3rd OMCC 2001


A1. A and B stand in a circle with 2001 other people. A and B are not adjacent. Starting with A they take turns in touching one of their neighbors. Each person who is touched must immediately leave the circle. The winner is the player who manages to touch his opponent. Show that one player has a winning strategy and find it.
A2. C and D are points on the circle diameter AB such that ∠AQB = 2 ∠COD. The tangents at C and D meet at P. The circle has radius 1. Find the distance of P from its center.
A3. Find all squares which have only two non-zero digits, one of them 3.
B1. Find the smallest n such that the sequence of positive integers a₁, a₂, ... , a_n has each term ≤ 15 and a₁! + a₂! + ... + a_n! has last four digits 2001.
B2. a, b, c are reals such that if p₁, p₂ are the roots of ax² + bx + c = 0 and q₁, q₂ are the roots of cx² + bx + a = 0, then p₁, q₁, p₂, q₂ is an arithmetic progression of distinct terms. Show that a + c = 0.
B3. 10000 points are marked on a circle and numbered clockwise from 1 to 10000. The points are divided into 5000 pairs and the points of each pair are joined by a segment, so that each segment intersects just one other segment. Each of the 5000 segments is labeled with the product of the numbers at its endpoints. Show that the sum of the segment labels is a multiple of 4.

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