HMMT 二月 2016 · 代数 · 第 3 题
HMMT February 2016 — Algebra — Problem 3
题目详情
英文原题
- Let A denote the set of all integers n such that 1 ≤ n ≤ 10000, and moreover the sum of the decimaldigits of n is 2. Find the sum of the squares of the elements of A .
解析
英文解析
- Let A denote the set of all integers n such that 1 ≤ n ≤ 10000, and moreover the sum of the decimaldigits of n is 2. Find the sum of the squares of the elements of A .
Proposed by: Evan Chen
Answer: 7294927
From the given conditions, we want to calculate
3 3
∑ ∑
i j 2
(10 + 10 ) .
i =0 j = i
6 5
By observing the formula, we notice that each term is an exponent of 10. 10 shows up 7 times, 10
4 3 2
shows up 2 times, 10 shows up 9 times, 10 shows up 4 times, 10 shows up 9 times, 10 shows 2 times,
1 shows up 7 times. Thus the answer is 7294927.