ABACUS International Math Challenge
for
7th and 8th graders
February, 2003
C.361. Emma and Ben are playing cards.
The loser paid $1 to the winner in the first game, $2 in the second game,
$4 in the third game, and so on. (The prize was always doubled.) Emma started
with $601, but lost all her money in 10 games. Which game did she win?
C.362. In a holiday resort at least 2 people
out of any three do not know each other, but there are always at least two
people who know each other out of any 7 people. Paul is staying in the resort.
Prove that he knows 6 people the most.
C.363. There are 10 numbered balls on a
sorter machine inclined downwards to the left. Their order from left to
right is: 8, 6, 2, 5, 10, 1, 3, 7, 9, 4. On your command, the machine can
lift up one, two or three consecutive balls, and put them on the right end
of the balls without changing their order, while all the other balls rolled
down to the left. Is it possible to put the balls in an increasing order
from left to right with 5 commands?
C.364. The teacher wrote the whole numbers
from 1 to something, but somebody erased one number so that the average
of the numbers on the board is 13 and 9/13. Which number was erased?
C.365. People with green eyes tell the
truth, people with blue eyes lie. There are 5 people in a group so that
everybody can see everybody. They say the following:
A: I see three people with green and one person with blue eyes.
B: I see 4 people with blue eyes.
C: I see three people with blue and one person with green eyes.
D: (He says nothing.)
E: I see 4 people with green eyes.
What are the colors of the eyes of these 5 people?
C.366. One clock maker designed an analog
clock on which one day has 10 hours, and one hour has 100 minutes. On this
clock the short hand goes around once a day, and the long hand goes around
in an "hour", which means that it goes around 10 times a day.
What is the angle between the two hands of this clock when it shows 6 hours
75 minutes counting from midnight? What time is it really then?
C.367. Bea usually takes the 5:00 pm train
from the city and gets to her village at 6:00 pm. Her father drives to the
train station from their home with a uniform speed to pick her up, and then
they drive back home with the same speed. One day Bea left the city one
hour early, so she got to her village at 5:00 pm. With a constant speed
she started walking home on the same road her father comes to pick her up.
Her father left their house to pick her up at the usual time, on his way
to the station they met, he picked her up, drove back home, so they got
home 20 minutes earlier than usual. (The time it took the car to turn around
is too small to count.) For how many minutes did Bea walk?
C.368. During practice every player takes
two shots in a row on the basket. This is called a double. Alessandra misses
2/3 of her first shots in the doubles, but makes 2/3 of her second shots.
If she makes the first shot in a double, then she makes the second shot,
also. In the next double which outcome has the greatest probability:
a) she misses both shots;
b) she makes only one of the shots
c) she makes both of the shots?