2025 HMMT November General Round Problem 1

Let ABCD be a rectangle. Let X and Y be points on segments \overline{BC} and \overline{AD}, respectively, such that \angle AXY = \angle XYC = 90^\circ. Given that AX : XY : YC = 1 : 2 : 1 and AB = 1, compute BC.

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2025 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.