ABACUS International Math Challenge
for
7th and 8th graders
October, 2004
C.441. Put down 12 identical circular coins
on the table. The coins may touch each other but they cannot cover one another
even partially. Prove that the number of mutual points is less than 36.
C.442. P is an inner point on side AC of
triangle ABC. Mark a point Q on the continuation of segment BP over P. Prove
that the smallest angle in triangle ABC is not at B if 
.
C.443. One side of a triangle is 7 cm,
its perimeter is greater than 22 cm but smaller than 37 cm. How long could
the other two sides be if one of them is twice as long as the other?
C.444. Pete left his house at 3:30 pm to
go to soccer practice. He checked his analog watch and realized that when
he arrived home that day from school the minute hand of his watch was also
in a "vertical" position, but it had a 15 degree less angle with
the hour hand than now at 3:30 pm. What time could Pete get home from school
that day?
C.445. Two students were bored in Science
class and they invented the following math game: taking turns, they draw
the diagonals of a regular 12-sided polygon. You are not allowed to draw
a diagonal that would intersect another diagonal already drawn. The person
who cannot draw such a new diagonal loses the game. Who has a winning strategy?
Who has a winning strategy if they play this game with a regular 2004-sided
polygon?
C.446. The least common multiple of 50
positive whole numbers is the same as the least common multiple of another
50 positive whole numbers. Is it possible that these 100 numbers are 100
consecutive numbers?
C.447. We are making monthly calendars
which do not indicate the names of the months, but next to every date we
indicate what day of the week that day is. At least how many of these calendars
do we have to make in order to be able to use one of them in any month of
any year for following the days of that month?
C.448. Between Dareville and Whoville there
is a milestones on the side of the road at every kilometer indicating how
many kilometers is still to Dareville (on one side) and to Whoville (on
the other side). At the borders of Dareville and Whoville are the first
and last such milestones. Marylou Who noticed that the sum of the digits
on every stone is 13. How far is Dareville from Whoville?