$p(x)$ is a polynomial of degree $n$ with leading coefficient $c$, and $q(x)$ is a polynomial of degree $m$ with leading coefficient $c$, such that \[ p(x)^2 = \left(x^2 - 1\right)q(x)^2 + 1 \] Show that $p'(x) = nq(x)$.

Consider a function $f:\mathbb{R}\rightarrow \mathbb{R}$. For $x\in \mathbb{R}$ we say that $f$ is [i]increasing in $x$[/i] if there exists $\epsilon_x > 0$ such that $f(x)\geq{f(a)}$, $\forall a\in (x-\epsilon_x,x)$ and $f(x)\leq f(b)$, $\forall b\in (x,x+\epsilon_x)$. $\textbf{(a)}$ Prove that if $f$ is increasing in $x$, $\forall x\in \mathbb{R}$ then $f$ is increasing over $\mathbb{R}$. $\textbf{(b)}$ We say that $f$ is [i]increasing to the left[/i] in $x$ if there exists $\epsilon_x > 0$ such that $f(x)\geq f(a) $, $ \forall a \in (x-\epsilon_x,x)$. Provide an example of a function $f: [0,1]\rightarrow \mathbb{R}$ for which there exists an infinite set $M \subset (0,1)$ such that $f$ is increasing to the left in every point of $M$, yet $f$ is increasing over no proper subinterval of $[0,1]$.

Determine the pairs of sets $X,Y\subset\mathbb{R}$ for which the following is true: if $f(x, y)$ is a function on $X\times Y{}$ such that for every $x\in X$ it is equal to a polynomial in $y$ on $Y$ and for every $y\in Y$ it is equal to a polynomial in $x$ on $X$ then $f$ is a bivariate polynomial on $X\times Y.$

A polynomial $P(x)$ of degree $3$ has three distinct real roots. Find the number of real roots of the equation $P'(x)^2 -2P(x)P''(x) = 0$.

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.

Let ${a_n}$ be a non-increasing sequence of positive numbers. Prove that if for $n \ge 2001$, $na_{n} \le 1$, then for any positive integer $m \ge 2001$ and $x \in \mathbb{R}$, the following inequality holds: $\left | \sum_{k=2001}^{m} a_{k} \sin kx \right | \le 1 + \pi$

Prove that if $f\colon \mathbb{R} \to \mathbb{R}$ is a continuous periodic function and $\alpha \in \mathbb{R}$ is irrational, then the sequence $\{n\alpha+f(n\alpha)\}_{n=1}^{\infty}$ modulo 1 is dense in $[0,1]$.

For any natural number $ m, \quad\lim_{n\to\infty } n^{1+m} \int_{0}^1 e^{-nx}\ln \left( 1+x^m \right) dx =m! . $ [i]Gheorghe Iurea[/i]

Consider a function $f:\mathbb{R}\rightarrow \mathbb{R}$. For $x\in \mathbb{R}$ we say that $f$ is [i]increasing in $x$[/i] if there exists $\epsilon_x > 0$ such that $f(x)\geq{f(a)}$, $\forall a\in (x-\epsilon_x,x)$ and $f(x)\leq f(b)$, $\forall b\in (x,x+\epsilon_x)$. $\textbf{(a)}$ Prove that if $f$ is increasing in $x$, $\forall x\in \mathbb{R}$ then $f$ is increasing over $\mathbb{R}$. $\textbf{(b)}$ We say that $f$ is [i]increasing to the left[/i] in $x$ if there exists $\epsilon_x > 0$ such that $f(x)\geq f(a) $, $ \forall a \in (x-\epsilon_x,x)$. Provide an example of a function $f: [0,1]\rightarrow \mathbb{R}$ for which there exists an infinite set $M \subset (0,1)$ such that $f$ is increasing to the left in every point of $M$, yet $f$ is increasing over no proper subinterval of $[0,1]$.

$f(x)$ is defined for $0 \leq x \leq 1$ and has a continuous derivative satisfying $|f'(x)| \leq C|f(x)|$ for some positive constant $C$. Show that if $f(0) = 0$, then $f(x)=0$ for the entire interval.

Let $p(n)\geq 0$ for all positive integers $n$. Furthermore, $x(0)=0, v(0)=1$, and \[x(n)=x(n-1)+v(n-1), \qquad v(n)=v(n-1)-p(n)x(n) \qquad (n=1,2,\dots).\] Assume that $v(n)\to 0$ in a decreasing manner as $n \to \infty$. Prove that the sequence $x(n)$ is bounded if and only if $\sum_{n=1}^{\infty}n\cdot p(n)<\infty$.

Let $C[-1,1]$ be the space of continuous real functions on the interval $[-1,1]$ with the usual supremum norm, and let $V{}$ be a closed, finite-codimensional subspace of $C[-1,1].$ Prove that there exists a polynomial $p\in V$ with norm at most one, which satisfies $p'(0)>2023.$

Let $p(x)$ be a polynomial of degree $n$ with real coefficients such that $p(x) \ge 0$ for all $x$. Prove that $p(x)+ p'(x)+ p''(x)+...+ p^{(n)}(x) \ge 0$.

Determine the values of $a, b, c$, so that the graphical representation of the function $$y = ax^3 + bx^2 + cx$$ has an inflection point at the point of abscissa $ x = 3$, with tangent at the point of equation $x - 4y + 1 = 0.$ Then draw the corresponding graph.

Let $n$ be a positive integer, $h$ a real number and $f(x)$ a polynomial (whole rational function) with real coefficients of degree n, which has no real zeros. Prove that then also the polynomial $$F(x) = f(x) + h f'(x) + h^2 f''(x) +... + h^n f^{(n)}(x)$$ has no real zeros.

The functions $f$ and $g$ are positive and continuous. $f$ is increasing and $g$ is decreasing. Show that \[ \int\limits_0^1 f(x)g(x) dx \leq \int\limits_0^1 f(x)g(1-x) dx \]

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]

Calculate the limit $$\lim_{n \to \infty} \frac{1}{n} \left(\frac{1}{n^k} +\frac{2^k}{n^k} +....+\frac{(n-1)^k}{n^k} +\frac{n^k}{n^k}\right).$$ (For the calculation of the limit, the integral construction procedure can be followed).

Draw the graph of the function $y = \frac{| x^3 - x^2 - 2x | }{3} - | x + 1 |$.

We say that two sequences $x,y \colon \mathbb{N} \to \mathbb{N}$ are [i]completely different[/i] if $x_n \neq y_n$ holds for all $n\in \mathbb{N}$. Let $F$ be a function assigning a natural number to every sequence of natural numbers such that $F(x)\neq F(y)$ for any pair of completely different sequences $x$, $y$, and for constant sequences we have $F \left((k,k,\dots)\right)=k$. Prove that there exists $n\in \mathbb{N}$ such that $F(x)=x_{n}$ for all sequences $x$.

Are there integers $m, n \geq 2$ such that the following property is always true? $$``\text{For any real numbers } x, y, \text{ if } x^m + y^m \text{ and } x^n + y^n \text{ are integers, then } x + y \text{ is an integer}".$$

Evaluate \[\frac{1/2}{1+\sqrt2}+\frac{1/4}{1+\sqrt[4]2}+\frac{1/8}{1+\sqrt[8]2}+\frac{1/16}{1+\sqrt[16]2}+\cdots\] [i]Proposed by Ethan Tan[/i]

Let ${a_n}$ be a non-increasing sequence of positive numbers. Prove that if for $n \ge 2001$, $na_{n} \le 1$, then for any positive integer $m \ge 2001$ and $x \in \mathbb{R}$, the following inequality holds: $\left | \sum_{k=2001}^{m} a_{k} \sin kx \right | \le 1 + \pi$

Write the equation of the line tangent to the graph of the function $y = x^4-x^2 + x$ to at least at two points.

Partition $\frac1{2002},\frac1{2003},\frac1{2004},\ldots,\frac{1}{2017}$ into two groups. Define $A$ the sum of the numbers in the first group, and $B$ the sum of the numbers in the second group. Find the partition such that $|A-B|$ attains it minimum and explains the reason.