ABACUS International Math Challenge
for
7th and 8th graders
February, 2004
C.417. A wine dealer, who carries his wine
in identical barrels, travels through three cities. Each city charges a
different tax which is collected upon entrance into the city. The first
city takes 1/3 of his wine as a tax, the second city takes 1/4 of his wine,
and the third city takes 1/5 of his wine. After paying all these taxes he
sold all of his remaining 72 barrels of wine. How many barrels of wine did
he have originally?
C.418. Prove that there is at least one
Friday the 13th every year.
C.419. A company sells its detergent in
three different size boxes: small, medium and large. The medium size is
50% more expensive than the small size, and contains 20% less detergent
than the large size. The large size is 50% heavier than the small size,
and 25% more expensive than the medium size. Which box is the most economical?
C.420. Let points X and Y divide the diagonal
BD of parallelogram ABCD into 3 congruent (equally long) sections. Prove
that quadrilateral AXCY is also a parallelogram. What is the ration of the
areas of ABCD and AXCY?
C.421. At a pizzeria you may buy circular
pizzas with diameters 20 cm and 25 cm. They deliver, also: the prices of
the two different size pizzas with ham are $5.70 and $6.80. The smaller
size pizza ala "Frutti de mare" (with all kinds of seafood on
top) costs $6.80. (These prices include delivery charges.) Suppose that
the delivery charge does not depend on the kind and size of the pizza, and
that the prices of the pizzas, without the delivery charges, are proportional
to the areas of the circular pizzas. How much does a larger pizza ala "Frutti
de mare" cost with delivery charges? (Use 3.14 for pi.)
C.422. Oleg drew the following diagram
in his new notebook. His older sister, Kate, kept on looking at the diagram
and then finally said that it is impossible to draw such a pentagon into
a circle. Find Oleg's mistake. (Side 5 is perpendicular to side 12. Which
number or numbers on the diagram are incorrect?)
C.423. There is a 6m by 8m sized rectangular
shaped barn standing in the middle of a meadow. They tied a dog to one of
the corners of the barn on a leash that is as log as the diagonal of the
barn. What is the area of the territory of the dog?
C.424. Prove that 
is divisible
by 186.