Difference between revisions of "2009 AIME II Problems/Problem 9"
God of Math (talk | contribs) (→Solution 1) |
God of Math (talk | contribs) (→Solution 1) |
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Brute force. It is actually reasonably easy to compute <math>m</math> and <math>n</math> exactly. | Brute force. It is actually reasonably easy to compute <math>m</math> and <math>n</math> exactly. | ||
− | First, note that if <math>4x+3y+2z=2009</math>, then <math>y</math> must be odd. Let <math>y=2y'-1</math>. We get <math>4x + 6y' - 3 + 2z = 2009, 4x + 3y' + 2z = 2012, and 2x + 3y' + z = 1006</math>. For any pair of positive integers <math>(x,y')</math> such that <math>2x + 3y' < 1006</math> we have exactly one <math>z</math> such that the equality holds. Hence we need to count the pairs <math>(x,y')</math>. | + | First, note that if <math>4x+3y+2z=2009</math>, then <math>y</math> must be odd. Let <math>y=2y'-1</math>. We get <math>4x + 6y' - 3 + 2z = 2009, 4x + 3y' + 2z = 2012, </math>and<math> 2x + 3y' + z = 1006</math>. For any pair of positive integers <math>(x,y')</math> such that <math>2x + 3y' < 1006</math> we have exactly one <math>z</math> such that the equality holds. Hence we need to count the pairs <math>(x,y')</math>. |
For a fixed <math>y'</math>, <math>x</math> can be at most <math>\left\lfloor \dfrac{1005-3y'}2 \right\rfloor</math>. Hence the number of solutions is | For a fixed <math>y'</math>, <math>x</math> can be at most <math>\left\lfloor \dfrac{1005-3y'}2 \right\rfloor</math>. Hence the number of solutions is |
Revision as of 19:53, 8 April 2009
Problem
Let be the number of solutions in positive integers to the equation , and let be the number of solutions in positive integers to the equation . Find the remainder when is divided by .
Solution
Solution 1
Brute force. It is actually reasonably easy to compute and exactly.
First, note that if , then must be odd. Let . We get and. For any pair of positive integers such that we have exactly one such that the equality holds. Hence we need to count the pairs .
For a fixed , can be at most . Hence the number of solutions is
Similarly, we can compute that , hence .
Solution 2
We can avoid computing and , instead we will compute directly.
Note that if and only if . Hence there is an almost 1-to-1 correspondence between the positive integer solutions of the two equations. The only exceptions are the solutions of the first equation in which at least one of the variables is equal to . The value is the number of such solutions.
If , we get the equation . The variable must be odd, and it must be between and , inclusive. For each such there is exactly one valid . Hence in this case there are valid solutions.
If , we get the equation , or equivalently . The variable must be between and , inclusive, and for each such there is exactly one valid . Hence in this case there are valid solutions.
If , we get the equation . The variable must be odd, thus let . We get , or equivalently, . Again, we see that must be odd, thus let . We get , which simplifies to . Now, we see that must be between and , inclusive, and for each such we have exactly one valid . Hence in this case there are valid solutions.
Finally, we must note that there are two special solutions: one with , and one with . We counted each of them twice, hence we have to subtract two from the total.
Therefore , and the answer is .
See Also
2009 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |