ABACUS International Math Challenge
for
5th and 6th graders
October, 2001
B.265. How many mutual points can a triangular
and a circular line possibly have?
B.266. Using the grid-lines of a square-grid,
we drew a rectangle, and then we counted how many little squares there are
inside the rectangle not touching the perimeter of the rectangle. How many
little squares are there inside of the rectangle, if:
a) 5 of them
b) 21 of them are not touching the perimeter of the rectangle?
B.267. Cut the following quadrilateral
with two straight lines into 6 pieces.
B.268. Andy, Ben, Charlie, and Danny are
stamp-collectors. Danny has more stamps than Charlie. Andy and Ben together
have the same amount of stamps than Charlie and Danny together. Ben and
Charlie together have more stamps than the other two boys together. What
is the order of the boys based on the sizes of their stamp-collections?
B.269. How many such positive 3-digit numbers
are there in which the sum of the digits is an odd number, and the sum of
the digits of the number one greater than the original number is odd, too?
B.270. A 4-member family is getting ready
for their vacation. The parents want to take 3 large suitcases each, and
the children want to take 2 small suitcases each. It takes a person one
hour to pack a large suitcase, and it takes a person a half an hour to pack
a small suitcase. What is the least amount of time this family can pack
all the suitcases, if any suitcase can be packed by only one person at a
time?
B.271. We write all the integers from 1
to 10 on a circular line in a random order. Then we pick a number on the
line and, going clockwise, we check the next number: if this number is less
than the first number then we switch the position of this number and the
first number. Then we move on to the next (the third number) and check it:
if the third number is less than the second number then we switch them.
And we keep on moving. After a full cycle (when we also compared the 10th
number with the first number) we keep on doing this. After how many cycles
will the positions of the numbers be the same as their original positions
were?
B.272. Tom is drawing on a paper with unit-square
grid lines on it. He starts from a grid point, always stays on the grid
lines without lifting his pencil, and never goes over the same section twice.
Prove that if he ends his drawing on the starting point then the length
of the line he drew is an even number.
Please, send your solutions to: