ABACUS International Math Challenge
for
3rd and 4th graders
September, 2006
A.537. On planet Kyra one-eyed, two-eyed
and three-eyed creatures are holding a meeting about interplanetary communications.
There are representatives from all three groups at the meeting. They counted
6 heads and 14 eyes. The one-eyed and the three-eyed group together have
twice as many representatives as the two-eyed group. How many representatives
did each group send? (Every creature has one head.)
A.538. On the main street of the capital
city Tiny of planet Mini there are a few important buildings such as the
Space Center, the School of Astronauts, and the Hotel of the Space Travelers,
not necessarily in this order. The Space Center is 85 meters from the school,
and 27 meters from the hotel. How far is the school from the hotel if the
main street is straight?
A.539. Everybody living on planet Dot develops
two more dots on his or her body on their birthdays. (For example: a 2-year
old has 4 dots, a 10-year old has 20 dots.) All together, how many dots
does a family with 3 children have if we know that Pete was born in June
2002, Fanny has 6 more dotes than Pete, and Paul was born 5 years earlier
than Fanny? Their mother had 48 dots when Fanny was born, and she is 3 years
younger than her husband. (One year is just as long on planet Dot as it
is on planet Earth and the two planets have the same calendar year.)
A.540. On planet Mini there are 4 shuttle
landing runways. How many different ways can the shuttles occupy the runways
if no more than one shuttle is allowed at a time on any of the runways,
and at least one runway has a shuttle on it at any time?
A.541. How many different ways can you
write all the whole numbers from 1 to 12 into the circles on the following
diagram so that the sum of the numbers on the inner circle is half of the
sum of the numbers on the outer circle?
A.542. An unknown spaceship sent the following
series of numbers to our planet:
0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, ...
How would you continue it, and what would the 15th, the 50th, and the
101st element of the series?
A.543. For the purpose of building interplanetary
relationships, Earth sent a spaceship with a few astronauts to Planet Zanza
for the first time. A few days later this spaceship returns to Earth with
the same number of astronauts, but with a few of Zanza's people. After the
spaceship landed on Earth, where can you find more of the other planet's
people: on Earth or on Zanza?
A.544. Planet Hesita has interesting people
living on it. There are two kinds of people: some of them always say the
truth, the others always lie. One day three of their representatives came
to Earth: Axa, Bory, and Cody. Before the first formal meeting the earthlings
wanted to figure out who belongs to which group. Therefore, they asked Axa:
"Are you a truthful guy?" Axa answered the question, but nobody
understood him. That is why they asked Cody: "What did Axa say?"
He said: "Axa said no." At that point Bory said: "Do not
believe Cody, he is a liar."
Who are the truthful guys among the representatives?
Please, send your solutions to: