HMMT 二月 2007 · 冲刺赛 · 第 10 题
HMMT February 2007 — Guts Round — Problem 10
题目详情
英文原题
- [ 8 ] Let A denote the answer to problem 12. There exists a unique triple of digits ( B, C, D ) such that
10 > A > B > C > D > 0 and 12
A BCD − DCBA = BDA C,12
12 12 12
where A BCD denotes the four digit base 10 integer. Compute B + C + D .
12
解析
英文解析
- [ 8 ] Let A denote the answer to problem 12. There exists a unique triple of digits ( B, C, D ) such that
10 > A > B > C > D > 0 and 12
A BCD − DCBA = BDA C,12
12 12 12
where A BCD denotes the four digit base 10 integer. Compute B + C + D .
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 12
Solving this system of four equations produces ( A , B, C, D ) = (7 , 6 , 4 , 1).
12