HMMT 十一月 2019 · 冲刺赛 · 第 10 题

HMMT November 2019 — Guts Round — Problem 10

[8] Let ABCD be a square of side length 5, and let E be the midpoint of side AB . Let P and Q be the feet of perpendiculars from B and D to CE , respectively, and let R be the foot of the perpendicular from A to DQ . The segments CE, BP, DQ , and AR partition ABCD into five regions. What is the median of the areas of these five regions?

解析

[8] Let ABCD be a square of side length 5, and let E be the midpoint of side AB . Let P and Q be the feet of perpendiculars from B and D to CE , respectively, and let R be the foot of the perpendicular from A to DQ . The segments CE, BP, DQ , and AR partition ABCD into five regions. What is the median of the areas of these five regions? Proposed by: Carl Schildkraut Answer: 5 ∼ ∼ We have DQ ⊥ CE and AR ⊥ DQ , so AR || CE . Thus, we can show that 4 ARD 4 DQC 4 CP B , = = so the median of the areas of the five regions is equal to the area of one of the three triangles listed above. √ √ BP EB 1 √ Now, note that 4 EBC ∼ 4 BP C , so = = . This means that BP = 5, so CP = 2 5. BC EC 5 Therefore, the area of 4 BP C , the median area, is 5.