Let Fn be the nth number in the Fibonacci Sequence. Consider the 3 points (F30,F31),(F32,F33),(F34,F35) in the Cartesian plane. You are allowed to repeatedly apply the following operation:

Let P be any one of the three points in the plane and let Q,R be the other points. Draw the perpendicular bisector of QR and let S be the closest point on this line to P. Move P to any point T in the plane such that S is also the closest point on the perpendicular bisector of QR to T.

After some sequence of operations, two of the points end up at (0,0) and (100,0). If the third point is in the first quadrant, the largest possible value of its y-coordinate can be expressed as ab where a and b are coprime positive integers. What is the value of a+b?

304

297 is true

wrong again

...............it's 101

1,710,448

To find the largest possible value of the y-coordinate of the third point, we need to understand the properties of the Fibonacci sequence and the given operation.

The Fibonacci sequence is a series of numbers in which each number (after the first two) is the sum of the two preceding ones. It starts with 0 and 1, so the first few numbers in the Fibonacci sequence are: 0, 1, 1, 2, 3, 5, 8, 13, ...

Let's analyze the given operation. We have three initial points: (F30, F31), (F32, F33), and (F34, F35). We can apply the operation by drawing the perpendicular bisector of the line connecting two points and finding the closest point on that line to the third point.

By repeating this operation, two of the points end up at (0,0) and (100,0). This means that after a series of operations, the x-coordinate of the third point becomes either 0 or 100. We want to find the largest possible value of its y-coordinate.

To solve this problem, we need to examine the pattern of the Fibonacci numbers. Notice that for every three consecutive Fibonacci numbers (F, F+1), (F+2, F+3), (F+4, F+5), the x-coordinate of the third point (F+4) becomes:

0, 100, 100, 0, -100, -100, 0, 100, 100, 0, -100, -100, 0, 100, 100, ...

The x-coordinate alternates between 0 and 100. Therefore, for the largest possible y-coordinate, we should focus on the cases where the x-coordinate is 100.

At each step, the y-coordinate of the next point in the Fibonacci sequence is the sum of the y-coordinates of the two previous points. Using this information, we can calculate the y-coordinates of the third point for the x-coordinates of 100:

For (F2, F3), the third point has y-coordinate F4 = 2 + 1 = 3.
For (F8, F9), the third point has y-coordinate F10 = 13 + 8 = 21.
For (F14, F15), the third point has y-coordinate F16 = 55 + 34 = 89.
For (F20, F21), the third point has y-coordinate F22 = 233 + 144 = 377.
For (F26, F27), the third point has y-coordinate F28 = 987 + 610 = 1597.
For (F32, F33), the third point has y-coordinate F34 = 4181 + 2584 = 6765.
For (F38, F39), the third point has y-coordinate F40 = 10946 + 6765 = 17711.

Thus, the largest possible value of the y-coordinate of the third point is 17711. The value of a+b is 17 + 711 = 728.