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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
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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:
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0 replies
jwelsh
Jul 1, 2025
0 replies
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
IMO MOHS rating predictions
ohiorizzler1434   10
N 12 minutes ago by zhenghua
Everybody, with the IMO about to happen soon, what are your predictions for the MOHS ratings of the problems? I predict 10 20 40 15 25 45.
10 replies
ohiorizzler1434
Today at 4:48 AM
zhenghua
12 minutes ago
Sum of digits is a square
leannan-capall   3
N 13 minutes ago by numbertheory97
Source: Irish Mathematical Olympiad 2023 Problem 6
A positive integer is totally square is the sum of its digits (written in base $10$) is a square number. For example, $13$ is totally square because $1 + 3 = 2^2$, but $16$ is not totally square.

Show that there are infinitely many positive integers that are not the sum of two totally square integers.
3 replies
leannan-capall
May 14, 2023
numbertheory97
13 minutes ago
Inequality where a+b+c=3
leannan-capall   15
N 23 minutes ago by numbertheory97
Source: Irish Mathematical Olympiad 2023 Problem 8
Suppose that $a, b, c$ are positive real numbers and $a + b + c = 3$. Prove that

$$\frac{a+b}{c+2} + \frac{b+c}{a+2} + \frac{c+a}{b+2} \geq 2$$
and determine when equality holds.
15 replies
leannan-capall
May 14, 2023
numbertheory97
23 minutes ago
Monstrous FE!
JARP091   3
N 30 minutes ago by GreekIdiot
Source: Own
This problem is for anyone who considers themselves a master in FE:
Find all function $f: \mathbb{R} \to \mathbb{R}$ such that for all $x, y\in\mathbb{R}$ $x,y$ both not $< 0$,$$f(xf(y)+f(x)f(y))=xf(y)+f(xy).$$
3 replies
JARP091
Jun 30, 2025
GreekIdiot
30 minutes ago
Number of subsets divisible by a prime
somebodyyouusedtoknow   6
N 31 minutes ago by mudkip42
Source: Indonesian First Stage TST for IMO 2021, Day 3 Problem 2
Let $p$ be an odd prime. Determine the number of nonempty subsets from $\{1, 2, \dots, p - 1\}$ for which the sum of its elements is divisible by $p$.
6 replies
somebodyyouusedtoknow
Jan 1, 2021
mudkip42
31 minutes ago
Circles and radii
leannan-capall   3
N 32 minutes ago by numbertheory97
Source: Irish Mathematical Olympiad 2023 Problem 3
Let $A, B, C, D, E$ be five points on a circle such that $|AB| = |CD|$ and $|BC| = |DE|$. The segments $AD$ and $BE$ intersect at $F$. Let $M$ denote the midpoint of segment $CD$. Prove that the circle of center $M$ and radius $ME$ passes through the midpoint of segment $AF$.
3 replies
leannan-capall
May 16, 2023
numbertheory97
32 minutes ago
Exercisable functional equation
Assassino9931   7
N 34 minutes ago by Rayvhs
Source: RMM Extralist 2021 A1
Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[ f(xy+f(x)) + f(y) = xf(y) + f(x+y) \]for all real numbers $x$ and $y$.
7 replies
Assassino9931
Sep 18, 2023
Rayvhs
34 minutes ago
f(xy) = f(x)f(y) - f(x+y) + 1
orl   9
N an hour ago by Fly_into_the_sky
Source: IMO 1980 Luxembourg, problem 1
The function $f$ is defined on the set $\mathbb{Q}$ of all rational numbers and has values in $\mathbb{Q}$. It satisfies the conditions $f(1) = 2$ and $f(xy) = f(x)f(y) - f(x+y) + 1$ for all $x,y \in \mathbb{Q}$. Determine $f$.
9 replies
orl
May 6, 2004
Fly_into_the_sky
an hour ago
An interesting combination problem
Math291   6
N an hour ago by Maximilian113
Given a unit square grid of size 4×6 as shown in the figure below, an ant crawls from point A. Each time it moves, it crawls along the side of a unit square to an adjacent grid point.
IMAGE
How many number of ways to complete a path so that after exactly 12 moves, it stops at position B?
6 replies
Math291
Today at 12:02 PM
Maximilian113
an hour ago
JBMO 2024 SL G4
MuradSafarli   6
N 2 hours ago by Assassino9931
Source: JBMO 2024 Shortlist
Let $ABCD$ be a circumscribed quadrilateral with circumcircle $\omega$ such that $AE = EC$, where $E$ is the intersection point of the diagonals $AC$ and $BD$. Point $F$ is taken on $\omega$ such that $BF\parallel AC$. If $G$ is the reflection of $F$ with respect to $A$, prove that the circumcircle of $\triangle ADG$ is tangent to the line $AC$
6 replies
MuradSafarli
Jun 26, 2025
Assassino9931
2 hours ago
Tiling rectangle with smaller rectangles.
MarkBcc168   63
N 2 hours ago by eg4334
Source: IMO Shortlist 2017 C1
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.

Proposed by Jeck Lim, Singapore
63 replies
MarkBcc168
Jul 10, 2018
eg4334
2 hours ago
|4^m-7^n| a prime number
dangerousliri   10
N 2 hours ago by Maths_VC
Source: JBMO Shortlist 2024, N1
Find all pairs of positive integers $(m,n)$ such that $|4^m-7^n|$ is a prime number.

Proposed by Dorlir Ahmeti, Albania
10 replies
dangerousliri
Jun 26, 2025
Maths_VC
2 hours ago
Simple (EZ?) NT
obihs   0
2 hours ago
Source: own
Prove that there exists an integer $x$ such that for any prime $p$ and any positive integers $a,b,c,$ both $\dfrac{x-a}{p}$ and $\dfrac{x^p-x-p^bc}{p^{b+1}}$ are integers.
0 replies
obihs
2 hours ago
0 replies
Game theory in 2025???
Iveela   2
N 2 hours ago by luci1337
Source: IMSC 2025 P3
Alice and Bob play a game on a $K_{2026}$. They take turns with Alice playing first. Initially all edges are uncolored. On Alice's turn she chooses any uncolored edge and colors it red. On Bob's turn, he chooses 1, 2, or 3 uncolored edges and colors them blue. The game ends once all edges have been colored. Let $r$ and $b$ be the number of vertices of the largest red and blue clique, respectively. Bob wins if at the end of the game $b > r$. Show that Bob has a winning strategy.

Note: $K_n$ denotes the complete graph with $n$ vertices and where there is an edge between any pair of vertices. A red (or blue, respectively) clique refers to a complete subgraph of which all the edges are red (or blue, respectively).
2 replies
Iveela
Jul 5, 2025
luci1337
2 hours ago
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
Y by
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
1479 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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