1. If a is 50% larger than c, and b is 25% larger than c, then a is
   what percent larger than b?

Answer:1/5


2. How many two-digit positive integers N have the property that
   the sum of N and the number obtained by reversing the order of
  the digits of N is a perfect square?

3. Medians BD and CE of triangle ABC are perpendicular,  |BD|=8, 
   and  |CE|=12. Find the area of triangle ABC.

Answer:64


4. A rising number, such as 34689, is a positive integer each digit
   of which is larger than each of the digits to its left. There are 
   C(9,5)=126 five-digit rising numbers.When these numbers are arranged 
   from smallest to largest,  the 97th number in the list does not
   contain the digit in  the set {4,5,6,7,8} 

5.How many of the integers between 1 and 1000, inclusive, can be
   expressed as the difference of the squares of two nonnegative integers?

6. Naim intended to multiply a two-digit number and a three-digit number, 
   but he left out the multiplication sign and simply placed the 
   two-digit number to the left of the three-digit number, thereby
   forming a five-digit number. This number is exactly nine times
   the product Naim should have obtained. What is the sum of the
   two-digit number and the three-digit number?

7. Let x=(The sum of cosn from n=1 to n=44)/(The sum of sin n
    from n=1 to n=44).  What is the greatest integer that does not 
    exceed 100x?

8. The function f defined by f(x)=(ax+b)/(cx+d), where a,b,c and d 
   are nonzero real numbers, has the properties f(19)=19, f(97)=97,
   and f(f(x))=x for all values of x except -d/c. Find the unique number
   that is not in the range of f.

9. How many positive factors of 36 are also multiples of 4? 

10. When walter drove up to the gasoline pump, he noticed that his 
   gasoline tank was 1/8 full. He purchased 7,5 gallons of gasoline
   for $10. With this additional gasoline, his gasoline tank was 
   then 5/8 full. What is the  number of gallons of gasoline his 
   tank holds when it is full ?

Answer:15


11. In the fall of 1996, a total of 800 students participated in 
   an annual school clean-up day. The organizers of the event except 
   that in each of the years 1997, 1998, and 1999, participation will
   increase by 50%  over the previous year. What is the number of
   participants the organizers except in the fall of 1999 ?

Answer:2700

12. The measure of angle ABC is 50 degree, [AD] bisects angle BAC,
    and [DC] bisects angle BCA. What is the measur of angle ADC?

13. Two mathematicians take a morning coffee break each day. 
    They arrive at the cafeteria independently, at random times 
    between 9 a.m. and 10 a.m., and stay for exactly m minutes.
    The probability that either one arrives while the other is
    in the cafeteria is 40%, and m=a-b.(square root c), where 
    a, b, and c  are positive integers, and c is not divisible
    by the square of any prime. Find a+b+c.

14. Let x be a real number such that sec x - tan x =2 . Find the value
    of  sec x + tan x

Answer:1/2 or 2 

15. Find the smallest prime that is the fifth term of an increasing 
   arithmetic sequence, all four preceding terms being also prime.

Answer:11

16. A speaker talked for sixty minutes to a full auditorium. 
    Twenty percent of the audience heard the entire talk and
    ten percent slept through the entire talk. Half of the remainder
    heard one third of the talk and the other half heard two thirds
    of the talk.  What was the average number of minutes of the talk 
    heard by members of the audience?

17. Naim rolls four standard six-sided dice and finds that the product of
    the numbers on the upper faces is 144. Which of the element of the set
     {14, 15, 16, 17, 18} could not be the sum of the upper four faces?

Answer:18

18. A piece of graph paper is folded once so that (0,2) is matched 
    with (4,0), and (7,3) is matched with (m,n). Find m+n

Answer:12

19. The strictly positive integers a and b are such that the numbers 
   15a+16b and 16a-15b are both the squares of positive integers. 
   What is the smallest value of the smallest of these two squares?
Answer: 49 (The second number is 76)

20. Let I be the incenter of triangle ABC. Let the incircle of ABC 
    touch the sides BC, CA, AND AB at K, L, and M, respectively.
    The line through B parallel to MK meets the lines LM and LK 
    at R and S, respectively. Prove that angle RIS is acute.(IMO-1998).