ABACUS International Math Challenge
for
7th and 8th graders
March, 2001
C.241. Can you divide all the positive
2-digit integers into two groups so that the product of the numbers in each
group is the same?
C.242. There are 10 liters of water in
the first container, a mixture of 5 liters of water and 5 liters of wine
in the second container, and 10 liters of wine in the third container. You
pour 1 liter from the first container into the second container, then you
pour 1 liter of the liquid from the second container into the third one,
and finally you pour 1 liter from the third container into the first. After
all of these, how much wine is there in each of the containers?
C.243. Write a digit 2 in front of a 2-digit
number, then write the digit 2 at the end of the same 2-digit number. This
way you get two 3-digit numbers with a difference of 81. Find this 2-digit
number.
C.244. How many zeros are there at the
end of the following product in its standard form:
C.245. The number 123 is shown on a computer
monitor. At the end of every minute the computer adds 102 to the number
on the screen. With his program, Smarty can interfere with this process
by changing the order of the digits of the number on the screen at any time,
as many times as he wants to. Could Smarty write such a program that would
guarantee that we will see only 3-digit numbers coming up on the screen
forever?
C.246. Find an N positive integer in which
the sum of the digits is 10, and the sum of the digits of the square of
N is 100.
C.247. Doc thought of a 200-digit number,
added its digits, and whispered the sum into Happy's ears. Happy calculated
the sum of the digits of the number he was just told, and whispered the
sum into Grumpy's ears. Grumpy figured out the sum of the digits of this
number, which happened to be a 2-digit number, and he whispered this number
into Sneezy's ears. Sneezy also calculated the sum of the digits of the
number he just heard and whispered it to Dopey.
Which number did Happy whisper to Grumpy, and which number did Sneezy
whisper to Dopey?
C.248. How many different ways can you
place 8 different books on 3 different shelves if the order of the books
on the shelves matters, too? (You may leave some of the shelves empty, too.)
Please, send your solutions to: