ABACUS International Math Challenge
for
7th and 8th graders
November, 2001
C.273. There are two cities on the same
river. There is a boat going back and forth between the two cities. Its
engines always work by the same power, so the boat always travels by the
same speed relative to the water in the river. It takes the boat 2 hours
downstream, and 4 hours upstream to get from one city to the other. If you
kick a soccer ball into the river in the city farther up the river, how
long does it take the ball floating down the river to get to the other city?
C.274. There is a simple way to calculate
in your head the square of a 2-digit number that ends by 5. You just have
to multiply the first digit by the number one greater than it, and stick
25 to the back of this product. Is there a similar method for more-than-2-digit
numbers ending by 5?
C.275. Timea, who is always in a hurry,
went up the escalator by making one step every second. This way it took
her 20 steps to get upstairs. Next days she made 2 steps every second on
her way up, and this time it took her 32 steps to get upstairs. How many
steps would it take her to go upstairs if the escalator did not work?
C.276. We have the following information
on the 33 passengers of a tour bus: 15 of them are boys or men, there are
twice as many women as boys, and there are two less women than girls. How
many men are there on the bus?
C.277. There were 60 dancers at a party.
Mary danced with the least number of boys, with 7 of them, Lucy danced with
8, Sara with 9, and so on, while the last girl danced with all the boys.
How many boys and how many girls were at the party?
C.278. The side of the small square is
3 cm. We extended the sides of the small square in one direction until they
intersected the sides of the bigger square, and then connected the points
of intersections to get the shaded area. How big is the shaded area, if
its measuring number is a 2-digit whole number, and the measuring number
of the side of the bigger square is a whole number, also?
C.279. Find positive whole number pairs
such that the difference of their squares is 2001.
C.280. Pick a few points inside of a convex
quadrilateral. Connect these points to each other and to the vertexes of
the quadrilateral by straight segments so that the connecting lines do not
intersect each other inside of the quadrilateral, and these connecting segments
break up the quadrilateral into small triangles and small quadrilaterals.
(Every point picked inside the quadrilateral is a vertex of a triangle or
a quadrilateral.) Could the number of triangles be 2001?