inequality with perimeter and sum of diagonals of inscribed quadrilateral

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0 replies

jlacosta

Apr 2, 2025

0 replies

Functional Equation

Keith50 1

N 37 minutes ago by jasperE3

Source: Own

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f(x+f(x)+2f(y))+f(2f(x)-y)=4x+f(y)$ holds for all reals $x$ and $y$ .

1 reply

Keith50

Jun 24, 2021

jasperE3

37 minutes ago

Cursed F.E. #3

EmilXM 6

N 38 minutes ago by jasperE3

Source: Own

Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ , such that:
$f(2f(x)y) = f(xy) + xf(y)$ $\forall x,y \in \mathbb{R}$ , and if $\exists t \in \mathbb{R}$ , such that $f(2^{2021} + 1 -t)= 2^{2021} + \frac{t-1}{2}$ , then $t=1$ .
Note

I am not really sure that there is a solution without second condition, that gives us a closed form for the $F.E.$ . So it would be awesome, that if someone gives a solution without second cursed condition, or if someone shows that there is indeed a pathological solution, which doesn't let us to find a closed form. But there is really nice solution with that condition. Enjoy the problem! :D

$\rule{100cm}{0.1px}$
[center]<Previous $\hspace{3.14in}$ Next>[/center]

6 replies

EmilXM

Jun 3, 2021

jasperE3

38 minutes ago

Functional equation in R^2

EmilXM 2

N 40 minutes ago by jasperE3

Source: Own

Find all functions $f:\mathbb{R}^2 \longrightarrow \mathbb{R}$ , such that: $\begin{align*} f(f(x,y),y^2) = f(x^2,0) + 2yf(x,y) \end{align*}$ For all $x,y \in \mathbb{R}$ .

@below fixed.

2 replies

EmilXM

Oct 14, 2020

jasperE3

40 minutes ago

Inequality with a,b,c

GeoMorocco 0

an hour ago

Source: Morocco Training 2025

Let $a,b,c$ be positive real numbers such that : $ab+bc+ca=3$ . Prove that : $\frac{a\sqrt{3+bc}}{b+c}+\frac{b\sqrt{3+ca}}{c+a}+\frac{c\sqrt{3+ab}}{a+b}\ge a+b+c$

0 replies

GeoMorocco

an hour ago

0 replies

No more topics!

inequality with perimeter and sum of diagonals of inscribed quadrilateral

parmenides51 2

N Jul 28, 2018 by mihaig

Source: Brazilian Mathematical Olympiad 1985 P3

A convex quadrilateral is inscribed in a circle of radius $1$ . Show that the its perimeter less the sum of its two diagonals lies between $0$ and $2$ .

2 replies

parmenides51

Jul 27, 2018

mihaig

Jul 28, 2018

inequality with perimeter and sum of diagonals of inscribed quadrilateral

G H J

Reveal topic tags

G H BBookmark kLocked kLocked NReply

Source: Brazilian Mathematical Olympiad 1985 P3

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parmenides51

30630 posts

parmenides51

#1 h Jul 27, 2018, 6:41 AM • 2 Y

Y by Adventure10, Mango247

A convex quadrilateral is inscribed in a circle of radius $1$ . Show that the its perimeter less the sum of its two diagonals lies between $0$ and $2$ .

Z K Y

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mihaig

7339 posts

mihaig

#2 h Jul 28, 2018, 12:08 PM • 2 Y

Y by Adventure10, Mango247

This is an exceptional problem.
https://leogiugiuc.wordpress.com/2018/07/28/geometry-20/

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mihaig

7339 posts

mihaig

#3 h Jul 28, 2018, 12:59 PM • 1 Y

Y by Adventure10

parmenides51 wrote:

A convex quadrilateral is inscribed in a circle of radius $1$ . Show that the its perimeter less the sum of its two diagonals lies between $0$ and $2$ .

...I'm interested in an elementary approach

Attachments:

This post has been edited 3 times. Last edited by mihaig, Jul 28, 2018, 5:57 PM

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