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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)!

Be sure to mark your calendars for the following events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]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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Problem 16
Nguyenhuyen AG   37
N 4 minutes ago by flower417477
Let $a,b,c >0 $ such that $abc\ge1 $. Prove that
$\frac{1}{a^4+b^3+c^2}+\frac{1}{b^4+c^3+a^2}+\frac{1}{c^4+a^3+b^2}\le 1$
37 replies
Nguyenhuyen AG
May 3, 2010
flower417477
4 minutes ago
J
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NEPAL TST 2025 DAY 2
Tony_stark0094   4
N 10 minutes ago by InterLoop
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.

$\textbf{Proposed by Kritesh Dhakal, Nepal.}$
4 replies
+1 w
Tony_stark0094
Yesterday at 8:40 AM
InterLoop
10 minutes ago
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Sets With a Given Property
oVlad   3
N 18 minutes ago by flower417477
Source: Romania TST 2025 Day 1 P4
Determine the sets $S{}$ of positive integers satisfying the following two conditions:
[list=a]
[*]For any positive integers $a, b, c{}$, if $ab + bc + ca{}$ is in $S$, then so are $a + b + c{}$ and $abc$; and
[*]The set $S{}$ contains an integer $N \geqslant 160$ such that $N-2$ is not divisible by $4$.
[/list]
Bogdan Blaga, United Kingdom
3 replies
oVlad
Apr 9, 2025
flower417477
18 minutes ago
J
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Number Theory Chain!
JetFire008   39
N 32 minutes ago by whwlqkd
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1
39 replies
+1 w
JetFire008
Apr 7, 2025
whwlqkd
32 minutes ago
J
No more topics!
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38th Austrian Mathematical Competition
FelixD   2
N Nov 15, 2009 by FelixD
Source: 3rd round
Determine all rhombuses $ ABCD$ with the given length $ 2a$ of ist sides by giving the angle $ \alpha = \angle BAD$, such that there exists a circle which cuts each side of the rhombus in a chord of length $ a$.
2 replies
FelixD
Feb 7, 2009
FelixD
Nov 15, 2009
J
38th Austrian Mathematical Competition
G H J
Reveal topic tags
geometry
rhombus
geometry unsolved
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G H BBookmark kLocked kLocked NReply
Source: 3rd round
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FelixD
588 posts
FelixD
#1 h  Feb 7, 2009, 7:41 PM • 2 Y
Y by Adventure10, Mango247
Determine all rhombuses $ ABCD$ with the given length $ 2a$ of ist sides by giving the angle $ \alpha = \angle BAD$, such that there exists a circle which cuts each side of the rhombus in a chord of length $ a$.
Z K Y
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Maharjun
109 posts
Maharjun
#2 h  Nov 15, 2009, 10:16 AM • 2 Y
Y by Adventure10, Mango247
please specify question clearly. it is a little ambiguous
Z K Y
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FelixD
588 posts
FelixD
#3 h  Nov 15, 2009, 10:37 AM • 2 Y
Y by Adventure10, Mango247
Let $ ABCD$ be a rhombus, such that $ AB=BC=CD=DA=2a$. Find all angles $ \alpha = \angle DAB$ such that there exists a circle in the plane of the rhombus that cuts each side of the rhombus in a chord of length $ a$.
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