ABACUS International Math Challenge
for
5th and 6th graders
November, 2005
B.505. Let's call a year perfect
if the sum of the digits of the number representing that year is a square
number. For example 2002 was a perfect year because 2+0+0+2=4 is
2x2, a square number. How many perfect years will there be in the 21st century?
B.506. The pages of a book are numbered,
starting on the first page with number 1.
a) How many pages in a row can you number the most if you may use the
digit 2 only 22 times? (You may use any number of the other digits.)
b) What is the last page you can number this way using the digit 2 exactly
25 times? (You may use any number of the other digits.)
B.507. Ali Baba's camel eats 1 kg of dates
while he walks 1 km, and he can carry 100 kg dates on his back. Ali Baba
grows dates at the edge of the desert, and 100 km away, across the desert,
is the market. There are oases in the desert every 25 km where you can store
dates, nowhere else. Ali Baba has 300 kg of dates and he wants to sell 50
kg on the market. Can he do it? (If yes, how; if not, why not?)
B.508. An apple, an orange, a pear, and
a banana are put an the table in a row. The orange is neither at the beginning
nor at the end of the row, and facing the row the orange is to the right
of the banana (not necessarily next to it). How many different orders can
these fruit be in?
B.509. Three friends are living in houses
next to one another: under numbers 34, 36 and 38. Every one of them has
a different hair color and a different hobby. The one with brown hair loves
diving. The blond man lives in the house which has a number divisible by
4. The one who loves to play soccer is happy because the sum of the digits
of his house number is exactly 11, which is the number of players in a team
of his favorite sport. In which house does the man whose hobby is music
live?
B.510. I wrote the number 1 on one piece
of paper, I wrote the number 2 on two pieces of paper, the number 3 on three
pieces of paper, ..., the number 50 on fifty pieces of paper. I put all
of them in a hat. How many of them do I have to pull out so that there are
10 pieces of paper among them with the same number written on them?
B.511. Every digit of a number in base
10 is 1. You subtract 10 and then divide it by 9? Which 9-digit number can
you get?
B.512. There is a modern picture on the
wall in Steve's room. (The dotted lines indicate a grid that you can place
over the picture.) What part of the area of the picture is white?
