Problem:
A piecewise linear periodic function is defined by
f(x)={x2−xif x∈[−1,1)if x∈[1,3)
and f(x+4)=f(x) for all real numbers x. The graph of f(x) has the sawtooth pattern depicted below.
The parabola x=34y2 intersects the graph of f(x) at finitely many points. The sum of the y-coordinates of these intersection points can be expressed in the form da+bc, where a,b,c, and d are positive integers, a,b, and d have greatest common divisor equal to 1, and c is not divisible by the square of any prime. Find a+b+c+d.
Solution:
The parabola intersects the segments from (4k+1,1) to (4k+3,−1) for 0≤k≤7 twice. Combining the equations x=4k+2−y and x=34y2 yields:
34y2=4k+2−y
which has sum of roots of −341 each time. It also intersects the segment from (33,1) to (35,−1) exactly once. We can solve for the intersection point by solving:
34−y=34y2
which yields:
y=685185−1
Similarly, it intersects the segments from (4k−1,−1) to (4k+1,1) twice for all 0≤k≤8. The equations are x=y+4k and x=34y2, so we have:
y+4k=34y2
which has a sum of roots of 341. Putting it all together, we get:
