ABACUS International Math Competition
for
7th and 8th graders
November, 1997
C.17. One day Felix went up to the backboard
and wrote the following statement on it: 1995+146=210+1117. It could be
made a true statement by moving two digits in this line from their places
to two new locations. How? (It is allowed to squeeze a digit in between
two other digits.)
C.18. Write the numbers 1, 2, 3, ..., 9
on the circumference of a circle in such a way that no two neighboring numbers
would give a sum that is divisible by either 3 or 5 or 7.
C.19. Find four different whole numbers
so that the sum of any three of them would be divisible by the fourth number.
C.20. A rectangle is twice as long as wide.
Cut this rectangle into pieces in such a way that you could make a square
out of the pieces.
C.21. John, Pete, Tom, George and Steve
are brothers. One day one of them broke a window. When their father asked
them who did it they gave the following answers:
John: "It was Pete or Tom."
Pete: "It was not George nor me."
Tom: "Both of you are lying."
Steve: "No, only one of them is lying."
George: "No, Steve, you are wrong."
Then their mother added: "Three of my sons are telling the truth,
but I do not believe what the two others said."
Who broke the window?
C.22. The product of my children's age
is 1664. The youngest one is at least half as many years old as the oldest
one. I am 50 years old. How many children do I have?
C.23. What is the last digit of the following
number?
C.24. Write the letters A, B, C, D, E in
the empty squares so that every row and every column would have only one
of each of these letters.
Please, send your solutions to:
Solutions of previous problems