Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.1 AIME, harder Find the answer

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

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

From the given conditions, we want to calculate $\sum_{i=0}^{3} \sum_{j=i}^{3}\left(10^{i}+10^{j}\right)^{2}$ By observing the formula, we notice that each term is an exponent of $10.10^{6}$ shows up 7 times, $10^{5}$ shows up 2 times, $10^{4}$ shows up 9 times, $10^{3}$ shows up 4 times, $10^{2}$ shows up 9 times, 10 shows 2 times, 1 shows up 7 times. Thus the answer is 7294927.

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