HMMT 十一月 2019 · GEN 赛 · 第 5 题
HMMT November 2019 — GEN Round — Problem 5
题目详情
- Let a, b, c be positive real numbers such that a ≤ b ≤ c ≤ 2 a . Find the maximum possible value of b c a
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- . a b c
解析
- Let a, b, c be positive real numbers such that a ≤ b ≤ c ≤ 2 a . Find the maximum possible value of b c a
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- . a b c Proposed by: Carl Schildkraut 7 Answer: 2 Fix the values of b, c. By inspecting the graph of b x f ( x ) = + , x c we see that on any interval the graph attains its maximum at an endpoint. This argument applies when we fix any two variables, so it suffices to check boundary cases in which b = a or b = c, and c = b or c = 2 a. All pairs of these conditions determine the ratio between a, b, c, except b = c and c = b, in which case the boundary condition on a tells us that a = b or 2 a = b = c. In summary, these cases are ( a, b, c ) ∈ { ( a, a, a ) , ( a, a, 2 a ) , ( a, 2 a, 2 a ) } . 7 The largest value achieved from any of these three is . 2