ABACUS International Math Challenge
for
7th and 8th graders
September, 2007
C.585. You put brand new tires on your
car. The tires can run 21000 km on the front wheels and 29000 km on the
rear wheels. (Let's suppose that the tires are getting used up proportionally
with the distance.) The car has a new spare tire also. What is the maximum
distance you can drive using the 5 new tires if you may rotate them at your
wish? (Give me the schedule of rotation of the tires, also.)
C.586. Henry goes to a casino in Las Vegas
with $5120 to play a game called "Double or Nothing". He always
puts up half of his money for a bet. He bets 10 times, out of which he wins
5 times and loses the other 5 bets but we do not know in what order. How
much money could he possibly have at the end?
C.587. A new laundry detergent is promoted
at the stores. They packed each bag of detergent with a small bucket. If
you buy more than one package, they give you a discount: if you buy two
packages you get a 10% discount, and for every additional package you buy
you get a 20 % discount. The full price of a package (a bag of detergent
+ the bucket) is $30. If you wanted to buy a bucket separately, it would
cost you $5. At least how many packages do you have to buy so that the buckets
would come out to be free?
C.588. The teacher gives the following
homework assignment: You have to write an essay by tomorrow. You may work
alone or in boy-girl pairs. As it turned out, 2/3 of the boys and 3/5 of
the girls worked in pairs. What fraction of the class worked alone?
C.589. How many rectangles are there on
a 4x4 chess board? How many of these are squares?
C.590. I have three boxes numbered 1, 2,
and 3; and three pieces of papers on which I wrote the number 1, 2, and
3 respectively. I put one piece of paper in each of the three boxes by the
following rule: if the number on a box is k, then the number on the paper
in this box shows you which box contains the paper with the number k on
it. How many different ways could I place the papers in these boxes following
this rule?
C.591. In a set of positive whole numbers
there is no such two elements of which the difference is a multiple of 2007.
How many elements could the set have the most?
C.592. Place a circle, a square, and an
equilateral triangle on top of each other so that their lines would have
the most intersection points. What is this number of intersection points?
(You may choose the size of each figure.)