How many words of length 10 using only the letters F and L are there which have no consecutive  L’s?

Let S₁ be the area of the region in the xy-plane bounded by the curve                    (x² +y² – 1)³ =x²y³ and let S₂ be the area of the region in the xy-plane bounded  by the curve  (x² +12y² – 4)³ = 24x³y².

Suppose you have a bucket with 3 balls, labelled  A, B and C. You draw a ball from the bucket at random and then return it, repeating this process until you have seen each of the three balls at least once. Find the average number of draws needed to see all three balls.

Into how many regions do 2020circles divide the plane, if each pair of circles intersects at two points and no point lies on three circles?

What curve is formed by projecting C onto the xy -plane?

Lockers labelled from 1 to 2020 are lined up in a row. Initially, all the lockers’ doors are open. A student walks by and closes all the lockers. Another student walks by and opens all lockers whose labels are divisible by two. Yet another student walks by and switches the state of lockers (that is, the students opens closed lockers and closes opened lockers) with labels divisible by three. This process continues, with the n^th student switching the state of lockers with labels divisible by n. After 2020 students have walked by, how many lockers are closed?

How many such functions f exist which satisfy

for all real numbers x, y, z?