Difference between revisions of "2025 AIME I Problems/Problem 11"
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The parabola <math>x^{2} = 34y</math> intersects the graph of <math>f(x)</math> at finitely many points. The sum of the <math>y</math>-coordinates of all these intersection points can be expressed in the form <math>\tfrac{a + b\sqrt{c}}{d}</math>, where <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are positive integers such that <math>a</math>, <math>b</math>, <math>d</math> have greatest common divisor equal to <math>1</math>, and <math>c</math> is not divisible by the square of any prime. Find <math>a + b + c + d</math>. | The parabola <math>x^{2} = 34y</math> intersects the graph of <math>f(x)</math> at finitely many points. The sum of the <math>y</math>-coordinates of all these intersection points can be expressed in the form <math>\tfrac{a + b\sqrt{c}}{d}</math>, where <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are positive integers such that <math>a</math>, <math>b</math>, <math>d</math> have greatest common divisor equal to <math>1</math>, and <math>c</math> is not divisible by the square of any prime. Find <math>a + b + c + d</math>. | ||
| + | |||
| + | ==Graph== | ||
| + | It may be helpful to graph certain parts of the graph to grasp a better understanding of what we need. I created an example diagram on Desmos here: https://www.desmos.com/calculator/ne8shyhyka | ||
| + | |||
| + | ~lprado | ||
==Solution== | ==Solution== | ||
Revision as of 19:29, 13 February 2025
Contents
Problem
A piecewise linear function is defined by ![\[f(x) = \begin{cases} x & \operatorname{if} ~ -1 \leq x < 1 \\ 2 - x & \operatorname{if} ~ 1 \leq x < 3\end{cases}\]](http://latex.artofproblemsolving.com/2/1/b/21bf20dc5e2dae2f8d5368cec6e395912388f875.png)
and 
for all real numbers 
. The graph of 
has the sawtooth pattern depicted below.
The parabola 
intersects the graph of at finitely many points. The sum of the 
-coordinates of all these intersection points can be expressed in the form 
, where 
, 
, 
, and 
are positive integers such that , , have greatest common divisor equal to 
, and is not divisible by the square of any prime. Find 
.
Graph
It may be helpful to graph certain parts of the graph to grasp a better understanding of what we need. I created an example diagram on Desmos here: https://www.desmos.com/calculator/ne8shyhyka
~lprado
Solution
Note that consists of lines of the form 
and 
for integers 
. In the first case, we get 
and the sum of the roots is 
by Vieta. In the second case, we similarly get a sum of 
Thus pairing 
and 
gives a -coordinate sum of 
This process of pairing continues until we get to 
. Then 
behaves exactly as we expect, with a sum of . However, 
is where things start becoming fishy, since there is one root with absolute value less than and one with absolute value greater than . We get ![\[34-34y^2 = y\]](http://latex.artofproblemsolving.com/8/5/3/8535871b8fda91d4c6d48f28f0c29f87aec2526c.png)
and solving with the quadratic formula (clear to take the positive root) gives ![\[y = \frac{-1 \pm \sqrt{68^2 + 1}}{68} = \frac{-1 + 5 \sqrt{185}}{68}.\]](http://latex.artofproblemsolving.com/7/f/9/7f9ea856344131d7827fbb7b9bcb355d7c7de6da.png)
Adding our from earlier gives the answer 
.
Solution credit: @EpicBird08
See Also
| 2025 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 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. 
