Found problems: 4776
Let
$f(x)=\frac{ax+b}{cx+d}$
$F_n(x)=f(f(f...f(x)...))$ (with $n\ f's$)
Suppose that $f(0) \not =0$, $f(f(0)) \not = 0$, and for some $n$ we have $F_n(0)=0$,
show that $F_n(x)=x$ (for any valid x).
Find all pairs of functions $ f : \mathbb R \to \mathbb R$, $g : \mathbb R \to \mathbb R$ such that \[f \left( x + g(y) \right) = xf(y) - y f(x) + g(x) \quad\text{for all } x, y\in\mathbb{R}.\]
Find the number of rational numbers $r$, $0<r<1$, such that when $r$ is written as a fraction in lowest terms, the numerator and denominator have a sum of $1000$.
Consider the graphs of $y=2\log{x}$ and $y=\log{2x}$. We may say that:
$ \textbf{(A)}\ \text{They do not intersect}$
$ \qquad\textbf{(B)}\ \text{They intersect at 1 point only}$
$\qquad\textbf{(C)}\ \text{They intersect at 2 points only}$
$\qquad\textbf{(D)}\ \text{They intersect at a finite number of points but greater than 2 }$
${\qquad\textbf{(E)}\ \text{They coincide} } $
Let $f:\mathbb R\to\mathbb R$ satisfy
\[|f(x+y)-f(x-y)-y|\leq y^2\]
For all $(x,y)\in\mathbb R^2.$ Show that $f(x)=\frac x2+c$ where $c$ is a constant.
Let $ n $ be an integer greater than or equal to $ 2 $. Prove that if the real numbers $ a_1 , a_2 , \cdots , a_n $ satisfy $ a_1 ^2 + a_2 ^2 + \cdots + a_n ^ 2 = n $, then
\[\sum_{1 \le i < j \le n} \frac{1}{n- a_i a_j} \le \frac{n}{2} \]
must hold.
Find all functions $ f:\mathbb{N} \rightarrow \mathbb{ N }_0 $ satisfy the following conditions:
i) $ f(ab)=f(a)+f(b)-f(\gcd(a,b)), \forall a,b \in \mathbb{N} $
ii) For all primes $ p $ and natural numbers $ a $, $ f(a)\ge f(ap) \Rightarrow f(a)+f(p) \ge f(a)f(p)+1 $
Let $a, b > 0$ be reals such that
\[ a^3=a+1\\ b^6=b+3a \]
Show that $a>b$
A positive number, if its fractional part, integeral part, and itself are geometric series, then the number is________.
Find the minimum positive integer $k$ such that there exists a function $f$ from the set $\Bbb{Z}$ of all integers to $\{1, 2, \ldots k\}$ with the property that $f(x) \neq f(y)$ whenever $|x-y| \in \{5, 7, 12\}$.
Suppose $f$ and $g$ are differentiable functions such that \[xg(f(x))f^\prime(g(x))g^\prime(x)=f(g(x))g^\prime(f(x))f^\prime(x)\] for all real $x$. Moreover, $f$ is nonnegative and $g$ is positive. Furthermore, \[\int_0^a f(g(x))dx=1-\dfrac{e^{-2a}}{2}\] for all reals $a$. Given that $g(f(0))=1$, compute the value of $g(f(4))$.
find all functions $f:\mathbb{Z} \to \mathbb{Z} $
such that $f(m+n)+f(m)f(n)=n^2(f(m)+1)+m^2(f(n)+1)+mn(2-mn)$ holds for all $m,n \in \mathbb{Z}$
Let $\mathbb{N}$ be the set of all positive integers. $f,g:\mathbb{N} \to \mathbb{N}$ be funtions such that $f$ is onto and $g$ is one-one and $f(n)\geq g(n)$ for all positive integers $n$. Prove that $f=g$.
Let $f:S\to\mathbb{R}$ be the function from the set of all right triangles into the set of real numbers, defined by $f(\Delta ABC)=\frac{h}{r}$, where $h$ is the height with respect to the hypotenuse and $r$ is the inscribed circle's radius. Find the image, $Im(f)$, of the function.
For the nonzero numbers $ a$, $ b$, $ c$, define
\[(a,b,c)\equal{}\frac{a}{b}\plus{}\frac{b}{c}\plus{}\frac{c}{a}.\]
Find $ (2,12,9)$.
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 7 \qquad
\textbf{(E)}\ 8$
Determine all functions $f: \mathbb R \to \mathbb R$ with the property
\[f(f(x)+y)=2x+f(f(y)-x), \quad \forall x,y \in \mathbb R.\]
How many of the first 1000 positive integers can be expressed in the form
\[ \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor, \]
where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?
Find, with proof, all real-valued functions $y=g(x)$ defined and continuous on $[0,\infty)$, positive on $(0,\infty)$, such that for all $x>0$ the $y$-coordinate of the centroid of the region
$$R_x=\{(s,t)\mid0\le s\le x,\enspace0\le t\le g(s)\}$$is the same as the average value of $g$ on $[0,x]$.
The problem is about real polynomial functions, denoted by $f$, of degree $\deg f$.
a) Prove that a polynomial function $f$ can`t be wrriten as sum of at most $\deg f$ periodic functions.
b) Show that if a polynomial function of degree $1$ is written as sum of two periodic functions, then they are unbounded on every interval (thus, they are "wild").
c) Show that every polynomial function of degree $1$ can be written as sum of two periodic functions.
d) Show that every polynomial function $f$ can be written as sum of $\deg f+1$ periodic functions.
e) Give an example of a function that can`t be written as a finite sum of periodic functions.
[i]Dan Schwarz[/i]
Let $c, s$ be real functions defined on $\mathbb{R}\setminus\{0\}$ that are nonconstant on any interval and satisfy
\[c\left(\frac{x}{y}\right)= c(x)c(y) - s(x)s(y)\text{ for any }x \neq 0, y \neq 0\]
Prove that:
$(a) c\left(\frac{1}{x}\right) = c(x), s\left(\frac{1}{x}\right) = -s(x)$ for any $x = 0$, and also $c(1) = 1, s(1) = s(-1) = 0$;
$(b) c$ and $s$ are either both even or both odd functions (a function $f$ is even if $f(x) = f(-x)$ for all $x$, and odd if $f(x) = -f(-x)$ for all $x$).
Find functions $c, s$ that also satisfy $c(x) + s(x) = x^n$ for all $x$, where $n$ is a given positive integer.
Let $f:[0,1)\to (0,1)$ a continous onto (surjective) function.
a) Prove that, for all $a\in(0,1)$, the function $f_a:(a,1)\to (0,1)$, given by $f_a(x) = f(x)$, for all $x\in(a,1)$ is onto;
b) Give an example of such a function.
Let $a$ be a real number in the open interval $(0,1)$. Let $n\geq 2$ be a positive integer and let $f_n\colon\mathbb{R}\to\mathbb{R}$ be defined by $f_n(x) = x+\frac{x^2}{n}$. Show that
\[\frac{a(1-a)n^2+2a^2n+a^3}{(1-a)^2n^2+a(2-a)n+a^2}<(f_n \circ\ \cdots\ \circ f_n)(a)<\frac{an+a^2}{(1-a)n+a}\] where there are $n$ functions in the composition.
Let $f,g:\mathbb R\to\mathbb R$ be continuous functions such that $g$ is differentiable. Assume that $(f(0)-g'(0))(g'(1)-f(1))>0$. Show that there exists a point $c\in (0,1)$ such that $f(c)=g'(c)$.
[i]Proposed by Fereshteh Malek, K. N. Toosi University of Technology[/i]
Let $S \subset [0, 1]$ be a set of 5 points with $\{0, 1\} \subset S$. The graph of a real function $f : [0, 1] \to [0, 1]$ is continuous and increasing, and it is linear on every subinterval $I$ in $[0, 1]$ such that the endpoints but no interior points of $I$ are in $S$.
We want to compute, using a computer, the extreme values of $g(x, t) = \frac{f(x+t)-f(x)}{ f(x)-f(x-t)}$ for $x - t, x + t \in [0, 1]$. At how many points $(x, t)$ is it necessary to compute $g(x, t)$ with the computer?
Let \(f:\{0, 1\}^n \to \{0, 1\} \subseteq \mathbb{R}\) be a function. Call such a function a Boolean function.
Let \(\wedge\) denote the component-wise multiplication in \(\{0,1\}^n\). For example, for \(n = 4\), \[(0,0,1,1) \wedge (0,1,0,1) = (0,0,0,1).\]
Let \(S = \left\{i_1,i_2,\ldots ,i_k\right\} \subseteq \left\{1,2,\ldots ,n\right\}\). \(f\) is called the oligarchy function over \(S\) if \[f (x) = x_{i_1},x_{i_2},\ldots,x_{i_k}\ \text{ (with the usual multiplication),}\]
where \(x_i\) denotes the \(i\)-th component of \(x\). By convention, \(f\) is called the oligarchy function over \(\emptyset\) if \(f\) is constantly 1.
(i) Suppose \(f\) is not constantly zero. Show that \(f\) is an oligarchy function [u]if and only if[/u] \(f\) satisfies \[f(x\wedge y)=f(x)f(y),\ \forall x,y\in\left\{0,1\right\}^n.\]
Let \(Y\) be a uniformly distributed random variable over \(\left\{0, 1\right\}^n\). Let \(T\) be an operator that maps Boolean functions to functions \(\left\{0, 1\right\}^n\to\mathbb{R}\), such that
\[(Tf)(x)=E_Y(f(x\wedge Y)),\ \forall x\in\left\{0,1\right\}^n\]
where \(E_Y()\) denotes the expectation over \(Y\). \(f\) is called an eigenfunction of \(T\) if \(\exists\lambda\in\mathbb{R}\backslash\left\{0\right\}\) such that
\[(Tf)(x)=\lambda f(x),\ \forall x\in\left\{0,1\right\}^n\]
(ii) Prove that \(f\) is an eigenfunction of \(T\) [u]if and only if[/u] \(f\) is an oligarchy function.