ABACUS International Math Challenge
for
7th and 8th graders
October, 1998
C.73. Is there a number x such that 
?
C.74. By completing a series of operations,
Pete got 250 as the result. Later he realized that in the last operation
he multiplied by 5 instead of dividing by 5, and in the one before last
operation he added -4 instead of subtracting -4. What will the result be
if he does the operations properly?
C.75. Find all those abcde 5-digit-numbers
that are divisible by 275, and their reverse numbers, edcba, are divisible
by 275, also.
C.76. How many zero digits are there at
the end of the following product:
C.77. The sum of the age of a ship and
its captain is 70 years. How old is the captain if the ship is twice as
old now as the captain was when the ship was the same age as the captain
now?
C.78. Somebody goes up the staircase by
either going up step-by-step, or skipping one step here and there. How many
different ways can he reach the 10th step?
C.79. Two kinds of people live on an island:
truthful people, who always say the truth, and liars, who always lie. Once
I was talking with 11 people from this island who knew each other, and I
asked every one of them,
"How many truthful people are there among the 11 of you?"
The first nine people gave the following answers: 4, 1, 6, 0, 1, 5,
7, 5, 6. What could the remaining two answers be?
C.80. In a running race there were three
runners: X, Y, and Z. X's start was the best, Y left the starting line second,
while Z had a late start. During the race Z's place changed 6 times, X's
place changed 5 times. Finally, Y finished before X. What were the final
places of the runners?
Please, send your solutions to: