If all ten sums are added together, the result will be the sum of the ten ages multiplied by 9 (since each person's age will have been counted nine times).

In the 9-element set{82,83,84,85,87,89,90,91,92}, one item must obviously be duplicated. The sum of the nine elements is 783, which is already a multiple of 9. So the duplicated element must also be a multiple of 9. The only possibility for this element is that it is 90. Thus 9 times the sum of the ages is 783+90 = 873.

That should be quite enough of a hint to enable you to solve the problem.

Edit.The problem says that "each person is asked to write the sum of ages of all the other 9 persons." The solution gives the age of the youngest person as 5. How many five-year-olds do you know who could correctly add nine numbers ranging from 6 to 15 and then write down the result? That makes the problem seem unrealistic to me.