Malkin M.I.

On the board, 101 numbers are written: $1^{2}, 2^{2}, \ldots, 101^{2}$. In one operation, it is allowed to erase any two numbers and write down the absolute value of their difference instead.

What is the smallest number that can result from 100 operations?
From four consecutive squares (in three operations), you can get the number 4: $(n+3)^{2}-(n+2)^{2}-((n+$ $\left.1)^{2}-n^{2}\right)=(2 n+5)-(2 n+1)=4$.

We can get 24 such fours from the numbers $6^{2}, 7^{2}, \ldots, 101^{2}$. 20 fours can be turned into zeros by pairwise subtraction. From the numbers $4,9,16,25$ we get

$14=(25-4)-(16-9)$. From the remaining numbers $(14,4,4,4,4,1)$ we get the unit: $4-(14-4-4-4)-1=1$. It is impossible to get 0, since the parity of the sum of all numbers is preserved.

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