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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 2, 2025
0 replies
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Parallelograms and concyclicity
Lukaluce   24
N 4 minutes ago by Assassino9931
Source: EGMO 2025 P4
Let $ABC$ be an acute triangle with incentre $I$ and $AB \neq AC$. Let lines $BI$ and $CI$ intersect the circumcircle of $ABC$ at $P \neq B$ and $Q \neq C$, respectively. Consider points $R$ and $S$ such that $AQRB$ and $ACSP$ are parallelograms (with $AQ \parallel RB, AB \parallel QR, AC \parallel SP$, and $AP \parallel CS$). Let $T$ be the point of intersection of lines $RB$ and $SC$. Prove that points $R, S, T$, and $I$ are concyclic.
24 replies
Lukaluce
Yesterday at 10:59 AM
Assassino9931
4 minutes ago
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Weird Inequality Problem
Omerking   2
N 17 minutes ago by sqing
Following inequality is given:
$$3\geq ab+bc+ca\geq \dfrac{1}{3}$$Find the range of values that can be taken by :
$1)a+b+c$
$2)abc$

Where $a,b,c$ are positive reals.
2 replies
Omerking
5 hours ago
sqing
17 minutes ago
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Fair division of loot among pirates
kiyoras_2001   0
18 minutes ago
Source: Kvant №4 1973, In a quite harbor
Three pirates share the loot, consisting of 10 piastres, 10 doubloons and a barrel of wine.

Though they have a container for pouring wine, each pirate has his own opinion about the comparative value of piastres, doubloons and wine. However, everyone agrees that a barrel of wine costs more than four piastres and more than four doubloons.

Prove that the pirates will be able to divide the loot so that each pirate gets a part worth (in his opinion) no less than the part of each of the others.
0 replies
kiyoras_2001
18 minutes ago
0 replies
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Inspired by my own results
sqing   1
N 26 minutes ago by sqing
Source: Own
Let $ a,b $ be reals such that $ a+b+a^2+b^2=1. $ Prove that
$$ \frac{1}{a^2+1}+\frac{1}{b^2+1} -ab\geq\frac{3(2-7\sqrt 3)}{26}$$$$ \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 }+ab\leq\frac{58+5\sqrt 3 }{26}$$$$ \frac{29+9\sqrt 3 }{13}\geq \frac{1}{a^2+1}+\frac{1}{b^2+1} -a-b\geq\frac{29-9\sqrt 3 }{13}$$$$ \frac{3+17\sqrt 3 }{13}\geq \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 }+a+b\geq\frac{3-17\sqrt 3 }{13}$$
1 reply
sqing
36 minutes ago
sqing
26 minutes ago
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No more topics!
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LM// BC wanted, median AK, angle bisectors KL, KM of AKB, AKC
parmenides51   2
N Jan 29, 2022 by Hopeooooo
Source: II Kolmogorov Memory Cup / Tournament 1999 p38 https://artofproblemsolving.com/community/c3031246_
Median $AK$ is drawn in $ \vartriangle ABC$. In $\vartriangle AKB$ and $\vartriangle AKC$, the angle bisectors $ KL$ and $KM$ were drawn. Prove that $LM\parallel BC$.

(S.L. Berlov)
2 replies
parmenides51
Jan 29, 2022
Hopeooooo
Jan 29, 2022
J
LM// BC wanted, median AK, angle bisectors KL, KM of AKB, AKC
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Reveal topic tags
geometry
parallel
angle bisector
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Source: II Kolmogorov Memory Cup / Tournament 1999 p38 https://artofproblemsolving.com/community/c3031246_
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parmenides51
30630 posts
parmenides51
#1 h  Jan 29, 2022, 9:26 AM • 1 Y
Y by moom
Median $AK$ is drawn in $ \vartriangle ABC$. In $\vartriangle AKB$ and $\vartriangle AKC$, the angle bisectors $ KL$ and $KM$ were drawn. Prove that $LM\parallel BC$.

(S.L. Berlov)
This post has been edited 1 time. Last edited by parmenides51, Apr 23, 2022, 10:17 AM
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Mahmood.sy
26 posts
Mahmood.sy
#3 h  Jan 29, 2022, 11:20 AM
Y by
only use law of angle bisector
$\frac{AK}{KC} =\frac{AM}{MC}$
$\frac{AK}{KB}=\frac{AL}{LB}$
$KB=KC$
$\longrightarrow$ $\frac{AM}{MC}=\frac{AL}{LB}$ $\longrightarrow$$\frac{AM}{AC}=\frac{AL}{AB}$
$\longrightarrow$$ LM \parallel BC $
This post has been edited 3 times. Last edited by Mahmood.sy, Jan 29, 2022, 11:22 AM
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Hopeooooo
819 posts
Hopeooooo
#4 h  Jan 29, 2022, 12:34 PM
Y by
$\frac{AK}{BK}=\frac{AD}{BD}$
$\frac{AK}{CK}=\frac{AE}{CE}$
$\frac{AB}{AC}=\frac{BK}{CK}$
$\rightarrow$ $\frac{AD}{AB}=\frac{AE}{AC}$
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