ABACUS International Math Competition
for
3rd and 4th graders
November, 1997
A.17. You want to paint the vertices of
a cube to red or blue so that there would be at least one red vertice on
every side. At least how many vertices do you have to paint red?
A.18. While making a number sequence starting
with 458, when creating the next number you can either double the previous
number or delete the last digit of the previous number. For example: 458,
45, 4, 8, 16, 1, 2, ...
With these rules, make a sequence which starts with 458, and 14 is one
of the numbers in the sequence.
A.19. Place 10 balls into 3 mugs so that
every mug would have an odd number of balls in it.
A.20. Are there seven such whole numbers
that have a sum of 100 and their product is an odd number?
A.21. What is the first digit of the smallest
positive whole number in which the sum of the digits is 1997?
A.22. What is the last digit of the following
number:
1992·1994·1996·1998+1991·1993·1995·1997
A.23. Start writing the positive whole
numbers one after another:
1234567891011121314...
What is the 1997th digit?
A.24. Three people, A, B, and C, are talking.
A says: "B is lying."
B says: "C is lying."
C says: "A and B are lying."
Who says the truth and who is lying?
Please, send your solutions to:
Solutions of previous problems