24th Swedish 1984


1. A, B are points inside a circle C. Show that there is a circle through A and B which lies entirely inside C.
2. Each point in a 3 x 7 array is colored yellow or blue. Show that it is always possible to find 4 points of the same color forming a rectangle with sides parallel to the side of the array.
3. Show that ( (a+1)/(b+1) )^b+1 ≥ (a/b)^b for positive reals a, b.
4. Find all positive integers m, n such that all roots of (x² - mx + n)(x² - nx + m) are positive integers.
5. Find all positive integer solutions to a³ - b³ - c³ = 3abc, a² = 2(a + b + c).
6. The positive integers a₁, a₂, ... , a₁₄ satisfy ∑ 3^a_i = 6558. Show that they must be two copies of each of 1, 2, ... , 7.

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