HMMT 十一月 2023 · 冲刺赛 · 第 10 题

HMMT November 2023 — Guts Round — Problem 10

[8] A real number x is chosen uniformly at random from the interval (0 , 10). Compute the probability √ √ √ that x , x + 7, and 10 − x are the side lengths of a non-degenerate triangle.

解析

[8] A real number x is chosen uniformly at random from the interval (0 , 10). Compute the probability √ √ √ that x , x + 7, and 10 − x are the side lengths of a non-degenerate triangle. Proposed by: Pitchayut Saengrungkongka 22 Answer: 25 Solution 1: For any positive reals a, b, c , numbers a, b, c is a side length of a triangle if and only if X 2 2 4 ( a + b + c )( − a + b + c )( a − b + c )( a + b − c ) > 0 ⇐⇒ (2 a b − a ) > 0 , cyc (to see why, just note that if a ≥ b + c , then only the factor − a + b + c is negative). Therefore, x works if and only if 2 2 2 2( x + 7)(10 − x ) + 2 x ( x + 7) + 2 x (10 − x ) > x + ( x + 7) + (10 − x ) 2 − 5 x + 46 x − 9 > 0 1 x ∈ , 9 , 5 22 giving the answer . 25 √ √ √ Solution 2: Note that x < x + 7, so x cannot be the maximum. Thus, x works if and only if the following equivalent inequalities hold. √ √ √ x > x + 7 − 10 − x p x > ( x + 7) + (10 − x ) − 2 ( x + 7)(10 − x ) p 4( x + 7)(10 − x ) > 17 − x 2 4( x + 7)(10 − x ) > x − 34 x + 289 2 2 4(70 + 3 x − x ) > x − 34 x + 289 2 0 > 5 x − 46 x + 9 0 > (5 x − 1)( x − 9) , 1 so the range is x ∈ , 9 , and the answer is 5 1 9 − 22 5 = . 10 25