HMMT 二月 2015 · 代数 · 第 5 题

HMMT February 2015 — Algebra — Problem 5

Let a , b , c be positive real numbers such that a + b + c = 10 and ab + bc + ca = 25. Let m = min { ab, bc, ca } . Find the largest possible value of m . 3

解析

Let a , b , c be positive real numbers such that a + b + c = 10 and ab + bc + ca = 25. Let m = min { ab, bc, ca } . Find the largest possible value of m . 25 Answer: Without loss of generality, we assume that c ≥ b ≥ a . We see that 3 c ≥ a + b + c = 10. 9 10 Therefore, c ≥ . 3 2 2 2 For instance, the O ( x ) refers to a function bounded by C | x | for some positive constant C , whenever x is close enough to − 100 0 (and as the 10 suggests, that’s all we care about). Algebra Since 2 0 ≤ ( a − b ) 2 = ( a + b ) − 4 ab 2 = (10 − c ) − 4 (25 − c ( a + b )) 2 = (10 − c ) − 4 (25 − c (10 − c )) = c (20 − 3 c ) , 20 we obtain c ≤ . Consider m = min { ab, bc, ca } = ab , as bc ≥ ca ≥ ab . We compute ab = 25 − c ( a + b ) = 3 2 10 20 25 25 25 − c (10 − c ) = ( c − 5) . Since ≤ c ≤ , we get that ab ≤ . Therefore, m ≤ in all cases and 3 3 9 9 5 5 20 the equality can be obtained when ( a, b, c ) = ( , , ). 3 3 3 3