ABACUS International Math Challenge
for
7th and 8th graders
March, 2005
C.481. Fill in the empty fields of the
following magic square:
(In a magic square the sum of the numbers in each column, row and diagonal
is the same.)
C.482. There are red and blue balls in
a box. At least 90% of the balls is red. Kate is taking the balls out of
the box one by one. Only one of the first 50 balls was blue. While she was
taking out the rest of them, every 8th ball was blue. How many balls could
there be in the box the most?
C.483. At a corporation the members of
the Board of Directors and the workers had to vote on an important issue.
19% of the members of the Board of Directors and 91% of the workers voted
yes. We know that every member of the Board of Directors and every worker
voted, and that 90% of all the votes were yes. What is the ration of the
number of members of the Board of Directors and the number of workers?
C.484. Pick any point P inside of rectangle
ABCD. The bisecting perpendiculars of segments PA, PB, PC, and PD create
a quadrilateral. Prove that the diagonals of this quadrilateral intercept
each other in the midpoint of rectangle ABCD.
C.485. Tie up a 6 cm diameter and an 18
cm diameter cylinder with a rope as shown on the diagram. How long is the
rope if we used 15 cm rope for the nut to tie the two ends of the rope together?
C.486. Two cities (A and B) are connected
by a straight road. There are 3 other cities on this road between cities
A and B. The distance between any two out of these 5 cities is a whole number
of kilometers. If somebody walks from one city to any other city, from the
number of kilometers walked you can tell exactly between which two cities
the person walked. What is the shortest possible distance between A and
B?
C.487. A red line is going upwards in a
spiral from the bottom of a cylinder to the top, rising by the same rate
all along, making 6 complete rotations around the cylinder. The radius of
the cylinder is 12 centimeter, and its height is 1 meter. How long is the
line. (Ignore the width of the line.)
C.488. We wrote down the 3-digit numbers
one after another in a row, so that the digits of the even numbers are written
in red, and the digits of the odd numbers are written in blue. What is the
2005th digit, and what is its color?