A square is divided into $25$ small squares, equal to each other, drawing lines parallel to the sides of the square. Some are drawn diagonals of small squares so that there are no two diagonals with a common point. What is the maximum number of diagonals that can be traced?

When we write the number $n > 2$ as the sum of some integers consecutive positives (at least two addends), we say that we have an elegant decomposition of $n$. Two elegant decompositions will be different if any of them contains some term that does not contains the other. How many different elegant decompositions does the number $3^{2004}$ have?

In an exam, $6$ problems were proposed. Every problem was solved by exactly $1000$ students, but in no case has it happened that two students together have solved the $6$ problems. Determine the smallest number of participants that could have been in said exam. original wordingEn un examen fueron propuestos 6 problemas. Cada problema fue resuelto por exactamente 1000 estudiantes, pero en ningun caso ha ocurrido que dos estudiantes en conjunto, hayan resuelto los 6 problemas. Determinar el menor numero de participantes que pudo haber en dicho exame

Determine all real solutions to the system of equations: $$x_1 + x_2 +...+ x_{2004 }= 2004$$$$x^4_1+ x^4_2+ ... + x^4_{2004} = x^3_1+x^3_2+... + x^3_{2004}$$

Write two ones, then a $2$ between them, then a $3$ between the numbers whose sum is $3$, then a $4$ between the numbers whose sum is $4$, as shown below: $$(1, 1), (1, 2, 1),(1, 3, 2, 3, 1), (1, 4, 3, 2, 3, 4, 1)$$and so on. Prove that the number of times $n$ appears, ($n\ge 2$), is equal to the number of positive integers less than $n$ and relative prime with $n$..

In the non-isosceles $\vartriangle ABC$, the interior bisectors of vertices $B$ and $C$ are drawn, which cut the sides $AC$ and $AB$ at $E$ and $F$ respectively.The line $EF$ cuts the extension of side $BC$ at $T$. In the side$ BC$ a point D is located, so that $\frac{DB}{DC} = \frac{TB}{TC}$. Prove that $AT$ is the exterior bisector of angle $A$.

Determine all pairs of natural numbers $ (x, y)$ for which it holds that $$x^2 = 4y + 3gcd (x, y).$$

Consider a circle $K$ and an inscribed quadrilateral $ABCD$, such that the diagonal $BD$ is not the diameter of the circle. Prove that the intersection of the lines tangent to $K$ through the points $B$ and $D$ lies on the line $AC$ if and only if $AB \cdot CD = AD \cdot BC$.

Given the equation $\frac{ax^2-24x+b}{x^2-1} = x$. Find all the real numbers $a$ and $b$ for which you have two real solutions whose sum is equal to $12$.

For real numbers, $a,b,c$ with $bc \ne 0$ we have to $\frac{1-c^2}{bc} \ge 0$. Prove that $$5( a^2+b^2+c^2 -bc^3) \ge ab.$$

Determine all functions $f : R_+ \to R_+$ such that: a) $f(xf(y))f(y) = f(x + y)$ for $x, y \ge 0$ b) $f(2) = 0$ c) $f(x) \ne 0$ for $0 \le x < 2$.

The angle $\angle XOY =\alpha $ and the points $A$ and $B$ on OY are given such that $OA = a$ and $OB = b$ with $a > b$. A circle passes through the points $A$ and $B$ and is tangent to $OX$. a) Calculate the radius of that circle in terms of $a, b$ and $\alpha $. b) If $a$ and $b$ are constants and $\alpha $ varies, show that the minimum value of the radius of the circle is $\frac{a-b}{2}$.