3rd CanMO 1971


1. A diameter and a chord of a circle intersect at a point inside the circle. The two parts of the chord are length 3 and 5 and one part of the diameter is length 1. What is the radius of the circle?
2. If two positive real numbers x and y have sum 1, show that (1 + 1/x)(1 + 1/y) ≥ 9.
3. ABCD is a quadrilateral with AB = CD and ∠ABC > ∠BCD. Show that AC > BD.
4. Find all real a such that x² + ax + 1 = x² + x + a = 0 for some real x.
5. A polynomial with integral coefficients has odd integer values at 0 and 1. Show that it has no integral roots.
6. Show that n² + 2n + 12 is not a multiple of 121 for any integer n.
7. Find all five digit numbers such that the number formed by deleting the middle digit divides the original number.
8. Show that the sum of the lengths of the perpendiculars from a point inside a regular pentagon to the sides (or their extensions) is constant. Find an expression for it in terms of the circumradius.
9. Find the locus of all points in the plane from which two flagpoles appear equally tall. The poles are heights h and k and are a distance 2a apart.
10. n people each have exactly one unique secret. How many phone calls are needed so that each person knows all n secrets? You should assume that in each phone call the caller tells the other person every secret he knows, but learns nothing from the person he calls.

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