MathDB

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

r_1

and

r_2

. A then chooses two integers

s

and

t

such that

r_1<s<r_1+xp

and

r_2<t<r_2+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

r_1<s<r_1+xp

and

r_2<t<r_2+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.

Problem Statement