USAMO joke!

Number Theory Level 3

The USAMO is a 6 question test. For each question, you submit a positive integer number p of pages on which your solution is written. On the ith page of this question, you write the fraction i/p to denote that this is the ith page out of p for this question. When you turned in your submissions for the 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. How many pages did you turn in?


The answer is 4028.

Syed Hamza Khalid by Syed Hamza Khalid , 17, France

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1 solution

Consider just one question, if you turned in n n pages, then the sum 1 n + 2 n + . . . + n − 1 n + n n \frac{1}{n} + \frac{2}{n} + ... + \frac{n-1}{n} + \frac{n}{n} is equal to n + 1 2 \frac{n+1}{2} .

Now, there are 6 questions, so let n 1 , n 2 , n 3 , n 4 , n 5 , n 6 n_1, n_2, n_3, n_4, n_5, n_6 be the number of pages you turned in for each question, then n 1 + 1 2 + n 2 + 1 2 + n 3 + 1 2 + n 4 + 1 2 + n 5 + 1 2 + n 6 + 1 2 = 2017 \frac{n_1 + 1}{2} + \frac{n_2 + 1}{2} + \frac{n_3 + 1}{2} + \frac{n_4 + 1}{2} + \frac{n_5 + 1}{2} + \frac{n_6 + 1}{2} = 2017 .

The total number of pages is n = n 1 + n 2 + n 3 + n 4 + n 5 + n 6 n = n_1 + n_2 + n_3 + n_4 + n_5 + n_6 , and from the equation above we get : n = 2017 ∗ 2 − 6 = 4028 n = 2017 * 2 - 6 = 4028 .

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