Found problems: 78
Let $N$ be a positive integer. The sequence $x_1, x_2, \ldots$ of non-negative reals is defined by $$x_n^2=\sum_{i=1}^{n-1} \sqrt{x_ix_{n-i}}$$ for all positive integers $n>N$. Show that there exists a constant $c>0$, such that $x_n \leq \frac{n} {2}+c$ for all positive integers $n$.
A polynomial $P(x)$ of degree $3$ has three distinct real roots.
Find the number of real roots of the equation $P'(x)^2 -2P(x)P''(x) = 0$.
$f(x)$ is continuous on the interval $[0, \pi]$ and satisfies
\[
\int\limits_0^\pi f(x)dx=0, \qquad \int\limits_0^\pi f(x)\cos x dx=0
\]
Show that $f(x)$ has at least two zeros in the interval $(0, \pi)$.
Justify if continuity can be affirmed, denied or cannot be decided in the point$ x = 0$ of a real function $f(x)$ of real variable, in each of the three (independent) cases .
a) It is known only that for all natural $n$: $f\left( \frac{1}{2n}\right)= 1$ and $f\left( \frac{1}{2n+1}\right)= -1$.
b) It is known that for all nonnegative real $x$ is $f(x) = x^2$ and for negative real $x$ is $f(x) = 0$.
c) It is only known that for all natural $n$ it is $f\left( \frac{1}{n}\right)= 1$.
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.
Let $a, b, c \in \mathbb{R}$ be such that
$$a + b + c = a^2 + b^2 + c^2 = 1, \hspace{8px} a^3 + b^3 + c^3 \neq 1.$$
We say that a function $f$ is a [i]Palić function[/i] if $f: \mathbb{R} \rightarrow \mathbb{R}$, $f$ is continuous and satisfies
$$f(x) + f(y) + f(z) = f(ax + by + cz) + f(bx + cy + az) + f(cx + ay + bz)$$
for all $x, y, z \in \mathbb{R}.$
Prove that any Palić function is infinitely many times differentiable and find all Palić functions.
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 $[0,1]$.
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 $[0,1]$.
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 $x$.
The real-valued function $f(x)$ is defined on the reals. It satisfies $|f(x)| \le A$, $|f''(x)| \le B$ for some positive $A, B$ (and all $x$). Show that $|f'(x)| \le C$, for some fixed$ C$, which depends only on $A$ and $B$. What is the smallest possible value of $C$?
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.
Construct a function $ f:[0,1]\longrightarrow\mathbb{R} $ that is primitivable, bounded, and doesn't touch its bounds.
[i]Dorian Popa[/i]
[b]4.[/b] Let $f(x)$ be a real- or complex-value integrable function on $(0,1)$ with $\mid f(x) \mid \leq 1 $. Set
$ c_k = \int_0^1 f(x) e^{-2 \pi i k x} dx $
and construct the following matrices of order $n$:
$ T= (t_{pq})_{p,q=0}^{n-1}, T^{*}= (t_{pq}^{*})_{p,q =0}^{n-1} $
where $t_{pq}= c_{q-p}, t^{*}= \overline {c_{p-q}}$ . Further, consider the following hyper-matrix of order $m$:
$
S= \begin{bmatrix}
E & T & T^2 & \dots & T^{m-2} & T^{m-1} \\
T^{*} & E & T & \dots & T^{m-3} & T^{m-2} \\
T^{*2} & T^{*} & E & \dots & T^{m-3} & T^{m-2} \\
\dots & \dots & \dots & \dots & \dots & \dots \\
T^{*m-1} & T^{*m-2} & T^{*m-3} & \dots & T^{*} & E
\end{bmatrix} $
($S$ is a matrix of order $mn$ in the ordinary sense; E denotes the unit matrix of order $n$).
Show that for any pair $(m , n) $ of positive integers, $S$ has only non-negative real eigenvalues. [b](R. 19)[/b]
Let $p(x)$ be a polynomial of degree $n$ with real coefficients such that $p(x) \ge 0$ for all $x$. Prove that $p(x)+ p'(x)+ p''(x)+...+ p^{(n)}(x) \ge 0$.
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).
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a countinuous and periodic function, of period $ T. $ If $ F $ is a primitive of $ f, $ show that:
[b]a)[/b] the function $ G:\mathbb{R}\longrightarrow\mathbb{R}, G(x)=F(x)-\frac{x}{T}\int_0^T f(t)dt $ is periodic.
[b]b)[/b] $ \lim_{n\to\infty}\sum_{i=1}^n\frac{F(i)}{n^2+i^2} =\frac{\ln 2}{2T}\int_0^T f(x)dx. $
For each $x>e^e$ define a sequence $S_x=u_0,u_1,\ldots$ recursively as follows: $u_0=e$, and for $n\ge0$, $u_{n+1}=\log_{u_n}x$. Prove that $S_x$ converges to a number $g(x)$ and that the function $g$ defined in this way is continuous for $x>e^e$.
Given a sphere of unit radius with the big circle (i.e of unit radius) that will be called "equator". We shall use the words "pole", "parallel","meridian" as self-explanatory.
a) Let $g(x)$, where $x$ is a point on the sphere, be the distance from this point to the equator plane. Prove that $g(x)$ has the property if $x_1, x_2, x_3$ are the ends of the pairwise orthogonal radiuses, then $$g(x_1)^2 + g(x_2)^2 + g(x_3)^2 = 1 \,\,\,\, (*)$$
Let function $f(x)$ be an arbitrary nonnegative function on a sphere that satisfies (*) property.
b) Let $x_1$ and $x_2$ points be on the same meridian between the north pole and equator, and $x_1$ is closer to the pole than $x_2$. Prove that $f(x_1) > f(x_2)$.
c) Let $y_1$ be closer to the pole than $y_2$. Prove that $f(y_1) > f(y_2)$.
d) Let $z_1$ and $z_2$ be on the same parallel. Prove that $f(z_1) = f(z_2)$.
e) Prove that for all $x , f(x) = g(x)$.
Prove that whatever the complex number $z$ is, it is true that
$$(1 + z^{2^n})(1-z^{2^n})= 1- z^{2^{n+1}}.$$
Writing the equalities that result from giving $n$ the values $0, 1, 2, . . .$ and multiplying them, show that for $|z| < 1$ holds
$$\frac{1}{1-z}= \lim_{k\to \infty}(1 + z)(1 + z^2)(1 + z^{2^2})...(1 + z^{2^k}).$$
$f(x)$ is defined for $x \geq 0$ and has a continuous derivative. It satisfies $f(0)=1$, $f'(0)=0$ and $(1+f(x))f''(x)=1+x$. Show that $f$ is increasing and that $f(1) \leq 4/3$.
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 $ I $ be an open real interval, and let be two functions $ f,g:I\longrightarrow\mathbb{R} $ satisfying the identity:
$$ x,y\in I\wedge x\neq y\implies\frac{f(x)-g(y)}{x-y} +|x-y|\ge 0. $$
[b]a)[/b] Prove that $ f,g $ are nondecreasing.
[b]b)[/b] Give a concrete example for $ f\neq g. $
A differentiable function $f$ with $f(0) = f(1) = 0$ is defined on the interval $[0,1]$.
Prove that there exists a point $y \in [0,1]$ such that $| f' (y)| = 4 \int _0^1 | f(x)|dx$.
Let $k$ be a positive integer and $a_1, a_2, ...$ be a sequence of terms from set $\{ 0, 1, ..., k \}$. Let
$b_n = \sqrt[n] {a_1^n + a_2^n + ... + a_n^n}$
for all positive integers $n$. Prove, that if in sequence $b_1, b_2, b_3, ...$ are infinitely many integers, then all terms of this series are integers.
Let $a_1<a_2<a_3$ be positive integers. Prove that there are integers $x_1,x_2,x_3$ such that $\sum_{i=1}^3 |x_i | >0$, $\sum_{i=1}^3 a_ix_i= 0$ and
$$\max_{1\le i\le 3} | x_i|<\frac{2}{\sqrt{3}}\sqrt{a_3}+1$$.