5

Part of 1985 Iran MO (2nd round)

Problems(2)

Find the derivative of f - [Iran Second Round 1985]

Source:

f: \mathbb R \to \mathbb R

g: \mathbb R \to \mathbb R

be two functions satisfying \forall x,y \in \mathbb R: \begin{cases} f(x+y)=f(x)f(y),\\ f(x)= x g(x)+1\end{cases} \text{and} \lim_{x \to 0} g(x)=1. Find the derivative of

f

in an arbitrary point

x.

calculusderivativefunctionlimitalgebra unsolvedalgebra

Archery - [Iran Second Round 1985]

Source:

In the Archery with an especial gun, the probability of goal is

90 \%.

If we continue our work until we goal.
i) What is the probability which exactly

3

balls consumed.
ii) What is the probability which at least

3

balls consumed.

probabilitycombinatorics proposedcombinatorics