ABACUS International Math Challenge
for
5th and 6th graders
January, 2001
B.225. Cut up an 8x8 grid into 4 congruent
decagons. (A decagon is a 10-sided polygon.)
B.226. One day 3 monkeys found a banana
tree. They picked all of the bananas off the tree, and went to sleep. When
the first monkey woke up, he divided the bananas into three equal groups,
but two bananas were left over. He ate these two and his third part of the
bananas, then he went back to sleep. When the second monkey woke up, he
divided the bananas into three equal groups, but two bananas were left over.
He ate these two and his third part of the bananas, then he went back to
sleep. When the third monkey woke up, he also divided the bananas into three
equal groups, but two bananas were left over. He also ate these two and
his third part of the bananas, then he went back to sleep. At least how
many bananas did the monkeys pick?
B.227. I created a number sequence by the
following rule: if an element n is odd then the next element is n+3, if
the element n is even then the next element is n/2. The first element is
odd, and the fourth element is 27. What is the first element?
B.228. There are 10 cards in a box with
the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 on them respectively. (There
is only one number on each card!) Adam, Ben, Cecil, Danny and Emily took
and kept 2 cards each out of the box. Everybody, except for Danny, announced
the sum of his/her two numbers: Adam said 5, Ben said 12, Cecil said 10,
and Emily said 12. What were Danny's numbers?
B.229. Five people are playing chess. Everybody
plays everybody once. After every match the winner gets 1 point, the loser
gets zero point, and both players get a half a point if the match is a tie.
After the end of the last match, we managed to put the players' points in
a strictly decreasing order. It was interesting that everybody won against
the person directly ahead of him in this order. How many matches did end
in a tie?
B.230. I thought of five numbers. When
I add any two of them in every possible way, I get the following sums: 0,
2, 4, 4, 6, 8, 9, 11, 13, 15. What are the five numbers?
B.231. Divide every side of a regular triangle
into 4 equal parts, then through the dividing points draw parallel lines
to the sides of the triangle. These lines and the sides of the original
triangle have 15 intersecting points on the perimeter and the inside of
the original triangle. How many regular triangles do these 15 points create?
B.232. Out of 11 different coins I want
to give away a few. When do I have more options: if I want to give away
an odd or an even number of coins?
Please, send your solutions to: