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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.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

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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 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
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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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Maximum angle ratio
miiirz30   2
N 6 minutes ago by zuzuzu222
Source: 2025 Euler Olympiad, Round 1
Given any arc $AB$ on a circle and points $C$ and $D$ on segment $AB$, such that $$CD = DB = 2AC.$$Find the ratio $\frac{CM}{MD}$, where $M$ is a point on arc $AB$, such that $\angle CMD$ is maximized.

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Proposed by Andria Gvaramia, Georgia
2 replies
miiirz30
Mar 31, 2025
zuzuzu222
6 minutes ago
J
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S must be the incentre of triangle ABC
WakeUp   3
N 9 minutes ago by Nari_Tom
Source: Baltic Way 2007
In triangle $ABC$ let $AD,BE$ and $CF$ be the altitudes. Let the points $P,Q,R$ and $S$ fulfil the following requirements:
i) $P$ is the circumcentre of triangle $ABC$.
ii) All the segments $PQ,QR$ and $RS$ are equal to the circumradius of triangle $ABC$.
iii) The oriented segment $PQ$ has the same direction as the oriented segment $AD$. Similarly, $QR$ has the same direction as $BE$, and $Rs$ has the same direction as $CF$.
Prove that $S$ is the incentre of triangle $ABC$.
3 replies
WakeUp
Nov 30, 2010
Nari_Tom
9 minutes ago
J
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Two midpoints and the circumcenter are collinear.
ricarlos   1
N an hour ago by Luis González
Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be a point on the perpendicular bisector of $AB$ (see figure) and $Q$, $R$ be the intersections of the perpendicular bisectors of $AC$ and $BC$, respectively, with $PA$ and $PB$. Prove that the midpoints of $PC$ and $QR$ and the point $O$ are collinear.

1 reply
ricarlos
Yesterday at 5:52 PM
Luis González
an hour ago
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one nice!
teomihai   1
N an hour ago by KAME06
3 girls and 4 boys must be seated at a round table. In how many distinct ways can they be seated so that the 3 girls do not sit next to each other and there can be a maximum of 2 girls next to each other. (The table is round so the seats are not numbered.)
1 reply
teomihai
Yesterday at 7:32 PM
KAME06
an hour ago
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J
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angle chasing with 3 circles, intersecting in pairs
parmenides51   0
Aug 10, 2018
Source: U sviti matematyky / In the World of Mathematics P252
The circles $\omega_1$ and $\omega_2$ with centres $O_1,O_2$ intersect at points $A$ and $B$. The circle $\omega$ passing through $O_1,O_2,A$ intersects $\omega_1$ and $\omega_2$ again in points $K,M$ respectively. Prove that $AB$ is a bisector of $\angle KAM$ or of angle adjacent to $\angle KAM$.

(T. Tymoshkevych, Kyiv)
0 replies
parmenides51
Aug 10, 2018
0 replies
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angle chasing with 3 circles, intersecting in pairs
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Source: U sviti matematyky / In the World of Mathematics P252
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parmenides51
30629 posts
parmenides51
#1 h  Aug 10, 2018, 7:39 AM • 1 Y
Y by Adventure10
The circles $\omega_1$ and $\omega_2$ with centres $O_1,O_2$ intersect at points $A$ and $B$. The circle $\omega$ passing through $O_1,O_2,A$ intersects $\omega_1$ and $\omega_2$ again in points $K,M$ respectively. Prove that $AB$ is a bisector of $\angle KAM$ or of angle adjacent to $\angle KAM$.

(T. Tymoshkevych, Kyiv)
This post has been edited 1 time. Last edited by parmenides51, Aug 10, 2018, 8:12 AM
Reason: added problem no
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