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Algebra Difficulty 7.1 National olympiad, round 2 Prove it Austria

The pages of a notebook are numbered consecutively such that the first sheet contains the numbers 11 and 22, the second sheet contains the numbers 33 and 44, and so on. One sheet is torn out of the notebook. The page numbers on the remaining sheets are added. The resulting sum equals 20212021.

a. How many pages can the notebook have had originally?

b. Which page numbers could be found on the sheet that has been torn out?

Solution

There is exactly one solution. The notebook had 6464 pages and the sheet with the page numbers 2929 and 3030 is ripped out.

Let b>0b > 0 be the number of sheets. The number of pages will be 2b2b. We are looking for a number 2b2b such that
1+2++(2b1)+2b=(2b)(2b+1)2>2021. 1 + 2 + \cdots + (2b - 1) + 2b = \frac{(2b) \cdot (2b + 1)}{2} > 2021.
Since 6022=1800\frac{60^2}{2} = 1800 has the right order of magnitude, we check the integers beginning with b=30b = 30 and find:
62632=1953<2021<64652=2080. \frac{62 \cdot 63}{2} = 1953 < 2021 < \frac{64 \cdot 65}{2} = 2080.
Therefore, the smallest possible number of pages is 6464.

The torn out sheet hh contains the page numbers 2h12h - 1 and 2h2h (first odd, then even).
This gives the equation
2h1+2h=20802021=59 2h - 1 + 2h = 2080 - 2021 = 59
which implies h=15h = 15.

So, one solution is that the book originally had 3232 sheets and the 1515th sheet with page numbers 2929 and 3030 has been torn out.

It remains to explain why this is the only solution. If the book has 6464 sheets, this is the only possibility because hh could be computed uniquely. Now, assume that the number bb of sheets is larger than 3232 and the number of pages at least 6666.

The sheet that has been torn out can have at most page numbers 2b12b - 1 and 2b2b, so the remaining sum is at least 1+2++63+64=2080>20211 + 2 + \cdots + 63 + 64 = 2080 > 2021. So there cannot be a solution with more than 3232 sheets.

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Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.