Competition problem — Number theory

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Number theory Difficulty 6.6 National olympiad Prove it

1. 135 We call natural numbers similar if they can be written using the same set of digits (for example, 112, 121, 211 are similar because they can all be written using the digit set $1,1,2$ ). Prove that there exist three similar 1995-digit numbers, whose digits do not contain 0, but the sum of two of these numbers equals the third.

Solution

[Solution] Since 1995 is a multiple of 3, and
$459+495=954$

therefore, there is a 1995-digit number
$\begin{array}{l} 459459459 \cdots 459, \\ 495495495 \cdots 495, \\ 954954954 \cdots 954, \end{array}$

which meets the requirements of the problem. Hence, the proposition is proved.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.