ABACUS International Math Challenge
for
5th and 6th graders
September, 2008
B.641. Using only 200 and 500 monetary
unit bills, what are all those values divisible by 100 that are payable
without any change necessary?
B.642. Kolmogorov, a Russian mathematician
once said: "You have to be a very good mathematician just to imagine,
with your eyes closed, a plain cross section of a cube where the plain,
containing the center of the cube, is perpendicular to one of the diagonals
of the solid." What polygon is this cross section?
B.643. A clock on the wall bells at every
hour and at every half an hour. (Once at the half hours and as many times
as the number of hour at the full hours.) One night I woke up for a bell.
How much longer do I have to stay awake in order to tell the exact time
for sure based on the number of bells only?
B.644. The opening dance at a ball is a
waltz. One of the main attractions of the dance is when the girls change
from a billiard formation into a flower formation. The beauty of the move
is that during the change only one girl moves at a time by having her skirt
roll around the skirt of another girl until her skirt touches her two neighbors'
skirts again. Create a choreography with four girls' such movements.
B.645. 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 4 layers?
B.646. Ten players are participating in
a ping-pong competition. If you lose 2 matches, you are out. (There is no
tie.) In each round they pick the opponents randomly from those who are
still in the competition. If somebody does not get an opponent, he advances
to the next round without a match. What is the least and what is the most
number of matches that can decide the winner of this competition?
B.647. We roll a prime number with each
of three regular dice so that the sum of those primes is a prime, also.
What could those 3 numbers be?
B.648. Let the length of a match be a unit.
Using 18 matches, create a polygon with an area of 9 units.