HMMT 十一月 2025 · THM 赛 · 第 4 题

HMMT November 2025 — THM Round — Problem 4

Compute the unique 5-digit integer EART H for which the following addition holds: H A T E R

H E A R T E A R T H The digits E , A , R , T , and H are not necessarily distinct, but the leading digits E and H must be nonzero.

解析

Compute the unique 5-digit integer EART H for which the following addition holds: H A T E R

H E A R T E A R T H The digits E , A , R , T , and H are not necessarily distinct, but the leading digits E and H must be nonzero. Proposed by: Derek Liu Answer: 99774 Solution: Looking at the thousands place, we see E = 0 or 9. The former is impossible by the given condition, so E = 9. Looking at the ten thousands place, we get H = 4. Looking at the ones place, R + T is either 4 or 14, so R and T are of the same parity. Looking at the tens place, either R ≡ T (mod 10), or R ≡ T + 1 (mod 10). Thus, R = T = 2, or R = T = 7. The former is impossible because the tens place addition requires a carry, so R = T = 7. The hundreds place then leads to A = 9, so EART H = 99774 , which can be checked directly to satisfy the addition.