Library / /383 of 1394

, 2016

Algebra Difficulty 5.0 AIME, harder Prove it United States

Problem:
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 $A$ .

Solution

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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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.