Let the numbers written in a circle be denoted as $a_1, a_2, \dots, a_{100}$. Vasya will track only ten numbers, which he will pair as follows: $(a_9, a_{18})$, $(a_{27}, a_{36})$, $\dots$, $(a_{90}, a_{99})$. In one move, Petya can decrease at most one of these $10$ numbers. If Petya decreases one number in a pair $(a_i, a_{i+1})$, Vasya will respond by adding $1$ to each of $a_i, a_{i+1}, \dots, a_{i+9}$. If Petya doesn't decrease any of these $10$ numbers, Vasya will make any allowed move.

Thus, after each pair of moves (Petya's and Vasya's), the sum of numbers in each of Vasya's five pairs will not decrease. Since initially all five pair sums are positive, after each of Vasya's moves the sum in each pair will remain positive, meaning each pair will contain at least one positive number. Therefore, after any of Vasya's moves there will be at least $5$ positive numbers, as required.