Difference between revisions of "1977 Canadian MO Problems/Problem 1"
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Thus, <math>4a^2 + 4a = b^2 + b</math>, or <math>(2a+1)^2 = b^2 + b + 1</math>. But <math>b^2 < b^2 + b + 1 < (b+1)^2</math>, so <math>b^2 + b + 1</math> cannot be a perfect square, contradiction. The conclusion follows. | Thus, <math>4a^2 + 4a = b^2 + b</math>, or <math>(2a+1)^2 = b^2 + b + 1</math>. But <math>b^2 < b^2 + b + 1 < (b+1)^2</math>, so <math>b^2 + b + 1</math> cannot be a perfect square, contradiction. The conclusion follows. | ||
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+ | ==Alternate Solutions?== | ||
{{alternate solutions}} | {{alternate solutions}} | ||
Revision as of 22:32, 21 September 2014
Problem
If prove that the equation has no solutions in positive integers and
Solution
Directly plugging and into the function, We now have a quadratic in
Applying the quadratic formula,
In order for both and to be integers, the discriminant must be a perfect square. However, since the quantity cannot be a perfect square when is an integer. Hence, when is a positive integer, cannot be.
Solution
Suppose there exist positive integral and such that .
Thus, , or . But , so cannot be a perfect square, contradiction. The conclusion follows.
Alternate Solutions?
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See also
1977 Canadian MO (Problems) | ||
Preceded by First question |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • | Followed by Problem 2 |