\section*{Problem 4 - 251034}

A natural number \(x\) is required to satisfy the following conditions (1) to (5):

(1) The number \(x\), when written in the binary system (base 2), has exactly ten digits.

(2) When \(x\) is written in the ternary system, the second digit is 1.

(3) When \(x\) is written in the quaternary system, the second digit is 0.

(4) The number \(x\), when written in the quinary system, has exactly four digits.

(5) When \(x\) is written in the decimal system, the last digit is 2.

Prove that there is exactly one natural number \(x\) that satisfies these conditions, and determine this number!
}

I. If a natural number \(x\) satisfies conditions (1) to (5), then:

Due to (1) and (84), \(2^{9} \leq x\) and \(x < 5^{4}\), i.e.,

\[
512 \leq x \leq 625
\]

Considering \(2 \cdot 3^{5} = 586\) and \(3^{6} = 729\), it follows from (6) that \(x\) has exactly six digits in the ternary system, with the first digit being 2. Due to (2), \(x \geq 2 \cdot 3^{5} + 1 \cdot 3^{4}\), i.e.,

\[
567 \leq x
\]

Considering \(2 \cdot 4^{4} = 512\) and \(3 \cdot 4^{4} = 768\), it follows from (6) that \(x\) has exactly five digits in the quaternary system, with the first digit being 2. Due to (3), \(x < 2 \cdot 4^{4} + 1 \cdot 4^{3}\), i.e.,

\[
x < 576
\]

The conditions (7), (8), and (5) are only satisfied by \(x = 572\). Therefore, only this number can satisfy conditions (1) to (5).

II. It satisfies these conditions; for it has the following representations:

Binary system: 1000111100 (10 digits)

Ternary system: 210012 (digit 1 at the second position)

Quaternary system: 20330 (digit 0 at the second position)

Quinary system: 4242 (4 digits)

Decimal system: 572 (digit 2 at the last position)

With I. and II., the required proof is provided; the number to be determined is \(x = 572\).