ABACUS International Math Competition
for
7th and 8th graders
January, 1998
C.33. In the following number behind the decimal
point we write the square numbers one after the other:
0.14916253649...
What is the 100th digit behind the decimal point?
C.34. Multiply 7 consecutive whole numbers
between 1 and 50, so that you would get two zeros at the end of the product.
How many such products are there?
C.35. From the list of the first 999 positive
whole numbers cross out those that have at least one prime digit. How many
numbers do you have left on the list?
C.36. In an isoceles triangle the line
bisecting one of the angles divides the triangle into two isoceles triangles.
How big are the angles of the original triangle?
C.37. In Math Country on the state lottery
you have to pick 5 out of the whole numbers between 1 and 90 . You have
to buy a separate ticket for every set of 5 numbers you want to play. On
the day of a drawing, a mathematician asked his colleague:
- Do you know what numbers they drew today?
- Can you imagine -answered the colleague-, one of the numbers is the
divisor of the sum of any two of the other numbers they drew!
- What is that number? -asked the more and more anxious mathematician.
- If I told you, you would know the winning numbers.
- At least tell me if that number is even or odd? - begged the mathematician,
and after the answer was given to him, he started jumping and screaming:
- I won! I won! I won!
What were the winning numbers if the mathematician, who had picked all
5 of the winning numbers, played with one lottery ticket only.
C.38. Show that the following equation
does not have a solution if a and b are positive whole numbers:
C.39. The average age of a family (mother,
father and children) is 18. Without the 38-year-old father the average age
is 14. How many children are there in the family?
C.40. Mrs. Ex proudly tells her colleagues:
- Both of my sons have their birthdays today. None of them are 10 years
old yet. Find out how old they are! Anna, I would tell you the product of
their ages. (Mrs. Ex whispers it in Anna's ear.)
But Anna says:
- From this you cannot determine their ages yet.
- Then I tell Lori the quotient of their ages. (She whispers it in Lori's
ear.)
- Even now it is not obvious. -Lori says.
- You are right, I should have told Lori their age-difference. Then
she would have been able to figure it out.
So, how old are Mrs. Ex's sons?
Please, send your solutions to:
Solutions of previous
problems