Difference between revisions of "1986 IMO Problems/Problem 1"
(Created page with "== Problem == Let <math>d</math> be any positive integer not equal to <math>2, 5</math> or <math>13</math>. Show that one can find distinct <math>a,b</math> in the set <math>...") |
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We do casework with mods. | We do casework with mods. | ||
| − | <math>d\equiv 0 \pmod{4}: 13d-1</math> is not a perfect square. | + | <math>d\equiv 0,3 \pmod{4}: 13d-1</math> is not a perfect square. |
<math>d\equiv 2\pmod{4}: 2d-1</math> is not a perfect square. | <math>d\equiv 2\pmod{4}: 2d-1</math> is not a perfect square. | ||
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Therefore, <math>d\equiv 1 \pmod{4}.</math> Now consider <math>d\pmod{16}.</math> | Therefore, <math>d\equiv 1 \pmod{4}.</math> Now consider <math>d\pmod{16}.</math> | ||
Revision as of 21:42, 29 April 2016
Problem
Let 
be any positive integer not equal to 
or 
. Show that one can find distinct 
in the set 
such that 
is not a perfect square.
Solution
We do casework with mods.
is not a perfect square.
is not a perfect square.
Therefore, 
Now consider 
is not a perfect square.
is not a perfect square.
As we have covered all possible cases, we are done.
| 1986 IMO (Problems) • Resources | ||
| Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
| All IMO Problems and Solutions | ||
