Question

The numbers $1,$ $2,$ $\dots,$ $10$ are to be entered into the 10 boxes shown below, so that each number is used exactly once: \[P = (\square + \square + \square + \square + \square)(\square + \square + \square + \square + \square).\]What is the maximum value of $P$? What is the minimum value of $P$?

Answer 1

Hint

If we define x to be the value of one of the factors, since 1 + 2 + 3 + 4 + 5 ... + 10 = 11(5) = 55, the value of the other factor has to be 55-x.

To maximize P, you'd like to make x as close as possible to the vertex you found. What if you want to minimize P? Remember x must be an integer.

I hope this helps :)