ABACUS International Math Challenge
for
7th and 8th graders
September, 2006
C.537. Out of 6 different whole numbers
we picked two every which way we could and added them together. We got the
following sums: 8; 21; 22; 23; 24; 37; 53; 55; 59; 61; 68; 69; 74; 75; 106.
Which of these sums is the one that we got when we added the middle two
numbers out of the six when they are put in an increasing order?
C.538. We multiply the sum of 5 consecutive
whole numbers by the sum of the next 5 consecutive numbers. Could this product
be 120635?
C.539. Two people are guessing the number
of marbles in a container. One person guesses 1200, the other person guesses
3600 marbles. Someone, who knows how many marbles there are in the box,
says: "One of the guesses is 40% off, the other guess is 80% off. How
many marbles could there be in the box?
C.540. You have a 27-digit number with
every digit being 1. Divide it by 27. What is the remainder?
C.541. We made a computer to draw and print
out all those rectangles whose sides measured in centimeters are whole numbers,
and whose areas are not more than 100 square centimeters. Inside the rectangles
we wrote their areas. How many such positive whole numbers are there between
1 and 100 that appear exactly twice on the rectangles?
C.542. We cut an isosceles triangle into
two isosceles triangles by one of its angle bisectors. How big are the angles
of the original triangle?
C.543. The coordinates of the vertices
of the quadrilateral ABCD are A(0;0), B(7;3), C(3;4), and D(0;8). What is
the area of quadrilateral ABCD?
C.544. The shape of Pete's room is a rectangle.
Its floor is covered by 93x231=21483 square tiles. Pete drew a straight
diagonal in his room. How many tiles does this line go through? (If the
line has only one mutual point with a tile then we do not consider the line
to be going through that tile.)