ABACUS International Math Challenge
for
5th and 6th graders
January, 1999
B.97. Two men and a woman met. They are members
of the Black, the White and the Brown families.
-"Interesting, our hair colors are the same as our family names."
-said the woman.
-"That is correct, but nobody has the same hair color as the meaning
of his or her family name."- said the person with black hair.
-"How could that be?"- asked the member of the White family.
The hair color of the woman in this group is not brown. What is the
color of her hair?
B.98. Pick four digits so that, with their
different orders, you could make three such 4-digit numbers, that the sum
of two of them is equal to the third one.
B.99. Write digits in a 3x4 grid, so that
the 4-digit numbers in the rows and the 3-digit numbers in the columns would
all be divisible by 92.
B.100. Out of the numbers 1, 2, 3, ...,
29, 30, pick nine different ones so that you could place them in a 3x3 grid
in such a way that the product of the numbers in every row and every column
is 270.
B.101. Within the first 1000 non-negative
integers, how many are divisible by
a) at least one
b) exactly one
c) no more than two
d) exactly two
of the following numbers: 2, 3, and 5?
B.102. Out of the digits 1, 2, 3, 4, 5,
and 6, (any of them may be used more then once) I wrote down a 4-digit number.
A few people tried to find out my number. The first guess was 4215. He hit
two digits, but only one of them is in the right place. The second guess
was 2365. Again, he hit two digits, but only one of them is in the right
place. The third guess was 5525. This time even the digits are incorrect.
What could my number be?
B.103. We have the following information
about a 4-digit number:
it is odd; every one of its digits is smaller than 7; its digits are
in a decreasing order from left to right; every sum of its two consecutive
digits is odd; it is divisible by 3 and 7.
Find this number.
B.104. We wrote -1 on two neighboring vertices
of a cube, on the rest of them we wrote 1. Then, on every edge we wrote
the sum of the numbers at the ends of this edge. Finally, on every side
we wrote the sum of the numbers on the bordering edges. What is the sum
of the numbers written on the sides of this cube?
Please, send your solutions to: