ABACUS International Math Challenge
for
7th and 8th graders
January, 2007
C.569. The sum of the three different edges
of a rectangular column is 35 cm. If we reduce the height of the column
by 3 cm, increase its width by 3 cm, and take only a third of its length,
we get a cube. How did the volume of this column change?
C.570. In a horse race at 3/4 of the distance
one of the horses lost its composure, threw his rider off, turned around
and started running back towards the start line. After running 2/3 of his
way back, the horse turned around yet again and started running towards
the finish line. The rider got up, dusted himself off, and when he saw his
horse coming towards him again, he started running towards his horse. When
they met he got back on his horse and rode his horse to the finish line.
This way the horse ran a total of 2400 meters. We know that the horse runs
half as fast without his rider than with him, and that the rider runs half
as fast as his horse without him.
a) How many meters did the rider run towards his horse?
b) How long did it take the rider to get from the start to the finish
line this way, if we know that normally it would take him 60 seconds to
ride his horse from start to finish?
C.571. There are two kinds of people live
in the country of King Arthur: those who always say the truth and those
who always lie. One day the king ordered that all the liars must be put
in jail. A woman, whose husband was put in jail, went to see the king and
said to him: "My husband is a truthful man, and he always says that
about himself, too. So, please, let him be free." What should the king
decide based on these?
C.572. I am twice as old today as you were
when I was as old as you are today. When you will be as old as I am today,
we will be 126 years old together. How old are we now?
C.573. The sum of the squares of three
consecutive odd numbers is a 4-digit number with identical digit. Find these
three odd numbers.
C.574. There are 100 black and 100 white
balls in a box. We take out three balls randomly, and depending on the colors
of those balls we put back some balls following the instructions on the
chart below. Somebody keeps on doing this until only two balls remain in
the box. The person does not tell us anything about how the process went.
Now we have to take one ball out of the box, but before that we have to
predict the color of this ball. What color should we predict?
C.575. We pick six consecutive one-digit
numbers and put them on the sides of 3 coins. We toss all three coins on
a table at once and we can see the numbers 6, 7, and 8. Then we toss the
3 coins four more times, and the sums of the numbers on top are: 16, 17,
20, and 23. Which numbers and how are they placed on the coins?
C.576. We have a copy machine, which is
able to enlarge an image up to 155% in one step. (It can use only whole
number percent steps.) How many different ways can we double the size of
an image with this machine in no more than 3 steps?