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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Wednesday at 3:18 PM
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Wednesday at 3:18 PM
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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Geometry problem
kjhgyuio   1
N 29 minutes ago by Mathzeus1024
Source: smo
In trapezium ABCD,AD is parallel to BC and points E and F are midpoints of AB and DC respectively. If
Area of AEFD/Area of EBCF =√3 + 1/3-√3 and the area of triangle ABD is √3 .find the area of trapezium ABCD
1 reply
kjhgyuio
Apr 1, 2025
Mathzeus1024
29 minutes ago
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D1018 : Can you do that ?
Dattier   1
N 41 minutes ago by Dattier
Source: les dattes à Dattier
We can find $A,B,C$, such that $\gcd(A,B)=\gcd(C,A)=\gcd(A,2)=1$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$.

For example :

$C=20$
$A=47650065401584409637777147310342834508082136874940478469495402430677786194142956609253842997905945723173497630499054266092849839$

$B=238877301561986449355077953728734922992395532218802882582141073061059783672634737309722816649187007910722185635031285098751698$

Can you find $A,B,C$ such that $A>3$ is prime, $C,B \in (\mathbb Z/A\mathbb Z)^*$ with $o(C)=(A-1)/2$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$?
1 reply
Dattier
Mar 24, 2025
Dattier
41 minutes ago
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2025 Caucasus MO Juniors P3
BR1F1SZ   1
N an hour ago by FarrukhKhayitboyev
Source: Caucasus MO
Let $K$ be a positive integer. Egor has $100$ cards with the number “$2$” written on them, and $100$ cards with the number “$3$” written on them. Egor wants to paint each card red or blue so that no subset of cards of the same color has the sum of the numbers equal to $K$. Find the greatest $K$ such that Egor will not be able to paint the cards in such a way.
1 reply
BR1F1SZ
Mar 26, 2025
FarrukhKhayitboyev
an hour ago
J
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1 area = 2025 points
giangtruong13   0
an hour ago
In a plane give a set $H$ that has 8097 distinct points with area of a triangle that has 3 points belong to $H$ all $ \leq 1$. Prove that there exists a triangle $G$ that has the area $\leq 1 $ contains at least 2025 points that belong to $H$( each of that 2025 points can be inside the triangle or lie on the edge of triangle $G$)X
0 replies
giangtruong13
an hour ago
0 replies
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D is incenter
Layaliya   3
N 2 hours ago by rong2020
Source: From my friend in Indonesia
Given an acute triangle \( ABC \) where \( AB > AC \). Point \( O \) is the circumcenter of triangle \( ABC \), and \( P \) is the projection of point \( A \) onto line \( BC \). The midpoints of \( BC \), \( CA \), and \( AB \) are \( D \), \( E \), and \( F \), respectively. The line \( AO \) intersects \( DE \) and \( DF \) at points \( Q \) and \( R \), respectively. Prove that \( D \) is the incenter of triangle \( PQR \).
3 replies
Layaliya
Yesterday at 11:03 AM
rong2020
2 hours ago
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Orthocenter is the midpoint of the altitude
plagueis   6
N 3 hours ago by FrancoGiosefAG
Source: Mexican Quarantine Mathematical Olympiad P4
Let $ABC$ be an acute triangle with orthocenter $H$. Let $A_1$, $B_1$ and $C_1$ be the feet of the altitudes of triangle $ABC$ opposite to vertices $A$, $B$, and $C$ respectively. Let $B_2$ and $C_2$ be the midpoints of $BB_1$ and $CC_1$, respectively. Let $O$ be the intersection of lines $BC_2$ and $CB_2$. Prove that $O$ is the circumcenter of triangle $ABC$ if and only if $H$ is the midpoint of $AA_1$.

Proposed by Dorlir Ahmeti
6 replies
plagueis
Apr 26, 2020
FrancoGiosefAG
3 hours ago
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IMO 2018 Problem 1
juckter   168
N 3 hours ago by Trasher_Cheeser12321
Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line.

Proposed by Silouanos Brazitikos, Evangelos Psychas and Michael Sarantis, Greece
168 replies
1 viewing
juckter
Jul 9, 2018
Trasher_Cheeser12321
3 hours ago
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An epitome of config geo
AndreiVila   9
N 4 hours ago by ihategeo_1969
Source: The Golden Digits Contest, December 2024, P3
Let $ABC$ be a scalene acute triangle with incenter $I$ and circumcircle $\Omega$. $M$ is the midpoint of small arc $BC$ on$\Omega$ and $N$ is the projection of $I$ onto the line passing through the midpoints of $AB$ and $AC$. A circle $\omega$ with center $Q$ is internally tangent to $\Omega$ at $A$, and touches segment $BC$. If the circle with diameter $IM$ meets $\Omega$ again at $J$, prove that $JI$ bisects $\angle QJN$.

Proposed by David Anghel
9 replies
AndreiVila
Dec 22, 2024
ihategeo_1969
4 hours ago
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Beautiful problem
luutrongphuc   0
4 hours ago
Let triangle $ABC$ be circumscribed about circle $(I)$, and let $H$ be the orthocenter of $\triangle ABC$. The circle $(I)$ touches line $BC$ at $D$. The tangent to the circle $(BHC)$ at $H$ meets $BC$ at $S$. Let $J$ be the midpoint of $HI$, and let the line $DJ$ meet $(I)$ again at $X$. The tangent to $(I)$ parallel to $BC$ meets the line $AX$ at $T$. Prove that $ST$ is tangent to $(I)$.
0 replies
luutrongphuc
4 hours ago
0 replies
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Square and equilateral triangle
m4thbl3nd3r   2
N 4 hours ago by m4thbl3nd3r
Let $ABCD$ be a square and a point $X$ lies on the interior of $ABCD$ such that triangle $BDX$ is equilateral. Evaluate $\angle AXD$
2 replies
m4thbl3nd3r
4 hours ago
m4thbl3nd3r
4 hours ago
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Olympiad Geometry problem-second time posting
kjhgyuio   7
N Today at 3:21 AM by kjhgyuio
Source: smo problem
In trapezium ABCD,AD is parallel to BC and points E and F are midpoints of AB and DC respectively. If
Area of AEFD/Area of EBCF =√3 + 1/3-√3 and the area of triangle ABD is √3 .find the area of trapezium ABCD
7 replies
kjhgyuio
Apr 2, 2025
kjhgyuio
Today at 3:21 AM
J
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Geometry problem-second time posting
kjhgyuio   0
Today at 3:18 AM
Source: smo roudn 2

A square is cut into several rectangles, none of which is a square ,so that the sides of each rectangles are parallel to the sides of a square .For each rectangle with sides a,b,a<b compute the ratio a/b Prove that the sum of these ratios is at least 1
0 replies
kjhgyuio
Today at 3:18 AM
0 replies
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Proving ZA=ZB
nAalniaOMliO   5
N Today at 3:17 AM by EmersonSoriano
Source: Belarusian National Olympiad 2025
Point $H$ is the foot of the altitude from $A$ of triangle $ABC$. On the lines $AB$ and $AC$ points $X$ and $Y$ are marked such that the circumcircles of triangles $BXH$ and $CYH$ are tangent, call this circles $w_B$ and $w_C$ respectively. Tangent lines to circles $w_B$ and $w_C$ at $X$ and $Y$ intersect at $Z$.
Prove that $ZA=ZH$.
Vadzim Kamianetski
5 replies
nAalniaOMliO
Mar 28, 2025
EmersonSoriano
Today at 3:17 AM
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pretty well known
dotscom26   2
N Today at 2:13 AM by Giant_PT
Let $\triangle ABC$ be a scalene triangle such that $\Omega$ is its incircle. $AB$ is tangent to $\Omega$ at $D$. A point $E$ ($E \notin \Omega$) is located on $BC$.

Let $\omega_1$, $\omega_2$, and $\omega_3$ be the incircles of the triangles $BED$, $ADE$, and $AEC$, respectively.

Show that the common tangent to $\omega_1$ and $\omega_3$ is also tangent to $\omega_2$.

2 replies
dotscom26
Yesterday at 2:03 AM
Giant_PT
Today at 2:13 AM
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concurrent wanted, incircle and angle bisectors
parmenides51   0
May 19, 2021
Source: 2013 XVI All-Ukrainian Tournament of Young Mathematicians, Final Battle p7
The inscribed circle $\omega$ from triangle $ABC$ touches its side $BC$ at point $D$. The bisector of the angle $ADB$ intersects the circle $\omega$ for the second time at the point $N$. The bisector of the angle $ADC$ intersects the circle $\omega$ for the second time at the point $M$. Prove that the lines $BM, CN$ and $AD$ intersect at one point.
0 replies
parmenides51
May 19, 2021
0 replies
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concurrent wanted, incircle and angle bisectors
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Reveal topic tags
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concurrent
concurrency
incircle
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G H BBookmark kLocked kLocked NReply
Source: 2013 XVI All-Ukrainian Tournament of Young Mathematicians, Final Battle p7
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parmenides51
30629 posts
parmenides51
#1 h  May 19, 2021, 6:37 PM • 2 Y
Y by centslordm, Mango247
The inscribed circle $\omega$ from triangle $ABC$ touches its side $BC$ at point $D$. The bisector of the angle $ADB$ intersects the circle $\omega$ for the second time at the point $N$. The bisector of the angle $ADC$ intersects the circle $\omega$ for the second time at the point $M$. Prove that the lines $BM, CN$ and $AD$ intersect at one point.
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