ABACUS International Math Challenge
for
5th and 6th graders
January, 2000
B.161. Multiply every positive whole number
that is not greater than 2000 by +1 or -1, and then add these products.
What is the smallest positive number you can get this way?
B.162. If you multiply the sum of the first
two digits and the sum of the last two digits of a 4-digit positive whole
number, you can get 187. How many such numbers are there?
B.163. Take 10 number cards with the following
numbers on them (one number on each card): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
Make a pile of them by putting one on top of another, and hold the pile
in your hands. Now put the top card on the table, put the next top card
on the bottom of the pile in your hands, put the next top card on the table,
the next top card on the bottom of the pile, and so on, until you run out
of cards. In what order do you have to stack the cards at the beginning
if you want the cards to be on the table at the end in the following order
: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10?
B.164. Two barrels contain a total of 96
liters of wine. If you pour as much wine from the first barrel into the
second one as the second barrel contained originally, and then pour back
as much as it remained in the first barrel, the two barrels will contain
the same amount of wine. How much wine was in each barrel originally?
B.165. Out of the numbers 1, 2, 3, ...,
200, pick one hundred numbers so that none of these numbers is divisible
by any other number in this new group.
B.166. Can you place 6 points on a plane
so that any three of them form an isosceles triangle?
B.167. Two kinds of people live on an island:
honest and liar. Honest people always say the truth, liars always lie. One
day we asked everybody in a group of 5 people from this island who know
each other: "How many of you are honest?" We received the following
answers: 0, 1, 2, 3, 4.
How many of them are honest?
B.168. What is the sum of all those positive
whole numbers that are smaller than 2000, and the sums of their digits are
even?
Please, send your solutions to: