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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
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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
J
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Geometry
MathsII-enjoy   7
N an hour ago by MathsII-enjoy
Given triangle $ABC$ inscribed in $(O)$, $M, N$ are respectively the midpoints of the major and minor arcs $BC$. Let $I$ be the center of the inscribed circle, $R$ be $A-mix$, $D$ is the intersection point of the line through $A$ parallel to $BC$ with $(O)$. $DI$ intersects $AR$ at $K$, take $L$ on $AK$ so that $LI//BC$. $NL$ intersects $(O)$ at $G$, $AG$ intersects $LI$ at $Z$. Prove that: $ZM$ is perpendicular to $KN$.
7 replies
MathsII-enjoy
May 23, 2025
MathsII-enjoy
an hour ago
J
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Find the Expression of Quadratic Expression
getequation000   1
N an hour ago by vanstraelen
Assume that the equation $2x^2-10x+3=0$ has two roots $x_1$ and $x_2$ in which $x_1>x_2$. Without solving the equation, find the value of the following expression:
$$T=\frac{\sqrt{24x_1-5}+2x_2+2026}{25-2x_1-8x_2}$$
1 reply
getequation000
Yesterday at 2:10 PM
vanstraelen
an hour ago
J
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Easy Geometry
pokmui9909   7
N an hour ago by RANDOM__USER
Source: FKMO 2025 P4
Triangle $ABC$ satisfies $\overline{CA} > \overline{AB}$. Let the incenter of triangle $ABC$ be $\omega$, which touches $BC, CA, AB$ at $D, E, F$, respectively. Let $M$ be the midpoint of $BC$. Let the circle centered at $M$ passing through $D$ intersect $DE, DF$ at $P(\neq D), Q(\neq D)$, respecively. Let line $AP$ meet $BC$ at $N$, line $BP$ meet $CA$ at $L$. Prove that the three lines $EQ, FP, NL$ are concurrent.
7 replies
pokmui9909
Mar 30, 2025
RANDOM__USER
an hour ago
J
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Misplaced Geometry (HHMO 2025 #6)
peace09   1
N an hour ago by Mathzeus1024
Suppose that both equilateral triangle $ABC{}$ and square $DEFG{}$ are inscribed in a circle with $\overline{AC}{}$ and $\overline{DG}{}$ being parallel. If $H$ is the intersection of $\overline{AC}{}$ and $\overline{DE}{}$, the ratio $DE:DH$ is $a+b\sqrt{c}$. Find $a^2+bc$.

(Here $AB>AD{}$.)
1 reply
peace09
Yesterday at 10:49 PM
Mathzeus1024
an hour ago
J
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Cute geometry
Rijul saini   6
N 2 hours ago by mathscrazy
Source: India IMOTC Practice Test 1 Problem 3
Let scalene $\triangle ABC$ have altitudes $BE, CF,$ circumcenter $O$ and orthocenter $H$. Let $R$ be a point on line $AO$. The points $P,Q$ are on lines $AB,AC$ respectively such that $RE \perp EP$ and $RF \perp FQ$. Prove that $PQ$ is perpendicular to $RH$.

Proposed by Rijul Saini
6 replies
Rijul saini
Yesterday at 6:51 PM
mathscrazy
2 hours ago
J
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2019 IGO Advanced P1
Dadgarnia   12
N 2 hours ago by fe.
Source: 6th Iranian Geometry Olympiad (Advanced) P1
Circles $\omega_1$ and $\omega_2$ intersect each other at points $A$ and $B$. Point $C$ lies on the tangent line from $A$ to $\omega_1$ such that
$\angle ABC = 90^\circ$. Arbitrary line $\ell$ passes through $C$ and cuts $\omega_2$ at points $P$ and $Q$. Lines $AP$ and $AQ$ cut $\omega_1$ for the second time at points $X$ and $Z$ respectively. Let $Y$ be the foot of altitude from $A$ to $\ell$. Prove that points $X, Y$ and $Z$ are collinear.

Proposed by Iman Maghsoudi
12 replies
Dadgarnia
Sep 20, 2019
fe.
2 hours ago
J
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Inequalities from SXTB
sqing   25
N 3 hours ago by DAVROS
T2811. Let $ a,b,c,d> 0 $ and $ a+b+c+d=1. $ Prove that$\frac{a}{a+1}+\frac{2b}{b+2}+\frac{3c}{c+3}+\frac{6d}{d+6} \leq \frac{12}{13}$
25 replies
sqing
Dec 5, 2024
DAVROS
3 hours ago
J
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Might be slightly generalizable
Rijul saini   5
N 3 hours ago by guptaamitu1
Source: India IMOTC Day 3 Problem 1
Let $ABC$ be an acute angled triangle with orthocenter $H$ and $AB<AC$. Let $T(\ne B,C, H)$ be any other point on the arc $\stackrel{\LARGE\frown}{BHC}$ of the circumcircle of $BHC$ and let line $BT$ intersect line $AC$ at $E(\ne A)$ and let line $CT$ intersect line $AB$ at $F(\ne A)$. Let the circumcircles of $AEF$ and $ABC$ intersect again at $X$ ($\ne A$). Let the lines $XE,XF,XT$ intersect the circumcircle of $(ABC)$ again at $P,Q,R$ ($\ne X$). Prove that the lines $AR,BC,PQ$ concur.
5 replies
Rijul saini
Yesterday at 6:39 PM
guptaamitu1
3 hours ago
J
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Orthocenters equidistant from circumcenter
Rijul saini   6
N 4 hours ago by guptaamitu1
Source: India IMOTC 2025 Day 1 Problem 2
In triangle $ABC$, consider points $A_1,A_2$ on line $BC$ such that $A_1,B,C,A_2$ are in that order and $A_1B=AC$ and $CA_2=AB$. Similarly consider points $B_1,B_2$ on line $AC$, and $C_1,C_2$ on line $AB$. Prove that orthocenters of triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equidistant from the circumcenter of $ABC$.

Proposed by Shantanu Nene
6 replies
Rijul saini
Yesterday at 6:31 PM
guptaamitu1
4 hours ago
J
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My Unsolved Problem
ZeltaQN2008   3
N 4 hours ago by Funcshun840
Source: IDK
Let \( ABC \) be an acute triangle inscribed in its circumcircle \( (O) \), and let \( (I) \) be its incircle. Let \( K \) be the point where the $A-mixtilinear$ incircle of triangle $ABC$ touches \((O)\). Suppose line \( OI \) intersects segment \( AK \) at \( P \), and intersects line \( BC \) at \( Q \). Let the line through \( I \) perpendicular to \( BC \) intersect line \( KQ \) at \( A' \). Prove that: \[AI \parallel PA'.\]
3 replies
ZeltaQN2008
Yesterday at 1:23 PM
Funcshun840
4 hours ago
J
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Onto the altitude'
TheUltimate123   4
N 5 hours ago by EpicBird08
Source: Extension of nukelauncher's and my Mock AIME #15 (https://artofproblemsolving.com/community/c875089h1825979p12212193)
In triangle $ABC$, let $D$, $E$, and $F$ denote the feet of the altitudes from $A$, $B$, and $C$, respectively, and let $O$ denote the circumcenter of $\triangle ABC$. Points $X$ and $Y$ denote the projections of $E$ and $F$, respectively, onto $\overline{AD}$, and $Z=\overline{AO}\cap\overline{EF}$. There exists a point $T$ such that $\angle DTZ=90^\circ$ and $AZ=AT$. If $P=\overline{AD}\cap\overline{ZT}$ and $Q$ lies on $\overline{EF}$ such that $\overline{PQ}\parallel\overline{BC}$, prove that line $AQ$ bisects $\overline{BC}$.
4 replies
TheUltimate123
May 19, 2019
EpicBird08
5 hours ago
J
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A beautiful collinearity regarding three wonderful points
math_pi_rate   10
N Today at 4:04 AM by alexanderchew
Source: Own
Let $\triangle DEF$ be the medial triangle of an acute-angle triangle $\triangle ABC$. Suppose the line through $A$ perpendicular to $AB$ meet $EF$ at $A_B$. Define $A_C,B_A,B_C,C_A,C_B$ analogously. Let $B_CC_B \cap BC=X_A$. Similarly define $X_B$ and $X_C$. Suppose the circle with diameter $BC$ meet the $A$-altitude at $A'$, where $A'$ lies inside $\triangle ABC$. Define $B'$ and $C'$ similarly. Let $N$ be the circumcenter of $\triangle DEF$, and let $\omega_A$ be the circle with diameter $X_AN$, which meets $\odot (X_A,A')$ at $A_1,A_2$. Similarly define $\omega_B,B_1,B_2$ and $\omega_C,C_1,C_2$.
1) Show that $X_A,X_B,X_C$ are collinear.
2) Prove that $A_1,A_2,B_1,B_2,C_1,C_2$ lie on a circle centered at $N$.
3) Prove that $\omega_A,\omega_B,\omega_C$ are coaxial.
4) Show that the line joining $X_A,X_B,X_C$ is perpendicular to the radical axis of $\omega_A,\omega_B,\omega_C$.
10 replies
math_pi_rate
Nov 8, 2018
alexanderchew
Today at 4:04 AM
J
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The line is a common tangent
Rijul saini   3
N Today at 3:22 AM by pingupignu
Source: India IMOTC 2025 Day 4 Problem 3
Let $ABCD$ be a cyclic quadrilateral with circumcentre $O$ and circumcircle $\Gamma$. Let $T$ be the intersection of tangents at $B$ and $C$ to $\Gamma$. Let $\omega$ be the circumcircle of triangle $TBC$ and let $M(\neq T)$, $N(\neq T)$ denote the second intersections of $TA,TD$ with $\omega$ respectively. Let $AD$ and $BC$ intersect at $E$ and $\Omega$ be the circumcircle of triangle $EMN$. If $AD$ intersects $\Omega$ again at $X \neq E$, prove that the line tangent to $\Omega$ at $X$ is also tangent to $\omega$.

Proposed by Malay Mahajan and Siddharth Choppara
3 replies
Rijul saini
Yesterday at 6:47 PM
pingupignu
Today at 3:22 AM
J
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One of P or Q lies on circle
Rijul saini   6
N Today at 3:21 AM by ZVFrozel
Source: LMAO 2025 Day 1 Problem 3
Let $ABC$ be an acute triangle with orthocenter $H$. Let $M$ be the midpoint of $BC$, and $K$ be the intersection of the tangents from $B$ and $C$ to the circumcircle of $ABC$. Denote by $\Omega$ the circle centered at $H$ and tangent to line $AM$.

Suppose $AK$ intersects $\Omega$ at two distinct points $X$, $Y$.
Lines $BX$ and $CY$ meet at $P$, while lines $BY$ and $CX$ meet at $Q$. Prove that either $P$ or $Q$ lies on $\Omega$.

Proposed by MV Adhitya, Archit Manas and Arnav Nanal
6 replies
Rijul saini
Yesterday at 6:59 PM
ZVFrozel
Today at 3:21 AM
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computational, intersections in a regular pentagon 2010 El Número de Oro p10
parmenides51   1
N Oct 28, 2020 by JustinLee2017
Given the regular pentagon $ABCDE$ constructed as indicated in the figure from side $1$, calculate the measure of $FG$ and $KL$.
IMAGE
1 reply
parmenides51
Oct 28, 2020
JustinLee2017
Oct 28, 2020
J
computational, intersections in a regular pentagon 2010 El Número de Oro p10
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Reveal topic tags
geometry
regular pentagon
pentagon
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parmenides51
30653 posts
parmenides51
#1 h  Oct 28, 2020, 3:29 PM • 1 Y
Y by son7
Given the regular pentagon $ABCDE$ constructed as indicated in the figure from side $1$, calculate the measure of $FG$ and $KL$.
//cdn.artofproblemsolving.com/images/e/3/9/e39d3354611b3f5029b8a72470f0099ccc55fbde.png
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JustinLee2017
1703 posts
JustinLee2017
#2 h  Oct 28, 2020, 4:11 PM • 3 Y
Y by Mango247, Mango247, Mango247
Similar Triangles (No trig)

FG, get KL similarly
This post has been edited 1 time. Last edited by JustinLee2017, Oct 28, 2020, 4:11 PM
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