12th Mexican Mathematical Olympiad Problems 1998

12th Mexican Mathematical Olympiad Problems 1998

A1.  Given a positive integer we can take the sum of the squares of its digits. If repeating this operation a finite number of times gives 1 we call the number tame. Show that there are infinitely many pairs (n, n+1) of consecutive tame integers.

A2.  The lines L and L' meet at A. P is a fixed point on L. A variable circle touches L at P and meets L' at Q and R. The bisector of ∠QPR meets the circle again at T. Find the locus of T as the circle varies.

A3.  Each side and diagonal of an octagon is colored red or black. Show that there are at least 7 triangles whose vertices are vertices of the octagon and whose sides are the same color.

B1.  Find all positive integers that can be written as 1/a₁ + 2/a₂ + ... + 9/a₉, where a_i are positive integers.

B2.  AB, AC are the tangents from A to a circle. Q is a point on the segment AC. The line BQ meets the circle again at P. The line through Q parallel to AB meets BC at J. Show that PJ is parallel to AC iff BC² = AC·QC.

B3.  Given 5 points, no 4 in the same plane, how many planes can be equidistant from the points? (A plane is equidistant from the points if the perpendicular distance from each point to the plane is the same.) 

Solutions

Problem A1

Given a positive integer we can take the sum of the squares of its digits. If repeating this operation a finite number of times gives 1 we call the number tame. Show that there are infinitely many pairs (n, n+1) of consecutive tame integers.

Solution

A little experimentation shows that we often get loops of numbers and chains leading into loops. There are some obvious tame numbers, such as 10, 13, and a few less obvious like 7, 10, 23. But the difficulty is finding any pair of consecutive tame integers. By brute force we can work up to 31 (→ 10 → 1) and 32 (→ 13 → 10 → 1). Now we get infinitely many by inserting arbitrarily many zeros between the 3 and the 1, to get the pairs 300...01, 300...02.