5
votes
1answer
104 views

find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that : $ f(f(x))=x^2-2 $ [duplicate]

This is a very hard functional equation. the problem is this : find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that : $ f(f(x))=x^2-2 $ to solve it i have no idea! can we solve it ...
8
votes
2answers
223 views

If $f$ is a strictly increasing function with $f(f(x))=x^2+2$, then $f(3)=?$

Bdmo 2014 regionals(a tweaked version of question): If $f$ is a strictly increasing function over the reals with $f(f(x))=x^2+2$, then $f(3)=?$ Obviously,$f(3)=f(1)^2+2$ but I can't see where we ...
6
votes
2answers
106 views

Problem about functional equation (Bulgarian selection team test)

This problem was taken from the bulgarian selection team test for the 47th IMO and appeared in a chinese magazine, I came across it in my own training. http://www.math.ust.hk/excalibur/v10_n4.pdf ...
3
votes
3answers
155 views

$1992$ IMO Functional Equation problem

The problem states: Let $\Bbb R$ denote the set of all real numbers. Find all functions $f : \Bbb R \rightarrow \Bbb R$ such that $$f(x^{2}+f(y))=y+(f(x))^{2} \space \space \space \forall x, y \in ...
4
votes
3answers
146 views

How find this function $f(x)$ such $f(a+f(b))=f(a+b)+f(b)$

let function $f:R_{+}\to R_{+}$,and such $$f(a+f(b))=f(a+b)+f(b),\forall a,b\in R_{+}$$ Find $f(x)$. my try: let $a=b=1$,then $$f(1+f(1))=f(2)+f(1)$$ $a=1,b=2$,then $$f(1+f(2))=f(3)+f(2)$$ then I ...
4
votes
1answer
94 views

determine all polynomials $P(x)$ such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial

Determine all polynomials $P(x)$ with real coefficients such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial. clearly we have to show $(x+1)P(x-1)-(x-1)P(x)=c$ for all values of $x$ ($c$ is a ...
10
votes
1answer
126 views

An IMO inspired problem

This problem from IMO 1988 is said to be one of the most elegant ones in functional equations. Problem : The function $f$ is defined on the set of all positive integers as follows: \begin{align} ...
2
votes
1answer
51 views

Functional equation of non-negative function

Find all $ f:[0,\infty)\rightarrow [0,\infty) $ such that $ f (2)=0 $, $ f (x)\not= 0 $ for $ x\in [0, 2) $ and $$ f (xf (y)) f (y)=f (x+y) $$ for all $ x, y\ge 0 $. I tried plugging in values ...
4
votes
2answers
119 views

Find all functions ${\rm f} :{ \mathbb R}_{+}\to{ \mathbb R}_{+}$

Find all functions ${\rm f}:{\mathbb R}_{+} \to {\mathbb R}_{+}$ , such that $\forall\ x,y \in \mathbb R_+$ the equation $$ \left[1 + y{\rm f}\left(x\right)\right]\left[1 - y{\rm f}\left(x + ...
7
votes
1answer
174 views

A functional relation which is satisfied by $\cos x$ and $\sin x$

Assume that the functions $f,g : \mathbb R\to \mathbb R$ satisfy the relations \begin{align} \left\{ \begin{array}{ll} f(x+y) &=& f(x)f(y)-g(x)g(y), \\ g(x+y) &=& f(x)g(y)+f(y)g(x), ...
18
votes
2answers
548 views

$\cos x\,$ is the only function satisfying $\,f(x)\,f(y)-f(x+y)=\sin x\,\sin y.$

I need to find all continuous functions $f$ which satisfy the functional equation $$ f(x)\,f(y)-f(x+y)=\sin x\,\sin y, $$ for all $x,y\in\mathbb R$. I can prove that ...
2
votes
1answer
128 views

Characterization of arithmetic mean

Let $f_m$: $\mathbb{R}_{\geq 0}^m \to \mathbb{R}_{\geq 0}$ be a series of functions that satisfy symmetry (when permuting indices), strong monotonicity (in every entry), homogeneity of degree 1, ...
4
votes
2answers
71 views

finding the value of $f(\frac{1}{7})$

$f$ is a function mapping positive reals between $0$ and $1$ to reals. Let $f$ be given by, $f( \frac{x+y}{2} ) = (1-a)f(x)+af(y)$ where $y > x$ and $a$ being a constant. Also,$f(0) = 0$ and $f(1) ...
6
votes
2answers
97 views

Functional Equation f(x) = f(x/2)

Find all functions $f$ satisfying the property that $$ f(x) = f(x/2) $$ for all $x \in \mathbb{R}$ So far I've come up with the following assumptions: -$f$ is periodic, i.e of form $f(x) = A ...
10
votes
2answers
246 views

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

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$$$f(x)f(yf(x))=f(x+y)$$ A start: set y=0 to get $f(x)f(0)=f(x)$. So $f(0)=1$ unless $f$ is identically zero.
3
votes
1answer
205 views

Minimum period of function such that $f\left(x+\frac{13}{42}\right)+f(x)=f\left(x+\frac{1}{6}\right)+f\left(x+\frac{1}{7}\right) $

Let $ f$ be a function from the set of real numbers $ \mathbb{R}$ into itself such for all $ x \in \mathbb{R},$ we have $ |f(x)| \leq 1,f(x)\neq constant $ and ...
11
votes
4answers
364 views

Find all functions $f(x+y)=f(x^{2}+y^{2})$ for positive $x,y$

Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}$ such that for any $x,y\in \mathbb{R}^{+}$ the following holds: $$f(x+y)=f(x^{2}+y^{2}).$$
11
votes
5answers
516 views

$g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$.

Find all functions $g:\mathbb{R}\to\mathbb{R}$ with $g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$. I think the solutions are $0, 2, x$. If $g(x)$ is not identically $2$, then $g(0)=0$. I'm trying ...
22
votes
7answers
3k views

Find $f(x)$ such that $f(f(x)) = x^2 - 2$

Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$. Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but ...
5
votes
1answer
85 views

Functional Equation help

Came across this problem a little while ago but can't seem to get beyond a certain point. Let $f:\mathbb{N} \rightarrow \mathbb{N}$ such that $f(n+1)>f(n)$ and $$f(f(n))=3n$$ for all $n$. ...
6
votes
2answers
219 views

Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$.

Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$. Some solutions I found are $f\equiv0,f\equiv1$, $f(x)=0$ if $x\neq0$ and $f(x)=1$ if $x=0$.
6
votes
2answers
115 views

Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ . . .

Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ such that $$(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$$ I've tried subbing in heaps of values but I keep getting things like ...
5
votes
2answers
87 views

Follow on from previous question: Functional Equation - a little tricky

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f[f(x)^2+f(y)]=xf(x)+y$ for all real numbers $x$ and $y$. The answer to this has already been posted, but it doesn't explain why ...
9
votes
2answers
268 views

Functional Equation: a little tricky

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f[f(x)^2+f(y)]=xf(x)+y$ for all real numbers $x$ and $y$. Clearly $f(x)=x$ is a solution, check by substitution. I'm at a loss as ...
2
votes
1answer
81 views

functional equation problem in competition

Find all $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that $f(1)=2$ and $f(xy)=f(x)f(y)−f(x+y)+1$. for all $x,y \in \mathbb{Q}$. thank you very much!
2
votes
1answer
348 views

Functional equation book for olympiad

what may be the good suggestions in olympiad functional equations for a beginner for . I have heard of this book by B.J.Venkatachala but do not whether it will be suitable for me or not. Anybody ...
23
votes
6answers
2k views

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$
2
votes
2answers
452 views

Functional Equation. $f(mn)=f(m)f(n)$ and …

I want to prove the following. We have a function $f: \mathbb{Z} \to\mathbb{R}$ s.t. (1) $f(mn) = f(m)f(n)$ (2) $f(m+n) \leq f(m) + f(n)$ (3) $0 \leq f(x) \leq 1$ then $f(m+n) \leq \max\big(f(m), ...
6
votes
3answers
178 views

Is there a non-constant function $f:\mathbb{R}^2 \to \mathbb{Z}/2\mathbb{Z}$ that sums to 0 on corners of squares?

A problem in the 2009 Putnam asks about functions $f:\mathbb{R}^2 \to \mathbb{R}$ such that whenever $A,B,C,D$ are corners of some square we have $f(A)+f(B)+f(C)+f(D)=0$. Without spoiling the problem ...
4
votes
2answers
171 views

Seeking a contest question on functional equation on $[-1,1]$

I vaguely remember a question going something like Let $f$ be a function on $[-1,1]$ with $f$ satisfying (something like) $$f(x^2-1)=(2x)f(x).$$ Show that $f$ is identically zero on $[-1,1]$. ...
8
votes
3answers
591 views

Help remembering a Putnam Problem

I recall that there was a Putnam problem which went something like this: Find all real functions satisfying $$f(s^2+f(t)) = t+f(s)^2$$ for all $s,t \in \mathbb{R}$. There was a cool trick to ...
15
votes
3answers
2k views

The easy(?) part of IMO 2011 Problem 3

Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies $$f(x + y) \leq yf(x) + f(f(x))$$ for all real numbers $x$ and $y$. How can I prove that ...
13
votes
4answers
772 views

Solving for the implicit function $f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$ and $f(1)=1$

How can I find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(1)=1$ and $$f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$$ for all real numbers $x$ and $y$ with $y\neq0$? PS. This is ...
1
vote
2answers
443 views

Functional equation $f(x^2+y)=f(x)+f(y^2)$ from Olympiad

How do i deal with stuff like that? tried to just write many equalities but it just doesnt help... The equation is $f(x^2+y)=f(x)+f(y^2)$. EDIT: the question is to find all functions such that this ...
14
votes
3answers
726 views

The Notorious Triangle Problem

I was told this question by a friend, who said that their friend had thought about it on and off for six months without any luck. I have then had it for a while without any luck either. It is in the ...