Problems(1)
A and B are playing a game on a blackboard. At the beginning, a positive real number x is fixed. First, A chooses a prime number p greater than or equal to 7. Then, B selects non-negative real numbers r1 and r2. A then chooses two integers s and t such that r1<s<r1+xp and r2<t<r2+xp, and writes on the blackboard the remainders of 0, s, t, and st when divided by p in this order.After that, A can repeatedly choose one of the following three operations as many times as they like:
[*] Choose an integer m between 1 and p−1 inclusive, and replace each of the four numbers on the blackboard with the remainder of adding m and then dividing by p.
[*] Choose an integer n between 2 and p−1 inclusive, and replace each of the four numbers on the blackboard with the remainder of multiplying by n and then dividing by p.
[*] Replace each of the four numbers on the blackboard with its remainder when squared and then divided by p, but this operation is not allowed if at least one of the four numbers is 0.A's objective is to make the numbers on the blackboard become 0, 2, 3, and 6 in that order. However, if there are no integers s and t that satisfy r1<s<r1+xp and r2<t<r2+xp, then A cannot achieve the objective.Find the maximum possible value of x such that regardless of A's actions, B can always prevent A from achieving the objective. number theory