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Problem 586

AMC 12 late, AIME early

Number theory Difficulty 5.0 Find the answer

## Task 1 - 040731

How many pages of a book are consecutively numbered from page 1 if a total of 1260 digits are printed?

Your answer

Official solution

First, there are pages 1-9, each with 1 digit $\Rightarrow 9$ digits.

Then the numbers from 10-99, each with 2 digits $\Rightarrow 180$ digits.

The following numbers are all three-digit. Up to page 99, 189 digits were used, subtracting this from the total 1260 digits leaves 1071 digits. Dividing this by three gives the number of three-digit numbers. This is 357.

This means that after page 99, there are 357 more pages. Calculating $99+357$ gives the total number of pages. Therefore, the book has 456 pages.

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