IT-CESENATICO 2012

On the sides of a triangle $ABC$ right angled at $A$ three points $D, E$ and $F$ (respectively $BC, AC$ and $AB$ ) are chosen so that the quadrilateral $AFDE$ is a square. If $x$ is the length of the side of the square, show that $$\frac{1}{x}=\frac{1}{AB}+\frac{1}{AC}$$

Determine all positive integers that are equal to $300$ times the sum of their digits.

Let $n$ be an integer greater than or equal to $2$ . There are $n$ people in one line, each of which is either a scoundrel (who always lie) or a knight (who always tells the truth). Every person, except the first, indicates a person in front of him/her and says "This person is a scoundrel" or "This person is a knight." Knowing that there are strictly more scoundrel than knights, seeing the statements show that it is possible to determine each person whether he/she is a scoundrel or a knight.

Let $x_1,x_2,x_3, \cdots$ be a sequence defined by the following recurrence relation: $$\begin{cases}x_{1}&= 4 x_{n+1}&= x_{1}x_{2}x_{3}\cdots x_{n}+5\text{ for }n\ge 1\end{cases}$$ The first few terms of the sequence are $x_1=4,x_2=9,x_3=41 \cdots$ Find all pairs of positive integers $\{a,b\}$ such that $x_a x_b$ is a perfect square.

$ABCD$ is a square. Describe the locus of points $P$ , different from $A, B, C, D$ , on that plane for which $$\widehat{APB}+\widehat{CPD}=180^\circ$$

Determine all pairs $\{a, b\}$ of positive integers with the property that, in whatever manner you color the positive integers with two colors $A$ and $B$ , there always exist two positive integers of color $A$ having their difference equal to $a$ or of color $B$ having their difference equal to $b$ .