ABACUS International Math Challenge
for
7th and 8th graders
January, 1999
C.97. There are 67 (white and red) balls in
a bowl. There are small ones and big ones among them. We know that:
a) the number of red balls is divisible by 5;
b) the number of big red balls is the same as the number of white balls;
c) the number of small white balls is the smallest;
d) the number of each kind of balls is a prime.
How many of each of the balls are there in the bowl?
C.98. Pick a 3-digit number, and divide
it by the sum of its digits. In case of which number will this quotient
be the greatest, and the smallest?
C.99. I picked four integers. Now, in all
possible ways, I pick 3 of them and every time I add those 3 numbers. Finally,
I add all these sums. The result is 51. The product of my 4 numbers is 216.
What are my numbers?
C.100. Out of the numbers 1, 2, 3, ...,
29, 30, pick 16 of them and place them in a 4x4 grid, so that the product
of the numbers in each row, column and diagonal is the same.
4x4 square
C.101. Let S(0) be a finite number sequence.
You can get the number sequence S(1) by replacing the elements of S(0) with
the number which indicates how many times that element appeared in S(0).
For example, if S(0)=(1,2,3,2,1), then S(1)=(2,2,1,2,2).
S(0) can be any finite number sequence. Out of the following number
sequences, which ones can be created as S(1)?
a) (1,1,2,2,2)
b) (1,1,1,2,2)
c) (1,1,2,2,3)
d) (1,3,3,3,3)
e) (2,2,2,3,3)
C.102. The coordinates of a triangle are
A(2;3), B(10;5), C(4;9). What is the area of the triangle?
C.103. Cut up a 13x13 square into less
than 13, whole-number-unit size squares.
C.104. The sum of 1999 different positive
whole numbers is 3 995 999. Prove that there are at least two even numbers
among them.
Please, send your solutions to: