A passing is defined when one skater passes another one

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 di fferent 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 diff er from the starting
4 point.) Find how many di fferent choices for L are there such that 117 passings
occur before the end of the race. (Note:  A passing is defi ned when one skater passes
another one. The beginning and the end of the race when all three skaters are at
together are not counted as a passing.)

A, B, P be three points on a circle

Let A, B, P be three points on a circle. Prove that if a and b are the distances from P to
the tangents at A and B and c is the distance from P to the chord AB, then c^2= ab.

Pixie-Vedansi has divided a square up into finitely many white and red rectangles

Pixie-Vedansi has divided a square up into finitely many white and red rectangles, each with sides
parallel to the sides of the square. Within each white rectangle, she writes down its width
divided by its height. Within each red rectangle, she writes down its height divided by
its width. Finally, she calculates x, the sum of these numbers. If the total area of the
white rectangles equals the total area of the red rectangles, what is the smallest possible
value of x?

70-digit numbers n

Consider 70-digit numbers n, with the property that each of the digits 1, 2, 3, . . . , 7
appears in the decimal expansion of n ten times (and 8, 9, and 0 do not appear).
Show that no number of this form can divide another number of this form

One mapping is selected at random

One mapping is selected at random from all mapping of the set S = {1, 2, 3,....., n} into itself.
if the probability that the mapping is Injective (One-One) is 3/32;
Then find the number of possible divisors of n.

FG is as large as possible and a five digit number is made

In the equation A + B + C + D + E = FG, where FG being the two digit number
whose value is (10F + G) and letters A, B, C, D, E, F & G each represents different
digits. if FG is as large as possible and a five digit number is made using letters
A, B, C, D, E, F & G  (repetition is not allowed) then
(i) find the probability that number divisible by 5;
(ii) find the probability that number divisible by 3;
(i) find the probability that number divisible by 4; 

cos(cos(cos(cos(cos(cos(cos(cos x)))))))

Let f(x) = cos(cos(cos(cos(cos(cos(cos(cos x))))))), and suppose that the number a
satisfies the equation a = cos a. Express f'(a) as a polynomial in a.