Convex quadrilateral divided

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0 replies

jlacosta

Apr 2, 2025

0 replies

Inspired by 2006 Romania

sqing 1

N 7 minutes ago by sqing

Source: Own

Let $a,b,c>0$ and $abc= \frac{1}{8} .$ Prove that
$\frac{a+b}{c+1}+\frac{b+c}{a+1}+\frac{c+a}{b+1}-a-b-c \geq \frac{1}{2}$

1 reply

sqing

11 minutes ago

sqing

7 minutes ago

Nice FE from Canada Winter Camp

AshAuktober 1

N 16 minutes ago by kokcio

Source: Canada Winter (please provide a link, I can't use search function well on a train)

Find all functions $f:\mathbb{R}\to\mathbb{Z}$ such that $f(x+y)<f(x)+f(y)$ and $f(f(x))=\lfloor x\rfloor+2$ for all reals $x,y$ .

1 reply

AshAuktober

an hour ago

kokcio

16 minutes ago

NEPAL TST DAY-2 PROBLEM 1

Tony_stark0094 1

N 28 minutes ago by ThatApollo777

Let the sequence $\{a_n\}_{n \geq 1}$ be defined by
$a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N}$ Prove that
$a_n^{2025} >n^{2024}$ for all positive integers $n \geq 2$ .

1 reply

Tony_stark0094

3 hours ago

ThatApollo777

28 minutes ago

A=b

k2c901_1 85

N 44 minutes ago by alexanderhamilton124

Source: Taiwan 1st TST 2006, 1st day, problem 3

Let $a$ , $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$ . Prove that $a=b$ .

Proposed by Mohsen Jamali, Iran

85 replies

k2c901_1

Mar 29, 2006

alexanderhamilton124

44 minutes ago

No more topics!

Convex quadrilateral divided

gobathegreat 2

N Apr 13, 2022 by Steve12345

Source: Bosnia and Herzegovina TST 2010 problem 4

Convex quadrilateral is divided by diagonals into four triangles with congruent inscribed circles. Prove that this quadrilateral is rhombus.

2 replies

gobathegreat

Mar 30, 2014

Steve12345

Apr 13, 2022

Convex quadrilateral divided

G H J

Reveal topic tags

geometry proposed

geometry

G H BBookmark kLocked kLocked NReply

Source: Bosnia and Herzegovina TST 2010 problem 4

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gobathegreat

741 posts

gobathegreat

#1 h Mar 30, 2014, 4:11 PM • 2 Y

Y by Adventure10, Mango247

Convex quadrilateral is divided by diagonals into four triangles with congruent inscribed circles. Prove that this quadrilateral is rhombus.

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delegat

652 posts

delegat

#2 h May 14, 2014, 12:57 AM • 2 Y

Y by Adventure10, Mango247

This problem is from V. Prasolov: Problems in plane and solid geometry. For those interested in solution I have attached one solution that I have extracted from Croatian 2010. booklet.

Attachments:

Bosnian TST 2010 problem 4.pdf (205kb)

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Steve12345

618 posts

Steve12345

#3 h Apr 13, 2022, 7:48 PM

Y by

Let the diagonals intersect at $S$ . Not hard to verify that $S$ must be the excircle touchpoint of all triangles $\triangle ABC,\triangle BCD,\triangle CDA,\triangle DAB$ (by Casey's theorem). Then since $S$ divides these segments in $s-a,a$ where $s$ is the semi perimeter, we must have that all sides are equal by writing out the simple equations.

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