ABACUS International Math Competition
for
7th and 8th graders
October, 1997
C.9. One day Felix wrote the following statement
on the board:
101-102=1
Find different ways of making this statement true by changing the location
of one of the characters in it. (For example, 1001-12=1, however, this move
did not make the statement true.) The lines in the "=" sign are
considered to be two separate characters.
C.10. Between what limits does the value
of the following expression change if you use parentheses in every possible
way?
1:2:3:4:5:6:7:8:9
C.11. Give four different whole numbers
so that any one of them would be a divisor of the higher neighbor of the
product of the three others.
C.12. How many such 4-digit numbers are
there inwhich there are at least two identical digits?
C.13. Three best friends were fishing together.
There were no two who caught the same number of fish. Next day they were
saying:
A: I caught the most fish and C caught the least.
B: I caught he most fish, more than A and C together.
C: I caught he most fish, B caught only half of what I did.
Who caught the most fish if exactly 3 out of the 6 statements made by
the children is true? Can you decide who caught the least fish?
C.14. Find such four consecutive odd numbers
that their product would be a square number.
C.15. Prove that 
is divisible by 10.
C.16. In the empty squares write one of
the letters A, B, C, D, or E, so that in every raw, every column and in
both of the diagonals there would be one of each letter. What letter goes
where the * is?
Please, send your solutions by the end of October to:
Solutions of previous problems