HMMT 二月 2008 · 冲刺赛 · 第 21 题

HMMT February 2008 — Guts Round — Problem 21

[ 10 ] Let ABC be a triangle with AB = 5, BC = 4 and AC = 3. Let P and Q be squares inside ABC with disjoint interiors such that they both have one side lying on AB . Also, the two squares each have an edge lying on a common line perpendicular to AB , and P has one vertex on AC and Q has one vertex on BC . Determine the minimum value of the sum of the areas of the two squares. C Q P A B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 11 HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUND 2

解析

[ 10 ] Let ABC be a triangle with AB = 5, BC = 4 and AC = 3. Let P and Q be squares inside ABC with disjoint interiors such that they both have one side lying on AB . Also, the two squares each have an edge lying on a common line perpendicular to AB , and P has one vertex on AC and Q has one vertex on BC . Determine the minimum value of the sum of the areas of the two squares. C Q P A B 144 Answer: Let the side lengths of P and Q be a and b , respectively. Label two of the vertices of 49 P as D and E so that D lies on AB and E lies on AC , and so that DE is perpendicular to AB . The 3 triangle ADE is similar to ACB . So AD = a . Using similar arguments, we find that 4 3 a 4 b

a + b + = AB = 5 4 3 so a b 5
= . 4 3 7 Using Cauchy-Schwarz inequality, we get ( ) ( ) 2 ( ) 1 1 a b 25 2 2 a + b + ≥ + = . 2 2 4 3 4 3 49 5 It follows that 144 2 2 a + b ≥ . 49 36 48 Equality occurs at a = and b = . 35 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 11 HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUND 2