ABACUS International Math Challenge
for
7th and 8th graders
January, 2000
C.161. How many 6-digit cube-numbers are
there?
C.162. Somebody calculated the exact values
of 
and 
in the decimal system. How many digits are there in
these two numbers all together?
C.163. Alex and Burt took their rabbits
to the market to trade them. Each of them got as many dollars for each of
their rabbits as many rabbits they each took to the market. But, because
their rabbits were so beautiful, they each got as many extra dollars for
their rabbits as many rabbits they each took to the market. This way Alex
received $202 more than Burt. How many rabbits did they each take to the
market?
C.164. How many {a, b, c} sets are there
containing three positive whole elements, where the product of a, b, and
c is 2310?
C.165. Let a, b, c, and d be different
digits. Find their values so that the sum 
has the least possible
number of divisors, but the sum itself is the greatest possible.
C.166. Fill in a 25x25 grid by using the
numbers +1 and -1. Create the products of the 25 numbers in each column
and in each row. Could the sum of these 50 numbers be:
a) 0
b) 10
c) 17?
C.167. Is there such a triangle in which
the heights are 1, 2, and 3 units long?
C.168. You put a plane on each side of
a regular, square-based pyramid. How many sections do these 5 planes divide
the space into?
Please, send your solutions to: