MathDB

3

Part of 2005 Pan African

Problems(2)

Want the line to half the area

Source: Pan African Maths Olympiad

8/11/2005
Let ABCABC be a triangle and let PP be a point on one of the sides of ABCABC. Construct a line passing through PP that divides triangle ABCABC into two parts of equal area.
geometry
Functional Equation on Z

Source: Pan African Maths Olympiad

8/11/2005
Let f:Z→Zf: \mathbb{Z} \rightarrow \mathbb{Z} be a function such that: For all aa and bb in Z−{0}\mathbb{Z} - \{0\}, f(ab)≥f(a)+f(b)f(ab) \geq f(a) + f(b). Show that for all a∈Z−{0}a \in \mathbb{Z} - \{0\} we have f(an)=nf(a)f(a^n) = nf(a) for all n∈Nn \in \mathbb{N} if and only if f(a2)=2f(a)f(a^2) = 2f(a)
functioninduction