Found problems: 1513
Find all binary operations $\diamondsuit: \mathbb R_{>0}\times \mathbb R_{>0}\to \mathbb R_{>0}$ (meaning $\diamondsuit$ takes pairs of positive real numbers to positive real numbers) such that for any real numbers $a, b, c > 0$,
[list]
[*] the equation $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ holds; and
[*] if $a\ge 1$ then $a\,\diamondsuit\, a\ge 1$.
[/list]
[i]Evan Chen[/i]
Prove that there is no polynomial $P(x)$ with real coefficients that satisfies
$$P'(x)P''(x)>P(x)P'''(x)\qquad\text{for all }x\in\mathbb R.$$Is this statement true for all of the thrice differentiable real functions?
Find all functions $f\colon\mathbb R\to\mathbb R$ such that for all real numbers $x$ and $y$,
\[f(xf(y))+f(y)=f(x+y)+f(xy).\]
[i]Milan Haiman[/i]
Find all functions $f:\mathbb{R^+} \to \mathbb{R^+}$ such that for all positive real $x, y$ holds
$$xf(x)+yf(y)=(x+y)f\left(\frac{x^2+y^2}{x+y}\right)$$.
[i]Fedir Yudin[/i]
Find all monotonous functions $ f: \mathbb{R} \to \mathbb{R}$ that satisfy the following functional equation:
\[f(f(x)) \equal{} f( \minus{} f(x)) \equal{} f(x)^2.\]
Does there exist a sequence $ F(1), F(2), F(3), \ldots$ of non-negative integers that simultaneously satisfies the following three conditions?
[b](a)[/b] Each of the integers $ 0, 1, 2, \ldots$ occurs in the sequence.
[b](b)[/b] Each positive integer occurs in the sequence infinitely often.
[b](c)[/b] For any $ n \geq 2,$
\[ F(F(n^{163})) \equal{} F(F(n)) \plus{} F(F(361)).
\]
Find all the continuous functions $f : \mathbb{R} \mapsto\mathbb{R}$ such that $\forall x,y \in \mathbb{R}$,
$(1+f(x)f(y))f(x+y)=f(x)+f(y)$.
Prove that if the function $f$ is defined on the set of positive real numbers, its values are real, and $f$ satisfies the equation
$$f\left( \frac{x+y}{2}\right) + f\left(\frac{2xy}{x+y} \right) =f(x)+f(y)$$
for all positive $x,y$, then
$$2f(\sqrt{xy})=f(x)+f(y)$$
for every pair $x,y$ of positive numbers.
Prove that all continuous solutions of the functional equation
$\left(f(x)-f(y)\right)\left(f\left(\frac{x+y}{2}\right)-f\left(\sqrt{xy}\right)\right)=0 \ , \ \forall x,y\in (0,+\infty)$
are the constant functions.
Find all functions $f : R \to R$ such that $f( 2x^3 + f (y)) = y + 2x^2 f (x)$ for all real numbers $x, y$.
Polynomial $P$ with non-negative real coefficients and function $f:\mathbb{R}^+\to \mathbb{R}^+$ are given such that for all $x, y\in \mathbb{R}^+$ we have
$$f(x+P(x)f(y)) = (y+1)f(x)$$
(a) Prove that $P$ has degree at most 1.
(b) Find all function $f$ and non-constant polynomials $P$ satisfying the equality.
Find all $f: R \longrightarrow R$ such that
\[f(xy+f(x))=xf(y)+f(x)\]
for every pair of real numbers $x,y$.
Let $f$ be a function from $\mathbb Z$ to $\mathbb R$ which is bounded from above and satisfies $f(n)\le\frac12(f(n-1)+f(n+1))$ for all $n$. Show that $f$ is constant.
Given an integer number $n \ge 2$, show that there exists a function $f : R \to R$ such that $f(x) + f(2x) + ...+ f(nx) = 0$, for all $x \in R$, and $f(x) = 0$ if and only if $x = 0$.
For which positive integers $n$ do there exist two polynomials $f$ and $g$ with integer coefficients of $n$ variables $x_1, x_2, \ldots , x_n$ such that the following equality is satisfied:
\[\sum_{i=1}^n x_i f(x_1, x_2, \ldots , x_n) = g(x_1^2, x_2^2, \ldots , x_n^2) \ ? \]
Determine all functions $f : R \to R$ such that
$$f(x)f(y) = 2f(x + y) + 9xy \ \ \forall x, y \in R.$$
Is there a function $f(x)$ defined for all $x$ and such that for some $a$ and all $x$ holds the equality
$$f(x) + f(2x^2 - 1) = 2x + a?$$
Given is a nonzero real number $\alpha$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy$$ for all $x, y \in \mathbb{R}$.
Find the function $\displaystyle{f : \mathbb{R}-\left\{ 0\,\right\} \rightarrow \mathbb{R} }$ such that $$f(x)+f(1-\frac{1}{x}) = \frac{1}{x},\,\,\, \forall x \in \mathbb{R}- \{ 0, 1\,\}$$
[i](-InnoXenT-)[/i]
Consider functions $f$ that satisfy $|f(x)-f(y)|\leq \frac{1}{2}|x-y|$ for all real numbers $x$ and $y$. Of all such functions that also satisfy the equation $f(300) = f(900)$, what is the greatest possible value of
$$f(f(800))-f(f(400))?$$
$ \textbf{(A)}\ 25 \qquad
\textbf{(B)}\ 50 \qquad
\textbf{(C)}\ 100 \qquad
\textbf{(D)}\ 150 \qquad
\textbf{(E)}\ 200$
Find all polynomials $p(x)$ with real coe ffcients such that
\[p(a + b - 2c) + p(b + c - 2a) + p(c + a - 2b) = 3p(a - b) + 3p(b - c) + 3p(c - a)\]
for all $a, b, c\in\mathbb{R}$.
[i](2nd Benelux Mathematical Olympiad 2010, Problem 2)[/i]
Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying
\[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\
1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\
+ f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\
+ f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases}
\]
for all nonnegative integers $ p$, $ q$, $ r$.
Find all $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, $f(x)+f(y) = f(x+y)$ and $f(x^{2013}) = f(x)^{2013}$.
[i]Proposed by Calvin Deng[/i]
$\definecolor{A}{RGB}{190,0,60}\color{A}\fbox{A1.}$ Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $$\definecolor{A}{RGB}{80,0,200}\color{A} x^4+y^4+z^4\ge f(xy)+f(yz)+f(zx)\ge xyz(x+y+z)$$holds for all $a,b,c\in\mathbb{R}$.
[i]Proposed by [/i][b][color=#FFFF00]usjl[/color][/b].
[color=#B6D7A8]#1733[/color]
Find all functions $f : R \to R$ such that $$xf(x+xy)= xf(x)+ f(x^2)f(y)$$ for all $x,y \in R$.