Let $T(n)$ be the number of dissimilar (non-degenerate) triangles with all side lengths integral and $\leq n$. Find $T(n+1)-T(n)$.

Prove that \[ \frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3} \] for all positive real numbers $a,b,c,d$.

What are the last two digits of ${7^{7^{7^7}}}$?

On the sides $BC, AC$ and $AB$ of the equilateral triangle $ABC$ mark the points $D, E$ and $F$ so that $\angle AEF = \angle FDB$ and $\angle AFE = \angle EDC$. Prove that $DA$ is the bisector of the angle $EDF$.

Some of the faces of a convex polyhedron $M$ are painted in blue, others are painted in white and there are no two walls with a common edge. Prove that if the sum of surfaces of the blue walls is bigger than half surface of $M$ then it may be inscribed a sphere in the polyhedron given $(M)$. [i](H. Lesov)[/i]

Suppose $f:\mathbb{R}\to \mathbb{R}$ is a two times differentiable function satisfying $f(0)=1,f^{\prime}(0)=0$ and for all $x\in [0,\infty)$, it satisfies \[ f^{\prime \prime}(x)-5f^{\prime}(x)+6f(x)\ge 0 \] Prove that, for all $x\in [0,\infty)$, \[ f(x)\ge 3e^{2x}-2e^{3x} \]

Three numbers are selected at random from the interval $[0,1]$. What is the probability that they form the lengths of the sides of a triangle?

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.)