2000 Greece Junior Math Olympiad

Part of Greece Junior Math Olympiad

Subcontests

(4)

1

A three-digit number.

A three-digit bumber

\overline{abc}

(in decimal representation) is such that (i) its hundreds digit is equal to the sum of the other two digits, and (ii)

b(c+1)=52-4a

. Find all such numbers.

1

A past mathematical olympiad

On a past Mathematical Olympiad the maximum possible score on a problem was 5. The average score of boys was 4, the average score of girls was 3.25, and the overall average score was 3.60. Find the total number of participants, knowing that it was in the range from 31 to 50.

1

Four pupils.

Four pupils decided to buy some mathematical books so that (a) everybody buys exactly three different books , and (b) every two of the pupils buy exactly one book in common. What are the greatest and smallest number of different books they can buy?

1

Three non-collinear points.

Given three non-collinear points in the plane, find a line which is equally distant from each of the points. How many such lines are there?