ABACUS International Math Challenge
for
7th and 8th graders
February, 2006
C.529. We built a solid bigger cube gluing
together unit cubes. Then we painted a few whole sides of this big cube.
After that we cut up the big cube into the unit cubes again and saw that
45 of the unit cubes have no paint on them at all. How many unit cubes did
we use to put the big cube together, and how many sides of the big cube
did we paint?
C.530. Prove that the last digit of the
sum of 10 consecutive cube numbers is 5.
C.531. Andi and Ben are twins, they run
by the same speed and they walk by the same speed relative to each other.
One day they both participated in a race. Andi was running half the distance
and walked the other half. Ben was running half of his time of racing and
walked the other half of his time. Andy or Ben got to the finish line first?
C.532. Using 1 and 2, you can get 4 as
a sum in 5 different ways: 1+1+1+1 = 1+1+2 = 1+2+1 = 2+1+1 = 2+2. How many
different ways can you get 9 this way?
C.533. Find the smallest positive even
number that can be written as the sum of 2 prime numbers in five different
ways. (The order of the two primes is not important.)
C.534. Write parentheses in the following
line so that the result of the operations is as great as possible:
3+3x3+3x3+3x3+3
C.535. Write parentheses in the following
line so that the result of the operations is either 6, 7, 8 or 9.
1 : 2 : 3 : 4 : 5 : 6 : 7 : 8 : 9 : 10
C.536. Write 1 as the sum of the reciprocals
of 4 different positive whole numbers.