ABACUS International Math Competition
for
7th and 8th graders
February, 1998
C.41. What is the greatest area a triangle
with sides a, b, and c could have if:
C.42. The sides of a right triangle are
3, 4, and 5 units long. How long is its shortest height? What is the radius
of its inscribed circle?
C.43. I thought of a number. If you divide
it by 8, the remainder is 3. If you divide it by 5, the quotient is 8 greater
and the remainder is 2. What number did I think of?
C.44. There are a few paper slips in a
box. There is a natural number written on each slip. We know that no matter
which 3 slips you take out of the box, the sum of the numbers on 2 of them
is always divisible by 5. What could the maximum number of slips be with
numbers on them that are not divisible by 5?
C.45. Three fortune-tellers, who look identical,
sit in a raw answering questions: Truth (who always says the truth), Lie
(who never says the truth), and Wise (who sometimes tells the truth, but
other times he lies). A philosopher arrived to decide who is who in the
group. He asked the person sitting on the left hand side: "Who is sitting
next to you?" The answer was: "Truth." Then he asked the
person sitting in the middle: "Who are you?" The answer was: "Wise."
Finally, he asked the person sitting on the right: "Who is sitting
next to you?" The answer was: "Lie."
So, who is sitting where?
C.46. "How many people live in this
house?" -asked a police man. "Three." - was the answer. "How
old are they?" "The product of their ages is 225, the sum of their
ages is the same as the number of the house in its address." The police
man looked at the address and said: "Are you the oldest in the house?"
"Yes." - was the answer. "This is enough!" - said the
police man.
Now, you have to determine the ages of the people living in this house.
C.47. A number sequence has 80 elements.
Any of its elements (except for the first and last element) is equal to
the product of its neighbors. The product of the first 40 elements, just
as the product of all the elements is 8. What are the first and the second
elements of this sequence?
C.48. Find the smallest positive whole
number that is equal to the product of the sum of its digits and 1998.
Please, send your solutions to:
Solutions of last year's problems