16th Brasil 1994


1. The edges of a cube are labeled from 1 to 12 in an arbitrary manner. Show that it is not possible to get the sum of the edges at each vertex the same. Show that we can get eight vertices with the same sum if one of the labels is changed to 13.
2. Given any convex polygon, show that there are three consecutive vertices such that the polygon lies inside the circle through them.
3. We are given n objects of identical appearance, but different mass, and a balance which can be used to compare any two objects (but only one object can be placed in each pan at a time). How many times must we use the balance to find the heaviest object and the lightest object?
4. Show that if the positive real numbers a, b satisfy a³ = a+1 and b⁶ = b+3a, then a > b.
5. Call a super-integer an infinite sequence of decimal digits: ...d_n...d₂d₁. Given two such super-integers ...c_n...c₂c₁ and ...d_n...d₂d₁, their product ...p_n...p₂p₁ is formed by taking p_n...p₂p₁ to be the last n digits of the product c_n...c₂c₁ and d_n...d₂d₁. Can we find two non-zero super-integers with zero product (a zero super-integer has all its digits zero).
6. A triangle has semi-perimeter s, circumradius R and inradius r. Show that it is right-angled iff 2R = s - r.

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