ABACUS International Math Challenge
for
7th and 8th graders
October, 2007
C.593. Eight swimmers participated in a
competition. The pool had eight lanes. We have the following information
about the outcome of the competition:
- Nobody finished in the same place as the number of his lane, and there
was no tie, either.
- Swimmers who used an even-number lane finished in an even-number place,
and those who used an odd-number lane finished in an odd-number place.
How many different ways could the swimmers finish this competition based
on these criteria?
C.594. Henry's mom made a chocolate-covered,
cube-shaped cake with a vanilla cream layer in the middle for his birthday.
Eight of Henry's best friends came over to celebrate with him. How can you
divide the cake between the 9 children so that everybody would get the same
amount of cake with the same amount of cream and chocolate cover?
C.595. A number is equal to the sum of
the factorials of its digits. Find all such 3-digit numbers.
(n factorial = n! = 1 x 2 x 3 x ... x n)
C.596. A shoe store owner bought 100 pairs
of shoes for $20 each pair, sold 75 of them for $25. Then he put the rest
of them on sale and sold them all. At the end he made a 20% profit. What
was the sale price of the 25 pairs of shoes?
C.597. We wrote 36 numbers in the following
6x6 grid. If you pick any 6 numbers from it so that no two numbers are from
the same column nor the same row, then the sum of those numbers will always
be the same. Why is that so?
C.598. I wrote down four numbers in a row.
The average of the first two numbers if 7, of the middle two is 2.3, and
of the last two is 8.4. What is the average of the first and last number?
C.599. A farmer has 749 sheep. He sold
700 of them for the same amount of money as he bought all of them for, then
he sold the rest of 49 sheep for them same higher price. What percent was
his profit?
C.600. If a car travels by a speed of 90
km/h it produces 20% less environmentally harmful gases in a unit of time
than if it traveled by 120 km/h. At which speed will the total environmentally
harmful gas production be less on a 210 km distance? What percent less is
this amount relative to the harmful gas production when traveling by the
other speed?