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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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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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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
11 minutes ago
Can somebody explain why it works.
0 replies
Jaxman8
11 minutes ago
0 replies
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NEPAL TST 2025 DAY 2
Tony_stark0094   9
N 12 minutes 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
Tony_stark0094
Apr 12, 2025
hectorleo123
12 minutes ago
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lines CV, BU intersect on the circumcircle of ABC
parmenides51   4
N 18 minutes 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
18 minutes ago
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one cyclic formed by two cyclic
CrazyInMath   33
N 20 minutes 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
20 minutes ago
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circumcenter of DEH_2 lies on altitude AH_1
parmenides51   0
Jun 26, 2021
Source: 2013 Olympiad of Rusanovsky Lyceum in Kyiv, Ukraine , IX-X Team p9
In triangle $ABC$ , altitudes $AH_1$, $BH_2$,$CH_3$ are drawn. On ray $H_3H_1$ behind point $H_1$, some point $D$ is selected. The circumscribed circle of the triangle $DH_1C$ intersects the segment $AC$ at point $E$. Prove that the center of the circumcircle of triangle $DEH_2$ lies on altitude $AH_1$.

(D. Khilko)
0 replies
parmenides51
Jun 26, 2021
0 replies
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circumcenter of DEH_2 lies on altitude AH_1
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Source: 2013 Olympiad of Rusanovsky Lyceum in Kyiv, Ukraine , IX-X Team p9
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parmenides51
30629 posts
parmenides51
#1 h  Jun 26, 2021, 7:21 AM
Y by
In triangle $ABC$ , altitudes $AH_1$, $BH_2$,$CH_3$ are drawn. On ray $H_3H_1$ behind point $H_1$, some point $D$ is selected. The circumscribed circle of the triangle $DH_1C$ intersects the segment $AC$ at point $E$. Prove that the center of the circumcircle of triangle $DEH_2$ lies on altitude $AH_1$.

(D. Khilko)
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