Tricky Problems in Algebra, Geometry and Mathematical Analysis

10th grade Which is greater exponent to the power of pi or pi to the power of exponent, where exponent is the base of the natural logarithm and pi is the ratio of the circumference of the circle to its diameter -- two well-known mathematical quanitities? Don't use the calculator!
Hint or Answer

10th grade You have a triangle with each side length known to be less than or equal to 1. Prove that the area of this triangle is less than or equal to the quarter of the square root of 3. I got this problem at the entrance exam into Moscow University.

Hint or Answer

8-10th grade Given a triangle ABC and a point O on its AB side not coinciding with either A or B, find the number of different lines that you can draw through O sich that the triangle resulting from disecting ABC is similar to ABC.

Hint or Answer

1-3 grade Given a square, how many different ways are there to cut it into two pieces of same shape with one straight cut of scissors? If you take 1 piece resulting from each of these cuts and measure their weight, would the weight be the same?

Hint or Answer

3-5 grade A piece of soap was used for 7 days which resulted in its height, width and length to reduce by a factor of 2 each. How long will the remaining piece of soap last if usedthe same way?

Hint or Answer

5-6 grade James Bond starts travelling from city A to city B on foot at a constant speed of 10 miles per hour. Exactly when he departs A, a fly whose speed is 30 miles per hour starts travelling from A to B. When the fly reaches B, it turns the opposite way and starts flying towards James, thus travelling back and forth until James reaches B. What distance will the fly travel if the distance between A and B is 50 miles.

Hint or Answer

7-8 grade You have a set of 32 domino pieces and the chess board. It is easy to show that you can cover the board with all pieces so that there is no overlap between the pieces and the whole board is covered. Now take the main diagonal, and remove 2 corner squares on it from the board, also discard one domino piece. Can the board still be covered with the remaining 31 pieces?

Hint or Answer

From http://www.ml-shopping.com/tricky_problems.html