Questions tagged [functional-equation]
Puzzles involving functional equations in mathematics. Use with [mathematics]
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Function that is 0 for all positive integers divisible by x and 1 otherwise
I am working on dice probabilities and I need a function where every xth item of the set of positive integers > 0 (n) is 0 and 1 otherwise.* x is also a positive integers.
So, can you, without the ...
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How is this correct? [duplicate]
How is the following equation is correct?$29$ - $1$ = $30$
Hint-
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Functional equation: composition to get quadratic
Consider the following functional equation: $$f(f(x))=x^2+x-7\quad\quad\forall\; x\in\mathbb{R}.$$ Does there exist a function $f:\mathbb{R}\to\mathbb{R}$ satisfying this, or not?
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Functional inequality?
Find all functions $f:\mathbb{R}\to\mathbb{R}$ s.t. for all $x,y\in\mathbb{R}$, we have $$yf(x)+f(y)\ge f(xy)$$
Problem from the my math olympiad training problem set few weeks before.
Functional ...
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All wrapped in functions
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(x)f\big(f(x)+y\big)=f\big(x^2\big)+f(xy)$$ for all $x,y\in\mathbb R$
Problem by me
Most elegant solution wins!
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5
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What should you substitute?
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$xf(x)-yf(y)=(x-y)f(x+y)$$ for all $x,y\in\mathbb{R}$.
Problem by me.
Most elegant solution gets the checkmark!
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11
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Functional equation by me!
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
$$f(x)f(x+y)=xf(x)+f\big(f(x)\big)f(y)$$
for all $x,y\in\mathbb R$.
Source: Problem by me.
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Floor functional equation
Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying the following functional equation: $$f(\lfloor x\rfloor y)=f(x)\lfloor f(y)\rfloor\quad\quad\text{for all }x,y\in\mathbb{R},$$ where $\lfloor\...
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A functional equation again!
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$(x-2)f(y)+f\big(y+2f(x)\big)=f\big(x+yf(x)\big)$$ for all $x,y\in\mathbb{R}$.
Source: Math Excalibur Volume 22, Number 1 ...
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Functional Equation: Squeeze it
Determine whether there exists a function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f\big(x^3+x\big)\le x\le\big(f(x)\big)^3+f(x)$$ for all $x\in\mathbb{R}$.
Source: Math Excalibur Volume 22 No....
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Functional equation involving 2020th powers
Find all functions $f:\mathbb{N}\to\mathbb{N}$ which satisfy
$$
(n-1)^{2020}<\prod_{k=1}^{2020}f^{(k)}(n)<n^{2020}+n^{2019}\quad\quad\text{for all }n\in\mathbb{N},
$$
where $\mathbb{N}$ is the ...
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What is $f(2016)$?
Let $f: \mathbb Z \to \mathbb Z $ be a function such that $$f(34f(x)+78)=57$$
What is $f(2016)$?
Do we know any other value of $f$ for sure?
This is one of my own problems.
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10
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Functional equation $f(f(f(x)f(y)))=f(x)f(y^2)$
Find all functions $f: \mathbb R_{>0} \rightarrow \mathbb R_{>0}$ such that $f(f(f(x)f(y)))=f(x)f(y^2)$ for all $x, y \in \mathbb R_{>0}$.
I made this problem myself.
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