There are $1000$ balls of dough $0.38$ and $5000$ balls of dough $0.038$ that must be packed in boxes. A box contains a collection of balls whose total mass is at most $1$. Find the smallest number of boxes that they are needed.

In a school with 5 different grades there are 250 girls and 250 boys. Each grade has the same number of students. for a competition of knowledge wants to form teams of a female and a male who are of the same grade. If in each grade there are at least $19$ females and $19$ males. Find the greatest amount of teams that can be formed.

On a $123 \times 123$ board, each square is painted red or blue according to the following conditions: a) Each square painted red that is not on the edge of the board has exactly $5$ blue squares among its $8$ neighboring squares. b) Each square painted blue that is not on the edge of the board has exactly $4$ red squares among its $8$ neighboring squares. Determine the number of red-painted squares on the board.

With $21$ pieces, some white and some black, a rectangle is formed of $3 \times 7$. Prove that there are always four pieces of the same color located at the vertices of a rectangle.

If $$\frac{x_1}{x_1+1} = \frac{x_2}{x_2+3} = \frac{x_3}{x_3+5} = ...= \frac{x_{1006}}{x_{1006}+2011}$$and $x_1+x_2+...+x_{1006} = 503^2$, determine the value of $x_{1006}$.

Given the triangle $ABC$, let $L$, $M$ and $N $be the midpoints of $BC$, $CA$ and $AB$ respectively. The lines $LM$ and $LN$ cut the tangent to the circumcircle at $A$ at $P$ and $Q$ respectively . Prove that $CP \parallel BQ$.

A mathematics teacher writes a quadratic equation on the blackboard of the form $$x^2+mx \star n = 0$$, with $m$ and $n$ integers. The sign of $n$ is blurred. Even so, Claudia solves it and obtains integer solutions, one of which is $2011$. Find all possible values of $m$ and $n$.

Let $x, y, z$ be positive reals. Prove that $$\frac{xz}{x^2 + xy + y^2 + 6z^2} + \frac{zx}{z^2 + zy + y^2 + 6x^2} + \frac{xy}{x^2 + xz + z^2 + 6y^2} \le \frac13$$

Find all pairs $(m, n)$ of positive integers such that $m^2 + n^2 =(m + 1)(n + 1).$

Let $ABC$ be a right triangle at $A$, and let $AD$ be the relative height to the hypotenuse. Let $N$ be the intersection of the bisector of the angle of vertex $C$ with $AD$. Prove that $$AD \cdot BC = AB \cdot DC + BD \cdot AN.$$

Find all the functions $f : R\to R$ such that $f(x^2 + f(y)) = y - x^2$ for all $x, y$ reals.

If the natural numbers $a, b, c, d$ verify the relationships: $$(a^2 + b^2)(c^2 + d^2) = (ab + cd)^2$$$$(a^2 + d^2)(b^2 + c^2) = (ad + bc)^2$$and the $gcd(a, b, c, d) = 1$, prove that $a + b + c + d$ is a perfect square.

Let $O$ be a point interior to triangle $ABC$ such that $\angle BAO = 30^o$, $\angle CBO = 20^o$ and $\angle ABO = \angle ACO = 40^o$ . Knowing that triangle $ABC$ is not equilateral, find the measures of its interior angles.