Found problems: 4776
Consider the functions $f$ defined on the set of integers such that
\[f(x)=f(x^2+x+1)\]
for all integer $x$. Find $(a)$ all even functions, $(b)$ all odd functions of this kind.
Define the function $f: \mathbb N \cup \{0\} \to \mathbb{Q}$ as follows: $f(0) = 0$ and \[ f(3n+k) = -\frac{3f(n)}{2} + k , \] for $k = 0, 1, 2$. Show that $f$ is one-to-one and determine the range of $f$.
We say that a function $f: \mathbb{N}\rightarrow\mathbb{N}$ has the $(\mathcal{P})$ property if, for any $y\in\mathbb{N}$, the equation $f(x)=y$ has exactly 3 solutions.
a) Prove that there exist an infinity of functions with the $(\mathcal{P})$ property ;
b) Find all monotonously functions with the $(\mathcal{P})$ property ;
c) Do there exist monotonously functions $f: \mathbb{Q}\rightarrow\mathbb{Q}$ satisfying the $(\mathcal{P})$ property ?
A student steps onto a stationary elevator and stands on a bathroom scale. The elevator then travels from the top of the building to the bottom. The student records the reading on the scale as a function of time.
At what time(s) does the student have maximum downward velocity?
$\textbf{(A)}$ At all times between $2 s$ and $4 s$
$\textbf{(B)}$ At $4 s$ only
$\textbf{(C)}$ At all times between $4 s$ and $22 s$
$\textbf{(D)}$ At $22 s$ only
$\textbf{(E)}$ At all times between $22 s$ and $24 s$
Does there exist a function $f: \mathbb R \to \mathbb R $ satisfying the following conditions:
(i) for each real $y$ there is a real $x$ such that $f(x)=y$ , and
(ii) $f(f(x)) = (x - 1)f(x) + 2$ for all real $x$ ?
[i]Proposed by Igor I. Voronovich, Belarus[/i]
Two distinct points $A$ and $B$ are chosen at random from 15 points equally spaced around a circle centered at $O$ such that each pair of points $A$ and $B$ has the same probability of being chosen. The probability that the perpendicular bisectors of $OA$ and $OB$ intersect strictly inside the circle can be expressed in the form $\frac{m}{n}$, where $m,n$ are relatively prime positive integers. Find $m+n$.
[i]Ray Li.[/i]
Let $k$ be a positive integer. Find all functions $f:\mathbb{N}\to \mathbb{N}$ satisfying the following two conditions:\\
• For infinitely many prime numbers $p$ there exists a positve integer $c$ such that $f(c)=p^k$.\\
• For all positive integers $m$ and $n$, $f(m)+f(n)$ divides $f(m+n)$.
[b](a)[/b] Sketch the diagram of the function $f$ if
\[f(x)=4x(1-|x|) , \quad |x| \leq 1.\]
[b](b)[/b] Does there exist derivative of $f$ in the point $x=0 \ ?$
[b](c)[/b] Let $g$ be a function such that
\[g(x)=\left\{\begin{array}{cc}\frac{f(x)}{x} \quad : x \neq 0\\ \text{ } \\ 4 \ \ \ \ \quad : x=0\end{array}\right.\]
Is the function $g$ continuous in the point $x=0 \ ?$
[b](d)[/b] Sketch the diagram of $g.$
Find all twice differentiable functions $f(x)$ such that $f^{\prime \prime}(x)=0$, $f(0)=19$, and $f(1)=99$.
The sequence $\{a_n\}$ is defined by: $a_1=\frac{21}{16}$, and for $n\ge2$,\[ 2a_n-3a_{n-1}=\frac{3}{2^{n+1}}. \]Let $m$ be an integer with $m\ge2$. Prove that: for $n\le m$, we have\[ \left(a_n+\frac{3}{2^{n+3}}\right)^{\frac{1}{m}}\left(m-\left(\frac{2}{3}\right)^{{\frac{n(m-1)}{m}}}\right)<\frac{m^2-1}{m-n+1}. \]
For all real numbers $x$, let \[ f(x) = \frac{1}{\sqrt[2011]{1-x^{2011}}}. \] Evaluate $(f(f(\ldots(f(2011))\ldots)))^{2011}$, where $f$ is applied $2010$ times.
Let be a surjective function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the property that if the sequence $ \left( f\left( x_n \right) \right)_{n\ge 1} $ is convergent, then the sequence $ \left( x_n \right)_{n\ge 1} $ is convergent. Prove that it is continuous.
$n$ is a natural number. for every positive real numbers $x_{1},x_{2},...,x_{n+1}$ such that $x_{1}x_{2}...x_{n+1}=1$ prove that:
$\sqrt[x_{1}]{n}+...+\sqrt[x_{n+1}]{n} \geq n^{\sqrt[n]{x_{1}}}+...+n^{\sqrt[n]{x_{n+1}}}$
Find the greatest integer less than the number
$1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{1000000}}$
The function $f(x) = ax + b$ satisfies the following equalities:
\begin{align*}
f(f(f(1))) &= 2023, \\
f(f(f(0))) &= 1996.
\end{align*}
Find the value of $a$.
Find the 3-digit number whose ratio with the sum of its digits it's minimal.
Let the Fourier series \[ \frac{a_0}{2}+ \sum _{k\geq 1}(a_k\cos kx+b_k \sin kx)\] of a function $ f(x)$ be
absolutely convergent, and let \[ a^2_k+b^2_k \geq a_{k+1}^2+b_{k+1}^2 \;(k=1,2,...)\ .\] Show that \[ \frac1h \int_0^{2\pi} (f(x+h)-f(x-h))^2dx \;(h>0)\] is uniformly bounded in $ h$. [K. Tandori]
Find all pairs of real numbers $(x, y)$ satisfying the following system of
equations
$2-x^{3}=y, 2-y^{3}=x$.
Let $P:\mathbb{R} \to \mathbb{R}$ be a continuous function such that $P(X)=X$ has no real solution. Prove that $P(P(X))=X$ has no real solution.
A function $f:\mathbb{Z}\implies\mathbb{Z}$ satisfies
$f(x+4)-f(x)=8x+20$
$f(x^2-1)=(f(x)-x)^2+x^2-2$
Find $f(0)$ and $f(1)$.
Let $c\ge 0$ be a real number. Give a complete description with proof of the set of all continuous functions $f: \mathbb{R}\to \mathbb{R}$ such that $f(x)=f(x^2+c)$ for all $x\in \mathbb{R}$.
[color=darkred]Find all differentiable functions $f\colon [0,\infty)\to [0,\infty)$ for which $f(0)=0$ and $f^{\prime}(x^2)=f(x)$ for any $x\in [0,\infty)$ .[/color]
Let $ a,b> 0$ such that $ a\ne b$. Prove that:
$ \sqrt {ab} < \dfrac{a \minus{} b}{\ln a \minus{} \ln b} < \dfrac{a \plus{} b}{2}$
Let $f:\mathbb{R}\to (0;+\infty )$ be a continuous function such that $\underset{x\to -\infty }{\mathop{\lim }}\,f(x)=\underset{x\to +\infty }{\mathop{\lim }}\,f(x)=0.$
a) Prove that $f(x)$ has the maximum value on $\mathbb{R}.$
b) Prove that there exist two sequeneces $({{x}_{n}}),({{y}_{n}})$ with ${{x}_{n}}<{{y}_{n}},\forall n=1,2,3,...$ such that they have the same limit when $n$ tends to infinity and $f({{x}_{n}})=f({{y}_{n}})$ for all $n.$
Let $x,y,z$ be complex numbers satisfying \begin{align*}
z^2 + 5x &= 10z \\
y^2 + 5z &= 10y \\
x^2 + 5y &= 10x
\end{align*}
Find the sum of all possible values of $z$.
[i]Proposed by Aaron Lin[/i]