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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
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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
J
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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
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Elegant DFT Practice
zqy648   0
22 minutes ago
Source: 2024 November 谜之竞赛-3
Given an odd prime \( p \), a permutation \( a_1, a_2, \ldots, a_{p-1} \) of \( 1, 2, \ldots, p-1 \) is called good if for any positive integer \( t \in \{1, 2, \ldots, p-2\} \) and any distinct integers \( i, j \in \{1, 2, \ldots, p-1\} \),
\[a_{i+t} - a_i \not\equiv a_{j+t} - a_j \pmod{p},\]where the subscripts are interpreted modulo \( p-1 \). Determine the number of good permutations.

Proposed by Hengye Zhang
0 replies
zqy648
22 minutes ago
0 replies
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Using Dilworth to Deal with Antichains
zqy648   0
28 minutes ago
Source: 2024 November 谜之竞赛-6
For prime number \( p \) and positive integer \( a \in \{1, 2, \cdots, p-1\} \), denote \( (a^{-1}\bmod{p}) \) by the unique integer \( m \) satisfying \( am \equiv 1 \pmod{p} \) with \( 0 \leq m \leq p-1 \).

Let \( k \) be a positive integer, and let \( p \) be a prime such that \( p > 100k^2 \). Prove that there exist positive integers \( x_1, x_2, \cdots, x_k \) satisfying:[list]
[*] \( 1 \leq x_1 < x_2 < \cdots < x_k \leq p-1 \);
[*]\( (x_1^{-1}\bmod{p}) > (x_2^{-1}\bmod{p}) > \cdots > (x_k^{-1}\bmod{p}). \) [/list]
Proposed by Cheng Jiang and Tianqin Li
0 replies
zqy648
28 minutes ago
0 replies
J
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Easy and Elegant Inequality
zqy648   0
35 minutes ago
Source: 2024 November 谜之竞赛-4
For positive integers \( n \) and positive real numbers \( a_1, a_2, \ldots, a_n \), define
\[[a_1, a_2, \cdots, a_n] = \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{\cdots + \frac{1}{a_n}}}}. \]Determine the minimal real number \( C \) such that for any positive real numbers \( x_1, x_2, \cdots, x_{2024} \),
\[[x_1, x_2, \cdots, x_{2024}] - [x_1^2, x_2^2, \cdots, x_{2024}^2] \leq C. \]Proposed by Ruixuan Zhu
0 replies
zqy648
35 minutes ago
0 replies
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Tangent to two Fixed Circles
zqy648   0
39 minutes ago
Source: 2024 November 谜之竞赛-1
Given a circle \(\Omega\) and its chord \(AB\), let \(C\) be a moving point on the segment \(AB\). \(D\) and \(E\) lie on \(\Omega\) such that \(\angle ADC = \angle BEC = 90^\circ\), \(AE\) and \(BD\) intersect at point \(F\).

Prove that as \(C\) moves along the open segment \(AB\) (excluding endpoints), the circumcircle of \(\triangle DEF\) is tangent to two fixed circles.

Proposed by Hongdao Chen
0 replies
zqy648
39 minutes ago
0 replies
J
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Bernoulli Site Percolation
zqy648   0
an hour ago
Source: 2025 Feb 谜之竞赛-2
Let \( \mathbb{Z}^2 \) denote the set of all integer lattice points in the plane with Cartesian coordinates. A graph \( G = (\mathbb{Z}^2, E) \) is constructed by connecting two lattice points with an edge if their Euclidean distance is $1{}{}{}$. For a positive integer \( n \), define \( f(n) \) as the number of connected subgraphs \( H = (V(H), E(H)) \) of \( G \) satisfying:
\[ \{(0, 0)\} \subseteq V(H) \subseteq \mathbb{Z}^2, \quad |V(H)| = n, \quad E(H) \subseteq E. \]Prove that there exists a positive constant \( C \) such that for any positive integer \( n \), \( f(n) \leq C \cdot 7^n. \)

Proposed by Hanqing Huang and Huankun Guo
0 replies
zqy648
an hour ago
0 replies
J
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Prove that KI₁=KI₂
zqy648   0
an hour ago
Source: 2025 February 谜之竞赛-6
In acute triangle \( ABC \), let \( H \) be the orthocenter. Denote the incenter of \( \triangle ABH \) by \( I_1 \) and its incircle by \( \omega_1 \); the incenter of \( \triangle ACH \) by \( I_2 \) and its incircle by \( \omega_2 \). Let \( l \) be the common external tangent of \( \omega_1 \) and \( \omega_2 \) closer to vertex \( A \), tangent to \( \omega_1 \) and \( \omega_2 \) at \( P \) and \( Q \), respectively. Lines \( BP \) and \( CQ \) intersect at \( K \). Prove that \( KI_1 = KI_2 \).

Created by Lolochen(Xiuyi Chen)
0 replies
zqy648
an hour ago
0 replies
J
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Find all function
Math2030   1
N an hour ago by Mathzeus1024
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation
\[ f\left(x^2 + xy^2 + y^2\right) = 2x^2 f(y) + 2x f(f(y)) + f\left(-x^2 - xy^2\right) + f(y^2) \]for all \( x, y \in \mathbb{R} \).
1 reply
Math2030
Friday at 1:50 PM
Mathzeus1024
an hour ago
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BK.CJ=BJ.CH
truongphatt2668   0
an hour ago
Let triangle $ABC$ has $D$ on $A$-angle bisector such that $\widehat{BDC} = 90^o$. A line passes $A$ perpendicular to $AD$ cuts $BD,CD$ at $E,F$, respectively. $I$ is the intersection of $AB$ and $(ADF)$. $J$ is defined similarly. $IC$ and $JB$ cuts $(ADF), (ADE)$ at $H,K$, respectively. Prove that: $BK.CJ = BJ.CH$.
0 replies
truongphatt2668
an hour ago
0 replies
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Using Nearly all Numbers
EthanWYX2009   0
an hour ago
Source: 2025 April 谜之竞赛-6
Let \(\varepsilon\) be a positive real number. Prove that there exists a positive integer \(M\) such that for any integer \(m > M\), if a positive integer \(N\) satisfies
\[ \frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{N} > m + \varepsilon, \]then there exist a positive integer \(k\) and \(k\) integers \( 1 \leq x_1 < x_2 < \cdots < x_k \leq N, \) satisfying
\[ \frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_k} = m. \]Created by Cheng Jiang and Hengxu Deng
0 replies
EthanWYX2009
an hour ago
0 replies
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A cyclic inequality
Nguyenhuyen_AG   3
N 2 hours ago by Victoria_Discalceata1
Let $a, \, b, \,c$ be positive real numbers. Prove that
\[(5a^3-b^3)(a-b)^2+(5b^3-c^3)(b-c)^2 + (5c^3-a^3)(c-a)^2 \geqslant 0.\]
3 replies
Nguyenhuyen_AG
Jul 11, 2025
Victoria_Discalceata1
2 hours ago
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d(2025^{a_i}-1) divides a_{n+1}
navi_09220114   3
N 2 hours ago by ihategeo_1969
Source: TASIMO 2025 Day 2 Problem 5
Let $a_n$ be a strictly increasing sequence of positive integers such that for all positive integers $n\ge 1$
\[d(2025^{a_n}-1)|a_{n+1}.\]Show that for any positive real number $c$ there is a positive integers $N_c$ such that $a_n>n^c$ for all $n\geq N_c$.

Note. Here $d(m)$ denotes the number of positive divisors of the positive integer $m$.
3 replies
navi_09220114
May 19, 2025
ihategeo_1969
2 hours ago
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AM-GM Problem
arcticfox009   11
N 2 hours ago by LayZee
Let $x, y$ be positive real numbers such that $xy \geq 1$. Find the minimum value of the expression

\[ \frac{(x^2 + y)(x + y^2)}{x + y}. \]
answer confirmation
11 replies
arcticfox009
Friday at 3:01 PM
LayZee
2 hours ago
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Easy geometry problem
menseggerofgod   3
N 4 hours ago by henryli3333
ABC is a right triangle, right at B, in which the height BD is drawn. E is a point on side BC such that AE = EC = 8. If BD is 6 and DE = k , find k
3 replies
menseggerofgod
Today at 2:47 AM
henryli3333
4 hours ago
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Trigonometry equation practice
ehz2701   3
N 6 hours ago by ehz2701
There is a lack of trigonometric bash practice, and I want to see techniques to do these problems. So here are 10 kinds of problems that are usually out in the wild. How do you tackle these problems? (I had ChatGPT write a code for this.). Please give me some general techniques to solve these kinds of problems, especially set 2b. I’ll add more later.

Leaderboard

problem set 1a

problem set 2a

problem set 2b
answers 2b

General techniques so far:

Trick 1: one thing to keep in mind is

[center] $\frac{1}{2} = \cos 36 - \sin 18$. [/center]

Many of these seem to be reducible to this. The half can be written as $\cos 60 = \sin 30$, and $\cos 36 = \sin 54$, $\sin 18 = \cos 72$. This is proven in solution 1a-1. We will refer to this as Trick 1.
3 replies
ehz2701
Yesterday at 8:48 AM
ehz2701
6 hours ago
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Easy Function
Darealzolt   1
N Jun 6, 2025 by alexheinis
Let \( f(x+y) = f(x^2y)\) for all real numbers \(x,y\), hence find the value of \(f(3)\) if \(f(2023)=26\).
1 reply
Darealzolt
Jun 6, 2025
alexheinis
Jun 6, 2025
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Easy Function
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Darealzolt
89 posts
Darealzolt
#1 h  Jun 6, 2025, 8:47 AM
Y by
Let \( f(x+y) = f(x^2y)\) for all real numbers \(x,y\), hence find the value of \(f(3)\) if \(f(2023)=26\).
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alexheinis
10735 posts
alexheinis
#2 h  Jun 6, 2025, 8:50 AM
Y by
Take $y=0$ then $f(x)=f(0)$ and $f$ is constant. Hence $f(3)=26$.
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