Tagged Questions
11
votes
4answers
366 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
2answers
242 views
Functions satisfying $f(m+f(n)) = f(m) + n$
I am a real newbie when it comes to funtions, and I don't understand what is supposed to happen or what I'm supposed to find when I get given an olympiad type question concerning functions. Could you ...
11
votes
1answer
291 views
Math Olympiad - pre-periodic function
Let $c \in \mathbb{Q}$, $f(x)=x^2+c$. Define
$$f^{0}(x)=x, \ \ f^{n+1}(x)=f(f^{n}(x)), \ \forall n \in \mathbb{N}$$
We say that $x \in \mathbb{R}$ is pre-periodic if $\{f^{n}(x), n \in \mathbb{N}\}$ ...
10
votes
2answers
256 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.
7
votes
1answer
145 views
$f(x)^2 ≥ f(x + y)(f(x) + y)$ for no $f$?
Prove that there is no function $f : \mathbb{R}^+ → \mathbb{R}^+$ such that
$$f(x)^2
≥ f(x + y)(f(x) + y)$$ for all $x,
y > 0$.
I can't think of a way of solving this.
6
votes
2answers
116 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 ...
6
votes
2answers
99 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 ...
6
votes
1answer
322 views
Proving or disproving $f(n)-f(n-1)\le n, \forall n \gt 1$, for a recursive function with floors.
The Olympiad-style question I was given was as follows:
A function $f:\mathbb{N}\to\mathbb{N}$ is defined by $f(1)=1$ and for $n>1$, by: ...
6
votes
2answers
228 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 $x=0$.
5
votes
2answers
166 views
Help with complicated functional equation
Problem: Let $T=\{(p,q,r)\mid p,q,r \in \mathbb{Z}_{\geq0}\}$. Find all functions $f:T\to \mathbb{R}$ such that:
$$f(p,q,r)=\\
=\begin{cases}
0, & \text{ if } pqr = 0 \\
1 + ...
5
votes
2answers
130 views
Showing $\{x\} + \{\frac{1}{x}\} \lt 1.5$ and other problems.
For any real number $x$, let $[x]$ be the greatest integer not exceeding $x$. We also define $\{x\}=x-[x]$. We now define the function:
$f(x)=\{x\}+\{\frac{1}{x}\}$.
(a) Prove that $f(x)<1.5$ for ...
4
votes
2answers
205 views
Prove that function is bijective
Let $n \in \mathbb{N} \setminus \{ 0 \} $ and $A \in M_n(\mathbb{R})$ with $m \in \mathbb{N} \setminus \{ 0 \}$ as $A^m= \alpha \times I_n$, with $ \alpha \in \mathbb{R} \setminus \{ -1,1 \}$.
...
3
votes
2answers
296 views
Modification of 5th question from BMO'81
First of all I will introduce original problem (Question 5 from British Mathematical Olympiad).
You can find complete list of BMO'81 there BMO'81.
Find, with proof, the smallest possible value ...
3
votes
1answer
211 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
...
2
votes
1answer
48 views
Find roots of a function
$f$ is a function defined on the whole real line which has the property that $f(1+x)=f(2-x)$ for all $x$. Assume that the equation $f(x)=0$ has $8$ distinct real roots. Find the sum of these roots. I ...
2
votes
1answer
37 views
Prove $f$ not continuous at SEEMOUS Contest
Let $n$ be a nonzero natural number and $f:\mathbb{R}\to\mathbb{R}\setminus\{0\}$ be a function such that $f(2014) = 1 − f(2013)$. Let $x_1,x_2,x_3,...,x_n$ be real numbers not equal to each other. ...
2
votes
4answers
99 views
Find the number of elements in the range$ f(x) =[x] + [2x] +[2x/3] +[3x] +[4x] +[5x]$ for $0\le x \le3$.
Find the number of elements in the range $f(x) =[x] + [2x] +[2x/3] +[3x] +[4x] +[5x]$ for $0\le x \le3.$
I cant understand...It will go very long if i keep breaking them into small intervals .
2
votes
1answer
58 views
Given $|f(x) - f(y)| \le \frac{1}{2}|x-y|$ what are the points of intersection of the graph of $y = f(x)$ and the line $y = x$?
Let $f(x)$ be a real-valued function, defined for all real numbers $x$ such that $$|f(x) - f(y)| \le \frac{1}{2}|x-y|$$
for all $x,y$.
Then the number of points of intersection of the graph of $y = ...
2
votes
0answers
113 views
Problem solution by model theory
Sorry if that's not the right place for asking this, but didn't have anywhere else to go. I was cheking out some math problems in the Mathematical Olympiad site, and I found this one:
Let $\mathbb ...
1
vote
1answer
68 views
how find all function $f:(0,+\infty)\to(0,+\infty)$ that satisfy in following conditions?
how find all function $f:(0,+\infty)\to(0,+\infty)$ such that $\forall w,x,y,z\in \mathbb R^+ ,wx=yz$$$\frac{f(w)^2+f(x)^2}{f(y^2)+f(z^2)}=\frac{w^2+x^2}{y^2+z^2}.$$Thanks for any hint .
0
votes
3answers
213 views
Find the function that satisfies the following
Let $f: \mathbb{R} \to \mathbb{R}$ inconstant so that $\exists \lim_{x \to +\infty} f(x) $ and for any arithmetical progression $(a_n)$ the sequence $(f(a_n))$ is an arithmetical progression.
...
0
votes
1answer
84 views
The number of functions $f: {\cal P}_n \to \{1, 2, \dots, m\}$ such that $f(A \cap B) = \min\{f(A), f(B)\}$ (Putnam 1993)
Let ${\cal P}_n$ be the set of subsets of $\{1, 2, \dots,
n\}$. Let $c(n, m)$ be the number of functions $f: {\cal P}_n \to \{1, 2,
\dots, m\}$ such that $f(A \cap B) = \min\{f(A), f(B)\}$. Prove that
...
0
votes
1answer
56 views
Function Combination on Computer Science
I read some material on Computational Function, every one could describe the result of following combination?
suppose $g_1(x)=3x$, $g_2(x)=4x$, $f(x,y)=x+y$, how we compute combination of $f$ with ...
