RU-ALLRUSSIAN 2023 Problem 102

A group of $100$ kids has a deck of $101$ cards numbered by $0, 1, 2,\dots, 100$ . The first kid takes the deck, shuffles it, and then takes the cards one by one; when he takes a card (not the last one in the deck), he computes the average of the numbers on the cards he took up to that moment, and writes down this average on the blackboard. Thus, he writes down $100$ numbers, the first of which is the number on the first taken card. Then he passes the deck to the second kid which shuffles the deck and then performs the same procedure, and so on. This way, each of $100$ kids writes down $100$ numbers. Prove that there are two equal numbers among the $10000$ numbers on the blackboard.