ABACUS International Math Challenge
for
7th and 8th graders
January, 2001
C.225. The sums of the 2-2 angles on the
same sides of a 7-sided polygon in order are: 257, 224, 243, 286, 303, 234,
and 253 degrees. How big is the smallest angle of this polygon?
C.226. There are five shipwrecked people
and a monkey on an island with no other food but coconuts. However, there
are a lot of coconuts on the island. The people agree that next day they
will divide the coconuts evenly among themselves so that everybody would
get the same whole number of coconuts, and they will give the leftover coconuts
to the monkey. Then they went to sleep. However, the first person, who woke
up hungry in the middle of the night, decided that he wants to eat all of
his share of the coconuts, so he divides them into 5 equal groups and sees
that one is left over. He gives one coconut to the monkey, eats a fifth
of the rest and goes back to sleep. A little while later the second person
wakes up hungry and does the same thing: he realizes that one coconut would
be left over, so he gives it to the monkey, eats one fifth of the rest and
goes back to sleep. The same thing happens to the third, the forth, and
the fifth person. In the morning they all wake up, but nobody says a word
about last night's activities. So they divide the coconuts into 5 equal
parts, and they find that one coconut is left over for the monkey. At least
how many coconuts did they have originally?
C.227. Find the smallest positive a, b,
c whole numbers so that (a,b)=4, (b,c)=6, and (a,c)=10.
Here (m,n) represents the greatest common factor of the numbers m and
n.
C.228. Color the fields of a 3x3 grid using
three colors so that every row and every column contains all three colors.
You may use only one color for a field. How many different colorings can
you produce?
C.229. Find 6-digit numbers in which the
number created from the first two digits is divisible by 2, the number created
from the first three digits is divisible by 3, the number created from the
first four digits is divisible by 4, the number created from the first five
digits is divisible by 5, the number itself is divisible by 6, and the number
created from the first two digits plus the number created from the third
and fourth digits is equal to the number created from the last two digits.
C.230. The number 3 can be written in 4
different ways as the sum of positive whole numbers: 3, 2+1, 1+2, and 1+1+1.
(The order of the addends is important!) How many different ways can you
write 20 as a sum of positive whole numbers?
C.231. Take all those 4-digit numbers that
contain only the digits 4 or 5. Create the sums of any two of these different
numbers. How many different sums will you have?
C.232. Find a 6-digit number in which the
digits will rotate by a few places when the number is multiplied by either
two, three, four, five or six!
by Christian Kiss-Tóth
Please, send your solutions to: