PUMaC 2013 · 数论（B 组） · 第 7 题
PUMaC 2013 — Number Theory (Division B) — Problem 7
题目详情
- [ 7 ] Define a ”digitized number” as a ten-digit number a a . . . a such that for k = 0 , 1 , . . . , 9, 0 1 9 a is equal to the number of times the digit k occurs in the number. Find the sum of all k digitized numbers.
解析
- [ 7 ] Define a ”digitized number” as a ten-digit number a a . . . a such that for k = 0 , 1 , . . . , 9, 0 1 9 a is equal to the number of times the digit k occurs in the number. Find the sum of all k digitized numbers. Solution From the condition, we need 9 9 ∑ ∑ ka = a = 10 . k k k =0 k =0 In particular, 9 − a 9 0 ∑ ∑ a = ( k − 1) a ≥ ( k − 1) 0 k k =2 k =2 yields a ≥ 6. By examining each case, we find the only solution 6210001000. 0 2