5
votes
2answers
151 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 + ...
0
votes
1answer
53 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 ...
2
votes
1answer
47 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
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
112 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 ...
2
votes
4answers
96 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
35 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. ...
6
votes
2answers
98 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
253 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
208 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
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: $$f(x+y)=f(x^{2}+y^{2}).$$
6
votes
2answers
226 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
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 ...
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 ...
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
290 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 $\{f^{n}(x), n \in \mathbb{N}\}$ ...
0
votes
3answers
212 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. ...
4
votes
2answers
203 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 \}$. ...
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 .
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 ...
6
votes
1answer
320 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: ...
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.