Problem
2025 AIME I Problem 11
A piecewise linear function is defined by f(x) = \begin{cases} x & \text{if} ~ -1 \leq x < 1 \\ 2 - x & \text{if} ~ 1 \leq x < 3\end{cases} and f(x + 4) = f(x) for all real numbers x. The graph of f(x) has the sawtooth pattern depicted below.
![[asy] import graph; size(300); Label f; f.p=fontsize(6); yaxis(-2,2,Ticks(f, 2.0)); xaxis(-6.5,6.5,Ticks(f, 2.0)); draw((-7,0)--(7,0), black+0.8bp); draw((0,-2.2)--(0,2.2), black+0.8bp); draw((-6,-0.1)--(-6,0.1), black); draw((-4,-0.1)--(-4,0.1), black); draw((-2,-0.1)--(-2,0.1), black); draw((0,-0.1)--(0,0.1), black); draw((2,-0.1)--(2,0.1), black); draw((4,-0.1)--(4,0.1), black); draw((6,-0.1)--(6,0.1), black); draw((-7,1)..(-5,-1), blue); draw((-5,-1)--(-3,1), blue); draw((-3,1)--(-1,-1), blue); draw((-1,-1)--(1,1), blue); draw((1,1)--(3,-1), blue); draw((3,-1)--(5,1), blue); draw((5,1)--(7,-1), blue); [/asy]](/static/aops/49a521bc363d72b42e5b898b55d91a3018b19434.png)
The parabola x = 34y^{2} intersects the graph of f(x) at finitely many points. The sum of the y-coordinates of all these intersection points can be expressed in the form \tfrac{a + b\sqrt{c}}{d}, where a, b, c, and d are positive integers such that a, b, d have greatest common divisor equal to 1, and c is not divisible by the square of any prime. Find a + b + c + d.
Leading zeroes must be inputted, so if your answer is 34, then input 034
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