HMMT 十一月 2013 · GEN 赛 · 第 7 题

英文解析

7. [ 6 ] Find the largest real number λ such that a + b + c + d ≥ ab + λbc + cd for all real numbersa, b, c, d .
   2 2 2 23
   Answer: Let f ( a, b, c, d ) = ( a + b + c + d ) − ( ab + λbc + cd ). For fixed ( b, c, d ), f is minimizedb c 2
   at a = , and for fixed ( a, b, c ), f is minimized at d = , so simply we want the largest λ such that
   2 2
   b c 3 2 2 3 3
   f ( , b, c, ) = ( b + c ) − λbc is always nonnegative. By AM-GM, this holds if and only if λ ≤ 2 = .
   2 2 4 4 2