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
Functional equation
shactal   1
N 4 minutes ago by Mathzeus1024
Source: Own
Hello, I found this functional equation that I can't solve, and I haven't got any hints. Could someone try and find the solution, it's actually quite difficult:
Find all continuous functions $f:\mathbb{R}\to \mathbb{R}$ such that, for all $x, y \in \mathbb{R} $,
$$
f(x + f(y)) + f(y + f(x)) = f(x \, f(y) + y \, f(x)) + f(x + y)$$Thank you.
1 reply
shactal
Yesterday at 11:15 PM
Mathzeus1024
4 minutes ago
Bonza functions
KevinYang2.71   64
N 5 minutes ago by hnkevin42
Source: 2025 IMO P3
Let $\mathbb{N}$ denote the set of positive integers. A function $f\colon\mathbb{N}\to\mathbb{N}$ is said to be bonza if
\[
f(a)~~\text{divides}~~b^a-f(b)^{f(a)}
\]for all positive integers $a$ and $b$.

Determine the smallest real constant $c$ such that $f(n)\leqslant cn$ for all bonza functions $f$ and all positive integers $n$.

Proposed by Lorenzo Sarria, Colombia
64 replies
KevinYang2.71
Jul 15, 2025
hnkevin42
5 minutes ago
Probability Inequality
EthanWYX2009   0
15 minutes ago
Source: 2024 June 谜之竞赛-5
Determine the minimum real number \(\lambda\) such that for any $2024$ real numbers \(a_1, a_2, \cdots, a_{2024}\) satisfying
\[\sum_{i=1}^{2024} a_i = 0,\quad\sum_{i=1}^{2024} a_i^2 = 1,\]there exists a non-empty subset \(I\) of \(\{1, 2, \cdots, 2024\}\) for which
\[\sum_{i\in I} a_i \leq \lambda \cdot \min\{|I|, 2024 - |I|\}.\]Proposed by Tianqin Li, High School Affiliated to Renmin University of China
0 replies
EthanWYX2009
15 minutes ago
0 replies
Elegant Geometry Problem
EthanWYX2009   0
21 minutes ago
Source: 2024 June 谜之竞赛-2
Let \( I \) be the incenter of \(\triangle ABC\). The incircle tangents to \( AC \), \( AB \) at \( E \), \( F \), respectively. Let \( EF \) intersect \( BC \) at \( P \). \(\odot BEP\) and \(\odot CFP\) intersect again at \( Q \). Let \( M \) be the midpoint of the arc \( BC \) of \(\odot ABC\). \(\odot MPQ\) intersects \(\odot ABC\) again at \( R \). Let \( H \) be the orthocenter of \(\triangle BIC\).

Prove that the intersection point of \( HR \) and \( QI \) lies on \(\odot MPQ\).

Proposed by Bohan Zhang, Shanghai Minban Huayu Middle School
0 replies
EthanWYX2009
21 minutes ago
0 replies
Group theory study and review
JerryZYang   1
N 4 hours ago by JerryZYang
I want to review and potentially teach group theory... So can anyone give me some nice resources for review and beginners. The review ones are for me with around Middle School reading level the beginner ones are for me to see how to teach. :) thanks!
1 reply
JerryZYang
5 hours ago
JerryZYang
4 hours ago
Finite Series and Sequences - AP, GP, Special Finite series and Telescoping seri
wimpykid   5
N 6 hours ago by mudkip42
Let a natural number, $n = 2^{p - 1}(2^p - 1)$, where $2^p - 1$ is a prime. Show that the sum of all positive divisors of $n$ is equal to $2n$.
5 replies
wimpykid
Yesterday at 1:28 PM
mudkip42
6 hours ago
[PMO25 Area Stage I.1] Parallelogram Area
Konigsberg   1
N Today at 2:49 AM by Siopao_Enjoyer
In parallelogram \( WXYZ \), the length of diagonal \( WY \) is 15, and the perpendicular distances from \( W \) to lines \( YZ \) and \( XY \) are 9 and 12, respectively. Find the least possible area of the parallelogram.

Answer Confirmation
1 reply
Konigsberg
Feb 15, 2025
Siopao_Enjoyer
Today at 2:49 AM
cool problem.
ChickensEatGrass   2
N Today at 2:39 AM by SpeedCuber7
if $x + y+z = 0$, prove
$ax^2 + by^2+cz^2+xyz=4abc$.
2 replies
ChickensEatGrass
Yesterday at 7:11 PM
SpeedCuber7
Today at 2:39 AM
Inequalities
sqing   19
N Today at 1:23 AM by sqing
Let $ a,b,c\geq 0  . $ Prove that
$$ \sqrt{ a^3+b^3+c^3+\frac{1}{4}} +  \frac{9}{5}abc+\frac{1}{2} \geq a+b+c$$O706
19 replies
sqing
Jul 19, 2025
sqing
Today at 1:23 AM
Complex numbers
preatsreard   3
N Yesterday at 10:26 PM by preatsreard
How many complex z exist, such that z,z²,z³,....,z²⁰²¹ form a perfect 2021-gon (in arbitary order) in a complex plane.
3 replies
preatsreard
Jul 20, 2025
preatsreard
Yesterday at 10:26 PM
Ez geo with 2 circiles and parallel lines
Tofa7a._.36   2
N Yesterday at 9:26 PM by Merlinaeus
Let $ABC$ be an acute triangle with circumcircle $\omega$. Let $D$ be the intersection of $\omega$ with the angle bisector of $\angle BAC$. Let $E$ be on $[DC)$ and assume that the circumcircle of the triangle $ACE$ meets $[BC]$ at $F$. Let $(DF)$ meet $\omega$ and $(ACF)$ at $G$ and $H$, respectively.
Prove that $(GC) \parallel (HE)$ .
2 replies
Tofa7a._.36
Jul 15, 2025
Merlinaeus
Yesterday at 9:26 PM
120 degrees arc
xeroxia   3
N Yesterday at 7:49 PM by xeroxia
Let $\Pi$ be a circle with center $O$, and $A,B$ be points on $\Pi$ such that $\angle AOB = 120^\circ$. Let $M$ be the midpoint of minor $\overarc {AB}$. Let $C$ be a point on segment $OA$ such that $AC = 5$ and $CO = 3$. Let the circumcircle of $\triangle OCM$ meet $\Pi$ at $N$ and $OB$ at $D$. Let $ND$ meet $\Pi$ at $E$.
$NE = \dfrac ab$, where $a,b$ are relatively prime positive integers. Find $a+b$.
3 replies
xeroxia
Yesterday at 5:35 AM
xeroxia
Yesterday at 7:49 PM
PRMO 2019 #26
molikpagaria   1
N Yesterday at 4:33 PM by nudinhtien
Positive integers x, y, z satisfy xy + z = 160. Compute the smallest possible value of x + yz.

I have seen many people doing this by hit and trial, but how do you solve it using an actual way?
1 reply
molikpagaria
Yesterday at 4:26 PM
nudinhtien
Yesterday at 4:33 PM
Mathematics
MathophobiticLad   0
Yesterday at 4:12 PM
Evaluate: $$\sum \sum_{0<r<s<n} \lfloor \frac{LCM(^nC_r , ^nC_s)}{^nC_r , ^nC_s} \rfloor $$Where $\lfloor . \rfloor$ represents Greatest integer function.
0 replies
MathophobiticLad
Yesterday at 4:12 PM
0 replies
D1036 : Composition of polynomials
Dattier   3
N May 26, 2025 by mohabstudent1
Source: les dattes à Dattier
Find all $A \in \mathbb Q[x]$ with $\exists Q \in \mathbb Q[x], Q(A(x))= x^{2025!+2}+x^2+x+1$ and $\deg(A)>1$.
3 replies
Dattier
May 24, 2025
mohabstudent1
May 26, 2025
D1036 : Composition of polynomials
G H J
G H BBookmark kLocked kLocked NReply
Source: les dattes à Dattier
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Dattier
1558 posts
#1
Y by
Find all $A \in \mathbb Q[x]$ with $\exists Q \in \mathbb Q[x], Q(A(x))= x^{2025!+2}+x^2+x+1$ and $\deg(A)>1$.
Z K Y
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Dattier
1558 posts
#3
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No one ?
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mohabstudent1
7 posts
#4
Y by
Coming to solve it!
Z K Y
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mohabstudent1
7 posts
#5
Y by
I have biology exam tomorrow, so I will solve it when I get up.
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