ABACUS International Math Challenge
for
7th and 8th graders
February, 2000
C.169. Find all those 3-digit numbers in
which every digit is a prime number and the number itself is divisible by
these primes.
C.170. How many such number-pairs are there
for which the greatest common factor is 7 and the least common multiple
is 16940?
C.171. Take n 2's, k 3's and m 5's (n,
k, and m are non-negative whole numbers), so that n+k+m=100. We know that
the last digit of the sum 
is 9. Prove, that the last digit
of the product 
cannot be zero.
C.172. How many different ways can you
break up 1 million into the product of three positive whole numbers if the
order of the numbers does not matter?
C.173. Lay a plane through every edge of
a regular tetrahedron so that every plane cuts the tetrahedron into two
identical parts. How many pieces did you cut the tetrahedron into this way?
C.174. A wanderer, on his way to Kazohinia,
arrives to a split of the road where three brothers live. Two of them always
tell the truth, but the third one is capricious: you never know when he
tells the truth or when he lies. The wanderer can learn only from these
three brothers which road goes to Kazohinia. If he is allowed to ask a question
from one person at a time, can he succeed for sure asking only two questions?
C.175. To a 2-digit number, add the number
you get if you reverse the order of the digits of the original number. If
the order of the digits of this sum is not symmetrical than add to it the
number you get by reversing the order of the digits of the sum. Following
this procedure as long as you needed to, can you always get to a number
in which the order of the digits is symmetrical?
C.176. Is it possible to break up a circle
into identical (so called congruent) pieces so that at least one piece does
not contain the center of the circle, not even on its border?
by Balázs Gerencsér, Hungary
Please, send your solutions to: