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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
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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power of a point
BekzodMarupov   0
a minute ago
Source: lemmas in olympiad geometry
Epsilon 1.3. Let ABC be a triangle and let D, E, F be the feet of the altitudes, with D on BC, E on CA, and F on AB. Let the parallel through D to EF meet AB at X and AC at Y. Let T be the intersection of EF with BC and let M be the midpoint of side BC. Prove that the points T, M, X, Y are concyclic.
0 replies
BekzodMarupov
a minute ago
0 replies
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Interesting inequalities
sqing   1
N an hour ago by sqing
Source: Own
Let $a,b,c \geq 0 $ and $ab+bc+ca- abc =3.$ Show that
$$a+k(b+c)\geq 2\sqrt{3 k}$$Where $ k\geq 1. $
Let $a,b,c \geq 0 $ and $2(ab+bc+ca)- abc =31.$ Show that
$$a+k(b+c)\geq \sqrt{62k}$$Where $ k\geq 1. $
1 reply
1 viewing
sqing
an hour ago
sqing
an hour ago
J
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euler function
mathsearcher   0
an hour ago
Prove that there exists infinitely many positive integers n such that
ϕ(n) | n+1
0 replies
mathsearcher
an hour ago
0 replies
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Mega angle chase
kjhgyuio   1
N 2 hours ago by jkim0656
Source: https://mrdrapermaths.wordpress.com/2021/01/30/filtering-with-basic-angle-facts/
........
1 reply
kjhgyuio
2 hours ago
jkim0656
2 hours ago
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Simple but hard
Lukariman   1
N 2 hours ago by Giant_PT
Given triangle ABC. Outside the triangle, construct rectangles ACDE and BCFG with equal areas. Let M be the midpoint of DF. Prove that CM passes through the center of the circle circumscribing triangle ABC.
1 reply
Lukariman
3 hours ago
Giant_PT
2 hours ago
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RMM 2013 Problem 3
dr_Civot   79
N 2 hours ago by Ilikeminecraft
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$. The lines $AB$ and $CD$ meet at $P$, the lines $AD$ and $BC$ meet at $Q$, and the diagonals $AC$ and $BD$ meet at $R$. Let $M$ be the midpoint of the segment $PQ$, and let $K$ be the common point of the segment $MR$ and the circle $\omega$. Prove that the circumcircle of the triangle $KPQ$ and $\omega$ are tangent to one another.
79 replies
dr_Civot
Mar 2, 2013
Ilikeminecraft
2 hours ago
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Collinearity with orthocenter
liberator   181
N 3 hours ago by Giant_PT
Source: IMO 2013 Problem 4
Let $ABC$ be an acute triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ is the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$. Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM$, and let $Y$ be the point such that $WY$ is a diameter of $\omega_2$. Prove that $X,Y$ and $H$ are collinear.

Proposed by Warut Suksompong and Potcharapol Suteparuk, Thailand
181 replies
liberator
Jan 4, 2016
Giant_PT
3 hours ago
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GMO 2024 P1
Z4ADies   5
N Yesterday at 11:11 PM by awesomeming327.
Source: Geometry Mains Olympiad (GMO) 2024 P1
Let \( ABC \) be an acute triangle. Define \( I \) as its incenter. Let \( D \) and \( E \) be the incircle's tangent points to \( AC \) and \( AB \), respectively. Let \( M \) be the midpoint of \( BC \). Let \( G \) be the intersection point of a perpendicular line passing through \( M \) to \( DE \). Line \( AM \) intersects the circumcircle of \( \triangle ABC \) at \( H \). The circumcircle of \( \triangle AGH \) intersects line \( GM \) at \( J \). Prove that quadrilateral \( BGCJ \) is cyclic.

Author:Ismayil Ismayilzada (Azerbaijan)
5 replies
Z4ADies
Oct 20, 2024
awesomeming327.
Yesterday at 11:11 PM
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Problem 5 (Second Day)
darij grinberg   78
N Yesterday at 6:38 PM by cj13609517288
Source: IMO 2004 Athens
In a convex quadrilateral $ABCD$, the diagonal $BD$ bisects neither the angle $ABC$ nor the angle $CDA$. The point $P$ lies inside $ABCD$ and satisfies \[\angle PBC=\angle DBA\quad\text{and}\quad \angle PDC=\angle BDA.\] Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$.
78 replies
darij grinberg
Jul 13, 2004
cj13609517288
Yesterday at 6:38 PM
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concyclic wanted, PQ = BP, cyclic quadrilateral and 2 parallelograms related
parmenides51   2
N Yesterday at 6:31 PM by SuperBarsh
Source: 2011 Italy TST 2.2
Let $ABCD$ be a cyclic quadrilateral in which the lines $BC$ and $AD$ meet at a point $P$. Let $Q$ be the point of the line $BP$, different from $B$, such that $PQ = BP$. We construct the parallelograms $CAQR$ and $DBCS$. Prove that the points $C, Q, R, S$ lie on the same circle.
2 replies
parmenides51
Sep 25, 2020
SuperBarsh
Yesterday at 6:31 PM
J
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Perpendicular passes from the intersection of diagonals, \angle AEB = \angle CED
NO_SQUARES   1
N Yesterday at 5:49 PM by mathuz
Source: 239 MO 2025 10-11 p3
Inside of convex quadrilateral $ABCD$ point $E$ was chosen such that $\angle DAE = \angle CAB$ and $\angle ADE = \angle CDB$. Prove that if perpendicular from $E$ to $AD$ passes from the intersection of diagonals of $ABCD$, then $\angle AEB = \angle CED$.
1 reply
NO_SQUARES
May 5, 2025
mathuz
Yesterday at 5:49 PM
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Angle Relationships in Triangles
steven_zhang123   2
N Yesterday at 5:30 PM by Captainscrubz
In $\triangle ABC$, $AB > AC$. The internal angle bisector of $\angle BAC$ and the external angle bisector of $\angle BAC$ intersect the ray $BC$ at points $D$ and $E$, respectively. Given that $CE - CD = 2AC$, prove that $\angle ACB = 2\angle ABC$.
2 replies
steven_zhang123
Wednesday at 11:09 PM
Captainscrubz
Yesterday at 5:30 PM
J
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Two circles, a tangent line and a parallel
Valentin Vornicu   105
N Yesterday at 5:25 PM by Fly_into_the_sky
Source: IMO 2000, Problem 1, IMO Shortlist 2000, G2
Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP = EQ$.
105 replies
Valentin Vornicu
Oct 24, 2005
Fly_into_the_sky
Yesterday at 5:25 PM
J
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Prove angles are equal
BigSams   51
N Yesterday at 5:21 PM by Fly_into_the_sky
Source: Canadian Mathematical Olympiad - 1994 - Problem 5.
Let $ABC$ be an acute triangle. Let $AD$ be the altitude on $BC$, and let $H$ be any interior point on $AD$. Lines $BH,CH$, when extended, intersect $AC,AB$ at $E,F$ respectively. Prove that $\angle EDH=\angle FDH$.
51 replies
BigSams
May 13, 2011
Fly_into_the_sky
Yesterday at 5:21 PM
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J
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angle wanted, 3 squares related
parmenides51   9
N Dec 26, 2022 by resources
Source: 2022 European Math Tournament - Senior Grand League - Math Battle 4.8
The figure shows three squares. Find the marked angle.
IMAGE
9 replies
parmenides51
Dec 19, 2022
resources
Dec 26, 2022
J
angle wanted, 3 squares related
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Reveal topic tags
geometry
angles
Squares
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G H BBookmark kLocked kLocked NReply
Source: 2022 European Math Tournament - Senior Grand League - Math Battle 4.8
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parmenides51
30652 posts
parmenides51
#1 h  Dec 19, 2022, 1:33 PM
Y by
The figure shows three squares. Find the marked angle.
https://cdn.artofproblemsolving.com/attachments/b/4/291ac111faa614ca2eac350806c03e1a670acf.png
This post has been edited 2 times. Last edited by parmenides51, Dec 19, 2022, 1:35 PM
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parmenides51
30652 posts
parmenides51
#2 h  Dec 19, 2022, 1:34 PM
Y by
posted for the image links
Attachments:
This post has been edited 2 times. Last edited by parmenides51, Dec 19, 2022, 9:16 PM
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RagvaloD
4918 posts
RagvaloD
#3 h  Dec 19, 2022, 1:43 PM
Y by
are there information about ratio of sides of squares? I think angle is depended on this ratio
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parmenides51
30652 posts
parmenides51
#4 h  Dec 19, 2022, 1:49 PM • 1 Y
Y by ETS1331
No other data was given, original wording source

My comment: The two small squares seem to be equal.
This post has been edited 2 times. Last edited by parmenides51, Dec 19, 2022, 1:53 PM
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DavyDuf
335 posts
DavyDuf
#5 h  Dec 19, 2022, 1:52 PM
Y by
RagvaloD wrote:
are there information about ratio of sides of squares? I think angle is depended on this ratio

surprisingly, it's not
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squareman
966 posts
squareman
#6 h  Dec 19, 2022, 5:22 PM • 2 Y
Y by rayfish, cosmicgenius
Spiral sim at $Q$ maps $AB$ to $CD$ $\implies (ABPQ), (CDPQ)$ cyclic $\implies \boxed{45^\circ}$
Attachments:
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AoPS111
109 posts
AoPS111
#7 h  Dec 19, 2022, 5:45 PM
Y by
alternatively, assume small squares have side length 0
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Arrowandbow426
448 posts
Arrowandbow426
#8 h  Dec 19, 2022, 5:52 PM • 1 Y
Y by Mango247
AoPS111 wrote:
alternatively, assume small squares have side length 0

Yes, if there isn't anything restraining the size of the small squares, and we can find the angle doubtless. It doesn't matter what the size of the square is. The angle is always the same no matter the size.

We can still find that the angle is 45 degrees.
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ryanbear
1056 posts
ryanbear
#9 h  Dec 25, 2022, 10:38 PM
Y by
sol
This post has been edited 2 times. Last edited by ryanbear, Dec 26, 2022, 2:20 AM
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resources
752 posts
resources
#10 h  Dec 26, 2022, 1:02 AM
Y by
ryanbear wrote:
whats wrong with this solution?
Click to reveal hidden text

if all squares are 1x1 then c is not symmetric to D so theres 2 different lengths
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