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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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MONT pg 31 example 1.10.40
Jaxman8   0
an hour ago
Can somebody explain why it works.
0 replies
Jaxman8
an hour ago
0 replies
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NEPAL TST 2025 DAY 2
Tony_stark0094   9
N an hour ago by hectorleo123
Consider an acute triangle $\Delta ABC$. Let $D$ and $E$ be the feet of the altitudes from $A$ to $BC$ and from $B$ to $AC$ respectively.

Define $D_1$ and $D_2$ as the reflections of $D$ across lines $AB$ and $AC$, respectively. Let $\Gamma$ be the circumcircle of $\Delta AD_1D_2$. Denote by $P$ the second intersection of line $D_1B$ with $\Gamma$, and by $Q$ the intersection of ray $EB$ with $\Gamma$.

If $O$ is the circumcenter of $\Delta ABC$, prove that $O$, $D$, and $Q$ are collinear if and only if quadrilateral $BCQP$ can be inscribed within a circle.

$\textbf{Proposed by Kritesh Dhakal, Nepal.}$
9 replies
1 viewing
Tony_stark0094
Apr 12, 2025
hectorleo123
an hour ago
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lines CV, BU intersect on the circumcircle of ABC
parmenides51   4
N an hour ago by ihategeo_1969
Source: 2019 Geo Mock - Olympiad by Tovi Wen #4 https://artofproblemsolving.com/community/c594864h1787237p11805928
Let $ABC$ be a triangle whose incircle is tangent to $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D, E, F$, respectively. Let $M$ and $N$ be the midpoints of $\overline{DE}$ and $\overline{DF}$, respectively. Suppose that points $U$ and $V$ lie on $\overline{MN}$ so that $BU = NU$ and $CV = MV$. Prove that lines $\overline{CV}$ and $\overline{BU}$ intersect on the circumcircle of $\triangle ABC$.
4 replies
parmenides51
Nov 26, 2023
ihategeo_1969
an hour ago
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one cyclic formed by two cyclic
CrazyInMath   33
N an hour ago by breloje17fr
Source: EGMO 2025/3
Let $ABC$ be an acute triangle. Points $B, D, E$, and $C$ lie on a line in this order and satisfy $BD = DE = EC$. Let $M$ and $N$ be the midpoints of $AD$ and $AE$, respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$, respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic.
33 replies
CrazyInMath
Apr 13, 2025
breloje17fr
an hour ago
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No more topics!
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Kazakhstan 2012 ( Grade 10 problem 4)
ts0_9   2
N Dec 23, 2022 by parmenides51
Let $k_{1},k_{2}, k_{3}$ -Excircles triangle $A_{1}A_{2}A_{3}$ with area $S$. $ k_{1}$ touch side $A_{2}A_{3} $ at the point $B_{1}$ Direct $A_{1}B_{1}$ intersect $k_{1}$ at the points $B_{1}$ and $C_{1}$.Let $S_{1}$ - area of ​​the quadrilateral $A_{1}A_{2}C_{1}A_{3}$ Similarly, we define $S_{2}, S_{3}$. Prove that $\frac{1}{S}\le \frac{1}{S_{1}}+\frac{1}{S_{2}}+\frac{1}{S_{2}}$
2 replies
ts0_9
May 20, 2012
parmenides51
Dec 23, 2022
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Kazakhstan 2012 ( Grade 10 problem 4)
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ts0_9
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ts0_9
#1 h  May 20, 2012, 5:38 PM • 2 Y
Y by Adventure10, Mango247
Let $k_{1},k_{2}, k_{3}$ -Excircles triangle $A_{1}A_{2}A_{3}$ with area $S$. $ k_{1}$ touch side $A_{2}A_{3} $ at the point $B_{1}$ Direct $A_{1}B_{1}$ intersect $k_{1}$ at the points $B_{1}$ and $C_{1}$.Let $S_{1}$ - area of ​​the quadrilateral $A_{1}A_{2}C_{1}A_{3}$ Similarly, we define $S_{2}, S_{3}$. Prove that $\frac{1}{S}\le \frac{1}{S_{1}}+\frac{1}{S_{2}}+\frac{1}{S_{2}}$
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RSM
736 posts
RSM
#2 h  May 23, 2012, 11:26 AM • 3 Y
Y by 125907, Adventure10, Mango247
Suppose, $ a_1,a_2,a_3 $ are the sides of $ A{}_1A{}_2A{}_3 $ and $ s $ denotes the semi-perimeter of $ \Delta A{}_1A{}_2A{}_3 $.
Then note that, $ A{}_1B{}_1.A{}_1C{}_1=s^2 $. $ \frac {S}{S_1}=\frac {A{}_1B{}_1}{A{}_1C{}_1}=\frac {A{}_1B{}_1^2}{s^2} $. So it suffices to prove hat, $ \sum A{}_1B{}_1^2\ge s^2 $.
By Stewart's theorem $ a_1(A{}_1B{}_1^2+(s-a_2)(s{}_a{}_3))=a_2^2(s-a_3)+a_3^2(s-a_2) $.
But note that, $ a_2^2(s-a_3)+a_3^2(s-a_2)\ge a{}_1a{}_2a{}_3 $.
So $ A{}_1B{}_1^2\ge s(a_2+a_3)-s^2 $.
So $ \sum A{}_1B{}_1^2\ge s^2 $. So done.
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parmenides51
30629 posts
parmenides51
#3 h  Dec 23, 2022, 10:06 PM • 3 Y
Y by Mango247, Mango247, Mango247
Let $k_{1},k_{2}, k_{3}$ be excircles of triangle $A_{1}A_{2}A_{3}$ with area $S$. $ k_{1}$ touch side $A_{2}A_{3} $ at the point $B_{1}$ Direct $A_{1}B_{1}$ intersect $k_{1}$ at the points $B_{1}$ and $C_{1}$.Let $S_{1}$ - area of the quadrilateral $A_{1}A_{2}C_{1}A_{3}$ Similarly, we define $S_{2}, S_{3}$. Prove that $$\frac{1}{S}\le \frac{1}{S_{1}}+\frac{1}{S_{2}}+\frac{1}{S_{2}}.$$
This post has been edited 2 times. Last edited by parmenides51, Jan 7, 2023, 11:58 PM
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