29th Spanish 1993


A1. There is a reunion of 201 people from 5 different countries. In each group of 6 people, at least two have the same age. Show that there must be at least 5 people with the same country, age and sex.
A2. In the triangle of numbers below, each number is the sum of the two immediately above: 0 1 2 3 4 ... 1991 1992 1993 1 3 5 7 ... 3983 3985 4 8 12 ... 7968 ... Show that the bottom number is a multiple of 1993.
A3. Show that for any triangle 2r ≤ R (where r is the inradius and R is the circumradius).
B1. Show that for any prime p ≠ 2, 5, infinitely many numbers of the form 11...1 are multiples of p.
B2. Given a 4 x 4 grid of points as shown below. The points at two opposite corners are marked A and D as shown. How many ways can we choose a set of two further points {B,C} so that the six distances between A, B, C, D are all distinct? . How many of the sets of 4 points are geometrically distinct (so that one cannot be obtained from another by a reflection, rotation etc)? Give the points coordinates (x,y) from (1,1) to (4,4). Take the grid-distance between (x,y) and (u,v) to be \|x-u\| + \|y-v\|. Show that the sum of the six grid-distances between the points is always the same.
B3. A casino game uses the diagram shown. At the start a ball appears at S. Each time the player presses a button, the ball moves to one of the adjacent letters (joined by a line segment) (with equal probability). If the ball returns to S the player loses. If the ball reaches G, then the player wins. Find the probability that the player wins and the expected number of button presses.

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