ABACUS International Math Challenge
for
7th and 8th graders
October, 2005
C.497. Two guards are serving a king. The
first guard makes 13/6 of a gold coin each day, the second guard makes 3/2
gold coins each day. The first guard owes the second guard 10 gold coins,
and he wants to give it back as soon as he possible can. They start working
for the king on the same day, and none of them has any money. How many days
do they have to work until both of them have the same money, if we suppose
that they do not spend any money until that day?
C.498. In a man's closet there are shoes,
pants, socks and shirts, the same number of each. He has only one of every
kind of clothing item. At least how many items should he have of each kind
of clothing if he wants to dress differently every day of the year? (That
is to say that he does not want to wear the same shoes-socks-pants-shirt
combination twice in the same year.)
C.499. Find all those rectangles whose
area and perimeter have the same measuring number.
C.500. One of 4 boys broke the window.
They said the following:
Andrew said: "It was Ben." Ben said: "It was Don."
Colin said: "It was not me." Don said: "Ben did not say the
truth."
Only one of them said the truth. Who broke the window?
C.501. Anne, Bea, Cecilia and Dori found
a secret staircase leading to the attic. Nobody used it for a long time,
so it was covered with dust. The girls ran down on it side-by-side as fast
as they could: Anne stepped on every other step, Bea used every 3rd step,
Cecilia used every 4th step, and Dori used only every 5th step. Everybody
started from the step on top. Once they all got downstairs, they realized
that their every step is shown in the dust, and that only the very top and
the very bottom steps have all of their shoe prints on them. How many steps
have only one shoe print on them?
C.502. Divide a 2-digit number by the sum
of its digits. What is the greatest possible remainder?
C.503. 729 is a very interesting number:
deleting just one digit from this number you can get three numbers whose
sum is 180, which is the sum of the inner angles of a triangle. (29+79+72=180)
Find all such 3-digit numbers.
C.504. The sum of the areas of the three
gardens on the diagram below is 1200 sq.meters. Find the area of each garden.
