Found problems: 4776
For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$.
Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.
Consider the equation $x^5+x=10$. Show that
(a) the equation has only one real root;
(b) this root lies between $1$ and $2$;
(c) this root must be irrational.
Let $f(x)$ be a function satisfying
\[xf(x)=\ln x \ \ \ \ \ \ \ \ \text{for} \ \ x>0\]
Show that $f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)$ where $f^{(n)}(x)$ denotes the $n$-th derivative evaluated at $x$.
Let $S = \{1,2,3,\ldots,n\}$. Consider a function $f\colon S\to S$. A subset $D$ of $S$ is said to be invariant if for all $x\in D$ we have $f(x)\in D$. The empty set and $S$ are also considered as invariant subsets. By $\deg (f)$ we define the number of invariant subsets $D$ of $S$ for the function $f$.
[b]i)[/b] Show that there exists a function $f\colon S\to S$ such that $\deg (f)=2$.
[b]ii)[/b] Show that for every $1\leq k\leq n$ there exists a function $f\colon S\to S$ such that $\deg (f)=2^{k}$.
Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation
\[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\]
for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.
a) Prove that the function $F:\mathbb{R}\rightarrow \mathbb{R},\ F(x)=2\lfloor x\rfloor-\cos(3\pi\{x\})$ is continuous over $\mathbb{R}$ and for any $y\in \mathbb{R}$, the equation $F(x)=y$ has exactly three solutions.
b) Let $k$ a positive even integer. Prove that there is no function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f$ is continuous over $\mathbb{R}$ and that for any $y\in \text{Im}\ f$, the equation $f(x)=y$ has exactly $k$ solutions $(\text{Im}\ f=f(\mathbb{R}))$.
In how many ways can a $2\times n$ grid be covered by
(a) 2 monominoes and $n-1$ dominoes
(b) 4 monominoes and $n-2$ dominoes.
Consider the function $ f:(-\infty,1]\longrightarrow\mathbb{R} $ defined as
$$ f(x)=\left\{ \begin{matrix} \frac{5}{2} +2^x-\frac{1}{2^x} ,& \quad x<-1 \\ 3^{\sqrt{1-x^2}} ,& \quad x\in [-1,1] \end{matrix} \right. . $$
[b]a)[/b] For a fixed parameter, find the roots of $ f-m. $
[b]b)[/b] Study the inversability of the restrictions of $ f $ to $ (-\infty,-1] $ and $ [-1,1] $ and find the inverses of these that admit them.
[i]D. Zaharia[/i]
Prove that there exists a nonconstant function $ f:\mathbb{R}^2\longrightarrow\mathbb{R} $ verifying the following system of relations:
$$ \left\{ \begin{matrix} f(x,x+y)=f(x,y) ,& \quad \forall x,y\in\mathbb{R} \\f(x,y+z)=f(x,y) +f(x,z) ,& \quad \forall x,y\in\mathbb{R} \end{matrix} \right. $$
Prove the following inequality.
\[\frac{1}{12}(\pi -6+2\sqrt{3})\leq \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \ln (1+\cos 2x) dx\leq \frac{1}{4}(2-\sqrt{3})\]
Let $f$ be a function defined on $(0, \infty )$ as follows:
\[f(x)=x+\frac1x\]
Let $h$ be a function defined for all $x \in (0,1)$ as
\[h(x)=\frac{x^4}{(1-x)^6}\]
Suppose that $g(x)=f(h(x))$ for all $x \in (0,1)$.
(a) Show that $h$ is a strictly increasing function.
(b) Show that there exists a real number $x_0 \in (0,1)$ such that $g$ is strictly decreasing in the interval $(0,x_0]$ and strictly increasing in the interval $[x_0,1)$.
Find all functions $f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+$ that satisfy the following conditions for all positive real numbers $x,y,z:$
$$f\left ( f(x,y),z \right )=x^2y^2f(x,z)$$
$$f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)$$
[i]Proposed by Mojtaba Zare, Ali Daei Nabi[/i]
[color=darkred]Let $\mathcal{C}$ be the set of integrable functions $f\colon [0,1]\to\mathbb{R}$ such that $0\le f(x)\le x$ for any $x\in [0,1]$ . Define the function $V\colon\mathcal{C}\to\mathbb{R}$ by
\[V(f)=\int_0^1f^2(x)\ \text{d}x-\left(\int_0^1f(x)\ \text{d}x\right)^2\ ,\ f\in\mathcal{C}\ .\]
Determine the following two sets:
[list][b]a)[/b] $\{V(f_a)\, |\, 0\le a\le 1\}$ , where $f_a(x)=0$ , if $0\le x\le a$ and $f_a(x)=x$ , if $a<x\le 1\, ;$
[b]b)[/b] $\{V(f)\, |\, f\in\mathcal{C}\}\ .$[/list] [/color]
Suppose $f : \mathbb{R} \longrightarrow \mathbb{R}$ be a function such that
\[2f (f (x)) = (x^2 - x)f (x) + 4 - 2x\]
for all real $x$. Find $f (2)$ and all possible values of $f (1)$. For each value of $f (1)$, construct a function achieving it and satisfying the given equation.
Let $ f: [0,1]\to\mathbb R$ be a derivable function, with a continuous derivative $ f'$ on $ [0,1]$. Prove that if $ f\left( \frac 12\right) \equal{} 0$, then
\[ \int^1_0 \left( f'(x) \right)^2 dx \geq 12 \left( \int^1_0 f(x) dx \right)^2.\]
Suppose $S$ is a finite subset of $\mathbb{R}$. If $f: S \rightarrow S$ is a function such that,
$$
\left|f\left(s_{1}\right)-f\left(s_{2}\right)\right| \leq \frac{1}{2}\left|s_{1}-s_{2}\right|, \forall s_{1}, s_{2} \in S
$$
Prove that, there exists a $x \in S$ such that $f(x)=x$
Triangle $ABC$ has $\angle BAC=60^\circ$, $\angle CBA \le 90^\circ$, $BC=1$, and $AC \ge AB$. Let $H$, $I$, and $O$ be the orthocenter, incenter, and circumcenter of $\triangle ABC$, respectively. Assume that the area of the pentagon $BCOIH$ is the maximum possible. What is $\angle CBA$?
$\textbf{(A)}\ 60 ^\circ \qquad
\textbf{(B)}\ 72 ^\circ\qquad
\textbf{(C)}\ 75 ^\circ \qquad
\textbf{(D)}\ 80 ^\circ\qquad
\textbf{(E)}\ 90 ^\circ$
Let $S = f\{0.1. 2.3,...\}$ be the set of the non-negative integers. Find all strictly increasing functions $f : S \to S$ such that $n + f(f(n)) \le 2f(n)$ for every $n$ in $S$
Let \[f(n)=\dfrac{5+3\sqrt{5}}{10}\left(\dfrac{1+\sqrt{5}}{2}\right)^n+\dfrac{5-3\sqrt{5}}{10}\left(\dfrac{1-\sqrt{5}}{2}\right)^n.\] Then $f(n+1)-f(n-1)$, expressed in terms of $f(n)$, equals:
$\textbf{(A)}\ \dfrac{1}{2}f(n) \qquad
\textbf{(B)}\ f(n)\qquad
\textbf{(C)}\ 2f(n)+1 \qquad
\textbf{(D)}\ f^2(n) \qquad
\textbf{(E)}\ \dfrac{1}{2}(f^2(n)-1)$
Let $ g : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $ x+g(x)$ is strictly monotone (increasing or
decreasing), and let $ u : [0,\infty) \rightarrow \mathbb{R}$ be a bounded and continuous function such that \[ u(t)+ \int_{t-1}^tg(u(s))ds\] is constant on $ [1,\infty)$. Prove that the limit $ \lim_{t\rightarrow \infty} u(t)$ exists.
[i]T. Krisztin[/i]
Let $x_1=1$, $x_2$, $x_3$, $\ldots$ be a sequence of real numbers such that for all $n\geq 1$ we have \[ x_{n+1} = x_n + \frac 1{2x_n} . \] Prove that \[ \lfloor 25 x_{625} \rfloor = 625 . \]
Find all $f : N \longmapsto N$ that there exist $k \in N$ and a prime $p$ that:
$\forall n \geq k \ f(n+p)=f(n)$ and also if $m \mid n$ then $f(m+1) \mid f(n)+1$
Let $ a(x)$ and $ r(x)$ be positive continuous functions defined on the interval $ [0,\infty)$, and let \[ \liminf_{x \rightarrow \infty} (x-r(x)) >0.\] Assume that $ y(x)$ is a continuous function on the whole real line, that it is differentiable on $ [0, \infty)$, and that it satisfies \[ y'(x)=a(x)y(x-r(x))\] on $ [0, \infty)$. Prove that the limit \[ \lim_{x \rightarrow \infty}y(x) \exp \left\{ -%Error. "diaplaymath" is a bad command.
\int_0^x a(u)du \right \}\] exists and is finite.
[i]I. Gyori[/i]
A natural number $k > 1$ is called [i]good[/i] if there exist natural numbers
$$a_1 < a_2 < \cdots < a_k$$
such that
$$\dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots + \dfrac{1}{\sqrt{a_k}} = 1$$.
Let $f(n)$ be the sum of the first $n$ [i][good[/i] numbers, $n \geq$ 1. Find the sum of all values of $n$ for which $f(n+5)/f(n)$ is an integer.
Let $f : \mathbb N \to \mathbb N$ be a function satisfying
\[f(f(m)+f(n))=m+n, \quad \forall m,n \in \mathbb N.\]
Prove that $f(x)=x$ for all $x \in \mathbb N$.