ABACUS International Math Challenge
for
5th and 6th graders
March, 2002
B.305. Seven cups are standing in a row.
One of them is made of Gold, the others are made of a mixture of Gold and
Copper, but they all look alike. Mark the cups in order by A, B, C, D, E,
F, and G. If you count them back and forth in the order of ABCDEFGFEDCBABCD...
starting with 1, then at 1000 you will be at the golden cup. What letter
marks the golden cup?
B.306. A small gear rolls around a big
gear. While it goes around once, it makes 4 complete rotations around its
own axis. What is the ration of the circumferences of the two gears?
B.307. Two people are playing cards with
a deck containing 16 pairs, a total of 32 cards. Each player gets 5 cards
at the beginning, and then they draw 1 card from the pack one after another.
If the deck runs out, they draw cards (one at a time) from each other. If
you have a pair of cards, you have to put it down on the table. (You can
put down more than 1 pair at a time.) Prove that at the end of the game
they will have the same number of pairs.
by Balázs Szalkai, Hungary
B.308. Start with the following series
of letters: AAAABBBB. In every step change the order of two neighboring
letters. At least how many steps do you need to create the ABABABAB series?
B.309. A number on the number line is 24
units away from another number, which is twice its opposite. Which number
is this?
B.310. On a 3 meter x 4 meter flag there
is a symmetrically placed cross with equally wide legs, as shown on the
diagram. How wide are the legs if the area of the cross is half of the area
of the whole flag?
B.311. There are 28 students in a class.
They want to divide 11 identical cakes among themselves equally without
cutting any of the cakes into 28 or more pieces. How can they do this?
B.312. What fraction of the area of the
rectangle is the area shaded?
Please, send your solutions to: