ABACUS International Math Challenge
for
7th and 8th graders
April, 1999
C.121. Find all those 5-digit numbers A that
have the following property: if you write down from left to right the remainders
A gives when divided by 2, 3, 4, 5, and 6, you get the original number A.
C.122. Divide the number 3000...007 (there
are 99 zeros in this number) by 37. What is the remainder? What are the
first nine and the last nine digits of the quotient?
C.123. A tile worker ordered tiles to cover
the floor of a squared shape hall. However, he was so absent minded that,
instead of the numbers of tiles needed along one side of the hall, he put
down his own age. This way he received 1111 more tiles than necessary. How
old is the tile worker?
C.124. We added three consecutive numbers,
then we added the next three consecutive numbers. Could the product of these
two numbers be 111 111 111?
C.125. The sides of a right triangle are
5, 12, and 13 units. What is the radius of the inscribed circle of this
triangle?
C.126. We divided a rectangle by two straight
lines parallel to its sides, so that the lines intersect on the diagonal
of the rectangle, as shown on the diagram. With the distances given, determine
the area of the section shaded.
C.127. A flee is jumping randomly to the
left and to the right on a long, thin stick. Every jump is 10 cm long. How
many different ways can it get 60 cm to the right of its starting point
using 10 jumps?
C.128. Find such a 6-digit number (in base
10) that if you multiply the number by either 2, 3, 4, 5, or 6, you get
a number that you could have gotten from the original number, too, just
by "sliding" its digits by a few places. "Sliding" here
means that you take a few digits from the end of the number and write them
in the front of the number in the same order. (For example: from abcdef,
you can get fabcde, or efabcd, or defabc, or ...)
Please, send your solutions to: