ABACUS International Math Challenge
for
7th and 8th graders
March, 2008
C.633. Where is the ration of the numbers
containing the digit 7 to those that do not greater: among the 2-digit numbers
or among the 4-digit numbers?
C.634. A and B are positive whole numbers.
None of them is divisible by 10, and their product is 10,000. What is their
sum?
C.635. Find the last two digits of:
C.636. There are many different roads between
Town A, Town K, and Town F. We know that between any two of these towns
the number of direct roads is at least 3 but no more than 10. You can get
from Town A to Town F directly or through Town K in a total of 33 different
ways. Similarly, you can get from Town K to Town F directly or through Town
A in a total of 23 different ways. In how many different ways can you get
from Town K to Town A?
C.637. Angle A of the convex quadrilateral
ABCD is 100 degrees. We know that diagonal AC breaks up the quadrilateral
into an equilateral and an isosceles triangle. How big are the inner angles
in ABCD?
C.638. We glue together 27 regular dice
into a 3x3x3 cube. What is the least amount of dots you can see on this
cube? (On the regular dice then number of dots are 1 to 6, and the dots
on the facing sides add up to 7.)
C.639. The sides of a rectangle are 14
cm and 24 cm. We draw the diagonal from one vertex, and we draw straight
segments from this vertex to the mid-, third-, and quarter-points of the
longer side facing this vertex, a total of 6 segments including the diagonal).
What are the areas of the triangles we created?
C.640. One year a monthly calendar looked
like the diagram below. The sum of the numbers in one of the 3x3 segments
of this calendar is 162. What is the smallest number in that 3x3 section?
