ABACUS International Math Challenge
for
7th and 8th graders
November, 2003
C.393. 95% of a 10 kg watermelon is water.
When you put it on the sun, it starts to dry out, so now only 90% of it
is water. How many kg is the mass of the watermelon now?
C.394. We drew 3 circles so that any two
are touching each other from the outside. The distances between the centers
of the circles are 2001 mm, 2002 mm, and 2003 mm. What are the radiuses
of these circles?
C.395. A number is called "beautiful"
if the product of its real divisors (the ones that are different from 1
and the number itself) is the number itself. Find the first 10 "beautiful"
numbers.
C.396. Find all those "n" positive
whole numbers for which the sum of all of their positive divisors is:
a) n+3
b) n+6.
C.397. We put 6 different color balls in
a row. If two balls were next to each other in an arrangement, then they
cannot be next to each other in the following arrangements. How many times
can we rearrange the balls this way?
C.398. We added three consecutive whole
numbers, and then we added the next three consecutive numbers following
them. Could the product of these two numbers be 111 111 111?
C.399. 126 can be written as the sum of
two prime numbers in a few ways. What is the greatest and what is the smallest
difference between two such prime numbers?
C.400. On a 1760 meter running competition
Andrew beat Ben by 330 meters, and Emma by 460 meters. If everybody ran
by a constant speed, by how many meters did Ben beat Emma?