ABACUS International Math Challenge
for
3rd and 4th graders
October, 1998
A.73. You write the following four numbers
on the blackboard: 1, 9, 9, 8. In every step you pick three of these numbers,
erase them and replace them with numbers one greater then the ones you erased.
With such steps, can you manage to have four identical numbers on the blackboard?
A.74. Three brothers distribute cats and
kittens evenly among themselves. 10 cats have 1 kitten each, 10 cats have
2 kittens each, and 10 cats have 3 kittens each. How should they distribute
the animals so that everybody should get the same number of cats and kittens,
and none of the kittens should be separated from its mother?
A.75. There are socks of the same size
in a box: enough white ones for exactly 5 pairs, enough black ones for exactly
10 pairs, and enough brown ones for exactly 15 pairs. At least how many
of them do you have to take out of the box (without looking), so that you
would have a pair for sure? (There is no difference between right and left
sock.)
A.76. Sneeze, Goofy, Sleepy, and Happy
said the following things to their father after a competition:
Sneeze: "I did not win the competition."
Goofy: "Happy won."
Sleepy: "Sneeze won."
Happy: "Sneeze did not win."
Who won the competition if only one of the four kids said the truth?
A.77. After computing a series of operations
Pete got 250 as the result. Later he realized that in the last operation
he multiplied by 5 instead of dividing by 5. What would his result be if
he did the last operation correctly?
A.78. There are two cups on the table,
A and B. One of them has 19 coins in it, and the other one has 32. We do
not know which cup has how many coins in it, but Charlotte, who know it,
multiplies the number of coins in cup A by 18, she multiplies the number
of coins in cup B by 13, and adds the products. The last digit of the result
is an even number. How many coins are in cup A and how many in cup B?
A.79. 3, 6, 12, 5, 10, 1, ... You can construct
the next element of this sequence from the previous element by doubling
the last digit of that element and to this number add the number you get
from the previous number by deleting its last digit. (For example, the next
element after 134 would be 2x4+13=21.) What is the 1998th element of the
above sequence?
A.80. During a dance the children create
a circle. Everybody in the circle gets a number in an order: 1, 2, 3, 4,
... The child with number 10 faces the child with number 43. How many children
are there in the circle?
Please, send your solutions to: