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]

Does there exist a function $f:\mathbb{R}\to \mathbb{R}$ such that $$f(x^2+f(y))=f(x)^2-y$$ for all $x, y\in \mathbb{R}$?

Find all functions $f: \mathbb R \to \mathbb R$ such that $$f(x+f(x+y))=x+f(f(x)+y)$$ holds for all real numbers $x$ and $y$.

Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y\in(1,\infty)$ and $x^2\le y\le x^3,$ then $(f(x))^2\le f(y) \le (f(x))^3.$

Find all functions $f : N \to N$ such that $f(m)+f(n)$ divides $m+n$ for all positive integers $m$ and $n$.

Prove that a function $f:\mathbb{R}_+\to\mathbb{R}$ satisfies \[f(x+y)-f(x)-f(y)=f\left(\frac{1}{x}+\frac{1}{y}\right)\]if and only if it satisfies $f(xy)=f(x)+f(y)$.

Find all functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, such that 1) $f(0,x)$ is non-decreasing ; 2) for any $x,y \in \mathbb{R}$, $f(x,y)=f(y,x)$ ; 3) for any $x,y,z \in \mathbb{R}$, $(f(x,y)-f(y,z))(f(y,z)-f(z,x))(f(z,x)-f(x,y))=0$ ; 4) for any $x,y,a \in \mathbb{R}$, $f(x+a,y+a)=f(x,y)+a$ .

Let $f : Z_{\ge 0} \to Z_{\ge 0}$ satisfy the functional equation $$f(m^2 + n^2) =(f(m) - f(n))^2 + f(2mn)$$ for all nonnegative integers $m, n$. If $8f(0) + 9f(1) = 2006$, compute $f(0)$.

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(2f(x)) = f(x - f(y)) + f(x) + y$$ for all $x, y \in \mathbb{R}$.

Find all functions $f : R^+ \to R^+$ such that for all $x, y > 0$ $$f(yf(x))(x + y) = x^2(f(x) + f(y)).$$

Find all functions $f:R \rightarrow R$ such that $$(x+y^2)f(yf(x))=xyf(y^2+f(x))$$, where $x,y \in \mathbb{R}$

Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that for every $x,y \in \mathbb{R}$ $$f(x^{4}-y^{4})+4f(xy)^{2}=f(x^{4}+y^{4})$$

Let $f(n)$ be defined on the positive integers and satisfy: $f(prime) = 1$, $f(ab) = a f(b) + f(a) b$. Show that $f$ is unique and find all $n$ such that $n = f(n)$.

Let $A$ be a set of functions $f : R\to R$. For all $f_1, f_2 \in A$ there exists a $f_3 \in A$ such that $f_1(f_2(y) - x)+ 2x = f_3(x + y)$ for all $x, y \in R$. Prove that for all $f \in A$, we have $f(x - f(x))= 0$ for all $x \in R$.

For some positive integer $n,$ Elmo writes down the equation \[x_1+x_2+\dots+x_n=x_1+x_2+\dots+x_n.\] Elmo inserts at least one $f$ to the left side of the equation and adds parentheses to create a valid functional equation. For example, if $n=3,$ Elmo could have created the equation \[f(x_1+f(f(x_2)+x_3))=x_1+x_2+x_3.\] Cookie Monster comes up with a function $f: \mathbb{Q}\to\mathbb{Q}$ which is a solution to Elmo's functional equation. (In other words, Elmo's equation is satisfied for all choices of $x_1,\dots,x_n\in\mathbb{Q})$. Is it possible that there is no integer $k$ (possibly depending on $f$) such that $f^k(x)=x$ for all $x$? [i]Srinivas Arun[/i]

a) Find at least two functions $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that $$\displaystyle{2f(x^2)\geq xf(x) + x,}$$ for all $x \in \mathbb{R}^+.$ b) Let $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ be a function such that $$\displaystyle{2f(x^2)\geq xf(x) + x,}$$ for all $x \in \mathbb{R}^+.$ Show that $ f(x^3)\geq x^2,$ for all $x \in \mathbb{R}^+.$ Can we find the best constant $a\in \Bbb{R}$ such that $f(x)\geq x^a,$ for all $x \in \mathbb{R}^+?$

Let $P$ be a given quadratic polynomial. Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that $$f(x+y)=f(x)+f(y)\text{ and } f(P(x))=f(x)\text{ for all }x,y\in\mathbb{R}.$$

Find all functions $f:\mathbb{R^+}\rightarrow\mathbb{R^+}$ such that $$\frac{x+f(y)}{xf(y)}=f(\frac{1}{y}+f(\frac{1}{x}))$$ for all positive real numbers $x$ and $y$.

Let $f : \mathbb Q \to \mathbb Q$ be a function such that for any $x,y \in \mathbb Q$, the number $f(x+y)-f(x)-f(y)$ is an integer. Decide whether it follows that there exists a constant $c$ such that $f(x) - cx$ is an integer for every rational number $x$. [i]Proposed by Victor Wang[/i]

Find all functions from $\mathbb{N}\cup\{0\}\to\mathbb{N}\cup\{0\}$ such that $f(m^2+mf(n))=mf(m+n)$, for all $m,n\in \mathbb{N}\cup\{0\}$.

Find all functions $f:\mathbb Z^+\to\mathbb R^+$ that satisfy $f(nk^2)=f(n)f^2(k)$ for all positive integers $n$ and $k$, furthermore $\lim\limits_{n\to\infty}\dfrac{f(n+1)}{f(n)}=1$.

Find all functions $f : \mathbb{N} \to \mathbb{N}$ with $$f(x) + yf(f(x)) < x(1 + f(y)) + 2021$$ holds for all positive integers $x,y.$

Find all functions $f : R- \{-1\} \to R$ such that $$f(x)f \left( f \left(\frac{1 - y}{1 + y} \right)\right) = f\left(\frac{x + y}{xy + 1}\right) $$ for all $x, y \in R$ that satisfy $(x + 1)(y + 1)(xy + 1) \ne 0$.

Let $n\in\mathbb{N}\setminus\left\{0\right\}$ be a positive integer. Find all the functions $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying that : $$f(x+y^{2n})=f(f(x))+y^{2n-1}f(y),(\forall)x,y\in\mathbb{R},$$ and $f(x)=0$ has an unique solution.

[b]a)[/b] Prove that not all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equality $$ f(x-1)+f(x+1) =\sqrt 5f(x) ,\quad\forall x\in\mathbb{R} , $$ are periodic. [b]b)[/b] Prove that that all functions $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equality $$ g(x-1)+g(x+1)=\sqrt 3g(x) ,\quad\forall x\in\mathbb{R} , $$ are periodic.