1. Arthur writes down five different positive integers less than $10 \mathrm{on}$. If you add any two of these five numbers, the sum is never 10.

Which number has Arthur certainly written down?
A) 1
B) 2
C) 3
D) 4
E) 5
A1. E) 5 Of the numbers 1 and 9, Arthur could have written down only one, because no two written numbers add up to 10. The same applies to 2 and 8, for 3 and 7, and also for 4 and 6. Of the eight numbers 1, 2, 3, 4, 6, 7, 8, and 9, Arthur has thus written down at most four. This means that he must have written down the remaining number 5.