IT-CESENATICO 1996

Among all the triangles which have a fixed side $l$ and a fixed area $S$ , determine for which triangles the product of the altitudes is maximum.

Show that the equation $a^2 + b^2 = c^2 + 3$ has infinetely many triples of integers $a, b, c$ that are solutions.

Given a cube of unit side. Let $A$ and $B$ be two opposite vertex. Determine the radius of the sphere, with center inside the cube, tangent to the three faces of the cube with common point $A$ and tangent to the three sides with common point $B$ .

There is a list of $n$ football matches. Determine how many possible columns, with an even number of draws, there are.

Given a circle $C$ and an exterior point $A$ . For every point $P$ on the circle construct the square $APQR$ (in counterclock order). Determine the locus of the point $Q$ when $P$ moves on the circle $C$ .

What is the minimum number of squares that is necessary to draw on a white sheet to obtain a square grid of side $n$ ?