Subject II

Consider eight distinct natural numbers. By summing any seven of these eight numbers, the results obtained are: $42, 47, 50, 52, 54, 55, 56, 57$. Determine the eight numbers.
Mathematical Gazette
Solution:

Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}, a_{8}$ be the eight numbers.
By adding the eight sums term by term, we obtain
$7 \cdot\left(a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}+a_{7}+a_{8}\right)=413$ ..... 2 points
$a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}+a_{7}+a_{8}=59$ ..... 2 points
The numbers are: $2,3,4,5,7,9,12,17$ ..... 3 points