= Solution
In a <Young-diagram hook>, the cell $(i,j)$ has $\lambda_i-j$ cells to its right and $\lambda'_j-i$ below it. Summing the arm lengths row by row and the leg lengths column by column gives
$$
\sum_{(i,j)\in\lambda}h_{i,j}
=\sum_{(i,j)\in\lambda}\bigl((j-1)+(i-1)+1\bigr)
=\sum_{(i,j)\in\lambda}(i+j-1).
$$
Adding the content $c_{i,j}=j-i$ turns the summand on the right into $2j-1$. Since $\sum_{j=1}^m(2j-1)=m^2$, summing each row proves
$$
\sum_{(i,j)\in\lambda}(h_{i,j}+c_{i,j})
=\sum_i\lambda_i^2.
$$
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