# 2021 CMC 12A Problems/Problem 7

The following problem is from both the 2021 CMC 12A #7 and 2021 CMC 10A #9, so both problems redirect to this page.

## Problem

It is known that every positive integer can be represented as the sum of at most $4$ squares. What is the sum of the $2$ smallest integers which cannot be represented as the sum of fewer than $4$ squares? $\textbf{(A) } 22\qquad\textbf{(B) } 23\qquad\textbf{(C) } 24\qquad\textbf{(D) } 25\qquad\textbf{(E) } 26\qquad$

## Solution \begin{align*} 1 &= 1^2 \\ 2 &= 1^2 + 1^2 \\ 3 &= 1^2 + 1^2 + 1^2 \\ 4 &= 2^2 \\ 5 &= 1^2 + 2^2 \\ 6 &= 1^2 + 1^2 + 2^2 \\ 7 \\ 8 &= 2^2 + 2^2 \\ 9 &= 3^2 \\ 10 &= 1^2 + 3^2 \\ 11 &= 1^2 + 1^2 + 3^2 \\ 12 &= 2^2 + 2^2 + 2^2 \\ 13 &= 2^2 + 3^2 \\ 14 &= 1^2 + 2^2 + 3^2 \\ 15 \\ \end{align*}

We cannot find a solution in fewer than $4$ squares for $7$ and $15$, so the answer is $7+15=\boxed{\textbf{(A) } 22}$.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 