Found problems: 4776
Let \(c\) be a real number. For all \(x\) and \(y\) real numbers we have,
\[f(x-f(y))=f(x-y)+c(f(x)-f(y))\]
and \(f(x)\) is not constant.
\(a)\) Find all possible values of \(c\).
\(b)\) Can \(f\) be periodic?
It is given the function $f:\mathbb{R} \to \mathbb{R}$ such that it holds $f(\sin x)=\sin (2011x)$. Find the value of $f(\cos x)$.
Let be two distinct continuous functions $ f,g:[0,1]\longrightarrow (0,\infty ) $ corelated by the equality $ \int_0^1 f(x)dx =\int_0^1 g(x)dx , $ and define the sequence $ \left( x_n \right)_{n\ge 0} $ as
$$ x_n=\int_0^1 \frac{\left( f(x) \right)^{n+1}}{\left( g(x) \right)^n} dx . $$
[b]a)[/b] Show that $ \infty =\lim_{n\to\infty} x_n. $
[b]b)[/b] Demonstrate that the sequence $ \left( x_n \right)_{n\ge 0} $ is monotone.
Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths
\[ a, f(b) \text{ and } f(b \plus{} f(a) \minus{} 1).\]
(A triangle is non-degenerate if its vertices are not collinear.)
[i]Proposed by Bruno Le Floch, France[/i]
For every $n\geq 3$, determine all the configurations of $n$ distinct points $X_1,X_2,\ldots,X_n$ in the plane, with the property that for any pair of distinct points $X_i$, $X_j$ there exists a permutation $\sigma$ of the integers $\{1,\ldots,n\}$, such that $\textrm{d}(X_i,X_k) = \textrm{d}(X_j,X_{\sigma(k)})$ for all $1\leq k \leq n$.
(We write $\textrm{d}(X,Y)$ to denote the distance between points $X$ and $Y$.)
[i](United Kingdom) Luke Betts[/i]
Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$.
[i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]
Find all $f$ functions from real numbers to itself such that for all real numbers $x,y$ the equation
\[f(f(y)+x^2+1)+2x=y+(f(x+1))^2\]
holds.
For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying
\[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\]
[i]Marko Radovanović, Serbia[/i]
Find all functions $f: \mathbb{Z}^+\to \mathbb{R}$, which satisfies $f(n+1)\geq f(n)$ for all $n\geq 1$ and $f(mn)=f(m)f(n)$ for all $(m,n)=1$.
Let $\mathfrak F$ be the family of all $k$-element subsets of the set $\{1, 2, \ldots, 2k + 1\}$. Prove that there exists a bijective function $f :\mathfrak F \to \mathfrak F$ such that for every $A \in \mathfrak F$, the sets $A$ and $f(A)$ are disjoint.
$\pi(n)$ is the number of primes that are not bigger than $n$. For $n=2,3,4,6,8,33,\dots$ we have $\pi(n)|n$. Does exist infinitely many integers $n$ that $\pi(n)|n$?
Prove that: there exists only one function $f:\mathbb{N^*}\to\mathbb{N^*}$ satisfying:
i) $f(1)=f(2)=1$;
ii)$f(n)=f(f(n-1))+f(n-f(n-1))$ for $n\ge 3$.
For each integer $m\ge 2$, find the value of $f(2^m)$.
Suppose that $n$ and $k$ are positive integers such that \[ 1 = \underbrace{\varphi( \varphi( \dots \varphi(}_{k\ \text{times}} n) \dots )). \] Prove that $n \le 3^k$.
Here $\varphi(n)$ denotes Euler's totient function, i.e. $\varphi(n)$ denotes the number of elements of $\{1, \dots, n\}$ which are relatively prime to $n$. In particular, $\varphi(1) = 1$.
[i]Proposed by Linus Hamilton[/i]
Consider the real numbers $ a\ne 0,b,c$ such that the function $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$ satisfies $ |f(x)|\le 1$ for all $ x\in [0,1]$. Find the greatest possible value of $ |a| \plus{} |b| \plus{} |c|$.
Let $a,b,c$ be positive real numbers such that $ab+bc+ca\le 3abc$. Prove that
\[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\le \sqrt{2} (\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})\]
In a village, there are $n$ houses with $n>2$ and all of them are not collinear. We want to generate a water resource in the village. For doing this, point $A$ is [i]better[/i] than point $B$ if the sum of the distances from point $A$ to the houses is less than the sum of the distances from point $B$ to the houses. We call a point [i]ideal[/i] if there doesn’t exist any [i]better[/i] point than it. Prove that there exist at most $1$ [i]ideal[/i] point to generate the resource.
Prove that if m,n are nonnegative integers and 0<=x<=1 then
$(1-x^n)^m + (1-(1-x)^m)^n \ge 1$
Let $ a < c < b$ be three real numbers and let $ f: [a,b]\rightarrow \mathbb{R}$ be a continuos function in $ c$. If $ f$ has primitives on each of the intervals $ [a,c)$ and $ (c,b]$, then prove that it has primitives on the interval $ [a,b]$.
Let $f$ be a polynomial function such that, for all real $x$,
\[f(x^2 + 1) = x^4 + 5x^2 + 3.\]
For all real $x$, $f(x^2-1)$ is
$ \textbf{(A)}\ x^4+5x^2+1\qquad\textbf{(B)}\ x^4+x^2-3\qquad\textbf{(C)}\ x^4-5x^2+1\qquad\textbf{(D)}\ x^4+x^2+3\qquad\textbf{(E)}\ \text{None of these} $
Let $f(x)$ be a real-valued function defined for $0<x<1.$ If
$$ \lim_{x \to 0} f(x) =0 \;\; \text{and} \;\; f(x) - f \left( \frac{x}{2} \right) =o(x),$$
prove that $f(x) =o(x),$ where we use the O-notation.
A function $f: \N\rightarrow\N$ is circular if for every $p\in\N$ there exists $n\in\N,\ n\leq{p}$ such that $f^n(p)=p$ ($f$ composed with itself $n$ times) The function $f$ has repulsion degree $k>0$ if for every $p\in\N$ $f^i(p)\neq{p}$ for every $i=1,2,\dots,\lfloor{kp}\rfloor$. Determine the maximum repulsion degree can have a circular function.
[b]Note:[/b] Here $\lfloor{x}\rfloor$ is the integer part of $x$.
Let $ f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} $ be a function such that for all $n > 1$ there is a prime divisor $p$ of $n$ such that \[ f(n)=f\left(\frac{n}{p}\right)-f(p). \]
Furthermore, it is given that $ f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 $. Determine $ f(2014^2)+f(2015^3)+f(2016^5) $.
Find the number of the subsets $B$ of the set $\{1,2,\cdots, 2005 \}$ such that the sum of the elements of $B$ is congruent to $2006$ modulo $2048$
Let $S$ be the set of all pairs $(m,n)$ of relatively prime positive integers $m,n$ with $n$ even and $m < n.$ For $s = (m,n) \in S$ write $n = 2^k \cdot n_o$ where $k, n_0$ are positive integers with $n_0$ odd and define \[ f(s) = (n_0, m + n - n_0). \] Prove that $f$ is a function from $S$ to $S$ and that for each $s = (m,n) \in S,$ there exists a positive integer $t \leq \frac{m+n+1}{4}$ such that \[ f^t(s) = s, \] where \[ f^t(s) = \underbrace{ (f \circ f \circ \cdots \circ f) }_{t \text{ times}}(s). \]
If $m+n$ is a prime number which does not divide $2^k - 1$ for $k = 1,2, \ldots, m+n-2,$ prove that the smallest value $t$ which satisfies the above conditions is $\left [\frac{m+n+1}{4} \right ]$ where $\left[ x \right]$ denotes the greatest integer $\leq x.$
For $a,b \in [1;\infty)$ show that \[ab\leq e^{a-1}+b\ln b\]