ABACUS International Math Challenge
for
5th and 6th graders
September, 2006
B.537. You cut a piece of paper into 5
or 7 pieces. You pick a piece and cut it again into 5 or 7 pieces. You keep
doing this. (Every time you may choose whether you want to cut the piece
you picked into 5 or 7 pieces.)
a) Can you get exactly 199 pieces?
b) Can you get exactly 200 pieces?
B.538. Four students handed in their homework
to a teacher, but none of them wrote their names on it. The teacher corrected
the four papers and gave them back randomly. (Every student received only
one paper.) How many different ways can it happen that nobody gets his/her
own paper back?
B.539. The shape of Pete's room is a rectangle.
Its floor is covered by 68x119=8092 square tiles. Pete drew a straight diagonal
in his room. How many tiles does this line go through? (If the line has
only one mutual point with a tile then we do not consider the line to be
going through that tile.)
B.540. Camille is keeping grasshoppers
and crickets is her terrarium. One day she counts them all and finds that
she has three times as many crickets as grasshoppers. However, next day
it turns out that she made a mistake and she counted one of the grasshoppers
as a cricket. So now she knows that in fact there are only twice as many
crickets as grasshoppers. How many crickets and grasshoppers are there in
the terrarium?
B.541. We write the sum of the first 123
positive whole numbers in the numerator of a fraction, and we write the
same sum in its denominator but we change the plus sign to a minus sign
in front of every even number. What is the value of this fraction?
B.542. The Junior Varsity Soccer Coach
is expecting his players for a season-opening party. If he wants to give
4 cookies to every player then he would be missing 8 cookies, but if he
gave 3 cookies to every player then the coach would have 6 cookies left
over. How many players does he have?
B.543. Write each of the numbers 1, 2,
3, and 4 on the vertices of each of several identical squares in any order
you want. Then place these squares one on top of the others. Is it possible
that the sum of the numbers in each of the 4 vertex columns is 2006?
B.544. You can arrange 10 coins in a triangular
shape, as shown on the diagram, but you cannot arrange them in a square
shape. Out of 80 coins I picked a few (more than one) so that they can be
arranged in a triangular and a square shape, also. How many coins did I
pick?
