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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
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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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Limit of a sequence involving the largest odd divisor
JackMinhHieu   1
N 9 minutes ago by mathreyes
Hi everyone,

I came across the following sequence and I’m curious about its behavior:

Let d(k) be the largest odd positive divisor of k. Define a sequence (x_n) by

x_n = (1/n) * sum_{k=1}^{n} (d(k)/k)

Question:
Does the sequence (x_n) converge? If so, what is its limit?

Any insights, proofs, or helpful observations would be appreciated. Thank you!
1 reply
1 viewing
JackMinhHieu
2 hours ago
mathreyes
9 minutes ago
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IMO 2010 Problem 4
mavropnevma   129
N 32 minutes ago by mahyar_ais
Let $P$ be a point interior to triangle $ABC$ (with $CA \neq CB$). The lines $AP$, $BP$ and $CP$ meet again its circumcircle $\Gamma$ at $K$, $L$, respectively $M$. The tangent line at $C$ to $\Gamma$ meets the line $AB$ at $S$. Show that from $SC = SP$ follows $MK = ML$.

Proposed by Marcin E. Kuczma, Poland
129 replies
mavropnevma
Jul 8, 2010
mahyar_ais
32 minutes ago
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JBMO 2014 #3 -- Inequality
gavrilos   42
N 38 minutes ago by kokos
For positive real numbers $a,b,c$ with $abc=1$ prove that $\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2}\geq 3(a+b+c+1)$
42 replies
gavrilos
Jun 23, 2014
kokos
38 minutes ago
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How about an AOPS MO?
MathMaxGreat   39
N an hour ago by JerryZYang
I am planning to make a $APOS$ $MO$, we can post new and original problems, my idea is to make an competition like $IMO$, 6 problems for 2 rounds
Any idea and plans?
39 replies
MathMaxGreat
Yesterday at 2:37 AM
JerryZYang
an hour ago
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Inequality with wx + xy + yz + zw = 1
Fermat -Euler   24
N an hour ago by SirAppel
Source: IMO ShortList 1990, Problem 24 (THA 2)
Let $ w, x, y, z$ are non-negative reals such that $ wx + xy + yz + zw = 1$.
Show that $ \frac {w^3}{x + y + z} + \frac {x^3}{w + y + z} + \frac {y^3}{w + x + z} + \frac {z^3}{w + x + y}\geq \frac {1}{3}$.
24 replies
Fermat -Euler
Nov 2, 2005
SirAppel
an hour ago
J
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Chinese Remainder Theorem
MathNerdRabbit103   0
an hour ago
Hi guys,
Lately i've been trying to understand the proof for the Chinese Remainder Theorem, however i have unfortunately had no luck. Can anybody post about how they understand the proof and please go step by step?
Appreciate it.
0 replies
MathNerdRabbit103
an hour ago
0 replies
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Good old functional equation on the reals
Tintarn   43
N an hour ago by Maths_VC
Source: Baltic Way 2020, Problem 4
Find all functions $f:\mathbb{R} \to \mathbb{R}$ so that
\[f(f(x)+x+y) = f(x+y) + y f(y)\]for all real numbers $x, y$.
43 replies
Tintarn
Nov 14, 2020
Maths_VC
an hour ago
J
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LCM of all pairwise sums
Philo-maths   0
an hour ago
Let $a_1,a_2,\dots,a_n$ be distinct positive integers. Prove that
\[ \operatorname{lcm}\bigl\{\,a_i + a_j : 1 \le i < j \le n\bigr\} \;\ge\; \sum_{i=1}^{n} a_i. \]
0 replies
Philo-maths
an hour ago
0 replies
J
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another geo in IMSC
ZeroHero   6
N 2 hours ago by hectorleo123
Given acute angled triangle $ABC$ with orthocenter $H$. Let $M$ be midpoint of arc $BC$ that containing $A$. Assume that $BH$ meet $MC$ at $F$ similarly $CH$ meet $MB$ at $E$, and $EF$ meet $BC$ at $K$. If $L$ is foot of perpendicular dropped from $M$ to line $KH$, prove that $AH=AL$.
6 replies
ZeroHero
Jul 6, 2025
hectorleo123
2 hours ago
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AOPS MO Introduce
MathMaxGreat   44
N 2 hours ago by JerryZYang
$AOPS MO$

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’
44 replies
MathMaxGreat
Today at 1:04 AM
JerryZYang
2 hours ago
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disjoint subsets
nayel   4
N 2 hours ago by chakrabortyahan
Source: Taiwan 2001
Let $n\ge 3$ be an integer and let $A_{1}, A_{2},\dots, A_{n}$ be $n$ distinct subsets of $S=\{1, 2,\dots, n\}$. Show that there exists $x\in S$ such that the n subsets $A_{i}-\{x\}, i=1,2,\dots n$ are also disjoint.

what i have is this
4 replies
nayel
Apr 18, 2007
chakrabortyahan
2 hours ago
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An Angle Trisector
bryanguo   3
N 2 hours ago by Sedro
Triangle $ABC$ has points $D$,$E$,$F$ on segment $BC$ in that order, where $D$ is between $B$ and $E$, and $AD$ and $AE$ trisect angle $BAF$. If $\angle BAF = 60^{\circ}$, $\frac{EF}{EC}=\frac{2}{3}$, and $\frac{AE}{AC} = 2$, find $\angle BAC$.

Individual #5
3 replies
bryanguo
Apr 11, 2024
Sedro
2 hours ago
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Great Inequality
EthanWYX2009   3
N 3 hours ago by arqady
Source: 2025 June 谜之竞赛-3
Given positive integer \( n \geq 2 \) and positive real \( t \). Let positive real numbers \( a_1, a_2, \ldots, a_n \) satisfy \( \sum_{i=1}^n a_i = t \). Denote \( S = \{1, 2, \cdots, n\} \). A non-empty subset \( I \) of \( S \) is called good if
\[\sum_{i\in I} a_i^3 \geq \left( \sum_{i\in I} a_i \right)^2.\]Determine the maximum possible number of good subsets of \( S \).

Created by Yuxing Ye
3 replies
EthanWYX2009
Today at 8:38 AM
arqady
3 hours ago
J
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10 Problems
Sedro   4
N 3 hours ago by Sedro
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 increasing sequence of positive integers $u_1, u_2, \dots, u_8$ has the property that the sum of its first $n$ terms is divisible by $n$ for every positive integer $n\le 8$. 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: 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: 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 $995$.
4 replies
Sedro
Jul 10, 2025
Sedro
3 hours ago
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Geometry
Exoticbuttersowo   1
N Jun 6, 2025 by sunken rock
In an isosceles triangle ABC with base AC and interior cevian AM, such that MC = 2 MB, and a point L on AM, such that BLC = 90 degrees and MAC = 42 degrees. Determine LBC.
1 reply
Exoticbuttersowo
Jun 5, 2025
sunken rock
Jun 6, 2025
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Geometry
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Exoticbuttersowo
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Exoticbuttersowo
#1 h  Jun 5, 2025, 7:28 PM
Y by
In an isosceles triangle ABC with base AC and interior cevian AM, such that MC = 2 MB, and a point L on AM, such that BLC = 90 degrees and MAC = 42 degrees. Determine LBC.
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sunken rock
4407 posts
sunken rock
#2 h  Jun 6, 2025, 11:27 AM • 1 Y
Y by Exoticbuttersowo
My solution at https://stanfulger.blogspot.com/2025/06/aops-httpsartofproblemsolvingcomcommuni.html

Best regards,
sunken rock
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