ABACUS International Math Challenge
for
7th and 8th graders
February, 2002
C.297. A farmer has a few columns, which
he wants to use to build a wire fence. He calculated that with them he could
fence in a big square-shaped garden, or two smaller adjacent square-shaped
gardens with equal areas, or three adjacent square-shaped gardens with equal
areas as shown on the diagram. (He would put the columns equally spaced
on the lines drawn on the diagram.) How many columns does he have at least?
C.298. Anne and Bea are playing a game
with a 10x5 square-sheet of chocolate. They are alternating in breaking
one piece off of the sheet of chocolate along the grid-lines. They agreed
that whoever breaks off a 1x1 piece first wins the whole chocolate. Anne
starts and they both play their best. Does Anne have a chance to win the
chocolate? Would Anne have a chance to win if the rule was that the first
person breaking off a 1x1 piece loses the whole chocolate?
C.299. There is a square-shaped land that
is divided equally into 100 smaller square-shaped gardens. Originally 9
of them are covered by weeds. The weeds spread into a garden if that garden
is adjacent to (shares a side with) two other gardens, which are covered
by weeds. Is there such an arrangement of the 9 initial weedy gardens that
would allow the weeds to cover the whole land after awhile?
C.300. Four people, A, B, C and D, stand
at the end of a bridge. It is very dark and they have only one torch. The
bridge can support the weight of no more than two people at a time, so when
a group of two people would reach the other end of the bridge one person
would have to carry the torch back. A, B, C and D take 5, 10, 20 and 25
minutes respectively to cross the bridge. Assuming that when two people
cross the bridge together they take the time of the slower person, find
the minimum time the four of them can cross the bridge.
by Anirban Bhattacharyya, Calcutta, India
C.301. The perimeter of a rectangular garden
is 100 meters. Its every side is a whole number of meters long. There is
a square-shaped garden attached to each of two neighboring sides of the
rectangle. The sides of these squares are equal to the sides of the rectangle
they are attached to.) Prove that the difference of the areas of these squares
is a multiple of 100.
C.302. John had less than 100 square-shaped
stamps. If he wanted to arrange all of them in a rectangle so that there
were at least two rows of stamps, he could make 4 different rectangles with
different dimensions. However, one stamp got blown out the window by the
wind, and now he can make only one rectangle with at least 2 rows. How many
stamps did he have originally?
C.303. Tim had 40 points on his last math
quiz, and it increased his average from 27 to 28 points. How many points
on the last quiz would have increased his average to 30 points per quiz?
C.304. In your roulette game you have only
$5 and $8 chips. What is the largest wager that cannot be placed?