ABACUS International Math Challenge
for
7th and 8th graders
February, 2007
(These is the last set of questions
for this year!
Please, send all your solutions in by April
30, 2007.)
C.577. Break up the first 100 positive
whole numbers into two groups any way you want to. Prove that at least one
of the groups contains two such numbers whose difference is in that group,
also.
C.578. Smart Alec participated in a Math
Competition. He solved 9 out of the first 12 problems perfectly, but did
not receive points for the other 3 problems. He solved a third of the rest
of the problems perfectly, but did not receive points for the others. This
way he received 50% of all possible points. How many problems did he have
to solve if each correct answer worth 6 points?
C.579. The phone numbers in New York are
7-digit numbers. They may start with any digit but zero. How many phone
numbers have strictly increasing digits from left to right?
C.580. In my candle holder one candle barely
finishes burning in one night. In the morning always collect the melted
wax leftover in the candle holder because from the remainder of 4 candles
I can make a new candle that is the same size as the original candles. Right
now I do not have any candles and any leftover wax. At least how many candles
should I buy if I need candle light for 100 nights?
C.581. I have a red and a white candle.
They have different heights and different thicknesses. I light them at the
same time and notice that after 45 minutes they are the same height. The
red candle burns down in 90 minutes, the white one in 2 hours. How many
times taller was the red candle originally than the white candle?
C.582. The sum of the abcde and
the edcba 5-digit numbers is divisible by 11111. (a, b, c, d
and e are different digits.) We also know that the number abcde
is divisible by 7, and that the sum of its digits is 10. Waht number
is abcde?
C.583. A merchant bought a truck-load of
items. After selling 2/3 of the items, he got 3/4 of his money back. What
% of profit can he expect if he sells the rest of the items by the same
rate of price increase as he had done sofar?
C.584. The shortest height of a right triangle
is 7 cm, and its smallest angle is 15 degrees. How long is its hypotenuse?