Found problems: 4776
Assuming that the roots of $x^3 + p \cdot x^2 + q \cdot x + r = 0$ are real and positive, find a relation between $p,q$ and $r$ which gives a necessary condition for the roots to be exactly the cosines of the three angles of a triangle.
Find the number of extrema of the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as
$$ f(x)=\prod_{j=1}^n (x-j)^j, $$
where $ n $ is a natural number.
Determine all possible (finite or infinite) values of
\[\lim_{x \to -\infty} f(x)-\lim_{x \to \infty} f(x),\]
if $f:\mathbb{R} \to \mathbb{R}$ is a strictly decreasing continuous function satisfying
\[f(f(x))^4-f(f(x))+f(x)=1\]
for all $x \in \mathbb{R}$.
Find the functions $f: \mathbb{R} \to (0, \infty)$ which satisfy $$2^{-x-y} \le \frac{f(x)f(y)}{(x^2+1)(y^2+1)} \le \frac{f(x+y)}{(x+y)^2+1},$$ for all $x,y \in \mathbb{R}.$
Let $f(x)$ be an even twice differentiable function such that $f''(0)\ne0$. Prove that $f(x)$ has a local extremum at $x=0$.
$f(x)$ is a function defined on $\mathbb{R}$. $f(1)=1$, and for all $x\in\mathbb{R}$,
$f(x+5)\geq x+5,f(x+1)\leq f(x)+1$.
If $g(x)=f(x)+1-x$, then $g(2002)=$________.
In a rhombus, $ ABCD$, line segments are drawn within the rhombus, parallel to diagonal $ BD$, and terminated in the sides of the rhombus. A graph is drawn showing the length of a segment as a function of its distance from vertex $ A$. The graph is:
$ \textbf{(A)}\ \text{A straight line passing through the origin.} \\
\textbf{(B)}\ \text{A straight line cutting across the upper right quadrant.} \\
\textbf{(C)}\ \text{Two line segments forming an upright V.} \\
\textbf{(D)}\ \text{Two line segments forming an inverted V.} \\
\textbf{(E)}\ \text{None of these.}$
Let $f$ be a real-valued function defined over ordered pairs of integers such that \[f(x+3m-2n, y-4m+5n) = f(x,y)\] for every integers $x,y,m,n$. At most how many elements does the range set of $f$ have?
$
\textbf{(A)}\ 7
\qquad\textbf{(B)}\ 8
\qquad\textbf{(C)}\ 15
\qquad\textbf{(D)}\ 49
\qquad\textbf{(E)}\ \text{Infinitely many}
$
Consider the function $f_k:\mathbb{Z}^{+}\rightarrow\mathbb{Z}^{+}$ satisfying
\[f_k(x)=x+k\varphi(x)\]
where $\varphi(x)$ is Euler's totient function, that is, the number of positive integers up to $x$ coprime to $x$. We define a sequence $a_1,a_2,...,a_{10}$ with
[list]
[*] $a_1=c$, and
[*] $a_n=f_k(a_{n-1}) \text{ }\forall \text{ } 2\le n\le 10$
[/list]
Is it possible to choose the initial value $c\ne 1$ such that each term is a multiple of the previous, if
(a) $k=2025$ ?
(b) $k=2065$ ?
[i]Proposed by chorn[/i]
Let $ f(x) \equal{} 1 \minus{} \cos x \minus{} x\sin x$.
(1) Show that $ f(x) \equal{} 0$ has a unique solution in $ 0 < x < \pi$.
(2) Let $ J \equal{} \int_0^{\pi} |f(x)|dx$. Denote by $ \alpha$ the solution in (1), express $ J$ in terms of $ \sin \alpha$.
(3) Compare the size of $ J$ defined in (2) with $ \sqrt {2}$.
Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$
Find all functions $ f \colon \mathbb{Z} \mapsto \mathbb{R}$ which satisfy:
i) $ f(x)f(y) \equal{} f(x \plus{} y) \plus{} f(x \minus{} y)$ for all integers $ x$, $ y$
ii) $ f(0) \neq 0$
iii) $ f(1) \equal{} \frac {5}{2}$
Find all function(s) $f:\mathbb R\to\mathbb R$ satisfying the equation
\[f(x+y)+f(x)f(y)=(1+y)f(x)+(1+x)f(y)+f(xy);\]
For all $x,y\in\mathbb R.$
For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.
Does there exist any continuous function $ f$ such that $ f(f(x))=-x^{2019}\ \forall\ x \in \mathbb{R}$
Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying
\begin{align*}
f(x+f(y)) &= g(x) + h(y) \\
g(x+g(y)) &= h(x) + f(y) \\
h(x+h(y)) &= f(x) + g(y)
\end{align*}
for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.)
[i]Proposed by Evan Chen[/i]
Let the function $f:N^*\to N^*$ such that
[b](1)[/b] $(f(m),f(n))\le (m,n)^{2014} , \forall m,n\in N^*$;
[b](2)[/b] $n\le f(n)\le n+2014 , \forall n\in N^*$
Show that: there exists the positive integers $N$ such that $ f(n)=n $, for each integer $n \ge N$.
(High School Affiliated to Nanjing Normal University )
A function $f:\mathbb{N} \rightarrow \mathbb{N} $ is called nice if $f^a(b)=f(a+b-1)$, where $f^a(b)$ denotes $a$ times applied function $f$.
Let $g$ be a nice function, and an integer $A$ exists such that $g(A+2018)=g(A)+1$.
a) Prove that $g(n+2017^{2017})=g(n)$ for all $n \geq A+2$.
b) If $g(A+1) \neq g(A+1+2017^{2017})$ find $g(n)$ for $n <A$.
Each positive integer is coloured red or blue. A function $f$ from the set of positive integers to itself has the following two properties:
(a) if $x\le y$, then $f(x)\le f(y)$; and
(b) if $x,y$ and $z$ are (not necessarily distinct) positive integers of the same colour and $x+y=z$, then $f(x)+f(y)=f(z)$.
Prove that there exists a positive number $a$ such that $f(x)\le ax$ for all positive integers $x$.
[i](United Kingdom) Ben Elliott[/i]
Show that there is no integer-valued function on the integers such that $f(m+f(n))=f(m)-n$ for all $m,n$.
Find all functions $f : Z_{>0} \to Z_{>0}$ with $a - f(b) | af(a) - bf(b)$ for all $a, b \in Z_{>0}$.
[i](Theresia Eisenkoelbl)[/i]
Find all functions $f(x)$ defined in the range $\left(-\frac\pi2,\frac\pi2\right)$ that are differentiable at $0$ and satisfy
$$f(x)=\frac12\left(1+\frac1{\cos x}\right)f\left(\frac x2\right)$$
for every $x$ in the range $\left(-\frac\pi2,\frac\pi2\right)$.
[i]L. Davidov[/i]
Find the natural numbers $ a $ that have the property that there exists a function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ such that $ f(f(n))=a+n, $ for any natural number $ n, $ and the function $ g:\mathbb{N}\longrightarrow\mathbb{N} $ defined as $ g(n)=f(n)-n $ is injective.
Let $f:X\rightarrow X$, where $X=\{1,2,\ldots ,100\}$, be a function satisfying:
1) $f(x)\neq x$ for all $x=1,2,\ldots,100$;
2) for any subset $A$ of $X$ such that $|A|=40$, we have $A\cap f(A)\neq\emptyset$.
Find the minimum $k$ such that for any such function $f$, there exist a subset $B$ of $X$, where $|B|=k$, such that $B\cup f(B)=X$.
Suppose a function $ f: \mathbb R\to \mathbb R$ satisfies $ f(f(x)) \equal{} \minus{} x$ for all $ x\in \mathbb R$. Prove that $ f$ has infinitely many points of discontinuity.