HMMT 二月 2024 · 冲刺赛 · 第 33 题

HMMT February 2024 — Guts Round — Problem 33

[20] Let p denote the proportion of teams, out of all participating teams, who submitted a negative response to problem 5 of the Team round (e.g. “there are no such integers”). Estimate P = ⌊ 10000 p ⌋ . An estimate of E earns max(0 , ⌊ 20 − | P − E | / 20 ⌋ ) points. If you have forgotten, problem 5 of the Team round was the following: “Determine, with proof, whether 2 2 3 3 there exist positive integers x and y such that x + y , x + y , and x + y are all perfect squares.” 6 2

解析

[20] Let p denote the proportion of teams, out of all participating teams, who submitted a negative response to problem 5 of the Team round (e.g. “there are no such integers”). Estimate P = ⌊ 10000 p ⌋ . An estimate of E earns max(0 , ⌊ 20 − | P − E | / 20 ⌋ ) points. If you have forgotten, problem 5 of the Team round was the following: “Determine, with proof, whether 2 2 3 3 there exist positive integers x and y such that x + y , x + y , and x + y are all perfect squares.” Proposed by: Arul Kolla Answer: 5568 Solution: Of the 88 teams competing in this year’s Team round, 49 of them answered negatively, 9 (correctly) provided a construction, 16 answered ambiguously or did not provide a construction, and 49 the remaining 14 teams did not submit to problem 5. Thus p = ≈ 0 . 5568 . 88 6 2