HMMT 十一月 2022 · THM 赛 · 第 4 题

HMMT November 2022 — THM Round — Problem 4

Alice and Bob stand atop two different towers in the Arctic. Both towers are a positive integer number of meters tall and are a positive (not necessarily integer) distance away from each other. One night, the sea between them has frozen completely into reflective ice. Alice shines her flashlight directly at the top of Bob’s tower, and Bob shines his flashlight at the top of Alice’s tower by first reflecting it off the ice. The light from Alice’s tower travels 16 meters to get to Bob’s tower, while the light from Bob’s tower travels 26 meters to get to Alice’s tower. Assuming that the lights are both shone from exactly the top of their respective towers, what are the possibilities for the height of Alice’s tower?

解析

Alice and Bob stand atop two different towers in the Arctic. Both towers are a positive integer number of meters tall and are a positive (not necessarily integer) distance away from each other. One night, the sea between them has frozen completely into reflective ice. Alice shines her flashlight directly at the top of Bob’s tower, and Bob shines his flashlight at the top of Alice’s tower by first reflecting it off the ice. The light from Alice’s tower travels 16 meters to get to Bob’s tower, while the light from Bob’s tower travels 26 meters to get to Alice’s tower. Assuming that the lights are both shone from exactly the top of their respective towers, what are the possibilities for the height of Alice’s tower? Proposed by: Eric Shen Answer: 7 , 15 Solution: Let Alice’s tower be of a height a , and Bob’s tower a height b . Reflect the diagram over the ice to 2 2 obtain an isosceles trapezoid. Then we get that by Ptolemy’s Theorem, 4 ab = 26 − 16 = 4 · 105, thus ab = 105. Hence a ∈ { 1 , 3 , 5 , 7 , 15 , 21 , 35 , 105 } . But max( a, b ) ≤ 26 + 16 = 42 by the Triangle inequality, so thus a ̸ ∈ { 1 , 105 } . Also, 3 , 5 , 21, and 35 don’t work because a + b < 26 and | a − b | < 16 .