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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
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Finding all integers with a divisibility condition
Tintarn   14
N a minute ago by Assassino9931
Source: Germany 2020, Problem 4
Determine all positive integers $n$ for which there exists a positive integer $d$ with the property that $n$ is divisible by $d$ and $n^2+d^2$ is divisible by $d^2n+1$.
14 replies
Tintarn
Jun 22, 2020
Assassino9931
a minute ago
J
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Geometry Handout is finally done!
SimplisticFormulas   2
N 29 minutes ago by parmenides51
If there’s any typo or problem you think will be a nice addition, do send here!
handout, geometry
2 replies
SimplisticFormulas
5 hours ago
parmenides51
29 minutes ago
J
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functional equation interesting
skellyrah   4
N 43 minutes ago by skellyrah
find all functions IR->IR such that $$xf(x+yf(xy)) + f(f(y)) = f(xf(y))^2 + (x+1)f(x)$$
4 replies
skellyrah
2 hours ago
skellyrah
43 minutes ago
J
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IMO ShortList 2002, number theory problem 2
orl   57
N an hour ago by Maximilian113
Source: IMO ShortList 2002, number theory problem 2
Let $n\geq2$ be a positive integer, with divisors $1=d_1<d_2<\,\ldots<d_k=n$. Prove that $d_1d_2+d_2d_3+\,\ldots\,+d_{k-1}d_k$ is always less than $n^2$, and determine when it is a divisor of $n^2$.
57 replies
orl
Sep 28, 2004
Maximilian113
an hour ago
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Find the side length and the area
April   1
N Apr 12, 2009 by Dr Sonnhard Graubner
Source: Vietnam NMO 1963 Problem 5
The triangle $ ABC$ has semiperimeter $ p$. Find the side length $ BC$ and the area $ S$ in terms of $ \angle A$, $ \angle B$ and $ p$. In particular, find $ S$ if $ p \approx 23.6$, $ \angle A \approx 52^{\circ}42'$, $ \angle B \approx 46^{\circ}16'$.
1 reply
April
Feb 2, 2009
Dr Sonnhard Graubner
Apr 12, 2009
J
Find the side length and the area
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Reveal topic tags
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Source: Vietnam NMO 1963 Problem 5
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April
1270 posts
April
#1 h  Feb 2, 2009, 1:08 AM • 2 Y
Y by Adventure10, Mango247
The triangle $ ABC$ has semiperimeter $ p$. Find the side length $ BC$ and the area $ S$ in terms of $ \angle A$, $ \angle B$ and $ p$. In particular, find $ S$ if $ p \approx 23.6$, $ \angle A \approx 52^{\circ}42'$, $ \angle B \approx 46^{\circ}16'$.
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Dr Sonnhard Graubner
16100 posts
Dr Sonnhard Graubner
#2 h  Apr 12, 2009, 8:41 PM • 2 Y
Y by Adventure10, Mango247
hello, let $ \alpha,\beta,\gamma$ be the angles of the given triangle, then we have
$ \gamma=\pi-\alpha-\beta$
$ \sin(\alpha)=\frac{a}{2R}$
$ \sin(\beta)=\frac{b}{2R}$
$ \sin(\gamma)=\frac{c}{2R}$
and from here we get $ R=\frac{p}{\sin(\alpha)+\sin(\beta)+\sin(\gamma)}$ and using the formula $ S=\frac{abc}{4R}$ we have $ S=\frac{2p^2\sin(\alpha)\sin(\beta)\sin(\gamma)}{(\sin(alpha)+\sin(\beta)+\sin(\gamma))^2}$
Sonnhard.
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