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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Wednesday at 3:18 PM
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
Wednesday at 3:18 PM
0 replies
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Bashing??
John_Mgr   0
6 minutes ago
I have learned little about what bashing mean as i am planning to start geo, feels like its less effort required and doesnt need much knowledge about the synthetic solutions?
what do you guys recommend ? also state the major difference of them... especially of bashing pros and cons..
0 replies
1 viewing
John_Mgr
6 minutes ago
0 replies
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1 area = 2025 points
giangtruong13   1
N 10 minutes ago by kiyoras_2001
In a plane give a set $H$ that has 8097 distinct points with area of a triangle that has 3 points belong to $H$ all $ \leq 1$. Prove that there exists a triangle $G$ that has the area $\leq 1 $ contains at least 2025 points that belong to $H$( each of that 2025 points can be inside the triangle or lie on the edge of triangle $G$)X
1 reply
giangtruong13
5 hours ago
kiyoras_2001
10 minutes ago
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A board with crosses that we color
nAalniaOMliO   2
N 13 minutes ago by CHESSR1DER
Source: Belarusian National Olympiad 2025
In some cells of the table $2025 \times 2025$ crosses are placed. A set of 2025 cells we will call balanced if no two of them are in the same row or column. It is known that any balanced set has at least $k$ crosses.
Find the minimal $k$ for which it is always possible to color crosses in two colors such that any balanced set has crosses of both colors.
2 replies
nAalniaOMliO
Mar 28, 2025
CHESSR1DER
13 minutes ago
J
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Geometry Finale: Incircles and concurrency
lminsl   173
N 21 minutes ago by Parsia--
Source: IMO 2019 Problem 6
Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA$, and $AB$ at $D, E,$ and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangle $PCE$ and $PBF$ meet again at $Q$.

Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$.

Proposed by Anant Mudgal, India
173 replies
lminsl
Jul 17, 2019
Parsia--
21 minutes ago
J
No more topics!
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J
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Geometry with known distances
nAalniaOMliO   1
N Mar 29, 2025 by RagvaloD
Source: Belarusian National Olympiad 2025
In a rectangle $ABCD$ two not intersecting circles $\omega_1$ and $\omega_2$ are drawn such that $\omega_1$ is tangent to $AB$ and $AD$ at points $P$ and $S$ respectively, and $\omega_2$ is tangent to $CB$ and $CD$ at $T$ and $Q$ respectively. It is known that $PQ=11, ST=10, BD=14$.
Find the distance between centers of circles $\omega_1$ and $\omega_2$.
1 reply
nAalniaOMliO
Mar 28, 2025
RagvaloD
Mar 29, 2025
J
Geometry with known distances
G H J
Reveal topic tags
geometry
L
G H BBookmark kLocked kLocked NReply
Source: Belarusian National Olympiad 2025
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nAalniaOMliO
294 posts
nAalniaOMliO
#1 h  Mar 28, 2025, 8:18 PM
Y by
In a rectangle $ABCD$ two not intersecting circles $\omega_1$ and $\omega_2$ are drawn such that $\omega_1$ is tangent to $AB$ and $AD$ at points $P$ and $S$ respectively, and $\omega_2$ is tangent to $CB$ and $CD$ at $T$ and $Q$ respectively. It is known that $PQ=11, ST=10, BD=14$.
Find the distance between centers of circles $\omega_1$ and $\omega_2$.
Z K Y
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RagvaloD
4894 posts
RagvaloD
#2 h  Mar 29, 2025, 1:00 PM
Y by
Easy to see that $d^2=PQ^2+ST^2-BD^2=25 \to d=5$
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