5. Vasya added four numbers in pairs. The four largest of the six sums obtained were $20, 16, 13, 9$. Find the two remaining sums and determine which numbers Vasya could have added.
Answer. The sums are 2 and 6. The numbers: $-0.5 ; 2.5 ; 6.5 ; 13.5$ or $-2.5; 4.5; 8.5; 11.5$.

Solution. Let the numbers be $a, b, c, d$; Assume $a \leq b \leq c \leq d$. The largest sum is obtained by adding $d$ and $c$, the next largest by adding $d$ and $b$. That is, $d+c=20, d+b=16$. There are further possibilities. 1) $d+a=13, c+b=9$ or 2) $d+a=9, c+b=13$.

In case 1) $c=20-d, b=16-d, c+b=20-d+16-d=9$, from which $d=13.5$. Then $c=20-13.5=6.5, b=16-13.5=2.5, a=13-13.5=-0.5$. The remaining sums are $a+b=2$ and $a+c=6$.

In case 2) $c=20-d, b=16-d, c+b=20-d+16-d=13$, from which $d=11.5$. Then $c=20-11.5=8.5, b=16-11.5=4.5, a=9-11.5=-2.5$. The remaining sums are $a+b=2$ and $a+c=6$.

Comment. A correct solution is worth 20 points. Points for a correct solution are distributed as follows: only one set of numbers found - 9 points, the second set found - 10 points, sums found - 1 point, these points are cumulative. Solution started, with some progress - 5 points. Solution started, but progress is insignificant - 1 point. Only the answer without a solution - 1 point.

## Variant 2