Consider the problem of expressing $42$ as \(42 = x^3 + y^3 + z^3 - w^2\), where \(x, y, z, w\) are integers. Determine the number of ways to represent $42$ in this form and prove your conclusion.

Suppose $a,b,c$ are positive numbers. Prove that $$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1\right)^2\ge (2a+b+c) \left(\frac{2}{a}+\frac{1}{b}+\frac{1}{c}\right)$$ with equality if and only if $a=b=c$.

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

Let $ABC$ be a triangle with incenter $I$ and $AB = 1400$, $AC = 1800$, $BC = 2014$. The circle centered at $I$ passing through $A$ intersects line $BC$ at two points $X$ and $Y$. Compute the length $XY$. [i]Proposed by Evan Chen[/i]

A convex pentagon $ABCDE$ has rational sides and equal angles. Show that it is regular.

Prove that there exists a positive $ c$ such that for every positive integer $ N$ among any $ N$ positive integers not exceeding $ 2N$ there are two numbers whose greatest common divisor is greater than $ cN$.

Let $n, m, k$ and $l$ be positive integers with $n \neq 1$ such that $n^k + mn^l + 1$ divides $n^{k+l} - 1$. Prove that [list] [*]$m = 1$ and $l = 2k$; or [*]$l|k$ and $m = \frac{n^{k-l}-1}{n^l-1}$. [/list]

Show that there exists a positive real $C$ such that for any naturals $H,N$ satisfying $H \geq 3, N \geq e^{CH}$, for any subset of $\{1,2,\ldots,N\}$ with size $\lceil \frac{CHN}{\ln N} \rceil$, one can find $H$ naturals in it such that the greatest common divisor of any two elements is the greatest common divisor of all $H$ elements.

Given is a triangle $ABC$ with the property that $|AB| + |AC| = 3|BC|$. Let $T$ be the point on segment $AC$ such that $|AC| = 4|AT|$. Let $K$ and $L$ be points on the interior of line segments $AB$ and $AC$ respectively such that $KL \parallel BC$ and $KL$ is tangent to the inscribed circle of $\vartriangle ABC$. Let $S$ be the intersection of $BT$ and $KL$. Determine the ratio $\frac{|SL|}{|KL|}$

Let $x_1,x_2,x_3\geq 0$ and $x_1+x_2+x_3=1$. Find the minimum value and the maximum value of $(x_1+3x_2+5x_3)\left(x_1+\frac{x_2}{3}+\frac{x_3}{5}\right).$

Let $P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{0}$, where $a_{0},\ldots,a_{n}$ are integers, $a_{n}>0$, $n\geq 2$. Prove that there exists a positive integer $m$ such that $P(m!)$ is a composite number.

Find all the pairs of the positive integers such that the product of the numbers of any pair plus the half of one of the numbers plus one third of the other number is three times less than $1004$.

Let $p_1, p_2, . . . , p_{30}$ be a permutation of the numbers $1, 2, . . . , 30.$ For how many permutations does the equality $\sum^{30}_{k=1}|p_k - k| = 450 $ hold?

The Pythagoras High School band has $100$ female and $80$ male members. The Pythagoras High School orchestra has $80$ female and $100$ male members. There are $60$ females who are members in both band and orchestra. Altogether, there are $230$ students who are in either band or orchestra or both. The number of males in the band who are NOT in the orchestra is $\text{(A)}\ 10 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 30 \qquad \text{(D)}\ 50 \qquad \text{(E)}\ 70$

Let $\mathcal{A}$ be the set of all $n\in\mathbb{N}$ for which there exist $k\in\mathbb{N}$ and $a_0,a_1,\dots,a_{k-1}\in \{1,2,\dots,9\}$ such that $a_0 \geq a_1 \geq \cdots \geq a_{k-1}$ and $n = a_0 +a_1 \cdot 10^1 +\cdots +a_{k-1}\cdot 10^{k-1}$. Let $\mathcal{B}$ be the set of all $m \in\mathbb{N}$ for which there exist $l \in\mathbb{N}$ and $b_0,b_1,\dots,b_{l-1} \in \{1,2,\dots,9\}$ such that $b_0 \leq b_1 \leq \cdots\leq b_{l-1}$ and $m = b_0 + b_1 \cdot 10^1 + \cdots+ b_{l-1}\cdot 10^{l-1}$. [list=a] [*] Are there infinitely many $n\in \mathcal{A}$ such that $n^2-3\in\mathcal{A} \ ?$ [*] Are there infinitely many $m\in \mathcal{B}$ such that $m^2-3\in\mathcal{B} \ ?$ [/list] [i]Proposed by Pakawut Jiradilok and Wijit Yangjit[/i]

We call a coloring of an $8\times 8$ board in three colors good if in any corner of five cells contains cells of all three colors. (A five-square corner is a shape made from a $3 \times 3$ square by cutting square $ 2\times 2$.) Prove that the number of good colorings is not less than than $68$.

Determine all triples $(a, b, c)$ of positive integers such that $$a! + b! = 2^{c!}.$$ [i](Walther Janous)[/i]

Fill in each cell of the grid with one of the numbers 1, 2, or 3. After all numbers are filled in, if a row, column, or any diagonal has a number of cells equal to a multiple of 3, then it must have the same amount of 1’s, 2’s, and 3’s. (There are 10 such diagonals, and they are all marked in the grid by a gray dashed line.) Some numbers have been given to you. [asy] defaultpen(linewidth(0.45)); real[][] arr = { {0, 2, 1, 0, 0, 0, 0, 0, 0}, {3, 0, 0, 2, 0, 0, 0, 0, 0}, {0, 0, 0, 2, 0, 0, 3, 2, 0}, {0, 2, 1, 0, 0, 1, 0, 0, 3}, {3, 0, 0, 0, 0, 3, 0, 0, 3}, {2, 0, 0, 0, 0, 0, 2, 3, 0}, {3, 2, 3, 2, 0, 2, 0, 0, 3}, {0, 0, 0, 0, 0, 3, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 1, 3, 0}}; for (int i=0; i<9; ++i){ for (int j=0; j<9; ++j){ draw((i,j)--(i+1,j)--(i+1, j+1)--(i,j+1)--cycle); if(arr[8-j][i] != 0){ label((string) arr[8-j][i], (i+0.5, j+0.5)); } } } draw((3,0)--(0,3), linetype(new real[] {4,4})+grey); draw((6,0)--(0,6), linetype(new real[] {4,4})+grey); draw((9,0)--(0,9), linetype(new real[] {4,4})+grey); draw((3,9)--(9,3), linetype(new real[] {4,4})+grey); draw((6,9)--(9,6), linetype(new real[] {4,4})+grey); draw((6,0)--(9,3), linetype(new real[] {4,4})+grey); draw((3,0)--(9,6), linetype(new real[] {4,4})+grey); draw((0,0)--(9,9), linetype(new real[] {4,4})+grey); draw((0,3)--(6,9), linetype(new real[] {4,4})+grey); draw((0,6)--(3,9), linetype(new real[] {4,4})+grey); [/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.)

One triangular face $F$ of a tetrahedron $\mathcal{T}$ has side lengths $\sqrt{5}$, $\sqrt{65}$, and $2\sqrt{17}$. The other three faces of $\mathcal{T}$ are right triangles whose hypotenuses coincide with the sides of $F$. There exists a sphere inside $\mathcal{T}$ tangent to all four of its faces. Compute the radius of this sphere.

Through a point located on a side of a triangle of area $1$, two straight lines are drawn parallel to the two remaining sides. They divided the triangle into three parts. Let $s$ be the largest of the areas of these parts. Find the smallest possible value of $s$.

If $x\in\left[-\frac{5\pi}{12},-\frac{\pi}{3}\right]$, then the maximum value of $y=\tan\left(x+\frac{2\pi}{3}\right)-\tan\left(x+\frac{\pi}{6}\right)+\cos\left(x+\frac{\pi}{6}\right)$ is $\text{(A)}\frac{12}{5}\sqrt2\qquad\text{(B)}\frac{11}{6}\sqrt2\qquad\text{(C)}\frac{11}{6}\sqrt3\qquad\text{(D)}\frac{12}{5}\sqrt3$

If $x,y,z$ are positive numbers, prove that $$\frac x{\sqrt{y+z}}+\frac y{\sqrt{z+x}}+\frac z{\sqrt{x+y}}\ge\sqrt{\frac32(x+y+z)}.$$

A trio of lousy salespeople charge increasing prices on tomatoes as you buy more. The first charges you $x^1_1$ dollars for the $x_1$[i]th [/i]tomato you buy from him, the second charges $x^2_2$ dollars for the $x_2$[i]th[/i] tomato, and the third charges $x^3_3$ dollars for the $x_3$[i]th [/i]tomato. If you want to buy $100$ tomatoes for as cheap as possible, how many should you buy from the first salesperson?

Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number $S$. What is the smallest possible value for the sum of the digits of $S$? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 21$

If there are several heaps of stones on the table, it is said that there are $\textit{many}$ stones on the table, if we can find $50$ piles and number them with the numbers from $1$ to $50$ so that the first pile contains at least one stone, the second - at least two stones,..., the $50$-th has at least $50$ stones. Let the table be initially contain $100$ piles of $100$ stones each. Find the largest $n \leq 10 000$ such that after removing any $n$ stones, there will still be $\textit{many}$ stones left on the table.