4th CanMO 1972


1. Three unit circles are arranged so that each touches the other two. Find the radii of the two circles which touch all three.
2. x₁, x₂, ... , x_n are non-negative reals. Let s = ∑_i<j x_ix_j. Show that at least one of the x_i has square not exceeding 2s/(n² - n).
3. Show that 10201 is composite in base n > 2. Show that 10101 is composite in any base.
4. Show how to construct a convex quadrilateral ABCD given the lengths of each side and the fact that AB is parallel to CD.
5. Show that there are no positive integers m, n such that m³ + 11³ = n³.
6. Given any distinct real numbers x, y, show that we can find integers m, n such that mx + ny > 0 and nx + my < 0.
7. Show that the roots of x² - 198x + 1 lie between 1/198 and 197.9949494949... . Hence show that √2 < 1.41421356 (where the digits 421356 repeat). Is it true that √2 < 1.41421356?
8. X is a set with n elements. Show that we cannot find more than 2^n-1 subsets of X such that every pair of subsets has non-empty intersection.
9. Given two pairs of parallel lines, find the locus of the point the sum of whose distances from the four lines is constant.
10. Find the longest possible geometric progression in {100, 101, 102, ... , 1000}.

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