ABACUS International Math Challenge
for
7th and 8th graders
January, 2006
C.521. A mysterious number has 246912 digits.
Each of the first 123455 digits is 3. The 123456th digit is unknown. Each
of the last 123456 digits is 6. The number is divisible by 7. What is the
mysterious number?
by Michael Do, Australia
C.522. You wrote the numbers from 1 to
90 on 90 different sheets of paper (one number on each paper), and put them
in a hat. You take papers out of the hat one-by-one, but you do not look
at them. After how many pieces of paper taken out can you be sure that there
are two among the pieces taken out where either the sum or the difference
of those two numbers is divisible by 11?
C.523. The sides of a rectangle in centimeters
are whole numbers, and one side is 1 cm longer than the other. The last
digit of the area of the rectangle in sq.centimeters is 6. Can the length
of any of the sides of the rectangle in centimeters be a square number?
C.524. The sum of the squares of the digits
of a positive whole number is 50. The digits of the number are increasing
from left to right. Find all of these numbers.
C.525. How many different ways can you
write 8 as the sum of 1s and/or 2s if the order of the addends is important?
(For example: 3 =1+1+1=1+2=2+1)
C.526. Two regular polygons have a total
of 17 sides and 53 diagonals. How many sides do they each have?
C.527. Two different size square areas
are covered with 1 dm x 1 dm square tiles. None of the tiles had to be cut!
The lengths of the sides of both squares are odd numbers in decimeters.
Using all these tiles, can you cover completely one square shaped area?
(None of the tiles should be cut!)
C.528. What is the sum of the digits of
all the numbers from 1 to 2006?