Three speed skaters have a friendly race on a skating oval. They all start from the same point and skate in the same direction, but with different speeds that they maintain throughout the race. The slowest skater does 1 lap a minute, the fastest one does 3.14 laps a minute, and the middle one does \( L \) laps a minute for some \( 1 < L < 3.14 \). The race ends at the moment when all three skaters again come together to the same point on the oval (which may differ from the starting

## Sun \(\mathscr{H}^{10}\)

## Life Financial

point.) Find how many minutes the race lasts such that 117 passings occur before the end of the race. (A passing is defined when one skater passes another one. The beginning and the end of the race when all three skaters are together are not counted as a passing.)
Assume that the length of the oval is one unit. Let \( x(t) \) be the difference of distances that the slowest and the fastest skaters have skated by time \( t \). Similarly, let \( y(t) \) be the difference between the middle skater and the slowest skater. The path \( (x(t), y(t)) \) is a straight ray \( R \) in \( \mathbb{R}^{2} \), starting from the origin, with slope depending on \( L \). By assumption, \( 0 < y(t) < x(t) \).

One skater passes another one when either \( x(t) \in \mathbb{Z}, y(t) \in \mathbb{Z} \) or \( x(t) - y(t) \in \mathbb{Z} \). The race ends when both \( x(t), y(t) \in \mathbb{Z} \).

Let \( (a, b) \in \mathbb{Z}^{2} \) be the endpoint of the ray \( R \). We need to find the number of such points satisfying:

(a) \( 0 < b < a \)

(b) The ray \( R \) intersects \( \mathbb{Z}^{2} \) at endpoints only.

(c) The ray \( R \) crosses 357 times the lines \( x \in \mathbb{Z}, y \in \mathbb{Z}, y - x \in \mathbb{Z} \).

The second condition says that \( a \) and \( b \) are relatively prime. The ray \( R \) crosses \( a-1 \) of the lines \( x \in \mathbb{Z}, b-1 \) of the lines \( y \in \mathbb{Z} \) and \( a-b-1 \) of the lines \( x-y \in \mathbb{Z} \). Thus, we need \( (a-1) + (b-1) + (a-b-1) = 357 \), or equivalently, \( 2a - 3 = 357 \). That is \( a = 180 \).

Now \( b \) must be a positive integer less than and relatively prime to 180. The number of such \( b \) can be found using the Euler's \( \phi \) function:

\[
\phi(180) = \phi(2^2 \cdot 3^2 \cdot 5) = (2-1) \cdot 2 \cdot (3-1) \cdot 3 \cdot (5-1) = 48.
\]

Thus the answer is 48.