MathDB

2

Part of 1985 IberoAmerican

Problems(2)

1st ibmo - colombia 1985./q2

Source: Spanish Communities

4/6/2006
Let P P be a point in the interior of the equilateral triangle △ABC \triangle{}ABC such that PA \equal{} 5, PB \equal{} 7, PC \equal{} 8. Find the length of the side of the triangle ABC ABC.
Columbiarotationtrigonometrygeometry solvedgeometry
1st ibmo - colombia 1985./q5

Source: Spanish Communities

4/6/2006
To each positive integer n n it is assigned a non-negative integer f(n)f(n) such that the following conditions are satisfied:
(1) f(rs) \equal{} f(r)\plus{}f(s) (2) f(n) \equal{} 0, if the first digit (from right to left) of n n is 3. (3) f(10) \equal{} 0.
Find f(1985)f(1985). Justify your answer.
Columbiafunctionpigeonhole principlenumber theoryrelatively primenumber theory unsolved