Questions tagged [functional-equations]

Question 1

Find all the solutions to $f(f(x) + 2020x + y) = f(2021x) + f(y)$ for all $x,y >0$ when: i) $f: \mathbb{N} \mapsto \mathbb{N}$ ii) $f: \mathbb{R}^{+} \mapsto \mathbb{R}^{+}$ The second part was a ...

Question 2

Find all funtions $f:R^+\to R^+$ satisfying: $f(f(x)+2020x+y)=f(2021x)+f(y),\forall x,y>0$

Question 3

A function $f: \mathbb{N} \to \mathbb{N}$ satisfies $$\underbrace{f(f(\dots f(n)\dots))}_{f(n) \text{ times}} = \frac{n^2}{f(f(n))}$$ for all positive integers $n$. What are all possible values of $f(...

Question 4

The problem is as follows Find every function $\mathbb{R}^+ \to \mathbb{R}^+$ satisfying $ f\left(x^{f(y)}\right) = {f(x)}^y $ for all $x, y \in \mathbb{R}^+$. There is the clear $f(x) = 1$ solution....

Question 5

This question arose from the simple observation that if $f(x)=\sin(x)$ $$\sin(2x)=2\sin(x)\cos(x)=2f(x)f'(x)$$ However a similar property does not hold for $\sin(3x)$ This came with the additional ...

Question 6

I was reading the solution provided for PB–Basic–007 in the DeepSeek-Math-V2 IMO-ProofBench-Basic dataset, and I am unsure whether one of its main steps is valid. I would like to confirm whether I am ...

Question 7

This problem is from the most recent USAMTS Round 2, which has ended. Let $\Bbb{Z}^+$ denote the set of positive integers. Determine, with proof, whether there exist functions $f,g:\Bbb{Z}^+\to\Bbb{Z}...

Question 8

Find all functions $\mathbb{R}^+$$\to$$\mathbb{R}^+$ such that: $$ \frac{f(xy)}{f(x+y)}+f(\frac{x}{y})=f(xy)+1 $$ Here is my solution: Let $P(x,y)$ denote the given assertion. Comparing $P(x,y)$ and $...

Question 9

Find all functions $f : \mathbb R \to \mathbb R$ such that: $f (x f ( y ))+y f ( x) = x f ( y ) + f ( x y )$, $\forall\ x , y \in \mathbb{R}$ and b) $\exists M \in \mathbb R$ such that $f(x)<M$ ...

Question 10

I wanna to prove If for all value of $x$ and $x'$ in real numbers we have; $$f(x) + f(x') = f(x+x')$$ Then $f(x) = Kx.$ I have not any counterexample for this but I can't prove it Thank you for your ...

Question 11

Consider a function $f : (0,\infty) \to (0,\infty)$ satisfying the identity $$ f(x^a y^b) \;=\; f(x)^{1/a}\, f(y)^{1/b} \qquad\text{for all } x,y>0 \text{ and all real } a,b\neq 0. $$ This can be ...

Question 12

Problem: Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(y^2+1)+f(xy)=f(x+y)f(y)+1$ holds for all $x,y \in \mathbb{R}$. My approach: I have got $f(0)=0,1$. Taking $f(0)=1$, and setting $...

Question 13

I am asking this question mainly to probe the knowledge of people already familiar with this problem, otherwise I would advise caution to the unfamiliar trying to use computation, this can be really ...

Question 14

I'm looking for a function, $f$, which satisfies \begin{align} tf(x)=f\left(x+\frac{1}{t}-1\right);\quad f(1)=1. \end{align} My attempt: Let $t\rightarrow 1/(1+\log t)$ and $x\rightarrow \log x$, ...

Question 15

Need to find out all the functions $f \colon \mathbb{R} \to \mathbb{R}$ such that $$f(x+y) \leq f(xy).$$ Since $f(x) \leq f(0)$ for every $x \in \mathbb{R}$ and $f(0) \leq f(-x^2)$, it follows that $f(...

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Questions tagged [functional-equations]

On solving the nested functional equation $f(f(x) + 2020x + y) = f(2021x) + f(y)$

Functional equation [closed]

USAMO 2019 P1: Functional equation $f^{f(n)}(n) = \frac{n^2}{f(f(n))}$

Finding real solutions to a functional equation with nested exponent

Find real functions $f$ such that $\forall n \in \mathbb{Z}^+, f(nx)=nf'(x)f^{n-1}(x)$

Is the PB–Basic–007 “solution” in DeepSeek-Math-V2 correct? Possible flaw in its convex-hull argument

USAMTS Inspired Function Problem

Functional Equation Problem Using Continuity

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f$ is bounded above and $f(xf(y))+yf(x)=xf(y)+f(xy)$

$f(x) + f(t) = f(x+t)$ then $f(x)=kx$ [duplicate]

Generalization of Cauchy's functional equation. What are the general solutions, $f$?

Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(y^2+1)+f(xy)=f(x+y)f(y)+1$ holds for all $x,y \in \mathbb{R}$

Is there an easy way to know if a rational function is an n-th (compositional) iteration of a power series with indices in $\mathbb{Z}$?

Are there any non-trivial functions satisfying $tf(x)=f\left(x+\frac{1}{t}-1\right)$?

Yet another nice functional inequality [duplicate]

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