A parallelopiped has vertices $A_1, A_2, ... , A_8$ and center $O$. Show that: \[ 4 \sum_{i=1}^8 OA_i ^2 \leq \left(\sum_{i=1}^8 OA_i \right) ^2 \]

The sum of the angles $A$ and $C$ of a convex quadrilateral $ABCD$ is less than $180^{\circ} .$ Prove that \[AB \cdot CD + AD \cdot BC < AC(AB + AD).\]

The lengths of the sides of a triangle with positive area are $\log_{10} 12$, $\log_{10} 75$, and $\log_{10} n$, where $n$ is a positive integer. Find the number of possible values for $n$.

Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.

Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds. Prove that $\sum_{j=1}^n |a_j| \leq 3$.

In trapezoid $ ABCD$, $ AB \parallel CD$, $ \angle CAB < 90^\circ$, $ AB \equal{} 5$, $ CD \equal{} 3$, $ AC \equal{} 15$. What are the sum of different integer values of possible $ BD$? $\textbf{(A)}\ 101 \qquad\textbf{(B)}\ 108 \qquad\textbf{(C)}\ 115 \qquad\textbf{(D)}\ 125 \qquad\textbf{(E)}\ \text{None}$

A number of robots are placed on the squares of a finite, rectangular grid of squares. A square can hold any number of robots. Every edge of each square of the grid is classified as either passable or impassable. All edges on the boundary of the grid are impassable. You can give any of the commands up, down, left, or right. All of the robots then simultaneously try to move in the specified direction. If the edge adjacent to a robot in that direction is passable, the robot moves across the edge and into the next square. Otherwise, the robot remains on its current square. You can then give another command of up, down, left, or right, then another, for as long as you want. Suppose that for any individual robot, and any square on the grid, there is a finite sequence of commands that will move that robot to that square. Prove that you can also give a finite sequence of commands such that all of the robots end up on the same square at the same time.

Natural numbers from $1$ to $200$ were divided into $50$ sets. Prove that one of them contains three numbers that are the lengths of the sides of some triangle

Let $n\geq 2$ be an integer, and let $0 < a_1 < a_2 < \cdots < a_{2n+1}$ be real numbers. Prove the inequality \[ \sqrt[n]{a_1} - \sqrt[n]{a_2} + \sqrt[n]{a_3} - \cdots + \sqrt[n]{a_{2n+1}} < \sqrt[n]{a_1 - a_2 + a_3 - \cdots + a_{2n+1}}. \] [i]Bogdan Enescu, Romania[/i]

Each point on the plane is colored either red or blue. Show that there are three points of the same color that form a triangle with side lengths $1, 2,\sqrt3$.

For positive integer $n$, define $S_n$ to be the minimum value of the sum \[ \sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2}, \] where $a_1,a_2,\ldots,a_n$ are positive real numbers whose sum is 17. There is a unique positive integer $n$ for which $S_n$ is also an integer. Find this $n$.

Faith has four different integer length segments. It turned out that any three of them can form a triangle. What is the smallest total length of this set of segments?

In a village, there are $n$ houses with $n>2$ and all of them are not collinear. We want to generate a water resource in the village. For doing this, point $A$ is [i]better[/i] than point $B$ if the sum of the distances from point $A$ to the houses is less than the sum of the distances from point $B$ to the houses. We call a point [i]ideal[/i] if there doesn’t exist any [i]better[/i] point than it. Prove that there exist at most $1$ [i]ideal[/i] point to generate the resource.

Let $ a\ge 3 $ and a polynom $ P. $ Show that: $$ \max_{1\le k\le \text{grad} P} \left| a^{k-1}-P(k-1) \right| \ge 1 $$

You have five segments of lengths $a_1, a_2, a_3, a_4,$ and $a_5$ such that it is possible to form a triangle with any three of them. Demonstrate that at least one of those triangles has angles that are all acute.

Let $ ABC$ be an equilateral triangle with side length 2, and let $ \Gamma$ be a circle with radius $ \frac {1}{2}$ centered at the center of the equilateral triangle. Determine the length of the shortest path that starts somewhere on $ \Gamma$, visits all three sides of $ ABC$, and ends somewhere on $ \Gamma$ (not necessarily at the starting point). Express your answer in the form of $ \sqrt p \minus{} q$, where $ p$ and $ q$ are rational numbers written as reduced fractions.

Show that the number $r(n)$ of representations of $n$ as a sum of two squares has $\pi$ as arithmetic mean, that is \[\lim_{n \to \infty}\frac{1}{n}\sum^{n}_{m=1}r(m) = \pi.\]

There are $n \ge 3$ positive integers written on a board. A [i]move[/i] consists of choosing three numbers $a, b, c$ written from the board such that there exists a non-degenerate non-equilateral triangle with sides $a, b, c$ and replacing those numbers with $a + b - c, b + c - a$ and $c + a - b$. Prove that a sequence of moves cannot be infinite.

Let $\triangle ABC$ be an acute triangle, $H$ is its orthocenter. $\overrightarrow{AH},\overrightarrow{BH},\overrightarrow{CH}$ intersect $\triangle ABC$'s circumcircle at $A',B',C'$ respectively. Find the range (minimum value and the maximum upper bound) of $$\dfrac{AH}{AA'}+\dfrac{BH}{BB'}+\dfrac{CH}{CC'}$$

Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that \[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0. \] Determine when equality occurs.

A convex quadrilateral is inscribed in a circle of radius $1$. Prove that the difference between its perimeter and the sum of the lengths of its diagonals is greater than zero and less than $2.$

Let $ x, y, z$ be positive real numbers such that $ x^n, y^n$ and $ z^n$ are side lengths of some triangle for all positive integers $ n$. Prove that at least two of x, y and z are equal.

The set of all points $P$ such that the sum of the (undirected) distances from $P$ to two fixed points $A$ and $B$ equals the distance between $A$ and $B$ is $\textbf{(A) }\text{the line segment from }A\text{ to }B\qquad$ $\textbf{(B) }\text{the line passing through }A\text{ and }B\qquad$ $\textbf{(C) }\text{the perpendicular bisector of the line segment from }A\text{ to }B\qquad$ $\textbf{(D) }\text{an elllipse having positive area}\qquad$ $\textbf{(E) }\text{a parabola}$

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle? $ \textbf{(A) } 43 \qquad \textbf{(B) } 44 \qquad \textbf{(C) } 45 \qquad \textbf{(D) } 46 \qquad \textbf{(E) } 47$

Find the number of all integer-sided [i]isosceles obtuse-angled[/i] triangles with perimeter $ 2008$. [16 points out of 100 for the 6 problems]