ABACUS International Math Challenge
for
7th and 8th graders
December, 2001
C.281. Pete bought 100 glass Christmas
ornaments, and he numbered them from 1 to 100. His younger brother broke
most of them while he tried to help to decorate the tree, only 21 of them
survived. "Don't worry, you still have four here, out of which the
sum of the numbers on two of them is the same as the sum of the numbers
on the other two." -said the younger brother to Pete without looking
at the numbers written on the ornaments that survived. Is this statement
true all the time, regardless which 21 decorations survived?
C.282. There are 3 times as many men on
a party as women. After 4 men left with their wives, there were 4 times
as many men as women. How many men and how many women came to the party?
C.283. The diagonals of a rectangle are
36 cm, and their angle is 30 degrees. How far is a vertex from the diagonal
that does not go through it?
C.284. Fill the fields of a 13x13 grid
with either +1 or 1 (negative one). Write the product of the numbers
in each column and each row under the columns and at the end of the rows.
This way you have 13 numbers under the grid and 13 numbers of the side of
the grid. Could the sum of these 26 numbers be zero?
C.285. Peter is making a 24cm x 24cm cake
for his sister. He covers the top of the cake with 4cm x 8cm caramel tiles
perfectly. (He used 18 tiles.) Then he puts whip cream lines on top of the
cake 4cm apart, breaking the top up into a 4cm x 4cm square-grid. Could
Peter's sister find a whip cream line such that if you cut the cake along
that line, you do not have to cut through any of the caramel tiles?
C.286. One class (we do not know how many
students there are in it) decided that everybody will buy a present for
everybody for Christmas, and that they will buy presents for their 11 teachers,
too. However, the exchange of presents got canceled, so they decided to
give all the presents to the 15 siblings of the classmates. They wanted
to give the same number of presents to each sibling. Is it possible?
C.287. The different digits of a digital
clock are shown on the diagram below. In every minute there are four digits
shown on the clock. The day starts at 00:00 and ends at 23:59. (It means
that half past 7 in the evening is shown as 19:30.) How many such minutes
are there in the day when the sum of the digits shown on the clock is the
same as the sum of the lines making up those digits?
C.288. We painted a few sides of a cube,
and then we cut it up to smaller but equally sized cubes. We got 45 smaller
cubes with no paint on them. How many sides of the original cube did we
paint?