_1^{A_1}+x_2^{A_2}+\ldots + x_n^{A_n}=x_{n+1}^{A_{n+1} }\] has an infinite set of solutions $(x_1,x_2,\ldots , x_{n+1})$ in positive integers.

Let $ABC$ be an acute, scalene triangle. Let $H$ be the orthocenter and $O$ be the circumcenter of triangle $ABC$, and let $P$ be a point interior to the segment $HO.$ The circle with center $P$ and radius $PA$ intersects the lines $AB$ and $AC$ again at $R$ and $S$, respectively. Denote by $Q$ the symmetric point of $P$ with respect to the perpendicular bisector of $BC$. Prove that points $P$, $Q$, $R$ and $S$ lie on the same circle.

A number of objects, each coloured in one of two given colours, are arranged in a line (there is at least one object having each of the given colours). It is known that each two objects, between which there are exactly $10$ or $15$ other objects, are of the same colour. What is the greatest possible number of such pieces?

Let $a, b, c$ be natural numbers such that $a+b+c = 2pq(p^{30}+q^{30}), p > q$ being two given positive integers. $(a)$ Prove that $k = a^3 + b^3 + c^3$ is not a prime number. $(b)$ Prove that if $a\cdot b\cdot c$ is maximum, then $1984$ divides $k$.

Given a nonconstant polynomial with real coefficients $f(x),$ let $S(f)$ denote the sum of its roots. Let p and q be nonconstant polynomials with real coefficients such that $S(p) = 7,$ $S(q) = 9,$ and $S(p-q)= 11$. Find, with proof, all possible values for $S(p + q)$.

Let $N$ be the number of functions $f: \mathbb{Z}/16\mathbb{Z} \to \mathbb{Z}/16\mathbb{Z}$ such that for all $a,b \in \mathbb{Z}/16\mathbb{Z}$: \[f(a)^2+f(b)^2+f(a+b)^2 \equiv 1+2f(a)f(b)f(a+b) \pmod{16}.\] Find the remainder when $N$ is divided by 2017. [i]Proposed by Zack Chroman[/i]

Every cell of a $m \times n$ chess board, $m\ge 2,n\ge 2$, is colored with one of four possible colors, e.g white, green, red, blue. We call such coloring good if the four cells of any $2\times 2$ square of the chessboard are colored with pairwise different colors. Determine the number of all good colorings of the chess board. [i]Proposed by N. Beluhov[/i]

On the board there are written in a row, the integers from $1$ until $2030$ (included that) in an increasing order. We have the right of ''movement'' $K$: [i]We choose any two numbers $a,b$ that are written in consecutive positions and we replace the pair $(a,b)$ by the number $(a-b)^{2020}$.[/i] We repeat the movement $K$, many times until only one number remains written on the board. Determine whether it would be possible, that number to be: (i) $2020^{2020}$ (ii)$2021^{2020}$

Start with a stack of $100$ cards. Now repeat the following: choose a stack of at least $2$ cards and split them into two smaller piles (at least $1$ card of each). Continue this until there are finally $100$ stacks of $1$ card each. Every time you split a pile into two stacks you get a number of points that is equal to the product of the number of cards in the two new stacks. What is the maximum number of points that you can earn in total?