Olympiad problems

1967 Spain Mathematical Olympiad, 8

To obtain the value of a polynomial of degree $n$, whose coefficients are $$a_0, a_1, . . . ,a_n$$ (starting with the term of highest degree), when the variable $x$ is given the value $b$, the process indicated in the attached flowchart can be applied, which develops the actions required to apply Ruffini's rule. It is requested to build another flowchart analogous that allows to express the calculation of the value of the derivative of the given polynomial, also for $x = b$. [img]https://cdn.artofproblemsolving.com/attachments/a/a/27563a0e97e74553a270fcd743f22176aed83b.png[/img]

analysis derivative polynomial algebra calculus differentiation polynomials real analysis

1987 Traian Lălescu, 1.3

Let be three polynomials of degree two $ p_1,p_2,p_3\in\mathbb{R} [X] $ and the function $$ f:\mathbb{R}\longrightarrow\mathbb{R} ,\quad f(x)=\max\left( p_1(x),p_2(x),p_3(x)\right) . $$ Then, $ f $ is differentiable if and only if any of these three polynomials dominates the other two.

function analysis real analysis polynomial algebra calculus polynomials

2015 IMAR Test, 4

(a) Show that, if $I \subset R$ is a closed bounded interval, and $f : I \to R$ is a non-constant monic polynomial function such that $max_{x\in I}|f(x)|< 2$, then there exists a non-constant monic polynomial function $g : I \to R$ such that $max_{x\in I} |g(x)| < 1$. (b) Show that there exists a closed bounded interval $I \subset R$ such that $max_{x\in I}|f(x)| \ge 2$ for every non-constant monic polynomial function $f : I \to R$.

function interval analysis conic real analysis polynomial algebra combinatorics conic sections polynomials