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abc(a+b+c)=3, show that prod(a+b)>=8 [Indian RMO 2012(b) Q4]
Potla   31
N an hour ago by sqing
Let $a,b,c$ be positive real numbers such that $abc(a+b+c)=3.$ Prove that we have
\[(a+b)(b+c)(c+a)\geq 8.\]
Also determine the case of equality.
31 replies
Potla
Dec 2, 2012
sqing
an hour ago
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Medium geometry with AH diameter circle
v_Enhance   93
N Yesterday at 10:36 AM by waterbottle432
Source: USA TSTST 2016 Problem 2, by Evan Chen
Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Denote by $M$, $N$ the midpoints of $\overline{AH}$, $\overline{BC}$. Suppose the circle $\gamma$ with diameter $\overline{AH}$ meets the circumcircle of $ABC$ at $G \neq A$, and meets line $AN$ at a point $Q \neq A$. The tangent to $\gamma$ at $G$ meets line $OM$ at $P$. Show that the circumcircles of $\triangle GNQ$ and $\triangle MBC$ intersect at a point $T$ on $\overline{PN}$.

Proposed by Evan Chen
93 replies
v_Enhance
Jun 28, 2016
waterbottle432
Yesterday at 10:36 AM
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Reflected point lies on radical axis
Mahdi_Mashayekhi   3
N Apr 19, 2025 by Mahdi_Mashayekhi
Source: Iran 2025 second round P4
Given is an acute and scalene triangle $ABC$ with circumcenter $O$. $BO$ and $CO$ intersect the altitude from $A$ to $BC$ at points $P$ and $Q$ respectively. $X$ is the circumcenter of triangle $OPQ$ and $O'$ is the reflection of $O$ over $BC$. $Y$ is the second intersection of circumcircles of triangles $BXP$ and $CXQ$. Show that $X,Y,O'$ are collinear.
3 replies
Mahdi_Mashayekhi
Apr 19, 2025
Mahdi_Mashayekhi
Apr 19, 2025
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A mediane as a radical axis
breloje17fr   0
Apr 19, 2025
Hello, ladies and gentlemen
Let ABC be a triangle, and D, E and F the middles of the sides BC, CA and AB respectively. The perpendicular bissector of CA intersects the line AB at E' and the bissector of the A angle at K, and the perpendicular bissector of AB intersects the line AC at F' and the bissector of the A angle at J. The two circles passing through J, F and E' and through K, E and F' intersect each other at P and Q.
Show that the radical axis of these circles is the A-mediane of ABC.
0 replies
breloje17fr
Apr 19, 2025
0 replies
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Concurrence in Cyclic Quadrilateral
GrantStar   38
N Apr 17, 2025 by wu2481632
Source: IMO Shortlist 2023 G3
Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$.

Prove that lines $AD, PM$, and $BC$ are concurrent.
38 replies
GrantStar
Jul 17, 2024
wu2481632
Apr 17, 2025
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Circles with same radical axis
Jalil_Huseynov   9
N Apr 17, 2025 by Nari_Tom
Source: DGO 2021, Individual stage, Day2 P3
Let $O$ be the circumcenter of triangle $ABC$. The altitudes from $A, B, C$ of triangle $ABC$ intersects the circumcircle of the triangle $ABC$ at $A_1, B_1, C_1$ respectively. $AO, BO, CO$ meets $BC, CA, AB$ at $A_2, B_2, C_2$ respectively. Prove that the circumcircles of triangles $AA_1A_2, BB_1B_2, CC_1C_2$ share two common points.

Proporsed by wassupevery1
9 replies
Jalil_Huseynov
Dec 26, 2021
Nari_Tom
Apr 17, 2025
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2011 Japan Mathematical Olympiad Finals Problem 1
Kunihiko_Chikaya   20
N Apr 14, 2025 by zhoujef000
Source: Japanese MO Finals 2011
Given an acute triangle $ABC$ with the midpoint $M$ of $BC$. Draw the perpendicular $HP$ from the orthocenter $H$ of $ABC$ to $AM$.
Show that $AM\cdot PM=BM^2$.
20 replies
Kunihiko_Chikaya
Feb 11, 2011
zhoujef000
Apr 14, 2025
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IMO 2008, Question 1
orl   154
N Apr 8, 2025 by eg4334
Source: IMO Shortlist 2008, G1
Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$.

Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic.

Author: Andrey Gavrilyuk, Russia
154 replies
orl
Jul 16, 2008
eg4334
Apr 8, 2025
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Lines pass through a common point
April   4
N Apr 7, 2025 by Nari_Tom
Source: Baltic Way 2008, Problem 18
Let $ AB$ be a diameter of a circle $ S$, and let $ L$ be the tangent at $ A$. Furthermore, let $ c$ be a fixed, positive real, and consider all pairs of points $ X$ and $ Y$ lying on $ L$, on opposite sides of $ A$, such that $ |AX|\cdot |AY| = c$. The lines $ BX$ and $ BY$ intersect $ S$ at points $ P$ and $ Q$, respectively. Show that all the lines $ PQ$ pass through a common point.
4 replies
April
Nov 23, 2008
Nari_Tom
Apr 7, 2025
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perpendicularity involving ex and incenter
Erken   19
N Apr 6, 2025 by Primeniyazidayi
Source: Kazakhstan NO 2008 problem 2
Suppose that $ B_1$ is the midpoint of the arc $ AC$, containing $ B$, in the circumcircle of $ \triangle ABC$, and let $ I_b$ be the $ B$-excircle's center. Assume that the external angle bisector of $ \angle ABC$ intersects $ AC$ at $ B_2$. Prove that $ B_2I$ is perpendicular to $ B_1I_B$, where $ I$ is the incenter of $ \triangle ABC$.
19 replies
Erken
Dec 24, 2008
Primeniyazidayi
Apr 6, 2025
J
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Geo with unnecessary condition
egxa   8
N Apr 4, 2025 by ErTeeEs06
Source: Turkey Olympic Revenge 2024 P4
Let the circumcircle of a triangle $ABC$ be $\Gamma$. The tangents to $\Gamma$ at $B,C$ meet at point $E$. For a point $F$ on line $BC$ which is not on the segment $BC$, let the midpoint of $EF$ be $G$. Lines $GB,GC$ meet $\Gamma$ again at points $I,H$ respectively. Let $M$ be the midpoint of $BC$. Prove that the points $F,I,H,M$ lie on a circle.

Proposed by Mehmet Can Baştemir
8 replies
egxa
Aug 6, 2024
ErTeeEs06
Apr 4, 2025
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J
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Something nice
KhuongTrang   25
N Apr 17, 2025 by KhuongTrang
Source: own
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
25 replies
KhuongTrang
Nov 1, 2023
KhuongTrang
Apr 17, 2025
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Something nice
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Reveal topic tags
inequalities
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G H BBookmark kLocked kLocked NReply
Source: own
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KhuongTrang
727 posts
KhuongTrang
#1 h  Nov 1, 2023, 12:56 PM • 1 Y
Y by Zuyong
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
This post has been edited 2 times. Last edited by KhuongTrang, Nov 19, 2023, 11:59 PM
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mihaig
7343 posts
mihaig
#2 h  Nov 1, 2023, 3:42 PM
Y by
Beauty. But difficult
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KhuongTrang
727 posts
KhuongTrang
#19 h  Nov 9, 2023, 1:09 PM • 2 Y
Y by MihaiT, Zuyong
Non sense post.
This post has been edited 1 time. Last edited by KhuongTrang, Dec 23, 2023, 1:30 PM
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KhuongTrang
727 posts
KhuongTrang
#31 h  Nov 19, 2023, 5:34 AM • 1 Y
Y by Zuyong
Something not relevant
This post has been edited 1 time. Last edited by KhuongTrang, Dec 16, 2023, 3:48 AM
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arqady
30207 posts
arqady
#32 h  Nov 19, 2023, 6:24 AM • 1 Y
Y by teomihai
KhuongTrang wrote:
Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=1.$ Prove that$$a\sqrt{bc+1}+b\sqrt{ca+1}+c\sqrt{ab+1}\ge 2\sqrt{a+b+c-1}.$$
Because $$\sum_{cyc}a\sqrt{bc+1}=\sqrt{\sum_{cyc}(a^2bc+a^2+2ab\sqrt{(bc+1)(ac+1)}}\geq\sqrt{\sum_{cyc}(a^2+2ab)}=a+b+c\geq2\sqrt{a+b+c-1}.$$
This post has been edited 1 time. Last edited by arqady, Nov 19, 2023, 6:25 AM
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KhuongTrang
727 posts
KhuongTrang
#34 h  Nov 20, 2023, 11:13 AM • 1 Y
Y by Zuyong
Something not relevant
This post has been edited 2 times. Last edited by KhuongTrang, Dec 16, 2023, 3:48 AM
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sqing
41776 posts
sqing
#35 h  Nov 20, 2023, 12:43 PM
Y by
Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=1.$ Prove that$$\sqrt{a+b+abc}+\sqrt{b+c+abc}+\sqrt{c+a+abc}\ge 2+\sqrt{2}$$
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KhuongTrang
727 posts
KhuongTrang
#43 h  Nov 30, 2023, 1:29 PM • 1 Y
Y by Zuyong
Something not relevant
This post has been edited 1 time. Last edited by KhuongTrang, Dec 16, 2023, 3:48 AM
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KhuongTrang
727 posts
KhuongTrang
#59 h  Jun 12, 2024, 1:29 PM • 1 Y
Y by Zuyong
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that
$$\color{blue}{\sqrt{\frac{a+bc}{a+1}}+\sqrt{\frac{b+ca}{b+1}}+\sqrt{\frac{c+ab}{c+1}}\le 1+\sqrt{2}. }$$
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mudok
3377 posts
mudok
#60 h  Jun 12, 2024, 3:00 PM • 1 Y
Y by arqady
arqady wrote:
KhuongTrang wrote:
Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=1.$ Prove that$$a\sqrt{bc+1}+b\sqrt{ca+1}+c\sqrt{ab+1}\ge 2\sqrt{a+b+c-1}.$$
Because $$\sum_{cyc}a\sqrt{bc+1}=\sqrt{\sum_{cyc}(a^2bc+a^2+2ab\sqrt{(bc+1)(ac+1)}}\geq\sqrt{\sum_{cyc}(a^2+2ab)}=a+b+c\geq2\sqrt{a+b+c-1}.$$
We can directly use: $\sum a\sqrt{bc+1}\ge \sum a$ :lol:
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KhuongTrang
727 posts
KhuongTrang
#65 h  Jun 22, 2024, 3:22 AM • 1 Y
Y by Zuyong
Problem. Given $a,b,c$ be non-negative real numbers such that $a+b+c=2.$ Prove that

$$\color{blue}{\sqrt{15a+1} +\sqrt{15b+1} +\sqrt{15c+1}\ge 3\sqrt{3}\cdot\sqrt{1+2(ab+bc+ca)}. }$$
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arqady
30207 posts
arqady
#66 h  Jun 22, 2024, 6:35 AM
Y by
KhuongTrang wrote:
Problem. Given $a,b,c$ be non-negative real numbers such that $a+b+c=2.$ Prove that

$$\color{blue}{\sqrt{15a+1} +\sqrt{15b+1} +\sqrt{15c+1}\ge 3\sqrt{3}\cdot\sqrt{1+2(ab+bc+ca)}. }$$
Holder with $(3a+1)^3$ and $uvw$.
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KhuongTrang
727 posts
KhuongTrang
#72 h  Jul 15, 2024, 1:47 AM • 2 Y
Y by ehuseyinyigit, Zuyong
KhuongTrang wrote:
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that
$$\color{blue}{\sqrt{\frac{a+bc}{a+1}}+\sqrt{\frac{b+ca}{b+1}}+\sqrt{\frac{c+ab}{c+1}}\le 1+\sqrt{2}. }$$

Problem. Given non-negative real numbers satisfying $ab+bc+ca=1.$ Prove that
$$\color{blue}{\sqrt{\frac{a+b}{c+1}}+\sqrt{\frac{c+b}{a+1}}+\sqrt{\frac{a+c}{b+1}}\le 2\sqrt{a+b+c}. }$$Equality holds iff $a=b=1,c=0$ or $a=b\rightarrow 0,c\rightarrow +\infty$ and any cyclic permutations.
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arqady
30207 posts
arqady
#73 h  Jul 15, 2024, 4:50 PM
Y by
KhuongTrang wrote:
Problem. Given non-negative real numbers satisfying $ab+bc+ca=1.$ Prove that
$$\color{blue}{\sqrt{\frac{a+b}{c+1}}+\sqrt{\frac{c+b}{a+1}}+\sqrt{\frac{a+c}{b+1}}\le 2\sqrt{a+b+c}. }$$Equality holds iff $a=b=1,c=0$ or $a=b\rightarrow 0,c\rightarrow +\infty$ and any cyclic permutations.
Because $$\sum_{cyc}\sqrt{\frac{a+b}{c+1}}\leq\sqrt{\sum_{cyc}(a+b)\sum_{cyc}\frac{1}{c+1}}\leq2\sqrt{a+b+c}.$$:-D
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bellahuangcat
253 posts
bellahuangcat
#74 h  Jul 15, 2024, 4:54 PM
Y by
KhuongTrang wrote:
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$

what why does that look so easy and difficult at the same time lol
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ehuseyinyigit
810 posts
ehuseyinyigit
#75 h  Jul 15, 2024, 5:38 PM
Y by
That's the beauty of it
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bellahuangcat
253 posts
bellahuangcat
#76 h  Jul 15, 2024, 6:01 PM
Y by
ehuseyinyigit wrote:
That's the beauty of it

yeah ig
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arqady
30207 posts
arqady
#78 h  Jul 16, 2024, 7:35 AM
Y by
sqing wrote:
Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=1.$ Prove that$$\sqrt{a+b+abc}+\sqrt{b+c+abc}+\sqrt{c+a+abc}\ge 2+\sqrt{2}$$
The following inequality is also true.
Let $a$, $b$ and $c$ be non-negative numbers such that $ab+ac+bc=1$. Prove that:
$$\sqrt{a+b+\frac{13}{14}abc}+\sqrt{b+c+\frac{13}{14}abc}+\sqrt{c+a+\frac{13}{14}abc}\ge 2+\sqrt{2}$$
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KhuongTrang
727 posts
KhuongTrang
#83 h  Aug 10, 2024, 3:56 PM • 1 Y
Y by Zuyong
Problem. Given $a,b,c$ be non-negative real numbers such that $a+b+c=3.$ Prove that

$$\color{blue}{\sqrt{\frac{4}{3}(ab+bc+ca)+5}\ge \sqrt{a}+\sqrt{b}+\sqrt{c}.}$$
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kiyoras_2001
674 posts
kiyoras_2001
#84 h  Aug 10, 2024, 5:59 PM
Y by
KhuongTrang wrote:
Problem. Given $a,b,c$ be non-negative real numbers such that $a+b+c=3.$ Prove that
$$\color{blue}{\sqrt{\frac{4}{3}(ab+bc+ca)+5}\ge \sqrt{a}+\sqrt{b}+\sqrt{c}.}$$
After homogenizing and squaring it becomes
\[\sum a^2+8\sum ab\ge 3\sum a\sum\sqrt{ab}.\]Changing \(a\to a^2, b\to b^2, c\to c^2\) it becomes a fourth degree inequality, so is linear in \(w^3\). Thus it remains to check only the cases \(c=0\) and \(b=c=1\) which is easy.
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KhuongTrang
727 posts
KhuongTrang
#92 h  Mar 26, 2025, 1:00 AM • 1 Y
Y by Zuyong
Problem. Let $a,b,c$ be non-negative real variables with $ab+bc+ca>0.$ Prove that$$\color{black}{\frac{a^2+2ab}{4ab+bc+ca}+\frac{b^2+2bc}{4bc+ca+ab}+\frac{c^2+2ca}{4ca+ab+bc}\ge \frac{3}{2}. }$$Equality holds iff $(a,b,c)\sim(t,t,t)$ or $(a,b,c)\sim\left(t,0,2t\right)$ where $t>0.$
This post has been edited 1 time. Last edited by KhuongTrang, Mar 28, 2025, 1:21 AM
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jokehim
1027 posts
jokehim
#93 h  Mar 26, 2025, 1:11 PM
Y by
KhuongTrang wrote:
Problem. Let $a,b,c$ be non-negative real variables with $a+b+c>0.$ Prove that$$\color{black}{\frac{a^2+2ab}{4ab+bc+ca}+\frac{b^2+2bc}{4bc+ca+ab}+\frac{c^2+2ca}{4ca+ab+bc}\ge \frac{3}{2}. }$$Equality holds iff $(a,b,c)\sim(t,t,t)$ or $(a,b,c)\sim\left(t,0,2t\right)$ where $t>0.$

Assume that $a+b+c=1$ and set $M=a^2b+b^2c+c^2a,\ \ ab+bc+ca=q,\ \ abc=r.$ The inequality becomes$$10 M^2 - 16 M q + 12 M r - 8 q^3 + 8 q^2 - 51 q r + 63 r^2 + 10 r\ge 0$$ưhere$$\Delta_M=8 (40 q^3 - 8 q^2 + 207 q r - r (297 r + 50))<0$$which ends the proofs :D
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KhuongTrang
727 posts
KhuongTrang
#94 h  Mar 27, 2025, 4:23 AM • 1 Y
Y by Zuyong
#93 Could you please check your solution again, jokehim? I think this inequality is very hard to think of a proof in normal way.
Hope to see some ideas. Btw, it is obviously true by BW.
This post has been edited 2 times. Last edited by KhuongTrang, Mar 29, 2025, 12:05 AM
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jokehim
1027 posts
jokehim
#95 h  Mar 27, 2025, 6:23 AM
Y by
KhuongTrang wrote:
#93 Could you please check your solution again, jokehim? I think this inequality is very hard to think of a proof in normal way.
Hope to see some ideas. Btw, it is obviously true by BW.
Problem. Let $a,b,c$ be positive real variables with $a+b+c+2\sqrt{abc}=1.$ Prove that$$\frac{\sqrt{a+ab+b}}{\sqrt{ab}+\sqrt{c}}+\frac{\sqrt{b+bc+c}}{\sqrt{bc}+\sqrt{a}}+\frac{\sqrt{c+ca+a}}{\sqrt{ca}+\sqrt{b}}\ge 3.$$Equality holds iff $a=b=c=\dfrac{1}{4}.$

I don't see what's wrong with my solution :|
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Nguyenhuyen_AG
3315 posts
Nguyenhuyen_AG
#96 h  Mar 27, 2025, 6:38 AM
Y by
KhuongTrang wrote:
Problem. Let $a,b,c$ be non-negative real variables with $a+b+c>0.$ Prove that$$\color{black}{\frac{a^2+2ab}{4ab+bc+ca}+\frac{b^2+2bc}{4bc+ca+ab}+\frac{c^2+2ca}{4ca+ab+bc}\ge \frac{3}{2}. }$$Equality holds iff $(a,b,c)\sim(t,t,t)$ or $(a,b,c)\sim\left(t,0,2t\right)$ where $t>0.$
We have the following estimate
\[\frac{12a(a+2b)}{4ab+bc+ca} \geqslant \frac{32a^3+3(33b+56c)a^2+3(26b^2+102bc+13c^2)a-4(4b+c)(b-2c)^2}{11[ab(a+b)+bc(b+c)+ca(c+a)]+51abc}.\]
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KhuongTrang
727 posts
KhuongTrang
#109 h  Apr 17, 2025, 8:33 AM • 2 Y
Y by arqady, Zuyong
Problem. Let $a,b,c$ be three non-negative real numbers with $ab+bc+ca=1.$ Prove that$$\frac{\sqrt{b+c}}{a+\sqrt{bc+1}}+\frac{\sqrt{c+a}}{b+\sqrt{ca+1}}+\frac{\sqrt{a+b}}{c+\sqrt{ab+1}}\ge \sqrt{2(a+b+c)}.$$When does equality hold?
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