ABACUS International Math Challenge
for
7th and 8th graders
January, 2004
C.409. Edit's grandmother's great grandmother's
age was 1/31 of her own birth year when she died. (Count her age in full
years.) How old was she in 1900?
C.410. Four men went to the theater. At
the end of the performance the lady at the coat check mixed up the coats
and the hats. The lady gave everybody somebody else's hat and yet another somebody
else's coat. Frank got the coat of the man who's hat was given to Daniel. Daniel's coat was given to the man who also got Frank's hat. Joe got Peter's coat. Who's coat and who's hat did Frank and Daniel get from the lady?
C.411. There are 33 pearls on a chain.
The one in the middle is the most expensive pearl. From left to right on the chain till the middle (including the middle
pearl) every pearl worth $100 more than the previous one. From right to left till the middle (including the middle
pearl) every pearl worth $150 more than the previous one. The whole chain worth $65000. What is the value if the pearl in the middle?
C.412. Pete's father never knows what the date is, so Pete decided to make a calendar for him out of two cubes. He wrote a digital looking number on each side of each cube, and put one cube next to the other in a clear container. From the front faces of the two cubes you can read a 2-digit number indicating today's date. (For example the 9th day of a month is indicated by 09.) You may switch, turn, and move the cubes whichever way you want to, and this way you are able to show the date of every day of every month. What numbers did Pete put on the sides of the cubes if we know that one of the cubes has the numbers 3, 4, and 5 on it? The digital numbers Pete used look like these:
C.413. The midpoint of the semicircle with diameter AB is C. D is another point on the semicircle. DE is perpendicular to AB. (E is on AB.) Prove that:
C.414. A group of friends bought a crystal bowl, but when they had to pay for it 4 of them were not present, therefore, everybody there had to give in an extra $36. (This way they paid for it fully.) Then 3 of the missing friends arrived and after dividing the cost evenly again among those who are currently present, each of those who paid for it the first time got $30 back. How many people are there in the group, and how much did the bowl cost?
C.415. Write down the greatest number by using four digit 2's without using any mathematical operation sign.
C.416. Andrew, Ben, and Calvin are discussing their ages and realized that their average age is 36 years, Andrew is 8 years older than Ben, and that average age of Andrew and Calvin is 40 years. How old are these men?