## Task 4 - 080924

Four people $A, B, C$, and $D$ make three statements each about the same number $x$. By agreement, each person must have at least one true statement and at least one false statement.

$A$ says:

(1) The reciprocal of $x$ is not less than 1.

(2) $x$ does not contain the digit 6 in its decimal representation.

(3) The cube of $x$ is less than 221.

$B$ says:

(1) $x$ is an even number.

(2) $x$ is a prime number.

(3) $x$ is an integer multiple of 5.

$C$ says:

(1) $x$ is irrational.

(2) $x$ is less than 6.

(3) $x$ is the square of a natural number.

$D$ says:

(1) $x$ is greater than 20.

(2) $x$ is a positive integer whose decimal representation has at least 3 digits.

(3) $x$ is not less than 10.

Determine $x$.
Since one of the statements from $\mathrm{B}$ is true, $x$ must be an integer.

If statement D2 were true, then D1 and D3 would also be true. Therefore, D2 is false, which means $x \neq 221$.

Thus, A2 is true. This leaves only the possibilities 25, 49, or 81 for $x$. Since $x$ is a perfect square, B2 is false. Therefore, $x$ must be even or divisible by 5. Thus, the only remaining option is $x = 25$. Indeed, $x = 25$ satisfies the requirements of the problem: A1, B1, C1, and D2 are false, and A2, B3, C3, and D1 are true.