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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
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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
J
H
Zack likes Moving Points
pinetree1   72
N an hour ago by endless_abyss
Source: USA TSTST 2019 Problem 5
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Gamma$. A line through $H$ intersects segments $AB$ and $AC$ at $E$ and $F$, respectively. Let $K$ be the circumcenter of $\triangle AEF$, and suppose line $AK$ intersects $\Gamma$ again at a point $D$. Prove that line $HK$ and the line through $D$ perpendicular to $\overline{BC}$ meet on $\Gamma$.

Gunmay Handa
72 replies
pinetree1
Jun 25, 2019
endless_abyss
an hour ago
J
H
Find all primes of the form n^n + 1 less than 10^{19}
Math5000   2
N 2 hours ago by SomeonecoolLovesMaths
Find all primes of the form $n^n + 1$ less than $10^{19}$

The first two primes are obvious: $n = 1, 2$ yields the primes $2, 5$. After that, it is clear that $n$ has to be even to yield an odd number.

So, $n = 2k \implies p = (2k)^{2k} + 1 \implies p-1 = (2k)^{k^2} = 2^{k^2}k^{k^2}$. All of these transformations don't seem to help. Is there any theorem I can use? Or is there something I'm missing?

2 replies
Math5000
Oct 15, 2019
SomeonecoolLovesMaths
2 hours ago
J
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IMO ShortList 2002, geometry problem 1
orl   47
N 2 hours ago by Avron
Source: IMO ShortList 2002, geometry problem 1
Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.
47 replies
orl
Sep 28, 2004
Avron
2 hours ago
J
H
2^2^n+2^2^{n-1}+1-Iran 3rd round-Number Theory 2007
Amir Hossein   5
N 2 hours ago by SomeonecoolLovesMaths
Prove that $2^{2^{n}}+2^{2^{{n-1}}}+1$ has at least $n$ distinct prime divisors.
5 replies
Amir Hossein
Jul 28, 2010
SomeonecoolLovesMaths
2 hours ago
J
No more topics!
H
J
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4th Romanian JBMO Team Selection Test, Problem 4
sunken rock   3
N May 29, 2017 by RagvaloD
Attached the subject problem and my solution.

Authors: Miguel Ochoa-Sanchez, Peru and Leonard Giugiuc, Romania

Best regards,
sunken rock
3 replies
sunken rock
May 28, 2017
RagvaloD
May 29, 2017
J
4th Romanian JBMO Team Selection Test, Problem 4
G H J
Reveal topic tags
geometry
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G H BBookmark kLocked kLocked NReply
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sunken rock
4380 posts
sunken rock
#1 h  May 28, 2017, 6:39 PM • 2 Y
Y by leonardg, Adventure10
Attached the subject problem and my solution.

Authors: Miguel Ochoa-Sanchez, Peru and Leonard Giugiuc, Romania

Best regards,
sunken rock
Attachments:
Problem 4 Test 4 BJMO Team Selection.pdf (81kb)
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tenplusten
1000 posts
tenplusten
#2 h  May 28, 2017, 6:46 PM • 2 Y
Y by Adventure10, Mango247
Where are other problems? Thanks!
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jayme
9775 posts
jayme
#3 h  May 29, 2017, 5:50 AM • 2 Y
Y by Adventure10, Mango247
Dear S.
can you please provides the problem in another format because I can't open you file...

Sincerely
Jean-Louis
Z K Y
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RagvaloD
4899 posts
RagvaloD
#4 h  May 29, 2017, 6:06 AM • 1 Y
Y by Adventure10
On the sides $AB, AC$ of the A-right-angled triangle $ABC$ one construct outwardly the equilateral triangles $ ABP$ and $ACQ$. Let $ AH $ be the altitude of the triangle. The median $AM$ intersects $PQ$ at $N$.
Prove that $m(\angle AHP) = m(\angle NHQ)$.
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