ABACUS International Math Challenge
for
7th and 8th graders
September, 2003
C.377. In Math class the teacher wrote
a number on the backboard. One student said: "This number is divisible
by 31." Another student said: "It is divisible by 30." The
next student said that it is divisible by 29, and so on. The 30th student
said that the number is divisible by 2. Then the teacher said that only
two of these statements are not correct, and that they were said one after
the other. Which two statements were incorrect?
C.378. 1/5 of the price of a jacket is
the profit of the salesman. If he increased the price by $20 then 1/3 of
the price would be his profit. What is the price of the jacket?
C.379. Mark 2003 different points on a
plane. Is it true that you can always draw a circle on this plane so that
1002 of these points would be inside of the circle, and none of them would
be on its circumference?
C.380. You know that half of the time spent
since midnight is the same as 3/4 of the time you have left until noon.
What time is it now?
C.381. Prove that the sum 
is divisible
by 100.
C.382. We wrote down 5 consecutive, positive,
2-digit numbers. The sum of their digits is 42. Find these 5 numbers.
C.383. John eats lunch in the school on
every school day mainly because it is cheap. He has to pay for his lunch
every month. The lunch costs the same every day. At the end of the school
year John checks on how much he paid in each month, and notices that there
were two months when the difference he paid for the lunches was $6.63, and
there were two months when the difference was $11.05. How much does the
lunch cost each day?
C.384. Ben is 100 meters, Colby is 300
meters ahead of Andrew. The three boys run by a uniform speed. Andrew catches
up with Ben in 6 minutes, and another 6 minutes he catches up with Colby,
too. How long does it take Ben to catch up with Colby?