MATHEMATICS OLYMPIAD

1. Insert "+" signs between some of the digits in the following sequence
    to obtain correct equality:
   1 2 3 4 5 6 7 = 100
 
    Solution:
   1 + 23 + 4 + 5 + 67 = 100


2. Jack made 3 quarts of fruit drink from orange and apple juice. 2/5 of
    his drink is orange juice and the rest is apple juice. Dick prefers
    more orange juice in the drink. How much orange juice should he add to the drink to obtain a drink composed of 3/5 of orange juice?

   Solution: Let V denote the amount of drink Jake made, A denote
   the amount of apple juice in the drink, U be the amount of drink
   after he adds more orange juice. Then A = {1 - (2/5)V} = (3/5) V .
   More-over, A = (2/5)U. Hence, U = {( 3/5 )/( 2/5 )}V = (3/2) V .
   So, Jack has to add 1/2 V orange juice to his drink, more exactly, 1.5 quarts of orange juice.

3. A train moving at 55 miles per hour meets and is passed by a train
    moving in the opposite direction at 35 miles per hour. A passenger
    in the firsrt train sees that the second train takes 8 seconds to pass
    him. How long is the second train?
    Solution: The speed of the passenger in the first train, in relation
    to the movement of the second train, is 55 + 35 = 90 miles per
    hour, or: {(90)/ (60 x 60)} = 1/40 miles per second. Therefore, the length of the
   second train is 8 x 1/ 40 = 1/5 mile.