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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
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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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 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
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Erasing a and b and replacing them with a - b + 1
jl_   1
N 12 minutes ago by maromex
Source: Malaysia IMONST 2 2023 (Primary) P5
Ruby writes the numbers $1, 2, 3, . . . , 10$ on the whiteboard. In each move, she selects two distinct numbers, $a$ and $b$, erases them, and replaces them with $a+b-1$. She repeats this process until only one number, $x$, remains. What are all the possible values of $x$?
1 reply
jl_
38 minutes ago
maromex
12 minutes ago
J
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Prove that sum of 1^3+...+n^3 is a square
jl_   2
N 15 minutes ago by NicoN9
Source: Malaysia IMONST 2 2023 (Primary) P1
Prove that for all positive integers $n$, $1^3 + 2^3 + 3^3 +\dots+n^3$ is a perfect square.
2 replies
jl_
an hour ago
NicoN9
15 minutes ago
J
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x^3+y^3 is prime
jl_   2
N 20 minutes ago by Jackson0423
Source: Malaysia IMONST 2 2023 (Primary) P3
Find all pairs of positive integers $(x,y)$, so that the number $x^3+y^3$ is a prime.
2 replies
1 viewing
jl_
an hour ago
Jackson0423
20 minutes ago
J
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Feet of perpendiculars to diagonal in cyclic quadrilateral
jl_   0
35 minutes ago
Source: Malaysia IMONST 2 2023 (Primary) P6
Suppose $ABCD$ is a cyclic quadrilateral with $\angle ABC = \angle ADC = 90^{\circ}$. Let $E$ and $F$ be the feet of perpendiculars from $A$ and $C$ to $BD$ respectively. Prove that $BE = DF$.
0 replies
jl_
35 minutes ago
0 replies
J
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Tangents forms triangle with two times less area
NO_SQUARES   0
2 hours ago
Source: Kvant 2025 no. 2 M2831
Let $DEF$ be triangle, inscribed in parabola. Tangents in points $D,E,F$ forms triangle $ABC$. Prove that $S_{DEF}=2S_{ABC}$. ($S_T$ is area of triangle $T$).
From F.S.Macaulay's book «Geometrical Conics», suggested by M. Panov
0 replies
1 viewing
NO_SQUARES
2 hours ago
0 replies
J
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Woaah a lot of external tangents
egxa   3
N 3 hours ago by NO_SQUARES
Source: All Russian 2025 11.7
A quadrilateral \( ABCD \) with no parallel sides is inscribed in a circle \( \Omega \). Circles \( \omega_a, \omega_b, \omega_c, \omega_d \) are inscribed in triangles \( DAB, ABC, BCD, CDA \), respectively. Common external tangents are drawn between \( \omega_a \) and \( \omega_b \), \( \omega_b \) and \( \omega_c \), \( \omega_c \) and \( \omega_d \), and \( \omega_d \) and \( \omega_a \), not containing any sides of quadrilateral \( ABCD \). A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle \( \Gamma \). Prove that the lines joining the centers of \( \omega_a \) and \( \omega_c \), \( \omega_b \) and \( \omega_d \), and the centers of \( \Omega \) and \( \Gamma \) all intersect at one point.
3 replies
egxa
Apr 18, 2025
NO_SQUARES
3 hours ago
J
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2023 Hong Kong TST 3 (CHKMO) Problem 4
PikaNiko   3
N 3 hours ago by lightsynth123
Source: 2023 Hong Kong TST 3 (CHKMO)
Let $ABCD$ be a quadrilateral inscribed in a circle $\Gamma$ such that $AB=BC=CD$. Let $M$ and $N$ be the midpoints of $AD$ and $AB$ respectively. The line $CM$ meets $\Gamma$ again at $E$. Prove that the tangent at $E$ to $\Gamma$, the line $AD$ and the line $CN$ are concurrent.
3 replies
PikaNiko
Dec 3, 2022
lightsynth123
3 hours ago
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Help my diagram has too many points
MarkBcc168   27
N 3 hours ago by Om245
Source: IMO Shortlist 2023 G6
Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. A circle $\Gamma$ is internally tangent to $\omega$ at $A$ and also tangent to $BC$ at $D$. Let $AB$ and $AC$ intersect $\Gamma$ at $P$ and $Q$ respectively. Let $M$ and $N$ be points on line $BC$ such that $B$ is the midpoint of $DM$ and $C$ is the midpoint of $DN$. Lines $MP$ and $NQ$ meet at $K$ and intersect $\Gamma$ again at $I$ and $J$ respectively. The ray $KA$ meets the circumcircle of triangle $IJK$ again at $X\neq K$.

Prove that $\angle BXP = \angle CXQ$.

Kian Moshiri, United Kingdom
27 replies
MarkBcc168
Jul 17, 2024
Om245
3 hours ago
J
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Geometry, SMO 2016, not easy
Zoom   18
N 4 hours ago by SimplisticFormulas
Source: Serbia National Olympiad 2016, day 1, P3
Let $ABC$ be a triangle and $O$ its circumcentre. A line tangent to the circumcircle of the triangle $BOC$ intersects sides $AB$ at $D$ and $AC$ at $E$. Let $A'$ be the image of $A$ under $DE$. Prove that the circumcircle of the triangle $A'DE$ is tangent to the circumcircle of triangle $ABC$.
18 replies
Zoom
Apr 1, 2016
SimplisticFormulas
4 hours ago
J
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A touching question on perpendicular lines
Tintarn   2
N 4 hours ago by pi_quadrat_sechstel
Source: Bundeswettbewerb Mathematik 2025, Round 1 - Problem 3
Let $k$ be a semicircle with diameter $AB$ and midpoint $M$. Let $P$ be a point on $k$ different from $A$ and $B$.

The circle $k_A$ touches $k$ in a point $C$, the segment $MA$ in a point $D$, and additionally the segment $MP$. The circle $k_B$ touches $k$ in a point $E$ and additionally the segments $MB$ and $MP$.

Show that the lines $AE$ and $CD$ are perpendicular.
2 replies
Tintarn
Mar 17, 2025
pi_quadrat_sechstel
4 hours ago
J
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Iran Team Selection Test 2016
MRF2017   9
N Today at 4:22 AM by SimplisticFormulas
Source: TST3,day1,P2
Let $ABC$ be an arbitrary triangle and $O$ is the circumcenter of $\triangle {ABC}$.Points $X,Y$ lie on $AB,AC$,respectively such that the reflection of $BC$ WRT $XY$ is tangent to circumcircle of $\triangle {AXY}$.Prove that the circumcircle of triangle $AXY$ is tangent to circumcircle of triangle $BOC$.
9 replies
MRF2017
Jul 15, 2016
SimplisticFormulas
Today at 4:22 AM
J
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Circumcircle excircle chaos
CyclicISLscelesTrapezoid   25
N Yesterday at 7:55 PM by bin_sherlo
Source: ISL 2021 G8
Let $ABC$ be a triangle with circumcircle $\omega$ and let $\Omega_A$ be the $A$-excircle. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_A$. Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_A$ at $X$ and $Y$ respectively. The tangent line at $P$ to the circumcircle of the triangle $APX$ intersects the tangent line at $Q$ to the circumcircle of the triangle $AQY$ at a point $R$. Prove that $\overline{AR} \perp \overline{BC}$.
25 replies
CyclicISLscelesTrapezoid
Jul 12, 2022
bin_sherlo
Yesterday at 7:55 PM
J
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Geometry Problem
Itoz   3
N Yesterday at 3:09 PM by Itoz
Source: Own
Given $\triangle ABC$. Let the perpendicular line from $A$ to $BC$ meets $BC,\odot(ABC)$ at points $S,K$, respectively, and the foot from $B$ to $AC$ is $L$. $\odot (AKL)$ intersects line $AB$ at $T(\neq A)$, $\odot(AST)$ intersects line $AC$ at $M(\neq A)$, and lines $TM,CK$ intersect at $N$.

Prove that $\odot(CNM)$ is tangent to $\odot (BST)$.
3 replies
Itoz
Apr 18, 2025
Itoz
Yesterday at 3:09 PM
J
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combinatorial geo question
SAAAAAAA_B   2
N Yesterday at 2:35 PM by R8kt
Kuba has two finite families $\mathcal{A}, \mathcal{B}$ of convex polygons (in the plane). It turns out that every point of the plane lies in the same number of elements of $\mathcal{A}$ as elements of $\mathcal{B}$. Prove that $|\mathcal{A}| = |\mathcal{B}|$.

\textit{Note:} We treat segments and points as degenerate convex polygons, and they can be elements of $\mathcal{A}$ or $\mathcal{B}$.
2 replies
SAAAAAAA_B
Apr 14, 2025
R8kt
Yesterday at 2:35 PM
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best source for inequalitys
Namisgood   2
N Apr 14, 2025 by Namisgood
I need some help do I am beginner and have completed Number theory and almost all of algebra (except inequalitys) can anybody suggest a book or resource from where I can study inequalitys
2 replies
Namisgood
Apr 13, 2025
Namisgood
Apr 14, 2025
J
best source for inequalitys
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Reveal topic tags
inequalities
number theory
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Namisgood
15 posts
Namisgood
#1 h  Apr 13, 2025, 8:18 AM
Y by
I need some help do I am beginner and have completed Number theory and almost all of algebra (except inequalitys) can anybody suggest a book or resource from where I can study inequalitys
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Jackson0423
42 posts
Jackson0423
#2 h  Apr 13, 2025, 8:33 AM
Y by
You can see here:
https://math.stackexchange.com/questions/2313176/number-theory-book-recommendation
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Namisgood
15 posts
Namisgood
#3 h  Apr 14, 2025, 3:42 PM
Y by
Jackson0423 wrote:
You can see here:
https://math.stackexchange.com/questions/2313176/number-theory-book-recommendation

thanks
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