SOLUTION: five positive integers ( not necessarily all different ) are written on five cards. Peter calculate the sum of the number of every pair of cards, he obtains only three different to

Question 1070109: five positive integers ( not necessarily all different ) are written on five cards. Peter calculate the sum of the number of every pair of cards, he obtains only three different totals. 57, 70 and 83. what is the largest integer on any card.
a. 35 b. 42 c. 48 d. 53 e. 82
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
The largest integer on any card cannot be 82,
because if 82 is written on one of the cards,
the number written on the other four cards has to be 1,
or else we would have at least a sum greater than 83.
Then, the sums would be and ,
and the sums given do not include 2, but instead include 57 and 70.

It cannot be 35,
because if there is no number greater than 35,
there would be no sum greater than ,
and one of the sums was 83.

To have 48 as the greatest of the integers,
since is not one of the sums,
meaning that there is only one 48 written in the cards,
we know that there will be at least one number smaller than 48.
The second largest number plus 48 would give 83, the greatest sum,
and .
That would require 35 to be the second greatest number,
and to be written in at least one card.
With 48 written on one card and 35 written on more than one card,
we would have the sums and ,
but we need at least one other number to get 57 as a sum.
That number is smaller than greatest number 48,
and smaller than second greatest number 35.
It would be added to 35 and 48 to yield
two different sums smaller than .
Those sums must be and
and tell us that would work.
So with 22, 35, and we can make the sums 57, 70 and 83.
We would have 35 written on three of the cards,
so the sums would use just those 3 numbers,
and the only possible sums would be
, , and .

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