HMMT 二月 2016 · 代数 · 第 3 题

HMMT February 2016 — Algebra — Problem 3

Let A denote the set of all integers n such that 1 ≤ n ≤ 10000, and moreover the sum of the decimal digits of n is 2. Find the sum of the squares of the elements of A .

解析

Let A denote the set of all integers n such that 1 ≤ n ≤ 10000, and moreover the sum of the decimal digits 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.