Puzzles

1. How would you find the center of a given circle with only a ruler and a pencil?
2. Why (3!)(5!)=(6!)? Find other three numbers with this property.
3. Someday my calculator lost the tangent key. Anyway, I managed to get the tangent of any angle, how? Another day I broke the cosine key, but I still calculate the cosine of any angle. Describe two methods to accomplish this. Assume you can't get another calculator.
4. Let Phi be the golden ratio. Show that Phi² = Phi + 1.
5. Given four consecutive numbers show that the product of all of them plus one is a perfect square.
6. Suppose that the hypotenuse of a right triangle ABC is divided in two segments, a and b, by a point D. If BAC is the right angle, express the length of the segment AD as a function of a and b. The angle BDA is equal to 90 degrees.
7. Find

   lim    tan 4x
   x->0  ________
             x
   

   without using L'Hospital rule.
8. Two friends had agreed to meet in a place between 3 o' clock and 4 o' clock. Find the time each one have to wait the other if the probability of their meeting should be 1/2.
9. Answer quickly: Is this true?

    /\1  2      /\1  3
    |   x   >   |   x
   \/ 0        \/ 0
   

10. Prove, by induction, that 9 is a divisor of the sum of three consecutive natural numbers.
11. Find:

     /\  
     | |x|dx 
    \/    
    

12. Calculate the volume of the intersection of two cilinders of radius r. The angle of intersection is t, and the axis of both cilinders lay in one plane.
13. Find the sum of the sequence a_n=n(1/2)ⁿ.