ABACUS International Math Challenge
for
7th and 8th graders
February, 2005
C.473. The edges of a cube measured in
centimeters are whole numbers. Its volume in cb.cm is a 6-digit number that
is divisible by 336. How big is this cube?
C.474. How many positive whole numbers
not greater than 2005 have at least one prime digit?
C.475. (1 + 1/2)(1 + 1/3)(1 + 1/4) ...
(1 + 1/2005) = ?
C.476. 2^2005 + 2^2004 + 2^2003 + 2^2002
+ 2^2001 = k(2^2001)
Find k.
C.477. Steve's rectangular shaped window
is divided into 3x3 smaller sections.
Steve paints an identical sized digital number-digit in the middle of
each of two of these sections so that the whole window looks just the same
from the outside than from the inside. How many different ways can he do
this?
The digital number-digits look like these:
C.478. 1! + 2! + 3! + ... + 2005! = N
Find the last two digits of N.
C.479. Is there such a positive whole power
of 5 in which all ten digits appear and exactly the same number of times?
C.480. In triangle ABC (see the diagram)
AB = AC. We know also that angle BAC is 20 degrees, angle QCB is 50 degrees,
and angle PBC is 40 degrees. How big is angle PQC?
