There is a board with $2010$ rows and $2001$ columns, on it there is a token located in the upper left box that can perform one of the following operations: (A) Walk 3 steps horizontally or vertically. (B) Walk 2 steps to the right and 3 steps down. (C) Walk 2 steps to the left and 2 steps up. With the condition that immediately after carrying out an operation on (B) or (C) it is mandatory to take a step to the right before perform the following operation. It is possible to exit the board, so count the number of steps necessary, entering through the other end of the row or column from which it exits, as if the board outside circular (example: from the beginning you can walk to the square located in row $1$ and column $1999$). Will it be possible that after $2011$ operations allowed the checker to land exactly on the bottom square right?

A cube of dimensions $20 \times 20 \times 20$ is constructed with blocks of $1 \times 2 \times 2$. Prove that there is a line that passes through the cube but not any block.

We have a board of $ 2011 \times 2011$, divided by lines parallel to the edges into $1 \times 1$ squares. Manuel, Reinaldo and Jorge (at that time order) play to form squares with vertices at the vertices of the grid. The one who forms the last possible square wins, so that its sides do not cut the sides of any unit square. Who can be sure that he will win?

Let $P(x) = x^3 + (t - 1)x^2 - (t + 3)x + 1$. For what values of real $t$ the sum of the squares and the reciprocals of the roots of $ P(x)$ is minimum?

Determine all the integer solutions of the equation $3x^4-2024y+1= 0$.

Let $n$ be a positive integer and let $$1 = d_1 < d_2 < d_3 < d_4$$the four smallest divisors of $n$. Find all$ n$ such that $$n^2 = d_1 + d_2^2+d_3^3 +d_4^4.$$

Let $x_1, x_2, ..., x_{24}$ be real numbers. prove that $$x_1 + 2x_2 + 3x_3 +...+ 24x_{24} - 439 \le \frac{x^2_1+x^2_2+... + x^2_{24}}{2}+ 2011.$$

Determine all functions $f : R \to R$ such that $$f(x)f(y) = 2f(x + y) + 9xy \ \ \forall x, y \in R.$$

Let $ABC$ be a triangle with circumcenter $O$. Let $\omega (O_1)$ be the circumference which passes through $A$ and $B$ and is tangent to $BC$ at $B$. $\omega (O_2)$ the circle that passes through $A$ and $C$ and is tangent to $BC$ at $C$. Let $M$ the midpoint of $O_1O_2$ and $D$ the symmetric point of $O$ with respect to $A$. Prove that $\angle O_1DM = \angle ODO_2$.

Find a set of positive integers with the greatest possible number of elements such that the least common multiple of all of them is less than $2011$.