How many ordered triples $(x,y,z)$ of integers satisfy the system of equations below? \[ \begin{array}{l} x^2-3xy+2yz-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \end{array} \] $\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ \text{a finite number greater than 2} \qquad \text{(E)} \ \text{infinately many}$

Given $n$ positive real numbers $x_1,x_2,x_3,...,x_n$ such that $$\left (1+\frac{1}{x_1}\right )\left(1+\frac{1}{x_2}\right)...\left(1+\frac{1}{x_n}\right)=(n+1)^n$$ Determine the minimum value of $x_1+x_2+x_3+...+x_n$. [i]Proposed by Loh Kwong Weng[/i]

Find all polynomials $p$ with integer coefficients such that the number $p(a) - p(b)$ is divisible by $a + b$ for all integers $a, b$, provided that $a + b \neq 0$.

Find all triples of positive integers $(a,b,c)$ such that: $a^2bc-2ab^2c-2abc^2+b^3c+bc^3+2b^2c^2=11$

Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the sum of the real solutions to the equation \[\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}.\] Find $a+b$.

Find all prime number $p$ such that there exists an integer-coefficient polynomial $f(x)=x^{p-1}+a_{p-2}x^{p-2}+…+a_1x+a_0$ that has $p-1$ consecutive positive integer roots and $p^2\mid f(i)f(-i)$, where $i$ is the imaginary unit.

Inside the square $ABCD$ mark the point $P$, for which $\angle BAP = 30^o$ and $\angle BCP = 15^o$. The point $Q$ was chosen so that $APCQ$ is an isosceles trapezoid ($PC\parallel AQ$). Find the angles of the triangle $CAM$, where $M$ is the midpoint of $PQ$.

An angle of size $n \times m$, where $m, n \ge 2$, is called a figure, resulting from a rectangle of size $n \times m$ cells by removing the rectangle size $(n - 1) \times (m - 1)$ cells. Two players take turns making moves consisting in painting in a corner an arbitrary non-zero number of cells forming a rectangle or square.

Prove that the composition of the sets of one of the following two forms is finite: (a) $2^{2^n}+1$ (b) $6^{2^n}+1$

If I roll three fair $4$-sided dice, what is the probability that the sum of the resulting numbers is relatively prime to the product of the resulting numbers?

Find the least positive integer $k$ for which the equation $\lfloor \frac{2002}{n}\rfloor = k$ has no integer solutions for $n.$ (The notation $\lfloor x \rfloor$ means the greatest integer less than or equal to $x.$)

Darwin takes an $11\times 11$ grid of lattice points and connects every pair of points that are 1 unit apart, creating a $10\times 10$ grid of unit squares. If he never retraced any segment, what is the total length of all segments that he drew? [i]Ray Li.[/i] [hide="Clarifications"][list=1][*]The problem asks for the total length of all *unit* segments (with two lattice points in the grid as endpoints) he drew.[/list][/hide]

Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$

Let $ABCD$ be a rectangle of sides $AB = 4$ and $BC = 3$. The perpendicular on the diagonal $BD$ drawn from $A$ cuts $BD$ at point $H$. We call $M$ the midpoint of $BH$ and $N$ the midpoint of $CD$. Calculate the measure of the segment $MN$.

Find the largest natural number $ k $ which has the property that there is a partition of the natural numbers $ \bigcup_{1\le j\le k} V_j, $ an index $ i\in\{ 1,\ldots ,k \} $ and three natural numbers $ a,b,c\in V_i, $ satisfying $ a+2b=4c. $

Let $f(x)=\tfrac{x+a}{x+b}$ for real numbers $x$ such that $x\neq -b$. Compute all pairs of real numbers $(a,b)$ such that $f(f(x))=-\tfrac{1}{x}$ for $x\neq0$.

Alice and Bob are playing a game on a graph with $n\ge3$ vertices. At each moment, Alice needs to choose two vertices so that the graph is connected even if one of them (along with the edges incident to it) is removed. Each turn, Bob removes one edge in the graph, and upon the removal, Alice needs to re-select the two vertices if necessary. However, Bob has to guarantee that after each removal, any two vertices in the graph are still connected via at most $k$ intermediate vertices. Here $0\le k\le n-2$ is some given integer. Suppose that Bob always knows which two vertices Alice chooses, and that initially, the graph is a complete graph. Alice's objective is to change her choice of the two vertices as few times as possible, and Bob's objective is to make Alive re-select as many times as possible. If both Alice and Bob are sufficiently smart, how many times will Alice change her choice of the two vertices? (usjl)

$ABCD$ is a scalene tetrahedron and let $G$ be its baricentre. A plane $\alpha$ passes through $G$ such that it intersects neither the interior of $\Delta BCD$ nor its perimeter. Prove that $$\textnormal{dist}(A,\alpha)=\textnormal{dist}(B,\alpha)+\textnormal{dist}(C,\alpha)+\textnormal{dist}(D,\alpha).$$ [i] (Adapted from folklore)[/i]

From the positive integers, $m,m+1,\dots,m+n$, only the sum of digits of $m$ and the sum of digits of $m+n$ are divisible by $8$. Find the maximum value of $n$.

Find the continuous functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following property: $$ f\left( x+\frac{1}{n}\right) \le f(x) +\frac{1}{n},\quad\forall n\in\mathbb{Z}^* ,\quad\forall x\in\mathbb{R} . $$

For which $n$, $ n \in \mathbb{N}$ is the number $1^n+2^n+3^n$ divisible by $7$?

Caroline wants to plant 10 trees in her orchard. Planting $n$ apple trees requires $n^2$ square meters, planting $n$ apricot trees requires $5n$ square meters, and planting $n$ plum trees requires $n^3$ square meters. If she is committed to growing only apple, apricot, and plum trees, what is the least amount of space in square meters, that her garden will take up?

Let $ABCD$ be a convex trapezoid such that $\angle BAD = \angle ADC = 90^o$, $AB = 20$, $AD = 21$, and $CD = 28$. Point $P \ne A$ is chosen on segment $AC$ such that $\angle BPD = 90^o$. Compute $AP$.

Let $m$ be a positive integer not divisible by 3. Prove that there are infinitely many positive integers $n$ such that $s(n)$ and $s(n+1)$ are divisible by $m$, where $s(x)$ is the sum of digits of $x$. [i]Dorel Miheț[/i]

A sequence of $X$s and $O$s is given, such that no three consecutive characters in the sequence are all the same, and let $N$ be the number of characters in this sequence. Maia may swap two consecutive characters in the sequence. After each swap, any consecutive block of three or more of the same character will be erased (if there are multiple consecutive blocks of three or more characters after a swap, then they will be erased at the same time), until there are no more consecutive blocks of three or more of the same character. For example, if the original sequence were $XXOOXOXO$ and Maia swaps the fifth and sixth character, the end result will be $$XXOOOXXO \to XXXXO \to O.$$ Find the maximum value $N$ for which Maia can’t necessarily erase all the characters after a series of swaps. Partial credit will be awarded for correct proofs of lower and upper bounds on $N$.