Plan ahead for the next school year. Schedule your class today!

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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.

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0 replies
jwelsh
Jul 1, 2025
0 replies
AOPS MO Introduce
MathMaxGreat   81
N 9 minutes ago by lbh_qys
$AOPS MO$

Problems: post it as a private message to me or @jerryZYang, please post it in $LATEX$ and have answers

6 Problems for two rounds, easier than $IMO$

If you want to do the problems or be interested, reply ’+1’
Want to post a problem reply’+2’ and message me
Want to be in the problem selection committee, reply’+3’
81 replies
MathMaxGreat
Jul 12, 2025
lbh_qys
9 minutes ago
Problem 4 of the HMO
GreekIdiot   13
N 15 minutes ago by Diaoest
Prove that no perfect cube is of the form $y^2+108$ where $y \in \mathbb{Z}$.
13 replies
GreekIdiot
Feb 22, 2025
Diaoest
15 minutes ago
Convex quad
MithsApprentice   83
N 21 minutes ago by mathprodigy2011
Source: USAMO 1993
Let $\, ABCD \,$ be a convex quadrilateral such that diagonals $\, AC \,$ and $\, BD \,$ intersect at right angles, and let $\, E \,$ be their intersection. Prove that the reflections of $\, E \,$ across $\, AB, \, BC, \, CD, \, DA \,$ are concyclic.
83 replies
1 viewing
MithsApprentice
Oct 27, 2005
mathprodigy2011
21 minutes ago
Areas of triangles AOH, BOH, COH
Arne   72
N an hour ago by mudkip42
Source: APMO 2004, Problem 2
Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Prove that the area of one of the triangles $AOH$, $BOH$ and $COH$ is equal to the sum of the areas of the other two.
72 replies
1 viewing
Arne
Mar 23, 2004
mudkip42
an hour ago
No more topics!
An algorithm for discovering prime numbers?
Lukaluce   4
N May 30, 2025 by alexanderhamilton124
Source: 2025 Junior Macedonian Mathematical Olympiad P3
Is there an infinite sequence of prime numbers $p_1, p_2, ..., p_n, ...,$ such that for every $i \in \mathbb{N}, p_{i + 1} \in \{2p_i - 1, 2p_i + 1\}$ is satisfied? Explain the answer.
4 replies
Lukaluce
May 18, 2025
alexanderhamilton124
May 30, 2025
An algorithm for discovering prime numbers?
G H J
Source: 2025 Junior Macedonian Mathematical Olympiad P3
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Lukaluce
286 posts
#1
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Is there an infinite sequence of prime numbers $p_1, p_2, ..., p_n, ...,$ such that for every $i \in \mathbb{N}, p_{i + 1} \in \{2p_i - 1, 2p_i + 1\}$ is satisfied? Explain the answer.
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grupyorum
1448 posts
#2
Y by
We first show that there is an $n_0$ and $\epsilon\in\{-1,1\}$ such that for every $n\ge n_0$, $p_{n+1} = 2p_n+\epsilon$.

To see this, suppose $p_1>3$. If $p_1\equiv 1\pmod{3}$ then $p_{n+1}=2p_n-1$ must hold necessarily (otherwise $3\mid 2p_n+1$ but $p_n>3$). Likewise if $p_1\equiv -1\pmod{3}$ then $p_{n+1}\equiv 2p_n+1$ must hold. If $p_1\le 3$, then $p_j>3$ for some $j>1$, so the same argument carries through. Shifting if necessary, we will analyze the sequence $p_{n+1} =2p_n-1$ and $p_{n+1}=2p_n+1$ for $p_1>3$.

Case 1. Let $p_{n+1} = 2p_n-1$ for $n\ge 1$. Set $b_n:=p_n-1$ to obtain $b_{n+1} = 2b_n$. Iterating, we find $b_n = 2^{n-1}b_1$. Consequently, $p_n = 2^{n-1}(p_1-1)+1$. Taking $n=k(p_1-1)+1$ for suitably large $k$, Fermat's theorem asserts $2^{n-1}\equiv 1\pmod{p_1}$. So, $p_1\mid p_n$ but $p_n>p_1$, hence $p_n$ cannot be a prime.

Case 2. Let $p_{n+1}=2p_n+1$ for $n\ge 1$. Set $b_n:=p_n+1$ to obtain $p_n = 2^{n-1}(p_1+1)-1$. The same choice of $n$ ensures $p_1\mid p_n$, a contradiction.

So, no such infinite sequence exists.

Remark. This is an old Bulgarian problem (between 2003-2010 I think), though I don't remember the exact year.
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Assassino9931
1477 posts
#3
Y by
@above Hm, haven't seen this in Bulgaria, but it is popular from Baltic Way 2004.
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TopGbulliedU
24 posts
#4 • 1 Y
Y by alexanderhamilton124
hahaha I was in the comp,after i got out I told everyone that nobody could solve this after the results came it was only me :-D
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alexanderhamilton124
404 posts
#5
Y by
TopGbulliedU wrote:
hahaha I was in the comp,after i got out I told everyone that nobody could solve this after the results came it was only me :-D

orz gj man
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