ABACUS International Math Competition
for
7th and 8th graders
April, 1998
C.57. Find three positive whole numbers such
that for any two of them, the number one less than their product is divisible
by the third number.
C.58. With a 1998 meter long fence, I want
to make the biggest rectangular shape garden with sides that are measured
in whole number of meters. What is the area of this garden?
by Bognár Ferencné, Hungary
C.59. Find the smallest positive whole
number that does not contain the digit 9, but it is divisible by 999.
C.60. What is the sum of all those 3-digit
numbers that are the products of four different prime numbers?
C.61. Find such a 5-digit number that is
equal to 45 times the product of its digits.
C.62. Once a mathematician was asked by
his colleagues about his family. He said: "I have three children, by
the way, they all have their birthdays today. The product of their ages
expressed in years is 36, but if I added these numbers I would get the number
of days in today's date." Soon the answer was: "This is not enough
to tell the ages of your children." Then the mathematician said: "That
is right, I forgot to tell you that when we were expecting our youngest
child we sent the two older children to their grandparents who live in the
country."
Find out how old the children are and what day of the month this conversation
took place?
C.63. E is the midpoint of side BC in rectangle
ABCD, and F is the midpoint of side CD. The area of triangle AEF is 3 area
units.What is the area of the rectangle?
C.64. The perpendicular bisector of a diagonal
of a rectangle intercepts the longer side of the rectangle in a ratio of
1:2. What is the angle between the two diagonals?
Please, send your solutions to:
Solutions of last year's
problems