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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
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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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2024 International Math Olympiad Number Theory Shortlist, Problem 3
brainfertilzer   8
N 20 minutes ago by shanelin-sigma
Source: 2024 ISL N3
Determine all sequences $a_1, a_2, \dots$ of positive integers such that for any pair of positive integers $m\leqslant n$, the arithmetic and geometric means
\[ \frac{a_m + a_{m+1} + \cdots + a_n}{n-m+1}\quad\text{and}\quad (a_ma_{m+1}\cdots a_n)^{\frac{1}{n-m+1}}\]are both integers.
8 replies
1 viewing
brainfertilzer
Today at 3:00 AM
shanelin-sigma
20 minutes ago
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IMO 2025 P2
sarjinius   57
N 21 minutes ago by khanhnx
Source: 2025 IMO P2
Let $\Omega$ and $\Gamma$ be circles with centres $M$ and $N$, respectively, such that the radius of $\Omega$ is less than the radius of $\Gamma$. Suppose $\Omega$ and $\Gamma$ intersect at two distinct points $A$ and $B$. Line $MN$ intersects $\Omega$ at $C$ and $\Gamma$ at $D$, so that $C, M, N, D$ lie on $MN$ in that order. Let $P$ be the circumcentre of triangle $ACD$. Line $AP$ meets $\Omega$ again at $E\neq A$ and meets $\Gamma$ again at $F\neq A$. Let $H$ be the orthocentre of triangle $PMN$.

Prove that the line through $H$ parallel to $AP$ is tangent to the circumcircle of triangle $BEF$.
57 replies
+2 w
sarjinius
Yesterday at 3:38 AM
khanhnx
21 minutes ago
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The inekoalaty game
sarjinius   18
N 24 minutes ago by blackidea
Source: 2025 IMO P5
Alice and Bazza are playing the inekoalaty game, a two‑player game whose rules depend on a positive real number $\lambda$ which is known to both players. On the $n$th turn of the game (starting with $n=1$) the following happens:
[list]
[*] If $n$ is odd, Alice chooses a nonnegative real number $x_n$ such that
\[ x_1 + x_2 + \cdots + x_n \le \lambda n. \][*]If $n$ is even, Bazza chooses a nonnegative real number $x_n$ such that
\[ x_1^2 + x_2^2 + \cdots + x_n^2 \le n. \][/list]
If a player cannot choose a suitable $x_n$, the game ends and the other player wins. If the game goes on forever, neither player wins. All chosen numbers are known to both players.

Determine all values of $\lambda$ for which Alice has a winning strategy and all those for which Bazza has a winning strategy.

Proposed by Massimiliano Foschi and Leonardo Franchi, Italy
18 replies
sarjinius
Today at 3:18 AM
blackidea
24 minutes ago
J
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Probably a Cool Optimization Problem
PhysicsIsChad   3
N 36 minutes ago by littleduckysteve
What i am trying to do is finding a general solution to this earlier problem (https://artofproblemsolving.com/community/u1056796h3610029p35343126) . Here i am trying to generalise such a solution for n points . Simply stated find the min value of \[ f(x, y) = \sum_{i=1}^{n} \sqrt{(x - a_i)^2 + (y - b_i)^2} \]. I would love to hear your insights on this problem .
My insights :-
\[ \text{Let } (x_1, y_1), (x_2, y_2), \dots, (x_n, y_n) \text{ be the points} \]\[ \text{Minimize } f(x, y) = \sqrt{(x - x_1)^2 + (y - y_1)^2} + \sqrt{(x - x_2)^2 + (y - y_2)^2} + \cdots + \sqrt{(x - x_n)^2 + (y - y_n)^2} \]\[ \text{Set } \frac{\partial f}{\partial x} = 0 \quad \text{and} \quad \frac{\partial f}{\partial y} = 0 \]\[ \frac{\partial f}{\partial x} = \frac{x - x_1}{\sqrt{(x - x_1)^2 + (y - y_1)^2}} + \frac{x - x_2}{\sqrt{(x - x_2)^2 + (y - y_2)^2}} + \cdots + \frac{x - x_n}{\sqrt{(x - x_n)^2 + (y - y_n)^2}} = 0 \]\[ \frac{\partial f}{\partial y} = \frac{y - y_1}{\sqrt{(x - x_1)^2 + (y - y_1)^2}} + \frac{y - y_2}{\sqrt{(x - x_2)^2 + (y - y_2)^2}} + \cdots + \frac{y - y_n}{\sqrt{(x - x_n)^2 + (y - y_n)^2}} = 0 \]I am not able to move ahead but probably we will find a constraint here through some inequality which will help us reach the solution .


3 replies
PhysicsIsChad
an hour ago
littleduckysteve
36 minutes ago
J
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Power Of Factorials
Kassuno   186
N 37 minutes ago by mudkip42
Source: IMO 2019 Problem 4
Find all pairs $(k,n)$ of positive integers such that \[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]Proposed by Gabriel Chicas Reyes, El Salvador
186 replies
Kassuno
Jul 17, 2019
mudkip42
37 minutes ago
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25% disscount
Mr.C   49
N 39 minutes ago by Bardia7003
Source: Iranian Third Round 2020 Algebra exam Problem1
find all functions from the reals to themselves. such that for every real $x,y$.
$$f(y-f(x))=f(x)-2x+f(f(y))$$
49 replies
Mr.C
Nov 20, 2020
Bardia7003
39 minutes ago
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Knight Yuri
RaymondZhu   5
N 40 minutes ago by math90
Source: ISL 2024 C3
Let $n$ be a positive integer. There are $2n$ knights sitting at a round table. They consist of $n$ pairs of partners, each pair of which wishes to shake hands. A pair can shake hands only when next to each other. Every minute, one pair of adjacent knights swaps places. Find the minimum number of exchanges of adjacent knights such that, regardless of the initial arrangement, every knight can meet her partner and shake hands at some time.
5 replies
RaymondZhu
Today at 3:00 AM
math90
40 minutes ago
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Difficulty in MOHS of IMO 2025 problems
carefully   0
40 minutes ago
What do you think about difficulty of IMO 2025 problems?

P1: 10M - typical P1, strightforward technique but with a case that some students might miss
P2: don't know
P3: 35M - on the easier side of P3
P4: 15-20M - quite difficult for P4, can even be a middle problem confortably, much harder than IMO 2005 P4
P5: 25-30M - a little bit on the harder side of P5, comparable to IMO 2016 P5
P6: 45M - on the harder side of P6, considerably harder than IMO 2022 P6
0 replies
carefully
40 minutes ago
0 replies
J
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Generalization of India TST
EthanWYX2009   0
43 minutes ago
Source: 2024 March 谜之竞赛-3, Generalization of aops.com/community/c6h3107633p28107117
Let rational number $r=\frac qp$ where $p$, $q$ are coprime positive integers. Show that for any integer $n\ge \min\{10\sqrt{pq},2q\}$, there exists ${n}$ positive integers $a_1$, $a_2$, $\cdots$, $a_n$ satisfying
\[\sum_{1\le i<j\le n}\frac 1{a_ia_j}=r.\]Created by Xianbang Wang, High School Affiliated to Renmin University of China
0 replies
EthanWYX2009
43 minutes ago
0 replies
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10 Problems
Sedro   20
N an hour ago by littleduckysteve
Title says most of it. I've been meaning to post a problem set on HSM since at least a few months ago, but since I proposed the most recent problems I made to the 2025 SSMO, I had to wait for that happen. (Hence, most of these problems will probably be familiar if you participated in that contest, though numbers and wording may be changed.) The problems are very roughly arranged by difficulty. Enjoy!

Problem 1: An sequence of positive integers $u_1, u_2, \dots, u_8$ has the property for every positive integer $n\le 8$, its $n^\text{th}$ term is greater than the mean of the first $n-1$ terms, and the sum of its first $n$ terms is a multiple of $n$. Let $S$ be the number of such sequences satisfying $u_1+u_2+\cdots + u_8 = 144$. Compute the remainder when $S$ is divided by $1000$.

Problem 2 (solved by fruitmonster97): Rhombus $PQRS$ has side length $3$. Point $X$ lies on segment $PR$ such that line $QX$ is perpendicular to line $PS$. Given that $QX=2$, the area of $PQRS$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

Problem 3: Positive integers $a$ and $b$ satisfy $a\mid b^2$, $b\mid a^3$, and $a^3b^2 \mid 2025^{36}$. If the number of possible ordered pairs $(a,b)$ is equal to $N$, compute the remainder when $N$ is divided by $1000$.

Problem 4: Let $ABC$ be a triangle. Point $P$ lies on side $BC$, point $Q$ lies on side $AB$, and point $R$ lies on side $AC$ such that $PQ=BQ$, $CR=PR$, and $\angle APB<90^\circ$. Let $H$ be the foot of the altitude from $A$ to $BC$. Given that $BP=3$, $CP=5$, and $[AQPR] = \tfrac{3}{7} \cdot [ABC]$, the value of $BH\cdot CH$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

Problem 5 (solved by maromex): Anna has a three-term arithmetic sequence of integers. She divides each term of her sequence by a positive integer $n>1$ and tells Bob that the three resulting remainders are $20$, $52$, and $R$, in some order. For how many values of $R$ is it possible for Bob to uniquely determine $n$?

Problem 6: There is a unique ordered triple of positive reals $(x,y,z)$ satisfying the system of equations \begin{align*} x^2 + 9 &= (y-\sqrt{192})^2 + 4 \\ y^2 + 4 &= (z-\sqrt{192})^2 + 49 \\ z^2 + 49 &= (x-\sqrt{192})^2 + 9. \end{align*}The value of $100x+10y+z$ can be expressed as $p\sqrt{q}$, where $p$ and $q$ are positive integers such that $q$ is square-free. Compute $p+q$.

Problem 7: Let $S$ be the set of all monotonically increasing six-term sequences whose terms are all integers between $0$ and $6$ inclusive. We say a sequence $s=n_1, n_2, \dots, n_6$ in $S$ is symmetric if for every integer $1\le i \le 6$, the number of terms of $s$ that are at least $i$ is $n_{7-i}$. The probability that a randomly chosen element of $S$ is symmetric is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Compute $p+q$.

Problem 8: For a positive integer $n$, let $r(n)$ denote the value of the binary number obtained by reading the binary representation of $n$ from right to left. Find the smallest positive integer $k$ such that the equation $n+r(n)=2k$ has at least ten positive integer solutions $n$.

Problem 9: Let $p$ be a quadratic polynomial with a positive leading coefficient. There exists a positive real number $r$ such that $r < 1 < \tfrac{5}{2r} < 5$ and $p(p(x)) = x$ for $x \in \{ r,1, \tfrac{5}{2r} , 5\}$. Compute $p(20)$.

Problem 10: Find the number of ordered triples of positive integers $(a,b,c)$ such that $a+b+c=995$ and $ab+bc+ca$ is a multiple of $995$.
20 replies
Sedro
Jul 10, 2025
littleduckysteve
an hour ago
J
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IMO 2025、start times and end times
cielblue   3
N an hour ago by GreenTea2593
What are the start times for Day 1, Day 2, and AEST UTC+10?

Is the following schedule correct?

\begin{tabular}{|c|c|c|} \hline \multicolumn{3}{|c|}{\textbf{Australia Local Time (AEST UTC+10)}} \\ \hline Date & Start Time (AEST) & End Time \\ \hline Tuesday, July 15 & 08:00 & 12:30 \\ Wednesday, July 16 & 08:00 & 12:30 \\ \hline \end{tabular}

3 replies
cielblue
5 hours ago
GreenTea2593
an hour ago
J
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Sequence with divisibility condition, but not polynomial?
cuden   1
N an hour ago by cuden
Source: 2025 Vinh Summer School selection test
I tried checking simple polynomial and non-polynomial sequences, but couldn't find a counterexample.
1 reply
cuden
5 hours ago
cuden
an hour ago
J
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Trigonometry equation practice
ehz2701   15
N an hour ago by vanstraelen
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 (1-10)

problem set 2a (1-20)

problem set 2b (1-20)
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.
15 replies
ehz2701
Jul 12, 2025
vanstraelen
an hour ago
J
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Concurrency of Lines Involving Altitudes and Circumcenters in a Triangle
JackMinhHieu   2
N 2 hours ago by JackMinhHieu
Hi everyone,
I recently came across an interesting geometry problem that I'd like to share. It involves a triangle inscribed in a circle, altitudes, points on arcs, and a surprising concurrency involving circumcenters. Here's the problem:
Problem:
Let ABC be an acute triangle inscribed in a circle (O). Let the altitudes AD, BE, CF intersect at the orthocenter H.
Points M and N lie on the minor arcs AB and AC of circle (O), respectively, such that MN // BC.
Let I be the circumcenter of triangle NEC, and J be the circumcenter of triangle MFB.
Prove that the lines OD, BI, and CJ are concurrent.

I find the configuration quite elegant, and I'm looking for different ways to approach the problem — whether it's synthetic, coordinate, vector-based, or inversion.
Any ideas, hints, or full solutions are appreciated. Thank you!
2 replies
JackMinhHieu
Jul 14, 2025
JackMinhHieu
2 hours ago
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J
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Polynomial optimization problem
ReticulatedPython   2
N Apr 2, 2025 by Mathzeus1024
Let $$p(x)=-ax^4+x^3$$, where $a$ is a real number. Prove that for all positive $a$, $$p(x) \le \frac{27}{256a^3}.$$
2 replies
ReticulatedPython
Mar 31, 2025
Mathzeus1024
Apr 2, 2025
J
Polynomial optimization problem
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734 posts
ReticulatedPython
#1 h  Mar 31, 2025, 2:51 PM
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Let $$p(x)=-ax^4+x^3$$, where $a$ is a real number. Prove that for all positive $a$, $$p(x) \le \frac{27}{256a^3}.$$
This post has been edited 1 time. Last edited by ReticulatedPython, Mar 31, 2025, 3:03 PM
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Lankou
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Lankou
#2 h  Mar 31, 2025, 7:09 PM
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The function has four real zeros: $0$ of multiplicity $3$ and $\frac{1}{a}$ of multiplicity $1$. The extremities of the graph both approache the $-\infty$ so the function has an absolute maximum between $0$ and $\frac{1}{a}$
$p'(x)=x^3(-ax+1)$
The graph has a turning point at $x=\frac {3}{4a}$
At this point the maximum is $=\frac {27}{256a^3}$
Hence $p(x) \le \frac{27}{256a^3}.$
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Mathzeus1024
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Mathzeus1024
#4 h  Apr 2, 2025, 12:38 PM
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Taking $a \in \mathbb{R}^{+}$, the derivative of $p(x)$ equal to zero yields the critical values:

$p'(x) = -4ax^3+3x^2 = x^2(3-4ax) = 0 \Rightarrow x = 0, \frac{3}{4a}$ (i).

A second derivative check of (i) shows that:

$p''(x) = -12ax^2+6x \Rightarrow p''(0) = 0$ (an inflection point) and $p''\left(\frac{3}{4a}\right) = -\frac{9}{4a} < 0$ (a global maximum).

Thus, $p_{MAX} = p\left(\frac{3}{4a}\right) = \textcolor{red}{\frac{27}{256a^3}} \le \frac{27}{256a^3}$.

QED
This post has been edited 2 times. Last edited by Mathzeus1024, Apr 2, 2025, 12:43 PM
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