interesting function equation (fe) in IR

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0 replies

jlacosta

May 1, 2025

0 replies

Numbers on circle

RagvaloD 1

N 18 minutes ago by Radin_

Source: St Petersburg Olympiad 2008, Grade 11, P4

There are $100$ numbers on circle, and no one number is divided by other. In same time for all numbers we make next operation:
If $(a,b)$ are two neighbors ( $a$ is left neighbor) , then we write between $a,b$ number $\frac{a}{(a,b)}$ and erase $a,b$
This operation was repeated some times. What maximum number of $1$ we can receive ?

Example: If we have circle with $3$ numbers $4,5,6$ then after operation we receive circle with numbers $\frac{4}{(4,5)}=4,\frac{5}{(5,6)}=5, \frac{6}{(6,4)}=3$ .

1 reply

RagvaloD

Aug 30, 2017

Radin_

18 minutes ago

3 var inequality

JARP091 5

N 19 minutes ago by Mathzeus1024

Source: Own

Let $x, y, z \in \mathbb{R}^+$ . Prove that
$\sum_{\text{cyc}} \frac{x^3}{y^2 + z^2} \geq \frac{x + y + z}{2}$ without using the Rearrangement Inequality or Chebyshev's Inequality.

5 replies

JARP091

2 hours ago

Mathzeus1024

19 minutes ago

Number Theory Chain!

JetFire008 63

N 26 minutes ago by GreekIdiot

I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1

Starting with the simplest
What is $2+3$ ?

63 replies

JetFire008

Apr 7, 2025

GreekIdiot

26 minutes ago

Computing functions

BBNoDollar 0

43 minutes ago

Let $f : [0, \infty) \to [0, \infty)$ , $f(x) = \dfrac{ax + b}{cx + d}$ , with $a, d \in (0, \infty)$ , $b, c \in [0, \infty)$ . Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
$f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.$ (For $n \in \mathbb{N}^*$ and $x \geq 0$ , the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$ . )

0 replies

BBNoDollar

43 minutes ago

0 replies

No more topics!

interesting function equation (fe) in IR

skellyrah 2

N Apr 23, 2025 by jasperE3

Source: mine

find all function F: IR->IR such that $xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy)$

2 replies

skellyrah

Apr 23, 2025

jasperE3

Apr 23, 2025

interesting function equation (fe) in IR

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Source: mine

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skellyrah

25 posts

skellyrah

#1 h Apr 23, 2025, 9:51 AM

Y by

find all function F: IR->IR such that $xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy)$

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CrazyInMath

459 posts

CrazyInMath

#2 h Apr 23, 2025, 11:37 AM

Y by

solution

$P(1,x)\Longrightarrow f(x)=xf(f(1))$
So $f(x)=cx$
Plug in to get $f(x)=x$ or $f(x)=0$

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jasperE3

11364 posts

jasperE3

#4 h Apr 23, 2025, 8:33 PM

Y by

skellyrah wrote:

find all function F: IR->IR such that $xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy)$

If IR means irrational numbers, and the problem is to find all $f$ such that $xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy)$ for all $x,y\in\mathbb{IR}$ then setting $x=f(y)^{-1}$ gives a contradiction

This post has been edited 1 time. Last edited by jasperE3, Apr 23, 2025, 9:43 PM

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