HMMT 二月 2022 · 冲刺赛 · 第 34 题

HMMT February 2022 — Guts Round — Problem 34

[20] Estimate A , the number of unordered triples of integers ( a, b, c ) so that there exists a nondegenerate triangle with side lengths a , b , and c fitting inside a 100 × 100 square. An estimate of E earns max(0 , ⌊ 20 − | A − E | / 1000 ⌋ ) points.

解析

[20] Estimate A , the number of unordered triples of integers ( a, b, c ) so that there exists a nondegenerate triangle with side lengths a , b , and c fitting inside a 100 × 100 square. An estimate of E earns max(0 , ⌊ 20 − | A − E | / 1000 ⌋ ) points. Proposed by: Daniel Zhu Answer: 194162 Solution: Let’s first count the number of such triangles with perimeter equal to p . By Stars and Bars, 2 p p there are ≈ ordered triples of positive integers that sum to p . Additionally, note that only about 2 2 2 p a quarter of them satisfy the triangle inequality, we have only possible triples. Dividing by 3! gives 8 2 p us approximately nondegenerate triangles with perimeter p . Summing this from p = 1 to n gives 48 3 n us approximately triangles with perimeter at most n . 144 Now, note that there are two “extremes” for our triangles. One extreme is a triangle that is very close √ to a line. In that case, we have that the maximum perimeter is 200 2 ≈ 283. In the other extreme, we have a triangle that is very close to an equilateral triangle, in which case we have the maximum 100 perimeter is 3 · ≈ 311. Thus, as a compromise between these extremes, we can plug in n = 300 ◦ cos 15 3 300 to get a value of = 187500 , which would have earned 13 points. 144