40th Swedish 2000 Problem 1 Each of the numbers 1, 2, ... , 10 is colored red or blue. 5 is red and at least one number is blue. If m, n are different colors and m+n ≤ 10, then m+n is blue. If m, n are different colors and mn ≤ 10, then mn is red. Find all the colors.
all blue except 5 and 10. Solution If 1 is red, take n to be blue (we are told n exists). Then n = 1·n is red. Contradiction. So 1 is blue. Suppose 4 is red, then 1+4 is blue. Contradiction, so 4 is blue. 1+5= 6 is blue. 4+5 = 9 is blue. If 2 is red, then 3=1+2 is blue, so 5=2+3 is blue. Contradiction, so 2 is blue. Hence 10=2·5 is red. If 8 is red, then 2+8 is blue. Contradiction, so 8 is blue. If 3 is red, then 2+3 is blue. Contradiction, so 3 is blue. If 7 is red, then 3+7 is blue. Contradiction. So 7 is blue. 40th Swedish 2000 © John Scholes jscholes@kalva.demon.co.uk 16 January 2004 Last corrected/updated 16 Jan 04