Summing a lot of fractions

Algebra Level 2

The USAMO is a 6 6 question test. For each question, you submit a positive integer number p p of pages on which your solution is written. On the i i th page of this question, you write the fraction i p \frac{i}{p} to denote that this is the i i th page out of p p for this question. When you turned in your submissions for the 2017 2017 USAMO, the bored proctor computed the sum of the fractions for all of the pages which you turned in. Surprisingly, this number turned out to be 2017 2017 . How many pages did you turn in?


The answer is 4028.

Sharky Kesa by Sharky Kesa , 19, United Kingdom

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Sharky Kesa
Nov 8, 2017

For Question n n , assume you submitted p n p_n pages for 1 ≤ n ≤ 6 1 \leq n \leq 6 . The sum of the fractions of each question would be ∑ i = 1 p n i p n = 1 p n ∑ i = 1 p n i = p n + 1 2 \sum \limits_{i=1}^{p_n} \frac{i}{p_n} = \frac{1}{p_n} \sum \limits_{i=1}^{p_n} i = \frac{p_n+1}{2} .

Thus, for the six questions, we have ∑ n = 1 6 p n + 1 2 = 2017 ⟺ ∑ n = 1 6 p n + 1 = 4034 ⟺ ∑ n = 1 6 p n = 4028 \sum \limits_{n=1}^6 \frac{p_n+1}{2} = 2017 \iff \sum \limits_{n=1}^6 p_n+1 = 4034 \iff \sum \limits_{n=1}^6 p_n = 4028 .

Therefore, you submitted 4028 \boxed{4028} pages.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...