Difference between revisions of "1999 AIME Problems/Problem 3"
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− | == Problem == | + | == Problem == |
− | Find the sum of all [[positive integer]]s <math> | + | Find the sum of all [[positive integer]]s <math>n</math> for which <math>n^2-19n+99</math> is a [[perfect square]]. |
− | |||
− | |||
− | + | == Solution 1== | |
+ | If <math>n^2-19n+99=x^2</math> for some positive integer <math>x</math>, then rearranging we get <math>n^2-19n+99-x^2=0</math>. Now from the quadratic formula, | ||
− | + | :<math>n=\frac{19\pm \sqrt{4x^2-35}}{2}</math> | |
− | + | Because <math>n</math> is an integer, this means <math>4x^2-35=q^2</math> for some nonnegative integer <math>q</math>. Rearranging gives <math>(2x+q)(2x-q)=35</math>. Thus <math>(2x+q, 2x-q)=(35, 1)</math> or <math>(7,5)</math>, giving <math>x=3</math> or <math>9</math>. This gives <math>n=1, 9, 10,</math> or <math>18</math>, and the sum is <math>1+9+10+18=\boxed{38}</math>. | |
− | + | ==Solution 2== | |
− | We have | + | Suppose there is some <math>k</math> such that <math>x^2 - 19x + 99 = k^2</math>. Completing the square, we have that <math>(x - 19/2)^2 + 99 - (19/2)^2 = k^2</math>, that is, <math>(x - 19/2)^2 + 35/4 = k^2</math>. Multiplying both sides by 4 and rearranging, we see that <math>(2k)^2 - (2x - 19)^2 = 35</math>. Thus, <math>(2k - 2x + 19)(2k + 2x - 19) = 35</math>. We then proceed as we did in the previous solution. |
+ | |||
+ | |||
+ | ==Solution 3== | ||
+ | |||
+ | When <math> n \geq 12 </math>, we have | ||
+ | <cmath> (n-10)^2 < n^2 -19n + 99 < (n-8)^2. </cmath> | ||
− | + | So if <math> n \geq 12</math> and <math> n^2 -19n + 99 </math> is a perfect square, then | |
+ | <cmath> n^2 -19n + 99 = (n-9)^2 </cmath> | ||
− | = | + | or <math> n = 18 </math>. |
− | + | For <math> 1 \leq n < 12 </math>, it is easy to check that <math> n^2 -19n + 99 </math> is a perfect square when <math> n = 1, 9 </math> and <math> 10 </math> ( using the identity <math> n^2 -19n + 99 = (n-10)^2 + n - 1.) </math> | |
+ | |||
+ | We conclude that the answer is <math>1 + 9 + 10 + 18 = \boxed{38}.</math> | ||
== See also == | == See also == | ||
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[[Category:Intermediate Number Theory Problems]] | [[Category:Intermediate Number Theory Problems]] | ||
[[Category:Intermediate Algebra Problems]] | [[Category:Intermediate Algebra Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 01:18, 29 January 2021
Problem
Find the sum of all positive integers for which
is a perfect square.
Solution 1
If for some positive integer
, then rearranging we get
. Now from the quadratic formula,
Because is an integer, this means
for some nonnegative integer
. Rearranging gives
. Thus
or
, giving
or
. This gives
or
, and the sum is
.
Solution 2
Suppose there is some such that
. Completing the square, we have that
, that is,
. Multiplying both sides by 4 and rearranging, we see that
. Thus,
. We then proceed as we did in the previous solution.
Solution 3
When , we have
So if and
is a perfect square, then
or .
For , it is easy to check that
is a perfect square when
and
( using the identity
We conclude that the answer is
See also
1999 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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.