HMMT 二月 2015 · 代数 · 第 5 题

英文解析

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 .
   Answer: Without loss of generality, we assume that c ≥ b ≥ a . We see that 3 c ≥ a + b + c = 10.25
   Therefore, c ≥ .109
   2 2 23
   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
   0 ≤ ( a − b )2 = ( a + b ) − 4 ab 2 = (10 − c ) − 4 (25 − c ( a + b ))2 = (10 − c ) − 4 (25 − c (10 − c ))2 = c (20 − 3 c ) ,
   we obtain c ≤ . Consider m = min { ab, bc, ca } = ab , as bc ≥ ca ≥ ab . We compute ab = 25 − c ( a + b ) =20
   2 10 20 25 253
   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