Question 124
Initially, we put $100$ cards onto a table, each card contains a positive integer. Exactly $43$ of these cards contain odd numbers. Then, on each minute the following operation has been performed. We compute the product of numbers on every set of three cards on the table, add up all these products, write this number onto a new card and put this card onto the table. After a year, it appears that the table contains a card with a number divisible by $2^{10000}$. Prove that some number divisible by $2^{10000}$ was already on the table a day after the process started.