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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 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
J
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A well-known geo configuration revisited
Tintarn   6
N 7 minutes ago by Primeniyazidayi
Source: Dutch TST 2024, 2.1
Let $ABC$ be a triangle with orthocenter $H$ and circumcircle $\Gamma$. Let $D$ be the reflection of $A$ in $B$ and let $E$ the reflection of $A$ in $C$. Let $M$ be the midpoint of segment $DE$. Show that the tangent to $\Gamma$ in $A$ is perpendicular to $HM$.
6 replies
Tintarn
Jun 28, 2024
Primeniyazidayi
7 minutes ago
J
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interesting inequality
pennypc123456789   2
N 7 minutes ago by User141208
Let \( a,b,c \) be real numbers satisfying \( a+b+c = 3 \) . Find the maximum value of
\[P = \dfrac{a(b+c)}{a^2+2bc+3} + \dfrac{b(a+c) }{b^2+2ca +3 } + \dfrac{c(a+b)}{c^2+2ab+3}.\]
2 replies
pennypc123456789
Yesterday at 9:47 AM
User141208
7 minutes ago
J
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Function Problem
Omerking   1
N 11 minutes ago by jasperE3
Find all functions $f:\mathbb {R}\rightarrow \mathbb {R}$ for all $x,y$ real numbers such that:

$$f(xf(y))+y+f(x)=f(x+f(y))+yf(x)$$
1 reply
Omerking
12 minutes ago
jasperE3
11 minutes ago
J
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incircle geometry
Tuguldur   0
an hour ago
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$. The diagonals $AC$ and $BD$ meet at $E$. The rays $CB$ and $DA$ meet at $F$. Prove that the line through the incenters of $\triangle ABE$ and $\triangle ABF$ and the line through the incenters of $\triangle CDE$ and $\triangle CDF$ meet at a point lying on $\omega$.
0 replies
Tuguldur
an hour ago
0 replies
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An interesting inequality
JK1603JK   5
N Apr 1, 2025 by JK1603JK
Source: unknown
Let a,b,c>=0 and a^2+b^2+c^2+abc=4 then prove \frac{1}{a+b+2}+\frac{1}{b+c+2}+\frac{1}{c+a+2} \le \frac{6-(a+b+c)}{4}
When does equality occur?
5 replies
JK1603JK
Mar 21, 2025
JK1603JK
Apr 1, 2025
J
An interesting inequality
G H J
Reveal topic tags
inequalities
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G H BBookmark kLocked kLocked NReply
Source: unknown
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JK1603JK
45 posts
JK1603JK
#1 h  Mar 21, 2025, 1:50 PM • 1 Y
Y by arqady
Let a,b,c>=0 and a^2+b^2+c^2+abc=4 then prove \frac{1}{a+b+2}+\frac{1}{b+c+2}+\frac{1}{c+a+2} \le \frac{6-(a+b+c)}{4}
When does equality occur?
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JK1603JK
45 posts
JK1603JK
#2 h  Mar 25, 2025, 7:56 AM
Y by
Can anyone help me? Thank you
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JK1603JK
45 posts
JK1603JK
#3 h  Mar 29, 2025, 1:04 PM
Y by
It is a very nice inequality. Anyone try?
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arqady
30184 posts
arqady
#4 h  Mar 29, 2025, 1:27 PM
Y by
JK1603JK wrote:
It is a very nice inequality. Anyone try?
I tried. It's not easy.
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Nguyenhuyen_AG
3307 posts
Nguyenhuyen_AG
#5 h  Mar 29, 2025, 1:32 PM
Y by
JK1603JK wrote:
Let a,b,c>=0 and a^2+b^2+c^2+abc=4 then prove \frac{1}{a+b+2}+\frac{1}{b+c+2}+\frac{1}{c+a+2} \le \frac{6-(a+b+c)}{4}
When does equality occur?
See here: https://artofproblemsolving.com/community/c6h1821941
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JK1603JK
45 posts
JK1603JK
#6 h  Apr 1, 2025, 8:15 AM
Y by
arqady wrote:
JK1603JK wrote:
It is a very nice inequality. Anyone try?
I tried. It's not easy.

Did you try the known substitution?
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