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A novel has 6 chapters. As usual, starting from the first page of the first chapter, the pages of the novel are numbered $1, 2, 3, 4, \ldots$. Also, each chapter begins on a new page. The last chapter is the longest and the page numbers of its pages add up to 2010. How many pages are there in the first 5 chapters?

I got $90$. I factored $4020$, which I got from the summation formula for arithmetic series. Now, $n(a_1+a_n)=4020$, where $n$ is the number of pages of the last chapter and the $a$'s are the starting and ending pages for that chapter. The expression in parenthesis became $(2a_1+n-1)$, because $a_n$ equals $a_1$ plus the number of pages minus one. I then set up some inequalities, tried the suitable factor pairings of $4020$, and got $n$ equal to $20$ and $a_1$ equal to $91$. Is this right?

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The approach seems fine --- I haven't checked the details. What contest was this from, please? – Gerry Myerson Nov 10 '13 at 5:17
Also from 2010 Fermat II of Pro2Serve at UTK. – Yadnarav3 Nov 10 '13 at 5:19
Note: it's best to show your work in the question body or as an answer to the question, rather than in a comment. I copied over your comment, which you can now delete. – dfeuer Nov 10 '13 at 6:57
up vote 1 down vote accepted

$91+92+93+\cdots + 110 = 2010$.

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