ABACUS International Math Challenge
for
5th and 6th graders
January, 2007
B.569. In Neverland's capital city the
phone numbers are 8-digit numbers. How many of them have their digits in
an increasing order from left to right? (None of the phone numbers starts
with a zero.)
B.570. How many 3-digit positive whole
numbers are greater when rounded to the nearest 10 than when rounded to
the nearest 100?
B.571. There are 12 locks on Santa's box.
Every lock can be opened by only one key, and every key opens only one lock.
Every key has a label on it indicating which lock it opens. On the 1st of
April the elves wanted to trick Santa, so they took all the labels off and
put them back on the keys randomly.
a) At least how many times does Santa have to try to open a lock with
a key until he opens all the locks?
b) At least how many times does Santa have to try to open a lock with
a key until he can put the labels back on all the keys correctly?
B.572. I have the following 10x23 sign
in my room. On the unit square grid I drew straight segments and parts of
unit-radius circles. What fraction of the area of the sign is white?
B.573. In a Math Competition 8 participants
received prizes. The organizers gave out a total of 50 sheets of chocolate.
Every winner received a whole number of sheets of chocolate. One of the
winners noticed that nobody received a square number of sheets of chocolate.
Prove that there were at least two winners who received the same number
of sheets of chocolate.
B.574. In a restaurant a costumer may eat
a certain number of slices of bread for free, but after that he/she has
to pay for each slice. (Each slice of bread costs the same.) Two people
together ate 13 slices and paid $3 for them. If a costumer ate this much
bread alone, he would have to pay $8 for it. How many slices of bread may
one costumer eat for free?
B.575. A positive whole number is 20 more
when rounded to the nearest ten than when rounded to the nearest 100. Add
all of these numbers below 1000. What is the sum?
B.576. When he died, a man left half of
his wealth to his wife, he left $400,000 to his daughter, half of the rest
of his wealth to his son, half of the rest to his grandson, and the remaining
$60,000 to charities. How much was his wealth?