1. Milla and Zhena came up with a number each and wrote down all the natural divisors of their numbers on the board. Milla wrote down 10 numbers, Zhena wrote down 9 numbers, and the largest number written on the board twice is 50. How many different numbers are written on the board?
Solution. Note that a number written twice is a common divisor of the original numbers; the largest such number is their GCD. Therefore, all numbers written twice are divisors of the number 50, that is, the numbers $1,2,5,10,25,50$. Thus, among the listed numbers, exactly 6 are repeated, and the number of different numbers is $10+9-6=13$.

Criteria.

An example of a 7-point solution: "The numbers 1,2,5,10,25,50 are written twice, and all other numbers are written once; in total, there are 13 different numbers." That is, we do not require justification that non-divisors of the number 50 cannot repeat, or even an explicit mention of this fact. We do not need to prove that the number 50 has no other divisors. Providing an example confirming the existence of such numbers is not required.

Errors. It is stated that the number 50 has six divisors, but it is not explained why there are six - minus one point.

One divisor of the number 50 is not accounted for - minus one point, two divisors are not accounted for - minus two points, more than two are not accounted for - the problem is considered unsolved. Similarly for extra divisors.

The person thinks that "different numbers" are only those that appear once (then the answer is 7) — minus one point.

Progress. It is stated that the divisors of the number 50 are repeated, but the divisors are not counted or there are more than two errors in the count - 2 points. If the word "divisors" is missing (for example, "1 and 50 are repeated, so there are 17 different numbers") — 0 points.

Only the answer $13-1$ point.