ABACUS International Math Challenge
for
7th and 8th graders
December, 2003
C.401. They are installing electric poles
between two villages, so that the distance between any to consecutive poles
is the same. If the workers place them 50 meters apart, then they will need
19 more poles than what they have. If they place them 55 meters apart, then
they will have 1 pole left over. How far are the two villages, if there
is a pole at the near end of each village already, so the workers have to
place the poles only between these two poles?
C.402. Extend sides AB and AC of triangle
ABC through B and C. Draw a circle that touches side BC and the extension
lines. Let the center of this circle be P. How big is angle BPC, if angle
BAC is 44 degrees?
C.403. The school's Student Council is
working in 4 subcommittees. Every member of the Student Council is working
in exactly 2 subcommittees. Any two subcommittees have exactly one mutual
member. How members does the Student Council have?
C.404. Mr. Smith, Mr. Taylor, and Mr. Elder
are teaching 6 different subjects (Biology, Geography, Math, History, English,
and Physics) in the school, each of them teaching two subjects. We have
the following information about them:
- The Physics and English teachers are neighbors.
- Mr. Smith is the youngest of them.
- Mr. Elder plays poker with the English teacher and the Biology teacher
every Sunday. The three of them love this game.
- The Biology teacher is older than the Math teacher.
- The Geography teacher, the Math teacher, and Mr. Smith will go to
bike together next weekend. Mr. Smith already rented the three bikes.
Which teacher teaches which subjects?
C.405. You can see two semicircles on the
diagram. The 24-unit-long straight segment is parallel to the diagonal of
the semicircles, and it touches the smaller semicircle in one point. How
big is the shaded area?
C.406. Prove that the product of 5 consecutive
numbers is always divisible by 240 if the fourth number is prime.
C.407. The sum of two different numbers
is 1. Peter added the square of the smaller number to the greater number.
Katie added the square of the greater number to the smaller number. Who
got a greater number, Peter or Katie?
C.408. Joe can build a brick wall alone
in 9 hours. Pete can build the same wall alone in 10 hours. If the two of
them work together they lay a total of 10 bricks less every hour than each
working alone, but they get the job done in 5 hours. How many bricks are
there in the wall?