ABACUS International Math Challenge
for
5th and 6th graders
October, 1998
B.73. How many isoceles triangle are there
with sides whose lengths are whole numbers, and the longest side is 1998
units long?
B.74. Find a group of numbers with 1998
as their sum, and their product, too.
B.75. Find seven consecutive positive integers
that can be divided into two groups so that the sum of the numbers in each
group is the same. Are there six such consecutive integers?
B.76. The sides of a rectangle are 8 and
18 units long. How do you have to cut it into 2 parts (not necessarily with
one straight cut) so that you could rearrange the two parts into a square?
B.77. Arrange the numbers 1, 2, 3, 4, 5,
6, 7, 8, 9 into two groups, so that non of the groups would have a number
in it that is the half of the sum of two other numbers in that group.
B.78. Andrew and Tom each thought of a
positive whole number, and they whispered it in Pete's ear. Pete told them
that the difference between the numbers is 1998. Andrew said that, based
on this, he cannot tell what Tom's number is. Then Tom said the same thing.
After all of these, Andrew said that now he can tell what Tom's number is,
but if both of them thought of a number one greater than their numbers,
he would still not be able to tell Tom's number.
What were the two numbers?
B.79. In the following multiplication,
substitute every "a" with a digit. (Different digits may be represented
by the letter "a".)
B.80. There are five people living in an
apartment: Mr. Smith, his wife, their adult son, Mr. Smith's older sister,
and his father. Everybody is working. One of them is a sales person, another
one is a lawyer, somebody is working in the post office, another one is
an engineer, and one is a teacher. The lawyer and the teacher are not blood-relatives.
The sales person is older then her sister-in-law and the teacher. The engineer
is older then the person working for the post office. Who has which profession?
Please, send your solutions to: