Key idea. A fixed sum and a non-negative square give a bound on the product.

1. The identity \((x-y)^2=(x+y)^2-4xy\) gives

   \[0\le(x-y)^2=100-4xy.\]

2. Rearranging, \(4xy\le100\), so \(xy\le25\).

3. The bound is attained when \(x-y=0\). Combining \(x=y\) with \(x+y=10\) gives \(x=y=5\), which are allowed positive numbers.

Remember. For two positive numbers with fixed sum, the product is largest when the numbers are equal.