Found problems: 5
Let $\mathcal{F}$ be the set of continuous functions $f : [0, 1]\to\mathbb{R}$ satisfying $\max_{0\le x\le 1} |f(x)| = 1$ and let $I : \mathcal{F} \to \mathbb{R}$,
\[I(f) = \int_0^1 f(x)\, \text{d}x - f(0) + f(1).\]
a) Show that $I(f) < 3$, for any $f \in \mathcal{F}$.
b) Determine $\sup\{I(f) \mid f \in \mathcal{F}\}$.
Let be two fixed points $ B,C. $ Find the locus of the spatial points $ A $ such that $ ABC $ is a nondegenerate triangle and the expression
$$ R^2 (A)\cdot\sin \left( 2\angle ABC\right)\cdot\sin \left( 2\angle BCA\right) $$
has the greatest value possible, where $ R(A) $ denotes the radius of the excirlce of $ ABC. $
Let be four real numbers. Find the polynom of least degree such that two of these numbers are some locally extreme values, and the other two are the respective points of local extrema.
Let $n$ be a positive integer. Let $S=\{a_1,\ldots, a_{k}\}$ be a finite collection of at least $n$ not necessarily distinct positive real numbers. Let
$$f(S)=\left(\sum_{i=1}^{k} a_{i}\right)^{n}$$ and
$$g(S)=\sum_{1\leq i_{1}<\ldots<i_{n} \leq k} a_{i_{1}}\cdot\ldots\cdot a_{i_{n}}.$$ Determine $\sup_{S} \frac{g(S)}{f(S)}$.
Among the spatial points $ A,B,C,D, $ at most two of are aparted at a distance greater than $ 1. $ Find the the maximum value of the expression:
$$ g(A,B,C,D) =AB+BC+ AD+CA+DB+DC. $$