inequality with perimeter and sum of diagonals of inscribed quadrilateral

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jlacosta

14 minutes ago

0 replies

k a My Retirement & New Leadership at AoPS

rrusczyk 1571

N Mar 26, 2025 by SmartGroot

I write today to announce my retirement as CEO from Art of Problem Solving. When I founded AoPS 22 years ago, I never imagined that we would reach so many students and families, or that we would find so many channels through which we discover, inspire, and train the great problem solvers of the next generation. I am very proud of all we have accomplished and I’m thankful for the many supporters who provided inspiration and encouragement along the way. I'm particularly grateful to all of the wonderful members of the AoPS Community!

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1571 replies

rrusczyk

Mar 24, 2025

SmartGroot

Mar 26, 2025

Functional equations

hanzo.ei 6

N 4 minutes ago by hanzo.ei

Source: Greekldiot

Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$

6 replies

+1 w

hanzo.ei

Mar 29, 2025

hanzo.ei

4 minutes ago

Polynomial

EtacticToe 2

N 32 minutes ago by yuribogomolov

Source: Own

Let $f(x)$ be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer $a,b,c,d$ such that $f(a)=…=f(d)=5$ .

Find all $k$ such that $f(k)=8$

2 replies

EtacticToe

Dec 14, 2024

yuribogomolov

32 minutes ago

Is this FE solvable?

Mathdreams 2

N 37 minutes ago by Mathdreams

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

2 replies

Mathdreams

Yesterday at 6:58 PM

Mathdreams

37 minutes ago

Prove that there are no tuples $(x, y, z)$ sastifying $x^2+y^2-z^2=xyz-2$

Anabcde 1

N 37 minutes ago by giangtruong13

Prove that there are no tuples $(x, y, z) \in \mathbb{Z}^3$ sastifying $x^2+y^2-z^2=xyz-2$

1 reply

Anabcde

2 hours ago

giangtruong13

37 minutes ago

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

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G H BBookmark kLocked kLocked NReply

Source: Brazilian Mathematical Olympiad 1985 P3

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parmenides51

30629 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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