11th Mexican 1997


A1. Find all primes p such that 8p⁴ - 3003 is a (positive) prime.
A2. ABC is a triangle with centroid G. P, P' are points on the side BC, Q is a point on the side AC, R is a point on the side AB, such that AR/RB = BP/PC = CQ/QA = CP'/P'B. The lines AP' and QR meet at K. Show that P, G and K are collinear.
A3. Show that it is possible to place the numbers 1, 2, ... , 16 on the squares of a 4 x 4 board (one per square), so that the numbers on two squares which share a side differ by at most 4. Show that it is not possible to place them so that the difference is at most 3.
B1. 3 non-collinear points in space determine a unique plane, which contains the points. What is the smallest number of planes determined by 6 points in space if no three points are collinear and the points do not all lie in the same plane?
B2. ABC is a triangle. P, Q, R are points on the sides BC, CA, AB such that BQ, CR meet at A', CR, AP meet at B', AP, BQ meet at C' and we have AB' = B'C', BC' = C'A', CA' = A'B'. Find area PQR/area ABC.
B3. Show that we can represent 1 as 1/5 + 1/a₁ + 1/a₂ + ... + 1/a_n (for positive integers a_i) in infinitely many different ways.

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