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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC 10 Problem Series[/list]
For those interested in Olympiad level training in math, computer science, physics, and chemistry, be sure to enroll in our WOOT courses before August 19th to take advantage of early bird pricing!

Summer camps are starting this month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have a transformative summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]June 5th, Thursday, 7:30pm ET: Open Discussion with Ben Kornell and Andrew Sutherland, Art of Problem Solving's incoming CEO Ben Kornell and CPO Andrew Sutherland host an Ask Me Anything-style chat. Come ask your questions and get to know our incoming CEO & CPO!
[*]June 9th, Monday, 7:30pm ET, Game Jam: Operation Shuffle!, Come join us to play our second round of Operation Shuffle! If you enjoy number sense, logic, and a healthy dose of luck, this is the game for you. No specific math background is required; all are welcome.[/list]
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0 replies
jlacosta
Jun 2, 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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m^2+3=n^3 no solutions over integers
blueprimes   7
N 8 minutes ago by eg4334
Source: Catherine Xu
Prove that there are no solutions over the integers to $m^2 + 3 = n^3$.

Catherine Xu
7 replies
1 viewing
blueprimes
Apr 23, 2024
eg4334
8 minutes ago
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Algebraic Number Theory book (from an olympiad perspective)
azerluigi   28
N 13 minutes ago by AR17296174
Hi!

This summer, I wrote a book about algebraic number theory from an olympiad perspective. There's basically no prerequisite, but it's better to have done some amount of olympiad number theory and polynomials beforehand.

I originally planned to write one more chapter and add hints, but uni is busier than I thought (I have so many classes, and most of them are not maths because France...) so I'm putting this off for a while. The most realistic scenario is that this last chapter will be written during the end-of-october holidays, and the hints during the Christmas holidays.

Well, that's about it I guess. For more info, you can see the table of content of the book and the read the foreword. https://www.dropbox.com/sh/lribbpkfooq96gs/AABJsHF8MuGZCkP8J0FxslD7a?dl=0

Tell me if the dropbox link somehow dies (it shouldn't). Also, feel free to send me any suggestion or comment you have. I would be particularly grateful if you could send me all the typos and mistakes you find, because I'm sure there are a lot.
28 replies
azerluigi
Oct 5, 2021
AR17296174
13 minutes ago
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IMO Shortlist 2012, Combinatorics 1
lyukhson   76
N 35 minutes ago by lpieleanu
Source: IMO Shortlist 2012, Combinatorics 1
Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x,y)$ by either $(y+1,x)$ or $(x-1,x)$. Prove that she can perform only finitely many such iterations.

Proposed by Warut Suksompong, Thailand
76 replies
lyukhson
Jul 29, 2013
lpieleanu
35 minutes ago
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Geometry IMO trianing 2025
hn111009   1
N 38 minutes ago by timon92
Source: somewhere
Let $I$ be the incenter of an acute triangle $ABC.$ Point $X$ lies on segment $BC,$ on the same side of line $AI$ as point $B.$ Point $Y$ lies on the shorter arc $AB$ of the circumcircle of triangle $ABC.$ The following equalities hold: $$\angle{AIX}=\angle{XYA}=120^\circ.$$Prove that line $YI$ is the angle bisector of $\angle{XYA}.$
1 reply
hn111009
2 hours ago
timon92
38 minutes ago
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easy substitutions for a functional in reals
Circumcircle   10
N an hour ago by dimi07
Source: Kosovo Math Olympiad 2025, Grade 11, Problem 2
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ with the property that for every real numbers $x$ and $y$ it holds that
$$f(x+yf(x+y))=f(x)+f(xy)+y^2.$$
10 replies
Circumcircle
Nov 16, 2024
dimi07
an hour ago
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Shortest sequence of coin flips
CyclicISLscelesTrapezoid   12
N an hour ago by pi271828
Source: USA TSTST 2023/7
The Bank of Pittsburgh issues coins that have a heads side and a tails side. Vera has a row of 2023 such coins alternately tails-up and heads-up, with the leftmost coin tails-up.

In a move, Vera may flip over one of the coins in the row, subject to the following rules:
[list=disc]
[*] On the first move, Vera may flip over any of the $2023$ coins.
[*] On all subsequent moves, Vera may only flip over a coin adjacent to the coin she flipped on the previous move. (We do not consider a coin to be adjacent to itself.)
[/list]
Determine the smallest possible number of moves Vera can make to reach a state in which every coin is heads-up.

Luke Robitaille
12 replies
CyclicISLscelesTrapezoid
Jun 26, 2023
pi271828
an hour ago
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Perpendiculars to the harmonic lines are also harmonic lines
menpo   5
N 2 hours ago by X.Luser
Source: Kazakhstan National Olympiad 2024 (10-11 grade), P6
The circle $\omega$ with center at point $I$ inscribed in an triangle $ABC$ ($AB\neq AC$) touches the sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$, respectively. The circumcircles of triangles $ABC$ and $AEF$ intersect secondary at point $K.$ The lines $EF$ and $AK$ intersect at point $X$ and intersects the line $BC$ at points $Y$ and $Z$, respectively. The tangent lines to $\omega$, other than $BC$, passing through points $Y$ and $Z$ touch $\omega$ at points $P$ and $Q$, respectively. Let the lines $AP$ and $KQ$ intersect at the point $R$. Prove that if $M$ is a midpoint of segment $YZ,$ then $IR\perp XM$.
5 replies
menpo
Mar 21, 2024
X.Luser
2 hours ago
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(2^n+2)/n being an integer
shobber   28
N 2 hours ago by lpieleanu
Source: APMO 1997
Find an integer $n$, where $100 \leq n \leq 1997$, such that
\[ \frac{2^n+2}{n} \]
is also an integer.
28 replies
shobber
Mar 17, 2006
lpieleanu
2 hours ago
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thanks u!
Ruji2018252   5
N 2 hours ago by math90
Let $a,b,c>0$. Prove
$\sqrt[3]{\dfrac{b^2+c^2}{a^2+bc}}+\sqrt[3]{\dfrac{a^2+c^2}{b^2+ac}}+\sqrt[3]{\dfrac{a^2+b^2}{c^2+ab}}\le \dfrac{a+b+c}{\sqrt[3]{abc}}$
5 replies
Ruji2018252
Yesterday at 3:35 PM
math90
2 hours ago
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τ, σ, φ connect all positive integers
MarkBcc168   16
N 2 hours ago by YaoAOPS
Source: ELMO 2020 P6
For any positive integer $n$, let
[list]
[*]$\tau(n)$ denote the number of positive integer divisors of $n$,
[*]$\sigma(n)$ denote the sum of the positive integer divisors of $n$, and
[*]$\varphi(n)$ denote the number of positive integers less than or equal to $n$ that are relatively prime to $n$.
[/list]
Let $a,b > 1$ be integers. Brandon has a calculator with three buttons that replace the integer $n$ currently displayed with $\tau(n)$, $\sigma(n)$, or $\varphi(n)$, respectively. Prove that if the calculator currently displays $a$, then Brandon can make the calculator display $b$ after a finite (possibly empty) sequence of button presses.

Proposed by Jaedon Whyte.
16 replies
MarkBcc168
Jul 28, 2020
YaoAOPS
2 hours ago
J
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Number Theory Marahon
Jupiterballs   9
N 2 hours ago by Svenskerhaor
Let's start a number theory marathon
Rules:-
just don't post >2 problems before a solution and be friendly :)

I'll start
P1
9 replies
Jupiterballs
Jun 23, 2025
Svenskerhaor
2 hours ago
J
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tangents circles , OA = OB + OC$
parmenides51   1
N 2 hours ago by HuongToiVMO
Source: 2008 Brazil Ibero & IMO Training List 2.3 https://artofproblemsolving.com/community/c3135350_
Let $O$ be a point inside the triangle $ABC$ such that $OA = OB + OC$. Let $B'$ and $C'$ be the midpoints of the arcs $AOC$ and $AOB$, respectively. Prove that the circumcircles of triangles $COC'$ and $BOB'$ are tangent to each other.
1 reply
parmenides51
Aug 9, 2024
HuongToiVMO
2 hours ago
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INMO 2022
Flying-Man   42
N 2 hours ago by Giant_PT
Let $D$ be an interior point on the side $BC$ of an acute-angled triangle $ABC$. Let the circumcircle of triangle $ADB$ intersect $AC$ again at $E(\ne A)$ and the circumcircle of triangle $ADC$ intersect $AB$ again at $F(\ne A)$. Let $AD$, $BE$, and $CF$ intersect the circumcircle of triangle $ABC$ again at $D_1(\ne A)$, $E_1(\ne B)$ and $F_1(\ne C)$, respectively. Let $I$ and $I_1$ be the incentres of triangles $DEF$ and $D_1E_1F_1$, respectively. Prove that $E,F, I, I_1$ are concyclic.
42 replies
Flying-Man
Mar 6, 2022
Giant_PT
2 hours ago
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Functional Equation Problem
dimi07   6
N 2 hours ago by dimi07
In the name of God, the Most Merciful and Compassionate.

Given a constant $a$.Find all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ so that
\[ f(x + y) = f(x) + f(y) + a \]holds for all $x, y \in \mathbb{R}^+$.

My solution:Let $P(x,y)$ be the assertion for the above mentioned function.Note that this function might look very similar to an additive equation but the only thing that is annoying us is that constant a,so what we do is simple,we seek to transform the equation,and such add a to both sides and the function changes to,
\[ f(x+y)+a=f(x)+a+f(y)+a \]
We substitute $g(x)=f(x)+a$,and thus the function becomes,and we define the domain on positive reals and the codomain on reals
\[ g(x+y)=g(x)+g(y) \]
Now we have this function as additive,now we have to do something to imply that the function is linear,so I thought to take advantage on the domain of the function,note that the domain of the function is all positive reals,which means that $g(x)>0$,which makes g bounded below,and such g is linear.Now maybe I am wrong and if so please tell me.So $g(x)=cx$,for some real c.

Now since g is linear then, $g(x)=f(x)+a$ so it means that the solution is $f(x)=cx-a$,for all real x.

And by the help of God we are done. $\blacksquare$
6 replies
dimi07
Today at 7:23 AM
dimi07
2 hours ago
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Concurrence
LiamChen   1
N Jun 5, 2025 by phonghatemath
Source: MOP1998
Problem:
1 reply
LiamChen
Jun 5, 2025
phonghatemath
Jun 5, 2025
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Concurrence
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Reveal topic tags
geometry
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Source: MOP1998
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LiamChen
35 posts
LiamChen
#1 h  Jun 5, 2025, 2:48 PM • 1 Y
Y by Exponent11
Problem:
Attachments:
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phonghatemath
16 posts
phonghatemath
#2 h  Jun 5, 2025, 4:50 PM
Y by
We see that: $QT, PS, RF, AA, BB', CC'$ are concurrent.

Let $I$ be the incenter of triangle $ABC$.

We claim that: $Q, I, T$ are collinear. By using Pascal's theorem. Similarly, we can also prove that $P, I, S$ are collinear and $F, I, R$ are collinear.

We obtain the required proof.
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