### Week 1

How many circles does the bisector of the two hands of the clock (hour and minute) trace out during a 24 hour day?

### Week 2

Take a standard sheet of paper. Can you make a hole in it large enough for an elephant to pass through?

### Week 3

Why does a mirror change left to right but not top to bottom?

### Week 4

You through a tennis ball directly upwards. Does the ball spend more time going up or down?

Hint: do not ignore air resistance.

### Week 5

Each point of the plane is colored red, blue or black. Show that there are two points of the same color 1 foot apart.

### Week 6

Find a 10-digit number N whose first digit equals the number of 0-s in N, the second digit equals the number of 1-s in N, …, the last digit equals the number of 9-s in N.

### Week 7

Three circles pass through one point. Let A,B,C be the other points of intersection of these circles. Find the sum of angles of the curvilinear triangle ABC.

### Week 8

One stands in front of a 100-story building with two identical glass marbles at hand. It is known that, being dropped from `n`th floor, a marble will break (but it won't break if you drop it from floor `k`th floor with `k`<`n`). One wants to find this `n` in the least possible number of tries; a try consists in dropping a marble from some floor (if the marble doesn't break, you can use it for the next try, but if it breaks it is not usable anymore). In the worst case scenario, what is the least number of tries?

### Week 9

Two diagonally opposite corner squares are cut out of a 8 by 8 chess board. Is it possible to tile the remaining board by dominoes that tile two adjacent squares?

### Week 10

Can one place a number of pennies on a table so that each penny touches exactly three others?

### Week 11

How does the speed of an animal depend on its size (say, a horse versus a rabbit)? Consider two situations: running on a flat surface and running uphill.

Hint: the answers are different.