ABACUS International Math Challenge
for
7th and 8th graders
September, 2008
C.641. Andrew, Ben, Charles and Daniel
are playing cards with a 32-card deck. Daniel distributes all the cards,
but not equally as he should have done it. To correct Daniel's mistake,
first Andrew distributes half of his cards equally among Ben and Charles,
then Ben does the same thing among Andrew and Charles. Finally, Charles
distributes half of his cards equally among Andrew and Ben. Now everybody
has the same number of cards. How many cards did each player get originally?
C.642. Pete, Agnes, Anne and Sofia divided
a rectangular shape area as shown on the diagram. One of the sides of this
rectangle is 48 meters. Pete received the only square shape piece of land,
which has half the area of Agnes' piece, and 2/3 of the area of Anne's piece.
Sofia received the piece with the greatest area, which has twice the area
of Anne's piece. What are the areas of the different pieces people received
here if we know that the dimensions of each piece are whole numbers in meters?
C.643. Take a 2-digit number, multiply
its digits, and do the same thing with the product until you get a one-digit
number. How many 2-digit numbers will produce zero at the end?
C.644. Every year they put up a Christmas
tree at the local train station. The station manager has 7 different color
lights, which can be turned on and off independently from one another, but
he does not want the pink and the purple lights to be on at the same time
ever. The station manager placed a few lights on the tree on December 7th.
How many different ways can he pick those light bulbs so that he could have
a different color combination light up on the tree for his liking every
day until January 6th? (The station manager does not want to take any chances,
so he does not want the pink and the purple lights to be on the tree at
the same time!)
C.645. Fiona glued together 19 regular
dice into a shape you can get by taking the corner pieces off of a 3x3x3
cube. She glued them together in such a way so that the number of dots on
the visible faces of the dice of this solid is as small as possible. How
many dots are there on this solid if we know that the sum of the dots on
the facing sides of all the dice is 7?
C.646. The diagonals of the 10 rectangles
on the diagram below have a 60 degrees angle to the horizontal. How long
is the shaded line if the total width of the 10 rectangles is 50 cm? (Below,
we show the shaded line again without the rectangles.)
C.647. How many positive 4-digit whole
numbers have all different digits and are divisible by 9 and 25?
C.648. One side of a parallelogram is twice
as long as its other side. Its perimeter is 24 cm, and its area is 16 square
cm. Find the heights and the measures of the angles of this parallelogram.