ABACUS International Math Challenge
for
7th and 8th graders
December, 2002
C.345. Find all those two-digit numbers
that are divisible by both the sum and the product of their digits.
C.346. Three hungry tourists meet at a
train station. One of them has 4 bagels, another has 5 bagels, and they
divide the bagels equally among the three of them because the third tourist
has no food. However, he gives $12 to the two of them for the food he received.
How should the two tourists with the bagels divide this money?
C.347. There were 5 Sundays in February,
1976. Which years of the 21st century will this happen again? (Every 4th
year is a leap year, 2000 was a leap year, too.)
C.348. Gene, while traveling on a bus in
New York City, sees Tom who walks on the sidewalk in the opposite direction
with the bus. When the bus stops a minute later, Gene gets off and starts
walking towards his friend. Gene walks twice as fast as Tom but still a
quarter as fast as the bus. How long after Gene saw Tom will he catch up
with Tom?
C.349. Peter has 100 books. One day he
rearranged them on his bookshelf. He put half of the books on the middle
self to the bottom shelf. Then he took a third of the books, which were
originally on the bottom shelf, and put one third of these on the middle
shelf, and the rest on the top shelf. Finally he took 10 books from the
top shelf and put half of them on the middle shelf, and the other half on
the bottom shelf. Now every shelf had the same numbers of books on them
as originally. How many books are on each shelf?
C.350. We make a class stand on a circular
line, and ask everybody to hold hands with the two people on their sides.
Let the number of boys be: b, the number of girls be: g, B is the number
how many times a boy's hand holds another boy's hand, and G is the number
how many times a girl's hand holds another girl's hand. Prove that: b-g=B-G.
C.351. The average of two three-digit whole
numbers is the same as the number that you can get by writing the two numbers
one after the other with a decimal point in between the two of them. What
are these two numbers?
C.352. Lenny and Pete are alternating while
taking penalty kicks on each other. There were a total of 9 shots taken,
and 5 times it was a goal. If you score you get 1 point, if you miss the
shot the person playing goalie gets a point. Lenny won 6:3 against Pete.
Who took the first shot?