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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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

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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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Beautiful problem
luutrongphuc   16
N a few seconds ago by Acorn-SJ
Let triangle $ABC$ be circumscribed about circle $(I)$, and let $H$ be the orthocenter of $\triangle ABC$. The circle $(I)$ touches line $BC$ at $D$. The tangent to the circle $(BHC)$ at $H$ meets $BC$ at $S$. Let $J$ be the midpoint of $HI$, and let the line $DJ$ meet $(I)$ again at $X$. The tangent to $(I)$ parallel to $BC$ meets the line $AX$ at $T$. Prove that $ST$ is tangent to $(I)$.
16 replies
luutrongphuc
Apr 4, 2025
Acorn-SJ
a few seconds ago
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Stereotypical Diophantine Equation
Mathdreams   2
N 2 minutes ago by grupyorum
Source: 2025 Nepal Mock TST Day 2 Problem 1
Find all solutions in the nonnegative integers to $2^a3^b5^c7^d - 1 = 11^e$.

(Shining Sun, USA)
2 replies
Mathdreams
27 minutes ago
grupyorum
2 minutes ago
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A very nice inequality
KhuongTrang   1
N 3 minutes ago by Mathdreams
Source: own
Problem. Let $a,b,c\in \mathbb{R}:\ a+b+c=3.$ Prove that $$\color{black}{\sqrt{5a^{2}-ab+5b^{2}}+\sqrt{5b^{2}-bc+5c^{2}}+\sqrt{5c^{2}-ca+5a^{2}}\le 2(a^2+b^2+c^2)+ab+bc+ca.}$$When does equality hold?
1 reply
1 viewing
KhuongTrang
24 minutes ago
Mathdreams
3 minutes ago
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Common tangent to diameter circles
Stuttgarden   1
N 12 minutes ago by jrpartty
Source: Spain MO 2025 P2
The cyclic quadrilateral $ABCD$, inscribed in the circle $\Gamma$, satisfies $AB=BC$ and $CD=DA$, and $E$ is the intersection point of the diagonals $AC$ and $BD$. The circle with center $A$ and radius $AE$ intersects $\Gamma$ in two points $F$ and $G$. Prove that the line $FG$ is tangent to the circles with diameters $BE$ and $DE$.
1 reply
1 viewing
Stuttgarden
Mar 31, 2025
jrpartty
12 minutes ago
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No more topics!
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APQ tangent to BC
topologicalsort   1
N Dec 2, 2024 by Mahdi_Mashayekhi
Source: Bulgarian Autumn Tournament 2024, 10.2
Let $ABC$ be a scalene acute triangle, where $AL$ $(L \in BC)$ is the internal bisector of $\angle BAC$ and $M$ is the midpoint of $BC$. Let the internal bisectors of $\angle AMB$ and $\angle CMA$ intersect $AB$ and $AC$ in $P$ and $Q$, respectively. Prove that the circumcircle of $APQ$ is tangent to $BC$ if and only if $L$ belongs to it.
1 reply
topologicalsort
Nov 16, 2024
Mahdi_Mashayekhi
Dec 2, 2024
J
APQ tangent to BC
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Reveal topic tags
homothety
geometry
circumcircle
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G H BBookmark kLocked kLocked NReply
Source: Bulgarian Autumn Tournament 2024, 10.2
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topologicalsort
27 posts
topologicalsort
#1 h  Nov 16, 2024, 8:14 PM
Y by
Let $ABC$ be a scalene acute triangle, where $AL$ $(L \in BC)$ is the internal bisector of $\angle BAC$ and $M$ is the midpoint of $BC$. Let the internal bisectors of $\angle AMB$ and $\angle CMA$ intersect $AB$ and $AC$ in $P$ and $Q$, respectively. Prove that the circumcircle of $APQ$ is tangent to $BC$ if and only if $L$ belongs to it.
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Mahdi_Mashayekhi
689 posts
Mahdi_Mashayekhi
#4 h  Dec 2, 2024, 6:13 AM
Y by
Note that $\frac{AP}{PB} = \frac{AM}{MB} = \frac{AM}{MC} = \frac{AQ}{QC}$ so $PQ \parallel BC$ so $APQ$ and $ABC$ are tangent. It's well-known that if $APQ$ is tangent to $BC$ at $L'$ then $AL'$ passes through midpoint of arc $BC$ so $L$ and $L'$ are the same. Now assume that $L$ belongs to $APQ$ and $APQ$ intersects $BC$ at $K$ for the second time. Let $T$ be the midpoint of arc $BC$. Let $TK$ meet $ABC$ again at $R$. Note that $ALKR$ would be cyclic so $APQ$ meets $ABC$ again at $R$ which gives contradiction since they are tangent.
This post has been edited 1 time. Last edited by Mahdi_Mashayekhi, Dec 2, 2024, 6:13 AM
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