32nd Spanish 1996


A1. The integers m, n are such that (m+1)/n + (n+1)/m is an integer. Show that gcd(m,n) ≤ √(m+n).
A2. G is the centroid of ABC. Show that if AB + GC = AC + GB, then the triangle is isosceles.
A3. p(x) = ax²+bx+c, q(x) = cx²+bx+a, and \|p(-1)\| ≤ 1, \|p(0)\| ≤ 1, \|p(1)\| ≤ 1. Show that \|p(x)\| ≤ 5/4 and \|q(x)\| ≤ 2 for x ∈ [-1,1].
B1. Discuss the existence of solutions to the equation √(x²-p) + 2√(x²-1) = x for varying values of the real parameter p.
B2. In Port Aventura there are 16 secret agents. Each of the agents watches some of his rivals. It is known that if agent A watches agent B, then agent B does not watch agent A. It is possible to find 10 agents such that the first watches the second, the second watches the third, ... , and the tenth watches the first. Show that it is possible to find a cycle of 11 such agents.
B3. Take a cup made of 6 regular pentagons, so that two such cups could be put together to form a regular dodecahedron. The edge length is 1. What volume of liquid will the cup hold?

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.