sin inequality

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0 replies

jlacosta

Apr 2, 2025

0 replies

Interesting inequalities

sqing 9

N a minute ago by sqing

Source: Own

Let $a,b\geq 0$ and $a+b+ab=3.$ Prove that
$ab^2( b +1) \leq 4$ $ab( b +1) \leq \frac{9}{4}$ $a^2b ( a+b^2 ) \leq \frac{76}{27}$ $a^2b( b +1 ) \leq \frac{3(69-11\sqrt{33})}{8}$ $a^2b^2( b +1 ) \leq \frac{2(73\sqrt{73}-595)}{27}$

9 replies

+1 w

sqing

Yesterday at 3:12 AM

sqing

a minute ago

Interesting inequality

sqing 2

N 7 minutes ago by sqing

Source: Own

Let $a,b,c>0 .$ Prove that
$\frac{a}{b}+ \frac{kb^2}{c^2} + \frac{c}{a}\geq 5\sqrt[5]{\frac{k}{16}}$ Where $k >0.$
$\frac{a}{b}+ \frac{16b^2}{c^2} + \frac{c}{a}\geq 5$ $\frac{a}{b}+ \frac{ b^2}{2c^2} + \frac{c}{a}\geq \frac{5}{2}$

2 replies

sqing

2 hours ago

sqing

7 minutes ago

A nice triangle center on line OI

proglote 23

N 9 minutes ago by Ilikeminecraft

Source: Brazil MO 2013, problem #6

The incircle of triangle $ABC$ touches sides $BC, CA$ and $AB$ at points $D, E$ and $F$ , respectively. Let $P$ be the intersection of lines $AD$ and $BE$ . The reflections of $P$ with respect to $EF, FD$ and $DE$ are $X,Y$ and $Z$ , respectively. Prove that lines $AX, BY$ and $CZ$ are concurrent at a point on line $IO$ , where $I$ and $O$ are the incenter and circumcenter of triangle $ABC$ .

23 replies

proglote

Oct 24, 2013

Ilikeminecraft

9 minutes ago

Balkan MO Shortlist official booklet

guptaamitu1 8

N 13 minutes ago by pythagorazz

These days I was trying to find the official booklet of Balkan MO Shortlist. But apparently, there's no big list of all Balkan shortlists for previous years. Through some sources, I have been able to find the official booklet for the following years. So if people have it for other years too, can they please put it on this thread, so that everything is in one place.
[list]
[*] 2021
[*] 2020
[*] 2019
[*] 2018
[*] 2017
[*] 2016
[/list]

8 replies

guptaamitu1

Jun 19, 2022

pythagorazz

13 minutes ago

No more topics!

sin inequality

Konigsberg 2

N Jul 2, 2021 by A.L.E.X

Source: MOP 2005 Homework - Red Group #21

Let $ABC$ be a triangle. Prove that $\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab} \ge 4\left(\sin^2\frac{A}{2}+\sin^2\frac{B}{2}+\sin^2\frac{C}{2}\right).$

2 replies

Konigsberg

May 6, 2014

A.L.E.X

Jul 2, 2021

sin inequality

G H J

Reveal topic tags

G H BBookmark kLocked kLocked NReply

Source: MOP 2005 Homework - Red Group #21

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Konigsberg

2211 posts

Konigsberg

#1 h May 6, 2014, 3:25 PM • 3 Y

Y by centslordm, Adventure10, Mango247

Let $ABC$ be a triangle. Prove that $\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab} \ge 4\left(\sin^2\frac{A}{2}+\sin^2\frac{B}{2}+\sin^2\frac{C}{2}\right).$

This post has been edited 1 time. Last edited by djmathman, Dec 7, 2016, 12:30 AM

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thugzmath10

708 posts

thugzmath10

#2 h May 6, 2014, 3:51 PM • 3 Y

Y by wiseman, centslordm, Adventure10

Proof

From $\cos A=\frac{b^2+c^2-a^2}{2bc}$ and the identity $\cos A=1-2\sin^2 \frac{A}{2}$ , we have $2\sin^2 \frac{A}{2}=\frac{a^2-(b-c)^2}{2bc}$ .

Thus, $4\sin^2 \frac{A}{2}=\frac{a^2-(b-c)^2}{bc}\leq \frac{a^2}{bc}$ (since $(b-c)^2\geq 0)$ . Similarly, $4\sin^2 \frac{B}{2}\leq \frac{b^2}{ca}$ and $4\sin^2 \frac{C}{2}\leq \frac{c^2}{ab}$ .

Therefore, adding these three inequalities, we get the desired inequality.

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A.L.E.X

133 posts

A.L.E.X

#3 h Jul 2, 2021, 8:26 PM • 1 Y

Y by centslordm

My Solution

We know that $:$
$\sin(\frac{\alpha}{2})=\sqrt{\frac{1-cos(\alpha)}{2}}$ Also $:cosA=\frac{b^2{2}+c^{2}-a^{2}}{2bc};cosB=\frac{c^{2}+a^{2}-b^{2}}{2ca};cosC=\frac{a^{2}+b^{2}-c^{2}}{2ab}$

So we have to prove that $: \sum \frac{a^{2}}{bc}\geq 4(\frac{3}{2}-\sum \frac{a^{2}+b^{2}-c^{2}}{4ab})\Leftrightarrow \sum \frac{b^{2}+c^{2}}{2bc}\geq 6$ Which is just $AM-GM$ $\blacksquare$

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