Found problems: 4776
Find all injective functions $f: \mathbb R \rightarrow \mathbb R$ such that for every real number $x$ and every positive integer $n$,$$ \left|\sum_{i=1}^n i\left(f(x+i+1)-f(f(x+i))\right)\right|<2016$$
[i](Macedonia)[/i]
Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(x)\geq 0\ \forall \ x\in \mathbb{R}$, $f'(x)$ exists $\forall \ x\in \mathbb{R}$ and $f'(x)\geq 0\ \forall \ x\in \mathbb{R}$ and $f(n)=0\ \forall \ n\in \mathbb{Z}$
Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$
Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be a function such that: For all $a$ and $b$ in $\mathbb{Z} - \{0\}$, $f(ab) \geq f(a) + f(b)$. Show that for all $a \in \mathbb{Z} - \{0\}$ we have $f(a^n) = nf(a)$ for all $n \in \mathbb{N}$ if and only if $f(a^2) = 2f(a)$
Answer the questions as below.
(1) Find the local minimum of $y=x(1-x^2)e^{x^2}.$
(2) Find the total area of the part bounded the graph of the function in (1) and the $x$-axis.
Find all functions $f : \mathbb N \to \mathbb N$ satisfying, for all $x \in \mathbb N$, \[ f(2f(x)) = x + 1998 . \]
Consider functions $f$ from the whole numbers (non-negative integers) to the whole numbers that have the following properties:
$\bullet$ For all $x$ and $y$, $f(xy) = f(x)f(y)$,
$\bullet$ $f(30) = 1$, and
$\bullet$ for any $n$ whose last digit is $7$, $f(n) = 1$.
Obviously, the function whose value at $n$ is $ 1$ for all $n$ is one such function. Are there any others? If not, why not, and if so, what are they?
Which integers can be represented as \[\frac{(x+y+z)^{2}}{xyz}\] where $x$, $y$, and $z$ are positive integers?
Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent:
[b]i)[/b] For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k,$
[b]ii)[/b] We have $ \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k$ for every $ s\in\left\{1,2,...,n\minus{}1\right\}$ and $ \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k$.
Find all functions $f:\mathbb R \to \mathbb R$ Such that for all real $x,y$:
$(x^2+xy+y^2)(f(x)-f(y))=f(x^3)-f(y^3)$
Find all the functions $f:R\to R$ such that \[f(x^2) + 4y^2f(y) = (f(x-y) + y^2)(f(x+y) + f(y))\] for every real $x,y$.
Let $f:[1,\infty)\to(0,\infty)$ be a continuous function. Assume that for every $a>0$, the equation $f(x)=ax$ has at least one solution in the interval $[1,\infty)$.
(a) Prove that for every $a>0$, the equation $f(x)=ax$ has infinitely many solutions.
(b) Give an example of a strictly increasing continuous function $f$ with these properties.
Let $g: R \to R$ be a linear function such that $g (1) = 0$. If $f: R \to R$ is a quadratic function such what $g (x^2) = f (x)$ and $f (x + 1) - f (x - 1) = x$ for all $x \in R$. Determine the value of $f (2019)$.
Find the number of positive integers $m$ for which there exist nonnegative integers $x_0,x_1,\ldots,x_{2011}$ such that \[ m^{x_0}=\sum_{k=1}^{2011}m^{x_k}. \]
Carl chooses a [i]functional expression[/i]* $E$ which is a finite nonempty string formed from a set $x_1, x_2, \dots$ of variables and applications of a function $f$, together with addition, subtraction, multiplication (but not division), and fixed real constants. He then considers the equation $E = 0$, and lets $S$ denote the set of functions $f \colon \mathbb R \to \mathbb R$ such that the equation holds for any choices of real numbers $x_1, x_2, \dots$. (For example, if Carl chooses the functional equation
$$ f(2f(x_1)+x_2) - 2f(x_1)-x_2 = 0, $$
then $S$ consists of one function, the identity function.
(a) Let $X$ denote the set of functions with domain $\mathbb R$ and image exactly $\mathbb Z$. Show that Carl can choose his functional equation such that $S$ is nonempty but $S \subseteq X$.
(b) Can Carl choose his functional equation such that $|S|=1$ and $S \subseteq X$?
*These can be defined formally in the following way: the set of functional expressions is the minimal one (by inclusion) such that (i) any fixed real constant is a functional expression, (ii) for any positive integer $i$, the variable $x_i$ is a functional expression, and (iii) if $V$ and $W$ are functional expressions, then so are $f(V)$, $V+W$, $V-W$, and $V \cdot W$.
[i]Proposed by Carl Schildkraut[/i]
Let $S$ denote the set of words $W = w_1w_2\ldots w_n$ of any length $n\ge0$ (including the empty string $\lambda$), with each letter $w_i$ from the set $\{x,y,z\}$. Call two words $U,V$ [i]similar[/i] if we can insert a string $s\in\{xyz,yzx,zxy\}$ of three consecutive letters somewhere in $U$ (possibly at one of the ends) to obtain $V$ or somewhere in $V$ (again, possibly at one of the ends) to obtain $U$, and say a word $W$ is [i]trivial[/i] if for some nonnegative integer $m$, there exists a sequence $W_0,W_1,\ldots,W_m$ such that $W_0=\lambda$ is the empty string, $W_m=W$, and $W_i,W_{i+1}$ are similar for $i=0,1,\ldots,m-1$. Given that for two relatively prime positive integers $p,q$ we have
\[\frac{p}{q} = \sum_{n\ge0} f(n)\left(\frac{225}{8192}\right)^n,\]where $f(n)$ denotes the number of trivial words in $S$ of length $3n$ (in particular, $f(0)=1$), find $p+q$.
[i]Victor Wang[/i]
Evaluate the following definite integral.
\[ 2^{2009}\frac {\int_0^1 x^{1004}(1 \minus{} x)^{1004}\ dx}{\int_0^1 x^{1004}(1 \minus{} x^{2010})^{1004}\ dx}\]
Let $n=p_1p_2\cdots p_s$, where $p_1,\ldots,p_s$ are distinct odd prime numbers.
(a) Prove that the expression
$$F_n(x)=\prod\left(x^{\frac n{p_{i_1}\cdots p_{i_k}}}-1\right)^{(-1)^k},$$where the product goes over all subsets $\{p_{i_1},\ldots,p_{i_k}\}$ or $\{p_1,\ldots,p_s\}$ (including itself and the empty set), can be written as a polynomial in $x$ with integer coefficients.
(b) Prove that if $p$ is a prime divisor of $F_n(2)$, then either $p\mid n$ or $n\mid p-1$.
Let $ \mathbb{R} ^{+} $ denote the set of all positive real numbers. Find all functions $ \mathbb{R} ^{+} \to \mathbb{R} ^{+} $ such that
\[ f(x+f(y)) = yf(xy+1)\]
holds for all $ x, y \in \mathbb{R} ^{+} $.
Let $M$ be an interior point of tetrahedron $V ABC$. Denote by $A_1,B_1, C_1$ the points of intersection of lines $MA,MB,MC$ with the planes $VBC,V CA,V AB$, and by $A_2,B_2, C_2$ the points of intersection of lines $V A_1, VB_1, V C_1$ with the sides $BC,CA,AB$.
[b](a)[/b] Prove that the volume of the tetrahedron $V A_2B_2C_2$ does not exceed one-fourth of the volume of $V ABC$.
[b](b)[/b] Calculate the volume of the tetrahedron $V_1A_1B_1C_1$ as a function of the volume of $V ABC$, where $V_1$ is the point of intersection of the line $VM$ with the plane $ABC$, and $M$ is the barycenter of $V ABC$.
Determine all functions $f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ that satisfies the equation
$$f\left(\frac{x+y+z}{3},\frac{a+b+c}{3}\right)=f(x,a)f(y,b)f(z,c)$$
for any real numbers $x,y,z,a,b,c$ such that $az+bx+cy\neq ay+bz+cx$.
Let $g:[2013,2014]\to\mathbb{R}$ a function that satisfy the following two conditions:
i) $g(2013)=g(2014) = 0,$
ii) for any $a,b \in [2013,2014]$ it hold that $g\left(\frac{a+b}{2}\right) \leq g(a) + g(b).$
Prove that $g$ has zeros in any open subinterval $(c,d) \subset[2013,2014].$
Let $A$ and $B$ be two subsets of $S = \{1, 2, . . . , 2000\}$ with $|A| \cdot |B| \geq 3999$. For a set $X$ , let $X-X$ denotes the set $\{s-t | s, t \in X, s \not = t\}$. Prove that $(A-A) \cap (B-B)$ is nonempty.
Find all functions $ f,g:\mathbb{Q}\longrightarrow\mathbb{Q} $ that verify the relations
$$ \left\{\begin{matrix} f(g(x)+g(y))=f(g(x))+y \\
g(f(x)+f(y))=g(f(x))+y\end{matrix}\right. , $$
for all $ x,y\in\mathbb{Q} . $
A real valued continuous function $f$ satisfies for all real $x$ and $y$ the functional equation
$$ f(\sqrt{x^2 +y^2 })= f(x)f(y).$$
Prove that
$$f(x) =f(1)^{x^{2}}.$$