7.5. Vasya had 101 real coins. One of them was replaced with a fake one (it differs in weight, but it is unknown whether it is lighter or heavier than a real coin). Vasya wants to find as many real coins as possible in one weighing on a balance scale without weights. What should he do and how many real coins will he be able to find?
Answer: 50 coins.

Divide the coins into piles of $25 ; 25 ; 51$ and place the piles of 25 coins on the scales. If the scales are in balance, then these piles contained genuine coins, and we have found 50 genuine coins. If the scales are not in balance, then the genuine coins are in the third pile, and we have found 51 genuine coins. Thus, in any case, it is possible to find 50 genuine coins. We will prove that more cannot be found. If the same number of coins, not more than 24, is placed on the scales, then in the case of equality, it is possible to reliably determine no more than 48 genuine coins. If the same number of coins, not less than 26, is placed on the scales, then in the case of inequality, it is possible to determine no more than 49 genuine coins - these are the coins that were not on the scales.

Comment. If a correct algorithm for determining a smaller number of genuine coins than 50 is proposed, the score is 1 point. If a correct algorithm for determining 50 genuine coins (without proving the maximum) is proposed, the score is 3 points.