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Weird geo

Source: Thailand MO 2023 Day 1 P5

May 11, 2023
geometry

Problem Statement

Let ℓ\ell be a line in the plane and let 90∘<θ<180∘90^\circ<\theta<180^\circ. Consider any distinct points P,Q,RP,Q,R that satisfy the following:
(i) PP lies on ℓ\ell and PQPQ is perpendicular to ℓ\ell
(ii) RR lies on the same side of ℓ\ell as QQ, and RR doesn’t lie on ℓ\ell
(iii) for any points A,BA,B on ℓ\ell, if ∠ARB=θ\angle ARB=\theta then ∠AQB≥θ\angle AQB \geq \theta.
Find the minimum value of ∠PQR\angle PQR.
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