**Problem 1 of USA Mathematical Olympiad 2010**

Let AXYZB be a convex pentagon inscribed in a semicircle of diameter AB. Denote by P, Q, R, S the feet of the perpendiculars from Y onto lines AX, BX, AZ, BZ, respectively. Prove that the acute angle formed by lines PQ and RS is half the size of ∠XOZ, where O is the midpoint of segment AB.

**Solution by Steve Dinh, a.k.a. Vo Duc Dien (dedicated to former classmate Đoàn Hiệp)**

Let AZ intercept BX at C, PQ and RS intercept at I. The acute angle formed by lines PQ and RS is

∠PIS = ∠PQY + ∠SRY - ∠QYR = ∠PQY + ∠SRY - ∠RCB

because angles QYR and RCB have pairwise perpendicular sides.

But ∠RCB subtends arcs AX and BZ; ∠PQY = ∠PXY subtends arc AY; ∠SRY = ∠SZY subtends arc BY.

Therefore, ∠PIS subtends the arc AY + BY – AX – BZ = arc XZ = ½ ∠XOZ.