ABACUS International Math Challenge
for
7th and 8th graders
March, 1999
C.113. The ages of a father and his two kids
are all different exponents of the same prime number. A year ago the ages
of all three of them were primes. How old are they now?
C.114. Find prime numbers x, y, and z,
for which 2x+3y+6z=78.
C.115. E and F are the midpoints of the
sides of square ABCD. What fraction of the area of the square is the shaded
area?
C.116. What is the sum of all the positive
3-digit whole numbers with even digits only?
C.117. Which one is a better bet:
- to get at least one 6 out of 4 rolls with one die;
or
- to get at least one double 6 out of 24 rolls with two dice?
C.118. There are 100 groups of pebbles
on a table containing 1, 2, 3, ..., 99, 100 pebbles, respectively. In one
step you may reduce the number of pebbles of any number of groups as long
as you take the same number of pebbles from each group you selected. What
is the least number of steps you can take all the pebbles off of the table?
C.119. Ten people are sitting around a
round table. Everybody thinks of a number and whispers it to his two neighbors.
Then, everybody announces the average of the two numbers he heard. These
results are shown on the diagram. What number did the person who said 6
think of?
C.120. Mr. and Mrs. Brown gave a party
for their friends they have not seen for a long time. Three couples came.
During the party, some of the people were so happy to see each other again,
that they even shook hands. (None of the men shook hands with their own
wives.) Later on, Mr. Brown asked everybody how many people he/she shook
hands with. He received seven different answers. How many guests did Mrs.
Brown shake hands with?
Please, send your solutions to: