ABACUS International Math Challenge
for
7th and 8th graders
November, 2005
C.505. Using three different digits, we
create all the possible 3-digit numbers that contain all three of these
digits. We would like to write the sum of these numbers as the product of
two consecutive numbers. What three digits should we use?
C.506. All 72 participants of a camp go
to have lunch to the same restaurant every day. Everybody eats the same
meal, so they get one check from the waitress for the whole group each day.
One day they could not read the first and the last digit of the total, they
could read only that the lunch costs $_67.9_. How much did one lunch cost
that day?
C.507. A tourist group of 86 people broke
up into two groups. The smaller group went to the beach, and the bigger
group went to see a water mill. The following statements are all true. How
many people were in each group?
a) A quarter of the people in one of the groups is more than the number
of people in the other group.
b) One of the groups contains 6 times as many women as men.
c) One of the groups contains only married couples.
d) The number of people in one of the groups is divisible by 4.
e) One of the groups contains 8 times as many people who are younger
than 50 as those who are at least 50 years old.
C.508. How many numbers can be shown on
a 7-digit calculator that are between 1/2 and 1/8 if we do not need a zero
to be shown in front of the decimal point, and the decimal point is not
considered to be a digit, of course?
C.509. Take three positive whole numbers
whose product is 720. Let x be the smallest one of the three numbers. What
is the greatest possible value of x?
C.510. What is the sum of all those positive
whole numbers below 1000 that are not multiples of neither 4 nor 5?
C.511. Prove that out of the abcd and the
dcba types of numbers none of them can be 7 times the other.
C.512. "Now I am twice as old as you
were when I was as old as you are now. When you will be as old as I am now,
then the sum of our ages will be 63." - said Lilly to Bea. How old
are they now?