IL-GILLIS 2014

Consider the number $\left(101^2-100^2\right)\cdot\left(102^2-101^2\right)\cdot\left(103^2-102^2\right)\cdot...\cdot\left(200^2-199^2\right)$ .

[list=a] [] Determine its units digit. [] Determine its tens digit. [/list]

Let $\Delta A_1A_2A_3, \Delta B_1B_2B_3, \Delta C_1C_2C_3$ be three equilateral triangles. The vertices in each triangle are numbered clockwise. It is given that $A_3=B_3=C_3$ . Let $M$ be the center of mass of $\Delta A_1B_1C_1$ , and let $N$ be the center of mass of $\Delta A_2B_2C_2$ .

Prove that $\Delta A_3MN$ is an equilateral triangle.

Let $ABCDEF$ be a convex hexagon. In the hexagon there is a point $K$ , such that $ABCK,DEFK$ are both parallelograms. Prove that the three lines connecting $A,B,C$ to the midpoints of segments $CE,DF,EA$ meet at one point.

We are given a row of $n\geq7$ tiles. In the leftmost 3 tiles, there is a white piece each, and in the rightmost 3 tiles, there is a black piece each. The white and black players play in turns (the white starts). In each move, a player may take a piece of their color, and move it to an adjacent tile, so long as it's not occupied by a piece of the same color. If the new tile is empty, nothing happens. If the tile is occupied by a piece of the opposite color, both pieces are destroyed (both white and black). The player who destroys the last two pieces wins the game.

Which player has a winning strategy, and what is it? (The answer may depend on $n$ )

Let $p$ be a polynomial with integer coefficients satisfying $p(16)=36,p(14)=16,p(5)=25$ . Determine all possible values of $p(10)$ .

Let $n$ be a positive integer. Find the maximal real number $k$ , such that the following holds:

For any $n$ real numbers $x_1,x_2,...,x_n$ , we have $\sqrt{x_1^2+x_2^2+\dots+x_n^2}\geq k\cdot\min(|x_1-x_2|,|x_2-x_3|,...,|x_{n-1}-x_n|,|x_n-x_1|)$

Find one real value of $x$ satisfying $\frac{x^7}{7}=1+\sqrt[7]{10}x\left(x^2-\sqrt[7]{10}\right)^2$ .