MathDB

3

Part of 1985 Iran MO (2nd round)

Problems(2)

Inequality of Functions - [Iran Second Round 1985]

Source:

12/28/2010
Let f:R→R,g:R→Rf: \mathbb R \to \mathbb R,g: \mathbb R \to \mathbb R and φ:R→R\varphi: \mathbb R \to \mathbb R be three ascendant functions such that f(x)≤g(x)≤φ(x)∀x∈R.f(x) \leq g(x) \leq \varphi(x) \qquad \forall x \in \mathbb R. Prove that f(f(x))≤g(g(x))≤φ(φ(x))∀x∈R.f(f(x)) \leq g(g(x)) \leq \varphi(\varphi(x)) \qquad \forall x \in \mathbb R.
Note. The function is k(x)k(x) ascendant if for every x,y∈Dk,x≤y x,y \in D_k, x \leq {y} we have g(x)≤g(y)g(x)\leq{g(y)}.
inequalitiesfunctionalgebra proposedalgebra
Find the angle - [Iran Second Round 1985]

Source:

12/29/2010
Find the angle between two common sections of the page 2x+y−z=02x+y-z=0 and the cone 4x2−y2+3z2=0.4x^2-y^2+3z^2=0.
geometry3D geometryvectortrigonometryalgebra proposedalgebra