 ## Irrational jumps Show HTML problem content Published on Saturday, 29th October 2016, 04:00 pm; Solved by 197;
Difficulty rating: 55%

### Problem 576

A bouncing point moves counterclockwise along a circle with circumference $1$ with jumps of constant length $l<1$, until it hits a gap of length $g<1$, that is placed in a distance $d$ counterclockwise from the starting point. The gap does not include the starting point, that is $g+d<1$.

Let $S(l,g,d)$ be the sum of the length of all jumps, until the point falls into the gap. It can be shown that $S(l,g,d)$ is finite for any irrational jump size $l$, regardless of the values of $g$ and $d$.
Examples:
$S(\sqrt{\frac 1 2}, 0.06, 0.7)=0.7071 \dots$, $S(\sqrt{\frac 1 2}, 0.06, 0.3543)=1.4142 \dots$ and
$S(\sqrt{\frac 1 2}, 0.06, 0.2427)=16.2634 \dots$.

Let $M(n, g)$ be the maximum of $\sum S(\sqrt{\frac 1 p}, g, d)$ for all primes $p \le n$ and any valid value of $d$.
Examples:
$M(3, 0.06) =29.5425 \dots$, since $S(\sqrt{\frac 1 2}, 0.06, 0.2427)+S(\sqrt{\frac 1 3}, 0.06, 0.2427)=29.5425 \dots$ is the maximal reachable sum for $g=0.06$.
$M(10, 0.01)=266.9010 \dots$

Find $M(100, 0.00002)$, rounded to 4 decimal places.