ABACUS International Math Challenge
for
7th and 8th graders
September, 2002
C.321. Find all those 2-digit numbers that
are divisible by both the sum and the product of their digits.
C.322. We put down 10 identical coins on
a circular line so that any two neighboring coins touch each other. 2 people
play the following game: you may take one coin off the table or two coins
that touch each other. Whoever takes the last coin off, wins the game. Does
the first or the second player have a winning strategy? What if we had 11
coins on the table?
C.323. There are 10 coins in a row on the
table. 9 coins show heads and one shows tail. In one step you may flip any
5 coins. Is it possible with such steps that after a certain number of steps
all coins would show heads?
C.324. Is it true that if you have 50 consecutive
positive whole numbers out of which the first number is divisible by 100,
you can always find a number among them in which the sum of the digits is
divisible by 14?
C.325. In a group of 10 people every body
is friends with at least 7 other people. At least how many mutual friends
do any two people in this group have?
C.326. The sum of a few numbers (not necessarily
whole, not necessarily different numbers) is 20. The sum of the 3 smallest
numbers is 5, and the sum of the 3 greatest numbers is 7. How many numbers
are we talking about?
C.327. You have an old 12-cm-long ruler,
but all of the marks faded away, except for the zero and the 12 cm marks.
At least how many marks do you have to place on the ruler, so that you could
measure any whole-number-long distance between zero and 12 cm?
C.328. A circle is drawn on a piece of
paper. A regular hexagon is inscribed in the circle and another regular
hexagon is described around the circle. We know that the area of the smaller
hexagon is 3 units. What is the area of the larger hexagon?