Olympiad problems

1986 Traian Lălescu, 2.3

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: $$ g(A,B,C,D) =AB+BC+ AD+CA+DB+DC. $$

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 $\sup\{I(f) \mid f \in \mathcal{F}\}$.

1975 Putnam, B3

Tags: inequalities , supremum

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 $\sup_{S} \frac{g(S)}{f(S)}$.

1986 Traian Lălescu, 1.4

Tags: trigonometry , geometry , excircle , 3d geometry , supremum

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. $

1986 Traian Lălescu, 1.3

Tags: algebra , real analysis , polynom , supremum , polynomial function , extremal

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.

1986 Traian Lălescu, 2.3

2018 District Olympiad, 1

1975 Putnam, B3

1986 Traian Lălescu, 1.4

1986 Traian Lălescu, 1.3