HMMT 二月 2007 · 冲刺赛 · 第 10 题

HMMT February 2007 — Guts Round — Problem 10

[ 8 ] Let A denote the answer to problem 12. There exists a unique triple of digits ( B, C, D ) such that 12 10 > A > B > C > D > 0 and 12 A BCD − DCBA = BDA C, 12 12 12 where A BCD denotes the four digit base 10 integer. Compute B + C + D . 12

解析

[ 8 ] Let A denote the answer to problem 12. There exists a unique triple of digits ( B, C, D ) such that 12 10 > A > B > C > D > 0 and 12 A BCD − DCBA = BDA C, 12 12 12 where A BCD denotes the four digit base 10 integer. Compute B + C + D . 12 Answer: 11 . Since D < A , when A is subtracted from D we must carry over from C . Thus, 12 D + 10 − A = C . Next, since C − 1 < C < B , we must carry over from the tens digit, so that 12 ( C − 1 + 10) − B = A . Now B > C so B − 1 ≥ C , and ( B − 1) − C = D . Similarly, A − D = B . 12 12 Solving this system of four equations produces ( A , B, C, D ) = (7 , 6 , 4 , 1). 12