HMMT 二月 2009 · 冲刺赛 · 第 19 题

HMMT February 2009 — Guts Round — Problem 19

[ 10 ] Shelly writes down a vector v = ( a, b, c, d ), where 0 < a < b < c < d are integers. Let σ ( v ) denote the set of 24 vectors whose coordinates are a , b , c , and d in some order. For instance, σ ( v ) contains ( b, c, d, a ). Shelly notes that there are 3 vectors in σ ( v ) whose sum is of the form ( s, s, s, s ) for some s . What is the smallest possible value of d ?

解析

[ 10 ] Shelly writes down a vector v = ( a, b, c, d ), where 0 < a < b < c < d are integers. Let σ ( v ) denote the set of 24 vectors whose coordinates are a , b , c , and d in some order. For instance, σ ( v ) contains ( b, c, d, a ). Shelly notes that there are 3 vectors in σ ( v ) whose sum is of the form ( s, s, s, s ) for some s . What is the smallest possible value of d ? Answer: 6 Solution: If k = a + b + c + d , first you notice 4 | 3 k , and k ≥ 10. So we try k = 12, which works with a, b, c, d = 1 , 2 , 3 , 6 and not 1 , 2 , 4 , 5.