ABACUS International Math Challenge
for
5th and 6th graders
February, 1999
B.105. Write six numbers on the circumference
of a circle so that every number is the sum of its neighbors.
B.106. Write the numbers 1, 2, 3, ...,
9 into the circles so that the sum of the numbers at the ends of each segment
is equal to the number written on that segment.
B.107. What is the smallest positive whole
number that ends with 1997 and divisible by 1999?
B.108. Can you get 16 as the sum of a few
consecutive integers?
B.109. Two barrels contain a certain amount
of water.If you poor from the first barrel as much water into the second
barrel as the second barrel had, then you poor from the second barrel as
much water into the first barrel as the first barrel just had, and finally
you poor from the first barrel as much water into the second barrel as the
second barrel just had, then both barrels will contain 160 liters of water.
How much water did the barrels contain originally?
B.110. A merchant used a 40 pound rock
to measure weights. Once he dropped the rock and it broke into 4 pieces.
The merchant was surprised when he realized that all four pieces have a
weight of whole number of pounds, and that with the help of these four pieces
he can measure any whole number of pounds weight up to 40 pounds. What are
the weights of the four pieces?
by Bachet de Meziriac (1581-1638)
B.111. How many 4-digit whole numbers with
at least one zero digit in them are there?
B.112. Andrew, Burt, Charlie, Danny, and
Eli are playing a game in which everybody is either a frog or a kangaroo.
The frogs' statements are always false, the kangaroos always tell the truth.
(1) Andrew says that Burt is a kangaroo.
(2) Charlie says that Danny is a frog.
(3) Eli says that Andrew is not a frog.
(4) Burt says that Charlie is not a kangaroo.
(5) Danny says that Eli and Andrew are different animals.
How many frogs are there among the 5 boys?
Please, send your solutions to: