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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 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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A weird problem
jayme   0
an hour ago
Dear Mathlinkers,

1. ABC a triangle
2. 0 the circumcircle
3. I the incenter
4. 1 a circle passing througn B and C
5. X, Y the second points of intersection of 1 wrt BI, CI
6. 2 the circumcircle of the triangle XYI
7. M, N the symetrics of B, C wrt XY.

Question : if 2 is tangent to 0 then, 2 is tangent to MN.

Sincerely
Jean-Louis
0 replies
jayme
an hour ago
0 replies
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NT game with products
Kimchiks926   4
N an hour ago by math-olympiad-clown
Source: Baltic Way 2022, Problem 20
Ingrid and Erik are playing a game. For a given odd prime $p$, the numbers $1, 2, 3, ..., p-1$ are written on a blackboard. The players take turns making moves with Ingrid starting. A move consists of one of the players crossing out a number on the board that has not yet been crossed out. If the product of all currently crossed out numbers is $1 \pmod p$ after the move, the player whose move it was receives one point, otherwise, zero points are awarded. The game ends after all numbers have been crossed out.

The player who has received the most points by the end of the game wins. If both players have the same score, the game ends in a draw. For each $p$, determine which player (if any) has a winning strategy
4 replies
Kimchiks926
Nov 12, 2022
math-olympiad-clown
an hour ago
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set with c+2a>3b
VicKmath7   49
N an hour ago by wangyanliluke
Source: ISL 2021 A1
Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$.

Proposed by Dominik Burek and Tomasz Ciesla, Poland
49 replies
VicKmath7
Jul 12, 2022
wangyanliluke
an hour ago
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interesting geo config (2/3)
Royal_mhyasd   8
N 2 hours ago by Royal_mhyasd
Source: own
Let $\triangle ABC$ be an acute triangle and $H$ its orthocenter. Let $P$ be a point on the parallel through $A$ to $BC$ such that $\angle APH = |\angle ABC-\angle ACB|$. Define $Q$ and $R$ as points on the parallels through $B$ to $AC$ and through $C$ to $AB$ similarly. If $P,Q,R$ are positioned around the sides of $\triangle ABC$ as in the given configuration, prove that $P,Q,R$ are collinear.
8 replies
Royal_mhyasd
Saturday at 11:36 PM
Royal_mhyasd
2 hours ago
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Problem 10
SlovEcience   4
N 2 hours ago by SlovEcience
Let \( x, y, z \) be positive real numbers satisfying
\[ xy + yz + zx = 3xyz. \]Prove that
\[ \sqrt{\frac{x}{3y^2z^2 + xyz}} + \sqrt{\frac{y}{3x^2z^2 + xyz}} + \sqrt{\frac{z}{3x^2y^2 + xyz}} \le \frac{3}{2}. \]
4 replies
SlovEcience
May 30, 2025
SlovEcience
2 hours ago
J
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IMO ShortList 2003, combinatorics problem 4
darij grinberg   39
N 3 hours ago by ThatApollo777
Source: Problem 5 of the German pre-TST 2004, written in December 03
Let $x_1,\ldots, x_n$ and $y_1,\ldots, y_n$ be real numbers. Let $A = (a_{ij})_{1\leq i,j\leq n}$ be the matrix with entries \[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\]Suppose that $B$ is an $n\times n$ matrix with entries $0$, $1$ such that the sum of the elements in each row and each column of $B$ is equal to the corresponding sum for the matrix $A$. Prove that $A=B$.
39 replies
darij grinberg
May 17, 2004
ThatApollo777
3 hours ago
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greatest volume
hzbrl   4
N 3 hours ago by hzbrl
Source: purple comet
A large sphere with radius 7 contains three smaller balls each with radius 3 . The three balls are each externally tangent to the other two balls and internally tangent to the large sphere. There are four right circular cones that can be inscribed in the large sphere in such a way that the bases of the cones are tangent to all three balls. Of these four cones, the one with the greatest volume has volume $n \pi$. Find $n$.
4 replies
hzbrl
May 8, 2025
hzbrl
3 hours ago
J
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Projective geo
drmzjoseph   1
N 3 hours ago by Luis González
Any pure projective solution? I mean no metrics, Menelaus, Ceva, bary, etc
Only pappus, desargues, dit, etc
Btw prove that $X',P,K$ are collinear, and $P,Q$ are arbitrary points
1 reply
drmzjoseph
Mar 6, 2025
Luis González
3 hours ago
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2019 Iberoamerican Mathematical Olympiad, P1
jbaca   9
N 3 hours ago by jordiejoh
For each positive integer $n$, let $s(n)$ be the sum of the squares of the digits of $n$. For example, $s(15)=1^2+5^2=26$. Determine all integers $n\geq 1$ such that $s(n)=n$.
9 replies
jbaca
Sep 15, 2019
jordiejoh
3 hours ago
J
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Conditional geo with centroid
a_507_bc   7
N 3 hours ago by Tkn
Source: Singapore Open MO Round 2 2023 P1
In a scalene triangle $ABC$ with centroid $G$ and circumcircle $\omega$ centred at $O$, the extension of $AG$ meets $\omega$ at $M$; lines $AB$ and $CM$ intersect at $P$; and lines $AC$ and $BM$ intersect at $Q$. Suppose the circumcentre $S$ of the triangle $APQ$ lies on $\omega$ and $A, O, S$ are collinear. Prove that $\angle AGO = 90^{o}$.
7 replies
a_507_bc
Jul 1, 2023
Tkn
3 hours ago
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People live in Kansas?
jj_ca888   13
N 3 hours ago by Ilikeminecraft
Source: SMO 2020/5
In triangle $\triangle ABC$, let $E$ and $F$ be points on sides $AC$ and $AB$, respectively, such that $BFEC$ is cyclic. Let lines $BE$ and $CF$ intersect at point $P$, and $M$ and $N$ be the midpoints of $\overline{BF}$ and $\overline{CE}$, respectively. If $U$ is the foot of the perpendicular from $P$ to $BC$, and the circumcircles of triangles $\triangle BMU$ and $\triangle CNU$ intersect at second point $V$ different from $U$, prove that $A, P,$ and $V$ are collinear.

Proposed by Andrew Wen and William Yue
13 replies
jj_ca888
Aug 28, 2020
Ilikeminecraft
3 hours ago
J
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Symmetric integer FE
a_507_bc   5
N 3 hours ago by Tkn
Source: Singapore Open MO Round 2 2023 P4
Find all functions $f: \mathbb{Z} \to \mathbb{Z}$, such that $$f(x+y)((f(x) - f(y))^2+f(xy))=f(x^3)+f(y^3)$$for all integers $x, y$.
5 replies
a_507_bc
Jul 1, 2023
Tkn
3 hours ago
J
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Channel name changed
Plane_geometry_youtuber   6
N 3 hours ago by Yiyj
Hi,

Due to the search handle issue in youtube. My channel is renamed to Olympiad Geometry Club. And the new link is as following:

https://www.youtube.com/@OlympiadGeometryClub

Recently I introduced the concept of harmonic bundle. I will move on to the conjugate median soon. In the future, I will discuss more than a thousand theorems on plane geometry and hopefully it can help to the students preparing for the Olympiad competition.

Please share this to the people may need it.

Thank you!
6 replies
Plane_geometry_youtuber
Yesterday at 9:31 PM
Yiyj
3 hours ago
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How many cases did you check?
avisioner   18
N 3 hours ago by ezpotd
Source: 2023 ISL N2
Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square.

Proposed by Tahjib Hossain Khan, Bangladesh
18 replies
avisioner
Jul 17, 2024
ezpotd
3 hours ago
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J
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Find the locus of P
xiaotabuta   1
N Jan 10, 2014 by Luis González
Give an equilateral triangle $ABC$. Find the locus of the point $P$ inside the triangle such that $\angle PAB + \angle PBC + \angle PCA = 90^\circ$
1 reply
xiaotabuta
Jan 10, 2014
Luis González
Jan 10, 2014
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Find the locus of P
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Reveal topic tags
geometry unsolved
geometry
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xiaotabuta
28 posts
xiaotabuta
#1 h  Jan 10, 2014, 5:08 PM • 2 Y
Y by Adventure10, Mango247
Give an equilateral triangle $ABC$. Find the locus of the point $P$ inside the triangle such that $\angle PAB + \angle PBC + \angle PCA = 90^\circ$
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Luis González
4150 posts
Luis González
#2 h  Jan 10, 2014, 6:51 PM • 2 Y
Y by Adventure10, Mango247
Posted several times before. Locus of P is the union of the three altitudes of ABC. More general, for an scalene triangle, the locus is the Mcay cubic of ABC (see the last link).

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=123822
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=141887
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=409326
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