HMMT 十一月 2025 · 冲刺赛 · 第 4 题
HMMT November 2025 — Guts Round — Problem 4
题目详情
- [6] A mercury thermometer reads the temperature using three temperature scales: C (Celsius), F ◦ (Fahrenheit), and S (Saengrungkongka). The conversions are as follows: 9 ◦ ◦ • A temperature of x C corresponds to ( x + 32) F . 5 9 ◦ ◦ • A temperature of x S corresponds to ( x + 32) C . 5 Given that the current temperature readings on all three scales are odd positive integers, compute the minimum possible value of the current temperature in degrees Celsius .
解析
- [6] A mercury thermometer reads the temperature using three temperature scales: C (Celsius), F ◦ (Fahrenheit), and S (Saengrungkongka). The conversions are as follows: 9 ◦ ◦ • A temperature of x C corresponds to ( x + 32) F . 5 ◦ 9 ◦ • A temperature of x S corresponds to ( x + 32) C . 5 Given that the current temperature readings on all three scales are odd positive integers, compute the minimum possible value of the current temperature in degrees Celsius . Proposed by: Jackson Dryg Answer: 95 Solution: Let the current temperature readings be f , c , and s in degrees Fahrenheit, Celsius, and Saengrungkongka, respectively. Write f and s in terms of c : 9 5 f = c + 32 and s = ( c − 32) . 5 9 We are given that f , c , and s are odd positive integers. For f to be an integer, c must be a multiple of 5. For s to be an integer, c − 32 must be a multiple of 9. Since c must also be odd, we get that c ≡ 0 (mod 5) c ≡ 32 ≡ 5 (mod 9) c ≡ 1 (mod 2) Note that c = 5 is a solution to this system of congruences. The least common multiple of the moduli 2, 5, and 9 is 90, so by the Chinese Remainder Theorem, all solutions are given by c ≡ 5 (mod 90). As c is positive, the smallest values we need to check are 5 and 95. © 2025 HMMT Observe that c = 5 fails since that leads to s = − 15 < 0. On the other hand, c = 95 works, since the corresponding values are s = 35 and f = 203, which are odd positive integers. Hence, the minimum possible value of the current temperature in degrees Celsius is 95 .