ABACUS International Math Challenge
for
7th and 8th graders
November, 2004
C.449. Pete has the same number of sisters
as brothers. However, each of her sisters has half as many sisters as brothers.
How many boys and how many girls are there in this family?
C.450. Connect an inner point of an equilateral
triangle to all three vertexes. Prove that for any inner point you can construct
a triangle using these three segments.
C.451. Two students were bored in Science
class and they invented the following math game: taking turns, they draw
the diagonals of a regular 7-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?
C.452. One of the outer angles of a triangle
is 135 degrees. The difference of two of its inner angles is 29 degrees.
What could the inner angles of this triangle be?
C.453. Place 2005 pieces on a 2005x2005
chess board, so that they are symmetrical to the center square. Prove that
there is a piece in the 1003rd row.
C.454. You can buy coats, shoes for women,
and shoes for men in a store. One time the inventory showed that 25% of
all the items are coats, and 60% of all the shoes are for men. What percent
of all the items in the store are shoes for women?
C.455. a) In the sum of the first 2004
positive whole numbers we change the sign of a few numbers. Could the sum
be 2005?
b) In the sum of the first 2005 positive whole numbers we change the
sign of a few numbers. Could the sum be 2005?
C.456. Start a number sequence with 14.
Calculate every next element by adding the cubes of the digits of the previous
element. What is the 2004th element? What would the 2004th element be if
the first element were 13?