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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.

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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
I miss Turbo
sarjinius   37
N 7 minutes ago by XaRoiMuaThu
Source: 2025 IMO P6
Consider a $2025\times2025$ grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile.

Determine the minimum number of tiles Matilda needs to place so that each row and each column of the grid has exactly one unit square that is not covered by any tile.

Proposed by Zhao Yu Ma and David Lin Kewei, Singapore
37 replies
+3 w
sarjinius
Jul 16, 2025
XaRoiMuaThu
7 minutes ago
Set of perfect powers is irreducible
Assassino9931   3
N 8 minutes ago by Just1
Source: Al-Khwarizmi International Junior Olympiad 2025 P4
For two sets of integers $X$ and $Y$ we define $X\cdot Y$ as the set of all products of an element of $X$ and an element of $Y$. For example, if $X=\{1, 2, 4\}$ and $Y=\{3, 4, 6\}$ then $X\cdot Y=\{3, 4, 6, 8, 12, 16, 24\}.$ We call a set $S$ of positive integers good if there do not exist sets $A,B$ of positive integers, each with at least two elements and such that the sets $A\cdot B$ and $S$ are the same. Prove that the set of perfect powers greater than or equal to $2025$ is good.

(In any of the sets $A$, $B$, $A\cdot B$ no two elements are equal, but any two or three of these sets may have common elements. A perfect power is an integer of the form $n^k$, where $n>1$ and $k > 1$ are integers.)

Lajos Hajdu and Andras Sarkozy, Hungary
3 replies
Assassino9931
May 9, 2025
Just1
8 minutes ago
shortlisted problems being used in undergraduate competition
enter16180   1
N 35 minutes ago by Happycaptain
Hello, I am posting here to let know ( clarified after a post in College Math forum) that Problem 10 of Open Mathematical Olympiad for University Students ( OMOUS-2025) held at Ashgabat, Turkmenistan on 13-18 April, 2025 is found to be A6 Shortlisted Problems IMO-2024.
Following is discussion on College Math Forum
https://artofproblemsolving.com/community/c7h3551018_omous2025_team_competition_p10


Image of problem from competition for reference below.
1 reply
+1 w
enter16180
2 hours ago
Happycaptain
35 minutes ago
Help me prove these lemmas
dimi07   1
N 36 minutes ago by blug
In the name of God, the most Merciful, the most Compassionate.
Let $a,b,c$ $\in$ $\mathbb{Z}$,then prove the following
\[
a\mid c, b\mid c \implies lcm(a,b)\mid c.
\]And also prove that
\[
c\mid a,c\mid b \implies c\mid gcd(a,b)
\]And by the help of God I finish this question.
1 reply
1 viewing
dimi07
3 hours ago
blug
36 minutes ago
Equilateral pentagon with four right angles
Miquel-point   1
N an hour ago by loup blanc
Source: KoMaL B. 5396
An equilateral pentagon in the three-dimensional space has four right angles. What can be its fifth angle?

Proposed by Péter Dombi, Pécs
1 reply
Miquel-point
Jun 11, 2024
loup blanc
an hour ago
2024 PMO Part II #2
orangefronted   3
N an hour ago by BinariouslyRandom
Determine all positive integers $k$ less than 2024 for which $4n+1$ and $kn+1$ are relatively prime for all integers $n$.
3 replies
orangefronted
Jan 16, 2024
BinariouslyRandom
an hour ago
Right tetrahedron of fixed volume and min perimeter
Miquel-point   1
N an hour ago by Mathzeus1024
Source: Romanian IMO TST 1981, Day 4 P3
Determine the lengths of the edges of a right tetrahedron of volume $a^3$ so that the sum of its edges' lengths is minumum.

1 reply
Miquel-point
Apr 6, 2025
Mathzeus1024
an hour ago
Functional equation
shactal   2
N an hour ago by shactal
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.
2 replies
+1 w
shactal
Yesterday at 11:15 PM
shactal
an hour ago
At least One pair with square of distance multiple of 2016
Johann Peter Dirichlet   5
N an hour ago by ja.
Source: 38th Brazilian MO (2016) - First Day, Problem 2
Find the smallest number \(n\) such that any set of \(n\) ponts in a Cartesian plan, all of them with integer coordinates, contains two poitns such that the square of its mutual distance is a multiple of \(2016\).
5 replies
Johann Peter Dirichlet
Nov 23, 2016
ja.
an hour ago
A symmetric inequality in n variables (3)
Nguyenhuyen_AG   1
N 2 hours ago by lbh_qys
Let $a_1,a_2,\ldots,a_n (n \geqslant 1)$ be non-negative real numbers. Prove that
\[\sum_{i=1}^n \frac{\displaystyle a_i^2 \left( \sum_{j=1}^n a_j - a_i \right)^2}{\displaystyle \sum_{j=1}^n a_j^2 - a_i^2} \geq \left(\sum_{i=1}^n a_i\right)^2 - \sum_{i=1}^n a_i^2.\]Assume all denominators are non-zero.
1 reply
Nguyenhuyen_AG
4 hours ago
lbh_qys
2 hours ago
Four variables
Nguyenhuyen_AG   0
2 hours ago
Let $a,\,b,\,c,\,d$ non-negative real numbers. Prove that
\[\frac{abc}{(a+b+c)^3}+\frac{bcd}{(b+c+d)^3}+\frac{cda}{(c+d+a)^3}+\frac{dab}{(d+a+b)^3} \leqslant \frac{(a+b+c+d)^2}{27(a^2+b^2+c^2+d^2)}.\]
0 replies
Nguyenhuyen_AG
2 hours ago
0 replies
Probability Inequality
EthanWYX2009   0
2 hours 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
2 hours ago
0 replies
Elegant Geometry Problem
EthanWYX2009   0
2 hours 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
2 hours ago
0 replies
Non-polynomial sequences satifying m+n|a_m+a_n?
TUAN2k8   0
2 hours ago
Source: own
Consider a sequence of integers \((a_n)_{n>0}\) such that for every pair of distinct positive integers \((m, n)\), \(m + n\) is a divisor of \(a_m + a_n\).

a) Prove that \(a_n\) is divisible by \(n\) for every positive integer \(n\).

b) Does there exist a sequence \((a_n)_{n>0}\) that is not a polynomial in \(n\) (i.e., there does not exist a polynomial \(P(X) \in \mathbb{R}[X]\) such that \(a_n = P(n)\) for all \(n \in \mathbb{Z}_+\)) and satisfies the given condition?
0 replies
TUAN2k8
2 hours ago
0 replies
Balanced grids
BR1F1SZ   2
N Jul 17, 2025 by Tamam
Source: 2025 Francophone MO Juniors/Seniors P2
Let $n \geqslant 2$ be an integer. We consider a square grid of size $2n \times 2n$ divided into $4n^2$ unit squares. The grid is called balanced if:
[list]
[*]Each cell contains a number equal to $-1$, $0$ or $1$.
[*]The absolute value of the sum of the numbers in the grid does not exceed $4n$.
[/list]
Determine, as a function of $n$, the smallest integer $k \geqslant 1$ such that any balanced grid always contains an $n \times n$ square whose absolute sum of the $n^2$ cells is less than or equal to $k$.
2 replies
BR1F1SZ
May 10, 2025
Tamam
Jul 17, 2025
Balanced grids
G H J
G H BBookmark kLocked kLocked NReply
Source: 2025 Francophone MO Juniors/Seniors P2
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BR1F1SZ
593 posts
#1
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Let $n \geqslant 2$ be an integer. We consider a square grid of size $2n \times 2n$ divided into $4n^2$ unit squares. The grid is called balanced if:
  • Each cell contains a number equal to $-1$, $0$ or $1$.
  • The absolute value of the sum of the numbers in the grid does not exceed $4n$.
Determine, as a function of $n$, the smallest integer $k \geqslant 1$ such that any balanced grid always contains an $n \times n$ square whose absolute sum of the $n^2$ cells is less than or equal to $k$.
Z K Y
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alfonsoramires
9 posts
#2
Y by
bumppppp
Z K Y
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Tamam
29 posts
#3
Y by
Lemme solve :yoda:

The answer is $k=n$

$1$) $k\geq n$

Just the first and the $n+1$ th row be $1$'s and other squares $0$'s. Every $n*n$ square has the sum $n$.

$2$) $k=n$ actually works

Assume otherwise(We just assume every squares sums absolute value is greater than $n$.). Let $S_i$ be the $n$ times $n$ square that has its most left highest square as the $i$ th square in the first row. WLOG let $S_1$ have a positive sum. Let $S_i$ be positive. Let $T_i$ and $T_{i+1}$ be the sums of $S_i$ and $S_{i+1}$ respectively. We can see that $\lvert T_{i+1}-T_i \rvert \leq 2n$ because the two squares differ by only two columns($2n$ numbers.). If $T_{i+1}$ was negative it could be at most $-n-1$.And since $T_i \geq n+1$ the inequality $\lvert T_{i+1}-T_i \rvert \leq 2n$ wouldn't hold if $T_{i+1}$ was negative. So all $T_i$ are positive. So the left upmost $n*n$ and the right up most $n*n$ has a postive sum. We can use the same argument for shifting rows instead of columns and get that the left down most $n*n$ is has a positive. And we can use the same argument to get that the right downmost $nxn$ has a positive sum. If we let sum the sums of the left/right upmost $n*n$ s and left/right downmost $n*n$s be $X$ , $X\geq 4(n+1)$ but this contradicts the second condition of the grid.Hence we are done $\square$
This post has been edited 6 times. Last edited by Tamam, Jul 17, 2025, 11:25 PM
Reason: dilly dilly dillin
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