ABACUS International Math Challenge
for
7th and 8th graders
November, 2006
C.553. Find all those ababab kind
of 6-digit numbers that are divisible by 34.
C.554. The base edge of a 4-cm tall square
based column is a whole number of centimeters long. We paint the whole column
green, and then we cut it into unit cubes with plains parallel to the sides
of the column. How many unit cubes have no green sides if the number of
unit cubes with one green side and the number of unit cubes with 2 green
sides are the same?
C.555. We make a series of numbers by writing
one more digit in the decimal form of 18/55 each time:
0.3; 0.32; 0.327; 0.3272; ...
a) What is the sum of the digits of the 123rd element of this series?
b) Is there an element in this series in which the sum of the digits
is 2006?
C.556. You have a 7x7x7 solid cube of cheese.
You cut 27 square based column-shaped tunnels into it , as shown on the
diagram. Each two intercepting tunnels have a unit cube intersection. What
is the mass of this cheese full of holes if the solid cheese had a mass
of 343 grams?
C.557. ABC is a regular triangle. Divide
each side of it into 3 equal parts. Connect the divider points on sides
BC, CA, and AB that are closer to vertex B, C, and A (in this order) to
each other. This way you get a smaller regular triangle. What is the area
of the smaller triangle if the area of the original triangle is 123 sq.cm?
C.558. Take the first 41 positive whole
powers of 2. Without calculating each of them, prove that there are two
among them whose difference is divisible by 100.
C.559. Which number is greater: 987 to
the 60th power or 987987 to the 30th power?
C.560. Find a series of 5 prime numbers
so that the difference between any two neighboring numbers is 6.