1. Hint:

Suppose the numbers are .

Consider the 10 sums

for j = 1, 2, 3, ...., 10.

If one of these sums is congruent to 0 mod 10 we are done. If not, all the sums must fall in the congruence classes 1, 2, 3, ... 9 mod 10. Since there are 10 sums and only 9 congruence classes, at least two of the sums must be in the same class. Then...