ABACUS International Math Challenge
for
7th and 8th graders
February, 2003
C.369. How many different ways can you
go from one vertex of a cube to the other vertex, which is the farthest
away from this, through the edges of the cube if you can visit every vertex
no more than once?
C.370. The ABCD quadrilateral is given.
Draw a parallel line to the AC diagonal through vertex B. It will intercept
line AD in point E. Prove that the area of triangle DCE is the same as the
area of quadrilateral ABCD.
C.371. There are 2003 squares (size 2cm
x 2cm) colored in by the following pattern. What is the total area of the
black parts?
C.372. Prove that 2003p+5 is not a prime
number if p is a prime number.
C.373. In a computer room there are 3 times
as many boys as girls. If 4 boys and 4 girls leave the room then 5 times
as many boys will be in the room as girls. How many students were there
in the room originally?
C.374. In the final round of a handyman
competition Peter and Paul had to make a 4cm x 4cm square grid. Peter received
five 8cm-long wires, and Paul received eight 5cm-long wires. The wires can
easily be bent, and soldered, but cannot be broken or cut. Is it possible
for Peter and for Paul to solve this problem?
C.375. Peter and Paul have a horse. They
agree that they will alternate walking and riding 2 km, so that the rider
will tie up the horse after 2 km, and continue walking. When the person
who started walking catches up with the horse, he gets on it and starts
riding. After 2 km he will tie up the horse and start walking. When walking,
they go by 4 km/h; when riding, they go by 12 km/h. The place they want
to go to is 40 km from where they are. What part of the time of the trip
is the horse resting if they started the trip at the same time: one of them
walking and the other on the horse riding?
C.376. A planes of the sides of a regular
polygon-based column cut up the space into 48 sections. How many sides does
this column have?