Difference between revisions of "1986 USAMO Problems/Problem 3"
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What is the smallest integer <math>n</math>, greater than one, for which the root-mean-square of the first <math>n</math> positive integers is an integer? | What is the smallest integer <math>n</math>, greater than one, for which the root-mean-square of the first <math>n</math> positive integers is an integer? | ||
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<math>\mathbf{Note.}</math> The root-mean-square of <math>n</math> numbers <math>a_1, a_2, \cdots, a_n</math> is defined to be | <math>\mathbf{Note.}</math> The root-mean-square of <math>n</math> numbers <math>a_1, a_2, \cdots, a_n</math> is defined to be | ||
<cmath>\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}</cmath> | <cmath>\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}</cmath> | ||
+ | ==Solution== | ||
Let's first obtain an algebraic expression for the root mean square of the first <math>n</math> integers, which we denote <math>I_n</math>. By repeatedly using the identity | Let's first obtain an algebraic expression for the root mean square of the first <math>n</math> integers, which we denote <math>I_n</math>. By repeatedly using the identity | ||
<math>(x+1)^3 = x^3 + 3x^2 + 3x + 1</math>, we can write | <math>(x+1)^3 = x^3 + 3x^2 + 3x + 1</math>, we can write |
Revision as of 13:49, 18 July 2016
Problem
What is the smallest integer , greater than one, for which the root-mean-square of the first positive integers is an integer?
The root-mean-square of numbers is defined to be
Solution
Let's first obtain an algebraic expression for the root mean square of the first integers, which we denote . By repeatedly using the identity , we can write and We can continue this pattern indefinitely, and thus for any positive integer , Since , we obtain Therefore, Requiring that be an integer, we find that where is an integer. Using the Euclidean algorithm, we see that , and so and share no factors greater than 1. The equation above thus implies that and is each proportional to a perfect square. Since is odd, there are only two possible cases:
Case 1: and , where and are integers.
Case 2: and .
In Case 1, . This means that for some integers and . We proceed by checking whether is a perfect square for . (The solution leads to , and we are asked to find a value of greater than 1.) The smallest positive integer greater than 1 for which is a perfect square is , which results in .
In Case 2, . We proceed by checking whether is a perfect square for . We find that is not a perfect square for , and when . Thus the smallest positive integers and for which result in a value of exceeding the value found in Case 1, which was 337.
In summary, the smallest value of greater than 1 for which is an integer is .
See Also
1986 USAMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.