PUMaC 2022 · 团队赛 · 第 14 题

PUMaC 2022 — Team Round — Problem 14

Let △ ABC be a triangle. Let Q be a point in the interior of △ ABC , and let X, Y, Z denote the feet of the altitudes from Q to sides BC , CA , AB , respectively. Suppose that BC = 15, ◦ ◦ ∠ ABC = 60 , BZ = 8, ZQ = 6, and ∠ QCA = 30 . Let line QX intersect the circumcircle p W Y of △ XY Z at the point W ̸ = X . If the ratio can be expressed as for relatively prime W Z q positive integers p, q , find p + q .

解析

Let △ ABC be a triangle. Let Q be a point in the interior of △ ABC , and let X, Y, Z denote the feet of the altitudes from Q to sides BC , CA , AB , respectively. Suppose that BC = 15, ◦ ◦ ∠ ABC = 60 , BZ = 8, ZQ = 6, and ∠ QCA = 30 . Let line QX intersect the circumcircle p W Y of △ XY Z at the point W ̸ = X . If the ratio can be expressed as for relatively prime W Z q positive integers p, q , find p + q . Proposed by Sunay Joshi 7 Answer: 11 Let θ = ∠ W Y Z and let φ = ∠ W ZY . By the Extended Law of Sines, W Y /W Z = sin φ/ sin θ . Since W Y XZ is cyclic, ∠ W XZ = θ , and since QXBZ is cyclic, ∠ W XZ = ∠ QBZ . Hence θ = ∠ QBZ . Since △ QBZ is right with sidelengths 6, 8, 10, we have sin θ = 3 / 5. Similarly, since ◦ ∠ W ZY = ∠ W XY = ∠ QCY = 30 , sin φ = 1 / 2. The desired ratio is therefore (1 / 2) / (3 / 5) = 5 / 6 and our answer is 5 + 6 = 11.