ABACUS International Math Challenge
for
5th and 6th graders
November, 1998
B.81. How many square numbers are there up
to 1998?
B.82. How many such rectangles are there
for which the lengths of the sides are whole numbers and the area is 1998
units?
B.83. I moved all of an encyclopedia's
many (more than 2) volumes from one shelf to another. Later I realized that
the volumes are in a totally wrong order:
1) none of the volumes are in the place where they should be,
2) none of them are followed by the right volume.
How many volumes does this encyclopedia have?
B.84. Find the missing digits of the 523abc
six-digit number, so that it would be divisible by 7, 8, and 9, also.
B.85. Is it true that if I pick 30 numbers
out of the first 50 positive whole numbers, then there is going to be two
such numbers among them that one of these two numbers is twice the other
one?
B.86. Pete chose two consecutive positive
integers, and wrote them down on two separate sheets of paper. He gave one
of the numbers to Andrew and gave the other number to Tom, who both knew
that the numbers they were given are consecutive positive integers. Then
the following highly intellectual conversation took place between Andrew
and Tom:
Andrew: " I do not know what number you got."
Tom: "I don't know what number you got either."
Andrew: " I do not know what number you got."
Tom: "I don't know what number you got either." ....
Andrew and Tom both said the same sentence 10 times. Then Andrew's 11th
sentence sounded like:
"Now I know what number you got."
What number did Tom get?
B.87. Find the smallest square number starting
with the digit-series 2468.
B.88. What is the sum of all the digits
of the following numbers: 1, 2, 3, ..., 1000?
Please, send your solutions to: