\section*{Problem 1 - 261031}

For a natural number \(x\), the following conditions must be satisfied:

(1) When written in the binary system, \(x\) has exactly seven digits.

(2) When \(x\) is written in the ternary system, no digit appears more than twice.

(3) When written in the quinary system, \(x\) has exactly four digits.

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

A number that is seven digits in the binary system is at least \([1000000]_{2}=64\) and at most \([1111111]_{2}=\) 127.

A number that is four digits in the quinary system is at least \([1000]_{5}=5^{3}=125\) and at most \([1111]_{5}=\) \(5^{3}+5^{2}+5^{1}+5^{0}=156\).

Since (1) and (3) must hold simultaneously, the desired number can only be 125, 126, or 127. For these 3 numbers, one determines the representation in the ternary system

\[
125=[11122]_{3} \quad ; \quad 126=[11200]_{3} \quad ; \quad 127=[11201]_{3}
\]

Accordingly, (2) the desired number is 126.