HMMT 二月 2023 · 冲刺赛 · 第 26 题

HMMT February 2023 — Guts Round — Problem 26

[20] Let P ABC be a tetrahedron such that ∠ AP B = ∠ AP C = ∠ BP C = 90 , ∠ ABC = 30 , and AP equals the area of triangle ABC . Compute tan ∠ ACB .

解析

[20] Let P ABC be a tetrahedron such that ∠ AP B = ∠ AP C = ∠ BP C = 90 , ∠ ABC = 30 , and 2 AP equals the area of triangle ABC . Compute tan ∠ ACB . Proposed by: Luke Robitaille √ Answer: 8 + 5 3 Solution: Observe that 1 2 · AB · AC · sin ∠ BAC = [ ABC ] = AP 2 1 2 2 2 = ( AB + AC − BC ) 2 = AB · AC · cos ∠ BAC, 1 ◦ √ so tan ∠ BAC = 2. Also, we have tan ∠ ABC = . Also, for any angles α, β, γ summing to 180 , 3 1 √ one can see that tan α + tan β + tan γ = tan α · tan β · tan γ . Thus we have tan ∠ ACB + 2 + = 3 √ 1 √ tan ∠ ACB · 2 · , so tan ∠ ACB = 8 + 5 3. 3