16th Chinese 2001 Problem B1 7 distinct primes can be written as a, b, c, a + b + c, a + b - c, a - b + c, -a + b + c, where two of a, b, c sum to 800. Let d be the difference between the largest and the smallest of the 7 primes. What is the largest possible value of d? Solution
: 1594. Suppose a + b = 800 and a < b. If c ≥ 800, then a + b - c <= 0, which is not possible, so c < 800. Clearly the largest of the 7 primes is a + b + c. The two largest primes under 800 are 787 and 797. So a + b + c ≤ 800 + 797 = 1597. Clearly a, b, c are all odd, because the only even prime is 2, but if one of a, b, c is even, then more than one of the seven primes is even. Hence all the 7 primes are odd. So the smallest possible is 3. Hence d ≤ 1597 - 3 = 1594. But that is achieved by a = 13, b = 787, c = 797, which gives a + b - c = 3, a = 13, a - b + c = 23, b = 787, c = 797, -a + b + c = 1571, a + b + c = 1597, all of which are prime. 16th Chinese 2001 © John Scholes jscholes@kalva.demon.co.uk 5 June 2002