Question

There are 5 cards. 5 positive integers (May be different or equal) are written on these cards, one on each card. The sum of the numbers on every pair of cards. are only three different totals 57, 70, 83. Largest integer written on the card?

Answer 1

If 5 different numbers were written on 5 cards then the total number of different pairs would be `""^5C_2=10` and we would have 10 different totals. But we have only three different totals.

If we have only three different number then we can get three three different pairs providing three different totals. So their must be three different numbers on the 5 cards and the possibilities are

(1)either each of the two numbers out of three gets repeated once or
(2)one of these three gets repeated thrice.

Again the totals obtained are `57,70and 83`. Among these only `70` is even.
As we know that odd number cannot be generated by summing two same number i.e doubling a number. We can say that sum `70` of two numbers is nothing but the sum of two same numbers. So we can say there exist at least two `35`s among 5 numbers.

So other numbers are `57-35=22` and `83-35=48`

So 4 possible numbers on the cards are `35,35,22,48`

Repetition of another `35` will satisfy all conditions and finally we get 5 numbers on the card as follows

`color(magenta)(35,35,35,)color(blue)22,color(green)48`

`color (green)"So largest integer on the card is 48"`