Problem in solving this book and pages question I was solving some old olympiad problems and I got that one. I m stuck at it.
"In a book with page numbers 1 to 100,some pages are torn off. The sum of the numbers on the remaining pages is 4949. How many pages are torn off??"
I tried to the summation formula(sum of all the pages is sum of all the natural numbers from 1 to 100,since the pages goes from 1 to 100)  and then try to eliminate some numbers (by hit and trial) to see whether the sum comes out to be 4949 or not but I wasn't so lucky. Hoping for help. Any suggestion is heartily welcome
 A: We have that $1+\cdots+100=5050.$ So, if the remaining pages sum $4949$ the pages which are not in the book add up $101.$ Moreover, if $2a-1$ is not in the book then $2a$ is also not in the book. They add up $4a-1.$ It's not possible to torn out only a page because $4a-1=101$ has no integer solution. If we turn out two pages we have to solve $4a-1+4b-1=101$ which also doesn't have integer solution. So, assume $n$ pages are torn out. We have $$4a_1-1+\cdots+4a_n-1=101 (\iff 4(a_1+\cdots+a_n)=101+n).$$ Thus the number of pages can be $3,7,11, \dots$ because $101+n$ must be a multiple of $4.$ Now, note that if $n\ge 7$ then $$\dfrac{n(n+1)}{2}=1+\cdots+n\le a_1+\cdots+a_n=\dfrac{101+n}{4}< \dfrac{n(n+1)}{2}$$ gives a contradiction. So, $n=3.$ 
Edit
Let's prove of the last inequality for $n\ge 7:$
$$\dfrac{101+n}{4}< \dfrac{n(n+1)}{2}\iff 2n^2+2n>101+n\iff 2n^2+n>101.$$ Now, $$n\ge 7\implies 2n^2+n\ge 98+7=105>101,$$ and we are done.
Edit 2
In order to determine the pages we have to solve $$a_1+a_2+a_3=26.$$ Then, the pages are $2a_1-1,2a_1;$ $2a_2-1,2a_2;$ and $2a_3-1,2a_3.$
Now, solving $a_1+a_2+a_3=26$ is to get the partitions of $26$ into distinct parts (that is, $a_1,a_2$ and $a_3$ are different). According to Wolfram Alpha there are $165$ solutions. (See https://www.wolframalpha.com/input/?i=PartitionsQ(26).)
A: Start by summing up all the pages from $1$ to $100$ to see how much it would add up if no pages are torn. Then it should become clear.
