Difference between revisions of "2008 AIME II Problems/Problem 15"
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(2 + \sqrt {3})^{2k + 1} = a_k + \sqrt {3}b_k | (2 + \sqrt {3})^{2k + 1} = a_k + \sqrt {3}b_k | ||
</cmath> | </cmath> | ||
− | where <math>k = 0,1,2,..</math>. So we can (with very little effort) obtain the following: <math>(k,a_k = 2n) = (0,2),(1,26),(2,362),(3,5042)</math>. It is an AIME problem so it is implicit that <math>n < 1000</math>, so <math>2n < 2000</math>. It is easy to see that <math>a_n</math> is strictly increasing by induction. Checking <math>2n = 362\implies n = 181</math> in the second condition works (we know <math>b_k</math> is odd so we don't need to find <math>m</math>). So we're done. | + | where <math>k = 0,1,2,..</math>. So we can (with very little effort) obtain the following: <math>(k,a_k = 2n) = (0,2),(1,26),(2,362),(3,5042)</math>. It is an AIME problem so it is implicit that <math>n < 1000</math>, so <math>2n < 2000</math>. It is easy to see that <math>a_n</math> is strictly increasing by induction. Checking <math>2n = 362\implies n =boxed{181}</math> in the second condition works (we know <math>b_k</math> is odd so we don't need to find <math>m</math>). So we're done. |
== See also == | == See also == |
Revision as of 18:55, 22 May 2017
Problem
Find the largest integer satisfying the following conditions:
- (i) can be expressed as the difference of two consecutive cubes;
- (ii) is a perfect square.
Solution
Solution 1
Write , or equivalently, .
Since and are both odd and their difference is , they are relatively prime. But since their product is three times a square, one of them must be a square and the other three times a square. We cannot have be three times a square, for then would be a square congruent to modulo , which is impossible.
Thus is a square, say . But is also a square, say . Then . Since and have the same parity and their product is even, they are both even. To maximize , it suffices to maximize and check that this yields an integral value for . This occurs when and , that is, when and . This yields and , so the answer is .
Solution 2
Suppose that the consecutive cubes are and . We can use completing the square and the first condition to get: where and are non-negative integers. Now this is a Pell equation, with solutions in the form . However, is even and is odd. It is easy to see that the parity of and switch each time (by induction). Hence all solutions to the first condition are in the form: where . So we can (with very little effort) obtain the following: . It is an AIME problem so it is implicit that , so . It is easy to see that is strictly increasing by induction. Checking in the second condition works (we know is odd so we don't need to find ). So we're done.
See also
2008 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last problem | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.