Points $X$ and $Y$ lie on side $\overline{AB}$ of $\vartriangle ABC$ such that $AX = 20$, $AY = 28$, and $AB = 42$. Suppose $XC = 26$ and $Y C = 30$. Find $AC + BC$.

Given a permutation $(a_{1}, a_{1},...,a_{n})$ of the numbers $1, 2,...,n$ one may interchange any two consecutive "blocks" - that is, one may transform ($a_{1}, a_{2},...,a_{i}$,$\underbrace {a_{i+1},... a_{i+p},}_{A} $ $ \underbrace{a_{i+p+1},...,a_{i+q},}_{B}...,a_{n}) $ into $ (a_{1}, a_{2},...,a_{i},$ $ \underbrace {a_{i+p+1},...,a_{i+q},}_{B} $ $ \underbrace {a_{i+1},... a_{i+p}}_{A}$$,...,a_{n}) $ by interchanging the "blocks" $A$ and $B$. Find the least number of such changes which are needed to transform $(n, n-1,...,1)$ into $(1,2,...,n)$

Each of the numbers $a_1,...,a_n$ is equal to $1$ or $-1$. If $a_1a_2 + a_2a_3 + ··· + a_{n-1}a_n + a_na_1 = 0$, proves that $n$ is divisible by $4$.

Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$, and $ x^{2}+y^{2}+xy+1\geq C(x+y)$ for all reals $x,y$.

Find all functions $f$ from integers to integers such that \[ f(m+n) + f(m-n) - 2f(m) = 6mn^2\] for all integers $m$ and $n$.

$(GBR 3)$ A smooth solid consists of a right circular cylinder of height $h$ and base-radius $r$, surmounted by a hemisphere of radius $r$ and center $O.$ The solid stands on a horizontal table. One end of a string is attached to a point on the base. The string is stretched (initially being kept in the vertical plane) over the highest point of the solid and held down at the point $P$ on the hemisphere such that $OP$ makes an angle $\alpha$ with the horizontal. Show that if $\alpha$ is small enough, the string will slacken if slightly displaced and no longer remain in a vertical plane. If then pulled tight through $P$, show that it will cross the common circular section of the hemisphere and cylinder at a point $Q$ such that $\angle SOQ = \phi$, $S$ being where it initially crossed this section, and $\sin \phi = \frac{r \tan \alpha}{h}$.

Let $N$ be an odd number, $N\geq 3$. $N$ tennis players take part in a championship. Before starting the championship, a commission puts the players in a row depending on how good they think the players are. During the championship, every player plays with every other player exactly once, and each match has a winner. A match is called [i]suprising[/i] if the winner was rated lower by the commission. At the end of the tournament, players are arranged in a line based on the number of victories they have achieved. In the event of a tie, the commission's initial order is used to decide which player will be higher. It turns out that the final order is exactly the same as the commission's initial order. What is the maximal number of suprising matches that could have happened.

Points $E$ and $F$ are taken on the diagonals $AB_1$ and $CA_1$ of the lateral faces $ABB_1A_1$ and $CAA_1C_1$ of a triangular prism $ABCA_1B_1C_1$ so that $EF\parallel BC_1$. Find the ratio of the lengths of $EF$ and $BC_1$.

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]

Which of the following expressions is never a prime number when $p$ is a prime number? $\textbf{(A) } p^2+16 \qquad \textbf{(B) } p^2+24 \qquad \textbf{(C) } p^2+26 \qquad \textbf{(D) } p^2+46 \qquad \textbf{(E) } p^2+96$

The decimal number $13^{101}$ is given. It is instead written as a ternary number. What are the two last digits of this ternary number?

Let $c$ be a fixed positive integer, and let ${a_n}^{\inf}_{n=1}$ be a sequence of positive integers such that $a_n < a_{n+1} < a_n+c$ for every positive integer $n$. Let $s$ denote the infinite string of digits obtained by writing the terms in the sequence consecutively from left to right, starting from the first term. For every positive integer $k$, let $s_k$ denote the number whose decimal representation is identical to the $k$ most left digits of $s$. Prove that for every positive integer $m$ there exists a positive integer $k$ such that $s_k$ is divisible by $m$.

Find all real functions $f$ defined on $ \mathbb R$, such that \[f(f(x)+y) = f(f(x)-y)+4f(x)y ,\] for all real numbers $x,y$.

Let $ABCDEF$ be a convex hexagon such that all of its vertices are on a circle. Prove that $AD$, $BE$ and $CF$ are concurrent if and only if $\frac {AB}{BC}\cdot\frac {CD}{DE}\cdot\frac {EF}{FA}= 1$.

Newton and Leibniz are playing a game with a coin that comes up heads with probability $p$. They take turns flipping the coin until one of them wins with Newton going first. Newton wins if he flips a heads and Leibniz wins if he flips a tails. Given that Newton and Leibniz each win the game half of the time, what is the probability $p$?

Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$. [i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]

Zachary tries to simplify the fraction $\frac{2020}{5050}$ by dividing the numerator and denominator by the same integer to get the fraction $\frac{m}{n}$ , where $m$ and $n$ are both positive integers. Find the sum of the (not necessarily distinct) prime factors of the sum of all the possible values of $m +n$

Given a prime $ p = 8k+1 $ for some integer $ k $. Let $ r $ be the remainder when $ \binom{4k}{k} $ is divided by $ p $. Prove that $ \sqrt{r} $ is not an integer. [i]Proposed by Evan Chen[/i]

The $ 8\times 18$ rectangle $ ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $ y$? [asy] unitsize(2mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; draw((0,4)--(18,4)--(18,-4)--(0,-4)--cycle); draw((6,4)--(6,0)--(12,0)--(12,-4)); label("$D$",(0,4),NW); label("$C$",(18,4),NE); label("$B$",(18,-4),SE); label("$A$",(0,-4),SW); label("$y$",(9,1)); [/asy]$ \textbf{(A) } 6\qquad \textbf{(B) } 7\qquad \textbf{(C) } 8\qquad \textbf{(D) } 9\qquad \textbf{(E) } 10$

We have a 4x4 board.All 1x1 squares are white.A move is changing colours of all squares of a 1x3 rectangle from black to white and from white to black.It is possible to make all the 1x1 squares black after several moves?

Fix integers $n\ge k\ge 2$. We call a collection of integral valued coins $n-diverse$ if no value occurs in it more than $n$ times. Given such a collection, a number $S$ is $n-reachable$ if that collection contains $n$ coins whose sum of values equals $S$. Find the least positive integer $D$ such that for any $n$-diverse collection of $D$ coins there are at least $k$ numbers that are $n$-reachable. [I]Proposed by Alexandar Ivanov, Bulgaria.[/i]

A square in the [i]xy[/i]-plane has area [i]A[/i], and three of its vertices have [i]x[/i]-coordinates $2,0,$ and $18$ in some order. Find the sum of all possible values of [i]A[/i].

Let $p$ be a prime number. Find all polynomials $P$ with integer coefficients with the following properties: $(a)$ $P(x)>x$ for all positive integers $x$. $(b)$ The sequence defined by $p_0:=p$, $p_{n+1}:=P(p_n)$ for all positive integers $n$, satisfies the property that for all positive integers $m$ there exists some $l\geq 0$ such that $m\mid p_l$.

How many positive integers less than $2022$ contain at least one digit less than $5$ and also at least one digit greater than $4$?

Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.