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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.

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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
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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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PAMO 2022 Problem 1 - Line Tangent to Circle Through Orthocenter
DylanN   5
N 11 minutes ago by Y77
Source: 2022 Pan-African Mathematics Olympiad Problem 1
Let $ABC$ be a triangle with $\angle ABC \neq 90^\circ$, and $AB$ its shortest side. Let $H$ be the orthocenter of $ABC$. Let $\Gamma$ be the circle with center $B$ and radius $BA$. Let $D$ be the second point where the line $CA$ meets $\Gamma$. Let $E$ be the second point where $\Gamma$ meets the circumcircle of the triangle $BCD$. Let $F$ be the intersection point of the lines $DE$ and $BH$.

Prove that the line $BD$ is tangent to the circumcircle of the triangle $DFH$.
5 replies
DylanN
Jun 25, 2022
Y77
11 minutes ago
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Conditional geo with centroid
a_507_bc   6
N an hour ago by LeYohan
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 $\angle AGO = 90^{o}$.
6 replies
a_507_bc
Jul 1, 2023
LeYohan
an hour ago
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Channel name changed
Plane_geometry_youtuber   0
an hour ago
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!
0 replies
Plane_geometry_youtuber
an hour ago
0 replies
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IMO Shortlist 2010 - Problem G1
Amir Hossein   134
N an hour ago by happypi31415
Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$

Proposed by Christopher Bradley, United Kingdom
134 replies
Amir Hossein
Jul 17, 2011
happypi31415
an hour ago
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Divisors on number
RagvaloD   34
N 2 hours ago by cubres
Source: All Russian Olympiad 2017,Day1,grade 10,P5
$n$ is composite. $1<a_1<a_2<...<a_k<n$ - all divisors of $n$. It is known, that $a_1+1,...,a_k+1$ are all divisors for some $m$ (except $1,m$). Find all such $n$.
34 replies
RagvaloD
May 3, 2017
cubres
2 hours ago
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IMO ShortList 2002, number theory problem 2
orl   59
N 2 hours ago by cubres
Source: IMO ShortList 2002, number theory problem 2
Let $n\geq2$ be a positive integer, with divisors $1=d_1<d_2<\,\ldots<d_k=n$. Prove that $d_1d_2+d_2d_3+\,\ldots\,+d_{k-1}d_k$ is always less than $n^2$, and determine when it is a divisor of $n^2$.
59 replies
orl
Sep 28, 2004
cubres
2 hours ago
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None of the circles contains the pentagon - ILL 1970, P34
Amir Hossein   1
N 2 hours ago by legogubbe
In connection with a convex pentagon $ABCDE$ we consider the set of ten circles, each of which contains three of the vertices of the pentagon on its circumference. Is it possible that none of these circles contains the pentagon? Prove your answer.
1 reply
Amir Hossein
May 21, 2011
legogubbe
2 hours ago
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interesting incenter/tangent circle config
LeYohan   0
2 hours ago
Source: 2022 St. Mary's Canossian College F4 Final Exam Mathematics Paper 1, Q 18d of 18 (modified)
$BC$ is tangent to the circle $AFDE$ at $D$. $AB$ and $AC$ cut the circle at $F$ and $E$ respectively. $I$ is the in-centre of $\triangle ABC$, and $D$ is on the line $AI$. $CI$ and $DE$ intersect at $G$, while $BI$ and $FD$ intersect at $P$. Prove that the points $P, F, G, E$ lie on a circle.
0 replies
LeYohan
2 hours ago
0 replies
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interesting geo config (2/3)
Royal_mhyasd   5
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.
5 replies
Royal_mhyasd
Yesterday at 11:36 PM
Royal_mhyasd
2 hours ago
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interesting geometry config (3/3)
Royal_mhyasd   2
N 2 hours ago by Royal_mhyasd
Let $\triangle ABC$ be an acute triangle, $H$ its orthocenter and $E$ the center of its nine point circle. Let $P$ be a point on the parallel through $C$ to $AB$ such that $\angle CPH = |\angle BAC-\angle ABC|$ and $P$ and $A$ are on different sides of $BC$ and $Q$ a point on the parallel through $B$ to $AC$ such that $\angle BQH = |\angle BAC - \angle ACB|$ and $C$ and $Q$ are on different sides of $AB$. If $B'$ and $C'$ are the reflections of $H$ over $AC$ and $AB$ respectively, $S$ and $T$ are the intersections of $B'Q$ and $C'P$ respectively with the circumcircle of $\triangle ABC$, prove that the intersection of lines $CT$ and $BS$ lies on $HE$.

final problem for this "points on parallels forming strange angles with the orthocenter" config, for now. personally i think its pretty cool :D
2 replies
Royal_mhyasd
Today at 7:06 AM
Royal_mhyasd
2 hours ago
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Convex Quadrilateral with Bisector Diagonal
matinyousefi   8
N 3 hours ago by lpieleanu
Source: Germany TST 2017
In a convex quadrilateral $ABCD$, $BD$ is the angle bisector of $\angle{ABC}$. The circumcircle of $ABC$ intersects $CD,AD$ in $P,Q$ respectively and the line through $D$ parallel to $AC$ cuts $AB,AC$ in $R,S$ respectively. Prove that point $P,Q,R,S$ lie on a circle.
8 replies
matinyousefi
Apr 11, 2020
lpieleanu
3 hours ago
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Kids in clubs
atdaotlohbh   0
3 hours ago
There are $6k-3$ kids in a class. Is it true that for all positive integers $k$ it is possible to create several clubs each with 3 kids such that any pair of kids are both present in exactly one club?
0 replies
atdaotlohbh
3 hours ago
0 replies
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Turbo's en route to visit each cell of the board
Lukaluce   22
N 3 hours ago by HamstPan38825
Source: EGMO 2025 P5
Let $n > 1$ be an integer. In a configuration of an $n \times n$ board, each of the $n^2$ cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate $90^{\circ}$ counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of $n$, the maximum number of good cells over all possible starting configurations.

Proposed by Melek Güngör, Turkey
22 replies
Lukaluce
Apr 14, 2025
HamstPan38825
3 hours ago
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n lamps
pohoatza   47
N 3 hours ago by yayyayyay
Source: IMO Shortlist 2006, Combinatorics 1, AIMO 2007, TST 2, P1
We have $ n \geq 2$ lamps $ L_{1}, . . . ,L_{n}$ in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp $ L_{i}$ and its neighbours (only one neighbour for $ i = 1$ or $ i = n$, two neighbours for other $ i$) are in the same state, then $ L_{i}$ is switched off; – otherwise, $ L_{i}$ is switched on.
Initially all the lamps are off except the leftmost one which is on.

$ (a)$ Prove that there are infinitely many integers $ n$ for which all the lamps will eventually be off.
$ (b)$ Prove that there are infinitely many integers $ n$ for which the lamps will never be all off.
47 replies
pohoatza
Jun 28, 2007
yayyayyay
3 hours ago
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hard ...
REGNA   2
N Apr 11, 2025 by KevinYang2.71
find function $f:\mathbb{R^+}\rightarrow \mathbb{R^+}$ such that :
$f()\frac{f(x)}{y}=yf(y)f(f(x)) \;\;\;\;\;(x,y> 0)$
2 replies
REGNA
Oct 21, 2022
KevinYang2.71
Apr 11, 2025
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hard ...
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Reveal topic tags
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REGNA
140 posts
REGNA
#1 h  Oct 21, 2022, 5:42 PM
Y by
find function $f:\mathbb{R^+}\rightarrow \mathbb{R^+}$ such that :
$f()\frac{f(x)}{y}=yf(y)f(f(x)) \;\;\;\;\;(x,y> 0)$
Z K Y
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ZETA_in_olympiad
2211 posts
ZETA_in_olympiad
#2 h  Oct 21, 2022, 6:21 PM
Y by
REGNA wrote:
find function $f:\mathbb{R^+}\rightarrow \mathbb{R^+}$ such that :
$f()\frac{f(x)}{y}=yf(y)f(f(x)) \;\;\;\;\;(x,y> 0)$

Let $P(x,y)$ denote (assuming) the assertion $f\left (\frac{f(x)}{y} \right )=yf(y)f(f(x))$. Clearly $f(1)=1$ and $P(1,x)$ gives $f\left (\frac 1x \right)=xf(x)$. $P(x,f(x))$ gives $f(f(x))=\frac 1{\sqrt{f(x)}}$, now $P\left (x, \frac 1x\right )$ gives $\frac{1}{f(x)}=f \left (\frac 1x \right )=xf(x)$ implying $f(x)=\sqrt{\frac 1x}$, which works.
Z K Y
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KevinYang2.71
428 posts
KevinYang2.71
#3 h  Apr 11, 2025, 3:44 PM • 3 Y
Y by bjump, megarnie, deduck
Let $P(x,\,y)$ be the given assertion. $P(1,\,1)$ gives $f(1)=1$ and $P(1,\,x)$ gives $f(1/x)=xf(x)$. Then $P(1/x,\,x)$ yields $f(f(x))=xf(x)f(f(1/x))$. Replacing $x$ with $\frac{1}{x}$ and comparing the two gives $xf(x)=\frac{f(f(x))}{f(f(1/x))}=\frac{x}{f(1/x)}$ so $f(1/x)=\frac{1}{f(x)}$. Thus $\frac{1}{f(x)}=xf(x)$ so $\boxed{f(x)=\frac{1}{\sqrt{x}}}$, which works. $\square$
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