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.

There is a dynamic illustration and a second solution.