HMMT 二月 2022 · 几何 · 第 3 题

HMMT February 2022 — Geometry — Problem 3

Let ABCD and AEF G be unit squares such that the area of their intersection is . Given that 21 ◦ a ∠ BAE < 45 , tan ∠ BAE can be expressed as for relatively prime positive integers a and b . Compute b 100 a + b .

解析

Let ABCD and AEF G be unit squares such that the area of their intersection is . Given that 21 a ◦ ∠ BAE < 45 , tan ∠ BAE can be expressed as for relatively prime positive integers a and b . Compute b 100 a + b . Proposed by: Benjamin Shimabukuro Answer: 4940 Solution: Suppose the two squares intersect at a point X ̸ = A . If S is the region formed by the 10 intersection of the squares, note that line AX splits S into two congruent pieces of area . Each of 21 20 these pieces is a right triangle with one leg of length 1, so the other leg must have length . Thus, 21 20 if the two squares are displaced by an angle of θ , then 90 − θ = 2 arctan . Though there is some 21 ◦ ambiguity in how the points are labeled, the fact that ∠ BAF < 45 tells us that ∠ BAF = θ . Therefore 2 20 1 − 1 41 2 21 tan ∠ BAF = = = . 20 20 840 tan(2 arctan ) 2 · 21 21