ABACUS International Math Competition
for
5th and 6th graders
September, 1997
B.1. The digits of a 2-digit number are the
same. When you multiply this number by 99, you get a 4-digit number in which
the third digit (the number of tens) is 5. What number did you get after
the multiplication?
B.2. What is the maximum number of months
with 5 Sundays that could occur in a year?
B.3. In a running race there were 12 racers
marked with numbers. What was their order at the finish line if, for every
racer, the product of the racing number and the place-number the runner
finished at was one more than a number divisible by 13?
B.4. Find such a number in which the sum
of the digits is divisible by 13, and the sum of the digits of the next
consecutive number is divisible by 13, also.
B.5. Using every digit only once, write
down 5 numbers with a ratio of 1:2:3:4:5.
B.6. Draw 6 points and connect some of
them with segments of a straight line in such a way that every point is
connected to 3 other points and the connecting segments do not intersect
each other.
B.7. How many such 4-digit numbers are
there in which the sum of the first two digits is the same as the sum of
the last two digits?
B.8. Between 1 and 1 million, what kind
of numbers are there more of: those that are divisible by 11 but not divisible
by 13, or those that are divisible by 13 but not divisible by 11? Why?
Please, send your solutions to:
Solutions of previous problems