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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:
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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
coloring in an array
AbhayAttarde01   1
N 5 minutes ago by AbhayAttarde01
ok lets see if i can make a good problem that actually has a good solution to it lol (i havent done it in a while)

i have $n$ distinct colors, and I want to color dots in an $n*n$ array of dots, where $n \geq 3$. However, I am not allowed to color in one dot a specific color, and then color in a dot that is adjacent to the previously colored spot (diagonally, horizontally, vertically). What is the smallest $n$ that enables me to color in all of the dots successfully, and how many ways can I color that array in?

an example of an unseccessful array is shown here:
A B C
A B C
A B C
the same colors are adjacent to each other vertically.
a sucessful array is shown here (i modified it so that it wont be alligned with the problem, but i think you get the idea)
A B C
D E A
B C D
as you can see, no 2 dots of the same color touch one another (diagonally, horizontally, vertically)
1 reply
AbhayAttarde01
43 minutes ago
AbhayAttarde01
5 minutes ago
Intermediate Algebra: How to Study
HamstPan38825   46
N 9 minutes ago by SwordAxe
I saw some users making comments about how to study from Introduction to Geometry, so I thought I'd put something down for Intermediate Algebra as well.
Chapters 1, 2, and 4 are extremely important review; if you think you have completely mastered the topic, these chapters are skippable but if you have not completed Introduction to Algebra, they don't take too long and are very good review.
Chapter 3 is not very important: reference Precalculus for deeper complex number studies.
Chapter 5, as stated, is mostly unimportant but quite interesting. Do this if you have time, (though 5.3 is the only really important section), but it's not vitally important.
Chapter 6 is one of those not negligible but not very important chapters: skim through this chapter lightly, since it really only gives quite basic information.
Chapters 7 and 8 are very important: Personally, Chapter 8 is the chapter more directed towards contests and Chapter 7 is a standard Algebra II chapter; the concepts are simple, but be sure to attempt all the challenge problems.
Chapter 9 is interesting but not very useful. Some of the concepts are interesting, but this chapter is not that important.
Chapters 10 and 11 are the most important chapters up to this point: Chapter 10 is arguably the most important chapter in the book unless you are preparing for contests. Chapter 11 is very general, a great tool for polishing up your bashing skills.
Chapter 12 is the hardest chapter in the book (excluding 17-20), and was my personal favorite chapter. Inequalities are important when training for olympiads, but if you're only aiming for AMC/AIME, it's not as important as Chapters 10, 11, etc. However, if you have time and enough persistence, the chapter is very useful (and in my opinion, one of the best-written chapters in the book.)
Chapters 13-16 all address a type of function: chapter 14 is the shortest chapter in the book, and the least important of the four; chapters 13 and 16 are the most important, with logarithmic identities and custom piecewise functions being a huge topic, especially in the AIME. Chapter 15 is a standard-oriented chapter, but some of the challenge problems are quite interesting.
Chapter 17 is the supplement to Chapter 10: if you're preparing for the AIME, Chapter 17 is just about as important as Chapter 10; the recursion section (17.1) is especially important.
Chapter 18 is the supplement to Chapter 12: the topics discussed here will likely not be used until the USAMO, so this chapter is probably the least important of the last four chapters. It is the second-hardest chapter in the entire book.
Chapter 19 is an interesting chapter; it is the only one of the last four chapters to bring up new material. Functional Equations are very high-level concepts: they frequently appear in the final AIME problems and various olympiads. Study this chapter, but you're not likely to completely understand it until you master the other chapters (and the AMC/AIME.)
Chapter 20 is the "monster" of all the chapters; this chapter is the most important and hardest of all the chapters when preparing for contests. Symmetry is probably most important here, though the problems given are insufficient for you to master symmetry. Attempt all problems in this chapter.

Final Note: Don't skim over sections and only do problems you think are sufficient. Do every single problem presented in the book, and all Review and Challenge Problems. 1,100 problems still might not be sufficient for you to master all the topics, and don't be intimidated by problems sourced from olympiads, USAMO, and IMO (personally, the last few challenge problems from Chapter 12 were way easier than their source: 12.77 was extremely easy, 12.80 was literally an easier version of 12.79).

Edit:
Most important chapters for AMC 10/12: Chapters 1, 2, 4, 8, 10, 13, 20 (maybe)
Most important chapters for AIME: Chapter 2, 8, 10, 12, 13, 16, 17, 20
Most important chapters for olympiads: Chapters 2, 8, 10, 11, 12, 13, 16, 17, 18, 19, 20

~HamstPan38825 (this took so long-)
46 replies
HamstPan38825
Nov 10, 2020
SwordAxe
9 minutes ago
AIMEification
ethanhansummerfun   12
N 17 minutes ago by aaravdodhia
(moved from MSM with modifications, if this is too hard go to MSM for an easier ver.)

(pls don’t flame me if this is too hard, I thought of it in the shower).

people this came up in the shower I was considering it and got a loosely defined statement on paper and copied it up here, please if there are errors don’t be mad. yall chilllll

To “AIMEify” a number $\frac{m}{n}$ is to convert it into $m+n$, given $gcd(m,n) = 1$ . To AIMEify a number $m\sqrt[k]{n}$ is to convert it into $m+k+n$, where $n$ is power free (i.e. not divisible by the power of any prime). Define a function $f(x)$ to AIMEify any square-root radical and any rational number such that it must always output a 3-digit number from $000$ to $999$, inclusive.

Find all $k$ such that there exists a real number $x$ such that $f(x^2) = f(x)^2 + k$, or prove none exist.

(I haven’t even solven this :rotfl: so feel free to cook)
12 replies
ethanhansummerfun
an hour ago
aaravdodhia
17 minutes ago
CMATH club selection test
justalonelyguy   7
N 18 minutes ago by aaravdodhia
Problem 1
a) Let \( a, b, c \in \mathbb{R} \) such that \( ab + bc + ca = 1 \). Compute the value of the expression:
\[
P = \frac{a}{1 + a^2} + \frac{b}{1 + b^2} + \frac{c}{1 + c^2} - \frac{2}{(a + b)(b + c)(c + a)}.
\]
b) Let \( x, y, z \in \mathbb{R} \) such that
\[
27(x + y + z)^3 = 240 + (3x + y - z)^3 + (3y + z - x)^3 + (3z + x - y)^3.
\]Prove that \( (x + 2y)(y + 2z)(z + 2x) = 10 \).

Problem 2

a) Let \( p \) be a prime number greater than 5. Prove that \( p^4 + 239 \) is divisible by 240.

b) Do there exist integers \( a, b \) such that
\[
(a + b\sqrt{2026})^2 = 2024 + 2025\sqrt{2026} \ ?
\]
\item[c)] Find all natural number pairs \( (x, y) \) such that
\[
2^x \cdot x^2 = 9y^2 + 6y + 88.
\]
Problem 3

a)Let \( a, b, c \geq 0 \) such that \( a + b + c = 3 \). Prove that
\[
\frac{a}{b^2 + 1} + \frac{b}{c^2 + 1} + \frac{c}{a^2 + 1} \geq \frac{3}{2}.
\]
b) Let \( a, b, c > 0 \). Prove that
\[
\frac{1 + bc + ca}{(1 + a + b)^2} + \frac{1 + ca + ab}{(1 + b + c)^2} + \frac{1 + ab + bc}{(1 + c + a)^2} \geq 1.
\]Problem 4

Let triangle \( ABC \) be acute with \( AB < CA < BC \). Let the altitudes \( BE \) and \( CF \) intersect at \( H \). The line through \( E \) perpendicular to \( EF \) intersects the extensions of \( BC \) and \( AB \) at points \( I \) and \( P \), respectively. The line through \( F \) perpendicular to \( EF \) intersects the extensions of \( BC \) and \( CA \) at points \( K \) and \( Q \), respectively. Let \( M \) be the midpoint of \( BC \).


a) Prove that triangles \( AEF \) and \( ABC \) are similar.

b) Prove that triangle \( MEF \) is isosceles and that \( BK = CI \).

c) Prove that \( \angle EHP = \angle FHQ \).


Problem 5

Mr. Quý chooses two distinct integers \( a \) and \( b \), then writes down the following six numbers:
\[
ab,\quad a(a + 4),\quad ab(a + 4),\quad b(a + 4),\quad b(b + 4),\quad (a + 4)(b + 4).
\]Among these six numbers, what is the maximum number of perfect squares that can appear? Explain why.
7 replies
justalonelyguy
Jul 28, 2025
aaravdodhia
18 minutes ago
Iran 3rd round, Problem 3, Final exam
Amirreza.J   0
29 minutes ago
Source: Iran 3rd round 2019
There are \( 2n \) red line and \( n \) blue lines in the plane in general position. Prove that there are at least
\[\frac{(n-1)(n-2)}{2}\]regions with a monochromatic boundary. (An unbounded region is also considered a region, where its boundary is formed by some line segments and half-lines.)
0 replies
Amirreza.J
29 minutes ago
0 replies
The perpendicular bisector of a side of a triangle
YII.I.   8
N 34 minutes ago by SuperBarsh
Source: 2022 Japan Junior MO Final P4
In an acute triangle $ABC$, $AB<AC$. The perpendicular bisector of the segment $BC$ intersects the lines $AB,AC$ at the points $D,E$ respectively. Denote the mid-point of $DE$ as $M$. Suppose the circumcircle of $\triangle ABC$ intersects the line $AM$ at points $P$ and $A$, and $M,A,P$ are arranged in order on the line. Prove that $\angle BPE=90^{\circ}$.
8 replies
YII.I.
Feb 12, 2022
SuperBarsh
34 minutes ago
IMO 2009, Problem 2
orl   151
N 44 minutes ago by lksb
Source: IMO 2009, Problem 2
Let $ ABC$ be a triangle with circumcentre $ O$. The points $ P$ and $ Q$ are interior points of the sides $ CA$ and $ AB$ respectively. Let $ K,L$ and $ M$ be the midpoints of the segments $ BP,CQ$ and $ PQ$. respectively, and let $ \Gamma$ be the circle passing through $ K,L$ and $ M$. Suppose that the line $ PQ$ is tangent to the circle $ \Gamma$. Prove that $ OP = OQ.$

Proposed by Sergei Berlov, Russia
151 replies
orl
Jul 15, 2009
lksb
44 minutes ago
Question 5
Valentin Vornicu   87
N an hour ago by youochange
Let $a$ and $b$ be positive integers. Show that if $4ab - 1$ divides $(4a^{2} - 1)^{2}$, then $a = b$.

Author: Kevin Buzzard and Edward Crane, United Kingdom
87 replies
Valentin Vornicu
Jul 26, 2007
youochange
an hour ago
Number theory, algebra
preatsreard   1
N an hour ago by preatsreard
Find all number systems in which number 3806130 is a 4 digit palindrom.(palindrom is a set of characters which is same reading front to back, and back to front)
1 reply
preatsreard
an hour ago
preatsreard
an hour ago
Iran 3rd combo 2
Yaghi   6
N an hour ago by Amirreza.J
Source: 2018 Iran national olympiad third round combinatorics exam
There are 8 points in the plane.we write down the area of each triangle having all vertices amoung these points(totally 56 numbers).Let them be $a_1,a_2,\dots a_{56}$.Prove that there is a choice of plus or minus such that:
$$\pm a_1 \pm a_2 \dots \pm a_{56}=0$$
6 replies
1 viewing
Yaghi
Aug 29, 2018
Amirreza.J
an hour ago
orang NT
KevinYang2.71   37
N an hour ago by eg4334
Source: ISL 2024 N1
Find all positive integers $n$ with the following property: for all positive divisors $d$ of $n$, we have $d+1\mid n$ or $d+1$ is prime.
37 replies
KevinYang2.71
Jul 16, 2025
eg4334
an hour ago
D1052 : How it's possible ?
Dattier   10
N an hour ago by AgileMongoose21
Source: les dattes à Dattier
Is it true for all $n$ natural integer : $(E\times 29^n \mod F) \mod 3\neq 0$ ?

E=163999081217965835070356295641931525591357567735624606830696386586994976172813816839262877
50922205042989559280142547393636527346272001339597483126086699049357460700008119117240578043
46281799731794620614941989125738298381362079843446150841376016501310942563338531951229469261
017554376486801


F=666165351866558215458553224258271230186695252725433417706426521946436303489813878149680284
24137540935869820330945301911165461108917069581547697978809314332789769417680417107249591988
71252061235265894516611712110379162326930843580773177773789047845826833190774483296276708089
431095213731040922452939281280
10 replies
Dattier
Jul 15, 2025
AgileMongoose21
an hour ago
Perpendicularity in Two Tangent Circles
steven_zhang123   3
N an hour ago by Royal_mhyasd
Source: 2025 Hope League Test 2 P3
Circle \(O_1\) and circle \(O_2\) are externally tangent at point \(T\). From a point \(X\) on circle \(O_2\), a tangent is drawn intersecting circle \(O_1\) at points \(A\) and \(B\). The line \(XT\) is extended to intersect circle \(O_1\) at point \(S\). A point \(C\) is taken on the arc \(TS\) of circle \(O_1\). The line \(SC\) is extended to intersect the angle bisector of \(\angle BAC\) at point \(I\). The circle passing through points \(A, T, X\) and the circle passing through points \(C, T, I\) intersect at another point \(E\). Prove that \(EO_2 \perp XI\).
Proposed by Luo Haoyu
3 replies
steven_zhang123
Jul 24, 2025
Royal_mhyasd
an hour ago
Israel Number Theory
mathisreaI   71
N an hour ago by heheman
Source: IMO 2022 Problem 5
Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]
71 replies
mathisreaI
Jul 13, 2022
heheman
an hour ago
Function Problem
Geometry285   4
N May 22, 2025 by maromex
The function $f(x)$ can be defined as a sequence such that $x=n$, and $a_n = | a_{n-1} | + \left \lceil \frac{n!}{n^{100}} \right \rceil$, such that $a_n = n$. The function $g(x)$ is such that $g(x) = x!(x+1)!$. How many numbers within the interval $0<n<101$ for the function $g(f(x))$ are perfect squares?
4 replies
Geometry285
Apr 11, 2021
maromex
May 22, 2025
Function Problem
G H J
G H BBookmark kLocked kLocked NReply
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Geometry285
902 posts
#1
Y by
The function $f(x)$ can be defined as a sequence such that $x=n$, and $a_n = | a_{n-1} | + \left \lceil \frac{n!}{n^{100}} \right \rceil$, such that $a_n = n$. The function $g(x)$ is such that $g(x) = x!(x+1)!$. How many numbers within the interval $0<n<101$ for the function $g(f(x))$ are perfect squares?
This post has been edited 1 time. Last edited by Geometry285, Apr 11, 2021, 11:13 PM
Z K Y
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Geometry285
902 posts
#3
Y by
Anyone? The notation shouldn’t be very scary at all....
This post has been edited 1 time. Last edited by Geometry285, Apr 15, 2021, 12:53 AM
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Saucepan_man02
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Geometry285 wrote:
The function $f(x)$ can be defined as a sequence such that $x=n$, and $a_n = | a_{n-1} | + \left \lceil \frac{n!}{n^{100}} \right \rceil$, such that $a_n = n$. The function $g(x)$ is such that $g(x) = x!(x+1)!$. How many numbers within the interval $0<n<101$ for the function $g(f(x))$ are perfect squares?

Could anyone kindly explain how $f(x)$ and $a_n$ are related?
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Saucepan_man02
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\bump help
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maromex
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This problem statement looks like nonsense to me. I think it is saying that $f$ is defined on the positive integers where $f(n) = a_n$ for all positive integers $n$. Also, the sequence $a_n$ is defined in two different ways that contradict each other if $n$ is allowed to be any positive integer. Maybe it is intended that we can only take $1 \le n \le 100$?

We can notice that $\left \lceil \dfrac{n!}{n^{100}} \right \rceil = 1$ for all positive integers $n$ such that $1 \le n \le 100$, which is probably a crucial idea for whatever this problem is meant to be.

So this problem should be to find how many positive integers $n$ there are such that $1 \le n \le 100$ and $n!(n+1)!$ is a perfect square. Everything else is just confusing.

Solution
This post has been edited 1 time. Last edited by maromex, May 22, 2025, 1:25 AM
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