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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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parallel wanted, right triangle, circumcircle, angle bisector related
parmenides51   6
N 7 minutes ago by Ianis
Source: Norwegian Mathematical Olympiad 2020 - Abel Competition p4b
The triangle $ABC$ has a right angle at $A$. The centre of the circumcircle is called $O$, and the base point of the normal from $O$ to $AC$ is called $D$. The point $E$ lies on $AO$ with $AE = AD$. The angle bisector of $\angle CAO$ meets $CE$ in $Q$. The lines $BE$ and $OQ$ intersect in $F$. Show that the lines $CF$ and $OE$ are parallel.
6 replies
parmenides51
Apr 26, 2020
Ianis
7 minutes ago
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IMO ShortList 2008, Number Theory problem 5
April   25
N an hour ago by awesomeming327.
Source: IMO ShortList 2008, Number Theory problem 5, German TST 6, P2, 2009
For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) = x$ for all $ x\in\mathbb{N}$.
[*] $ f(xy)$ divides $ (x - 1)y^{xy - 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list]

Proposed by Bruno Le Floch, France
25 replies
April
Jul 9, 2009
awesomeming327.
an hour ago
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IMO Shortlist 2014 N2
hajimbrak   32
N an hour ago by ezpotd
Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\]
Proposed by Titu Andreescu, USA
32 replies
hajimbrak
Jul 11, 2015
ezpotd
an hour ago
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An easy geometry problem in NEHS Mock APMO
chengbilly   2
N 2 hours ago by MathLuis
Source: own
Let $ABC$ be a triangle with circumcenter $O$ and orthocenter $H$. $AD,BE,CF$ the altitudes of $\triangle ABC$. A point $T$ lies on line $EF$ such that $DT \perp EF$. A point $X$ lies on the circumcircle of $\triangle ABC$ such that $AX,EF,DO$ are concurrent. $DT$ meets $AX$ at $R$. Prove that $H,T,R,X$ are concyclic.
2 replies
chengbilly
May 23, 2021
MathLuis
2 hours ago
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The reflection of AD intersect (ABC) lies on (AEF)
alifenix-   62
N 2 hours ago by Rayvhs
Source: USA TST for EGMO 2020, Problem 4
Let $ABC$ be a triangle. Distinct points $D$, $E$, $F$ lie on sides $BC$, $AC$, and $AB$, respectively, such that quadrilaterals $ABDE$ and $ACDF$ are cyclic. Line $AD$ meets the circumcircle of $\triangle ABC$ again at $P$. Let $Q$ denote the reflection of $P$ across $BC$. Show that $Q$ lies on the circumcircle of $\triangle AEF$.

Proposed by Ankan Bhattacharya
62 replies
alifenix-
Jan 27, 2020
Rayvhs
2 hours ago
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PAMO 2022 Problem 1 - Line Tangent to Circle Through Orthocenter
DylanN   5
N 2 hours 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
2 hours ago
J
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Conditional geo with centroid
a_507_bc   6
N 3 hours 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
3 hours ago
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Channel name changed
Plane_geometry_youtuber   0
3 hours 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
3 hours ago
0 replies
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IMO Shortlist 2010 - Problem G1
Amir Hossein   134
N 3 hours 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
3 hours ago
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interesting incenter/tangent circle config
LeYohan   0
4 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
4 hours ago
0 replies
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interesting geo config (2/3)
Royal_mhyasd   5
N 4 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
Saturday at 11:36 PM
Royal_mhyasd
4 hours ago
J
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interesting geometry config (3/3)
Royal_mhyasd   2
N 4 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
Yesterday at 7:06 AM
Royal_mhyasd
4 hours ago
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Convex Quadrilateral with Bisector Diagonal
matinyousefi   8
N 5 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
5 hours ago
J
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collinear wanted, toucpoints of incircle related
parmenides51   2
N 5 hours ago by Tamam
Source: 2018 Thailand October Camp 1.2
Let $\Omega$ be the inscribed circle of a triangle $\vartriangle ABC$. Let $D, E$ and $F$ be the tangency points of $\Omega$ and the sides $BC, CA$ and $AB$, respectively, and let $AD, BE$ and $CF$ intersect $\Omega$ at $K, L$ and $M$, respectively, such that $D, E, F, K, L$ and $M$ are all distinct. The tangent line of $\Omega$ at $K$ intersects $EF$ at $X$, the tangent line of $\Omega$ at $L$ intersects $DE$ at $Y$ , and the tangent line of $\Omega$ at M intersects $DF$ at $Z$. Prove that $X,Y$ and $Z$ are collinear.
2 replies
parmenides51
Oct 15, 2020
Tamam
5 hours ago
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Some geometry
RagvaloD   1
N Sep 10, 2023 by Bexultan
Source: St Petersburg Olympiad 2012, Grade 9, P3
$ABCD$ is inscribed. Bisector of angle between diagonals intersect $AB$ anc $CD$ at $X$ and $Y$. $M,N$ are midpoints of $AD,BC$. $XM=YM$ Prove, that $XN=YN$.
1 reply
RagvaloD
Sep 29, 2017
Bexultan
Sep 10, 2023
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Some geometry
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Reveal topic tags
geometry
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G H BBookmark kLocked kLocked NReply
Source: St Petersburg Olympiad 2012, Grade 9, P3
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RagvaloD
4922 posts
RagvaloD
#1 h  Sep 29, 2017, 12:45 PM • 1 Y
Y by Adventure10
$ABCD$ is inscribed. Bisector of angle between diagonals intersect $AB$ anc $CD$ at $X$ and $Y$. $M,N$ are midpoints of $AD,BC$. $XM=YM$ Prove, that $XN=YN$.
This post has been edited 1 time. Last edited by RagvaloD, Jun 22, 2018, 9:32 AM
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Bexultan
178 posts
Bexultan
#2 h  Sep 10, 2023, 8:43 AM • 1 Y
Y by SolveForChocolate
In case $AB\parallel CD$, symmetry wrt perpendicular bisector of $AB$ solves the problem. Now we will assume $AB$ and $CD$ are not parallel. Let $Q=AB\cap CD$. By simple angle chasing, $QX=QY$ or $Q$ lies on the perpendicular bisector of $XY$. Since $MX=MY$, $M$ also lies on this perpendicular bisector. Note that $BC$ and $AD$ are antiparallel wrt $\angle AQD$. This means that the line $QM$, which is the median of $\triangle QAB$ is the symmedian of triangle $QBC$. In other words, lines $QM$ and $QN$ are symmetric wrt bisector of line $AQD$. If we look at it at another angle, $QM$ is the angle bisector of $\angle XQY$ (due to symmetry wrt perpendicular bisector of $XY$). So reflection of line $QM$ wrt angle bisector $\angle XQY$ should be $QM$ itself. But earlier, we found it is $QN$. Thus, line $QM$ and $QN$ coincide or $Q,M,N$ are collinear. Since $Q$ and $M$ lie on the perpendicular bisector of $XY$, $N$ also does. So $NX=NY$
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