ABACUS International Math Challenge
for
7th and 8th graders
November, 2008
C.657. Oliver has a collection of 100 chocolates
for the winter. All of them are either milch or dark chocolates, and all
of them are either rectangular or circular. 1/5 of all the milch chocolates
are circular, and 1/3 of all the circular chocolates are milch chocolates.
He has 30 rectangular dark chocolates. How many circular milch chocolates
does Oliver have?
C.658. Kate's measuring strip has centimeter
divisions on both sides but in opposite directions. On the yellow side it
starts at one end, and on the green side it starts at the other end. The
measuring numbers are written on the centimeter division lines. One night
out of curiosity Kate measured the dimensions of her sofa. Its length was
a whole strip and 90 cm, its width was 10 cm. From these Kate realized the
she used the wrong side while taking each measurement. How long is the measuring
strip if the width of the sofa is 2/3 of its length?
C.659. Every inner angle of the hexagon
below is 120 degrees. Prove that AB + AF = CD +DE.
C.660. In a library 3 children pick from
5 different books the following way: one copy of each of the 5 different
books are placed on the table in a row. The children put their library cards
on the book they want to borrow. More children may borrow the same book
since the library has multiple copies, but each child has to borrow only
one book. How many different ways can the cards be placed on the books?
(Two placements are different if at least one child did not put his card
on the same book in the two arrangements.)
C.661. Fiona has 11 cylinders. If she puts
them in an order of their heights (all of them standing on their circular
bases) then there is a 2 cm height difference between any two consecutive
cylinders. The highest cylinder has the same height as the middle one and
one of the neighbors of the middle one put one on top of the other. How
tall is the column you can build by putting all of these cylinder on top
of one another?
C.662. Olga typed the following addition
into her calculator: 1+2+3+4+...+100. Every time she added a number the
sum appeared on the indicator of the calculator. How many times did she
see a number divisible by 97 on the indicator?
C.663. Adam and Eve are playing Head or
Tail with a coin. If it is a head then Eve is the winner and gets 2 pieces
of candy from Adam. If it is a tail then Adam is the winner and gets 3 pieces
of candy from Eve. After 30 games they each have the same numbers of candies
that they started with. How many times did Adam win?
C.664. We have a subscription for a daily
newspaper, but our mailbox is not big enough for the newspaper to fit in
without folding it. Our mailman, just for his own entertainment, decided
that he will fold the newspaper differently every day. He always folds the
paper parallel to its shorter side so that the layers cover each other completely.
On the diagram below you can see all possible mainly different ways of folding
the paper into 3 layers. How many mainly different ways can you fold the
paper into 5 layers?
