MathDB

2

Part of 1987 Iran MO (2nd round)

Problems(2)

The function g is bounded - [Iran Second Round 1987]

Source:

12/19/2010
Let ff be a real function defined in the interval [0,+∞)[0, +\infty ) and suppose that there exist two functions f′,f′′f', f'' in the interval [0,+∞)[0, +\infty ) such that f′′(x)=1x2+f′(x)2+1andf(0)=f′(0)=0.f''(x)=\frac{1}{x^2+f'(x)^2 +1} \qquad \text{and} \qquad f(0)=f'(0)=0. Let gg be a function for which g(0)=0andg(x)=f(x)x.g(0)=0 \qquad \text{and} \qquad g(x)=\frac{f(x)}{x}. Prove that gg is bounded.
functionalgebra proposedalgebra
Find all functions - [Iran Second Round 1987]

Source:

12/23/2010
Find all continuous functions f:R→Rf: \mathbb R \to \mathbb R such that f(x^2-y^2)=f(x)^2 + f(y)^2,   \forall x,y \in \mathbb R.
functionalgebra proposedalgebra