ABACUS International Math Challenge
for
7th and 8th graders
December, 2007
C.609. Similarly to a magic square, we
made a magic 3x3x3 cube by writing the numbers from 1 to 27 on the unit
cubes so that the sum of the numbers in every 1x3 column or row is the same.
(Not the diagonals!) If you made vertical 3x3 slices of the cube, you would
see the following:
How did we place the rest of the numbers on the cube?
C.610. There are 52 cards in a deck of
French cards. There are 4 suits: spades (black leaf), clubs (black leaf
of clover), hearts (red heart), and diamonds (red rhombus). In each suit
there are J, Q, K and A cards and the numbered cards go from 2 to 10. We
took 5 cards out of them: at least one out of every suit; all 5 cards are
numbered cards; the sums of the numbers on the odd- and on the even-numbered
cards are the same; the sum of the spade cards is 14; the sum of the red
cards is 10; the smallest value card is a heart. Which 5 cards did we take
out?
C.611. Are there 21 different positive
whole numbers such that the sum of their reciprocals is 1?
C.612. How many positive 3-digit whole
numbers are there which are less than twice the product of their digits?
C.613. Frank and Bonnie bought the same
package of letter papers and envelopes. Frank used 1 sheet, and Bonnie used
3 sheets of paper for each letter. Frank used up all of the envelopes and
had 50 sheets left over, Bonnie used up all the sheets and had 50 envelopes
left over. How many sheets and how many envelopes were there in the package?
C.614. The midpoint of side AB of triangle
ABC is M. You know that BM = BC = CM. How big is side AC if BC = a?
C.615. Find the smallest positive prime
number that can be written as the sums of 2, 3, 4, and even 5 different
prime numbers.
C.616. The diagram below is a number pyramid.
Take the sums of the products of the numbers in each row. Can we ever get
a square number (greater than 1)? (The product of the numbers in the first
row is 1.)
