ABACUS International Math Challenge
for
7th and 8th graders
January, 2008
C.617. Six soccer teams are in a league.
(Every team has to play against every team.) The season is not over yet,
and the diagram below shows the games played already, and their outcomes.
(An arrow points from the winner to the loser of that game. A dot on a line
indicates a tie.) A team gets 3 points for a win, 1 point for a tie, and
0 point for a loss. Teams with the same number of points finish in the same
place.
a) How many games are left to play this season?
b) What is the current order of the teams?
c) Which teams won the remaining games if we know that the team that
is the current leader, did not win the championship afterall; and that the
team that is in last place currently, did not finish last afterall?
C.618. The diagonals AC and BE of an ABCDEFGH
regular octagon intercept each other in point P. Find the angles of triangle
APE.
C.619. On the diagram below you have to
get from the marked field containing 2 to the marked field containing 8.
You can step on each square field no more than once. From each field you
can step only horizontally or vertically. (No diagonal steps are allowed.)
Add up the numbers on the fields you step on. What is the greatest sum you
can get?
C.620. Can you find 2008 different positive
odd numbers so that the sum of their reciprocals is 1?
C.621. A quadrilateral in a coordinate
system is given by the coordinates of its vertexes: A(0;0), B(5;0), C(3;2),
and D(0;1). Prove that the angle of its diagonals is 45 degrees.
C.622. Find two consecutive prime numbers
with exactly 2008 consecutive composite numbers in between them.
C.623. How many different ways can a King
get from c5 to g2 using the least number of steps possible? (A King may
take a step to the neighboring fields horizontally, vertically or diagonally.)
C.624. The sum of the ages of Tamara and
her mother is 49 years. 16 years from now Tamara is going to be the same
age as her mother was when she (Tamara) was born. 20 years from now Tamara
is going to be the same age as her father was when she (Tamara) was born.
How old are the family members now?