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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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following events:
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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
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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 books
T.Mousavidin   1
N 2 minutes ago by martianrunner
Hello, I wanted to ask if anybody knows some good books for geometry that has these topics in:
Desargues's Theorem, Projective geometry, 3D geometry,
1 reply
T.Mousavidin
2 hours ago
martianrunner
2 minutes ago
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Geometry with orthocenter config
thdnder   1
N an hour ago by thdnder
Source: Own
Let $ABC$ be a triangle, and let $AD, BE, CF$ be its altitudes. Let $H$ be its orthocenter, and let $O_B$ and $O_C$ be the circumcenters of triangles $AHC$ and $AHB$. Let $G$ be the second intersection of the circumcircles of triangles $FDO_B$ and $EDO_C$. Prove that the lines $DG$, $EF$, and $A$-median of $\triangle ABC$ are concurrent.
1 reply
thdnder
an hour ago
thdnder
an hour ago
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n = a*b , numbers of the form a^b
falantrng   3
N an hour ago by MuradSafarli
Source: Azerbaijan NMO 2023. Senior P1
The teacher calculates and writes on the board all the numbers $a^b$ that satisfy the condition $n = a\times b$ for the natural number $n.$ Here $a$ and $b$ are natural numbers. Is there a natural number $n$ such that each of the numbers $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ is the last digit of one of the numbers written by the teacher on the board? Justify your opinion.
3 replies
falantrng
Aug 24, 2023
MuradSafarli
an hour ago
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Inequality with 3 variables and a special condition
Nuran2010   1
N an hour ago by arqady
Source: Azerbaijan Al-Khwarizmi IJMO TST 2024
For positive real numbers $a,b,c$ we have $3abc \geq ab+bc+ca$.
Prove that:

$\frac{1}{a^3+b^3+c}+\frac{1}{b^3+c^3+a}+\frac{1}{c^3+a^3+b} \leq \frac{3}{a+b+c}$.

Determine the equality case.
1 reply
Nuran2010
an hour ago
arqady
an hour ago
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find all functions
DNCT1   4
N an hour ago by jasperE3
Find all functions $f:\mathbb{R^+}\rightarrow \mathbb{R^+}$ such that
$$f(2f(x)+2y)=f(2x+y)+y\quad\forall x,y,\in\mathbb{R^+} $$
4 replies
+1 w
DNCT1
Oct 10, 2020
jasperE3
an hour ago
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Inequalities
sqing   16
N an hour ago by martianrunner
Let $ a,b \in [0 ,1] . $ Prove that
$$\frac{a}{ 1-ab+b }+\frac{b }{ 1-ab+a } \leq 2$$$$ \frac{a}{ 1+ab+b^2 }+\frac{b }{ 1+ab+a^2 }+\frac{ab }{2+ab } \leq 1$$$$\frac{a}{ 1-ab+b^2 }+\frac{b }{ 1-ab+a^2 }+\frac{1 }{1+ab }\leq \frac{5}{2}$$$$\frac{a}{ 1-ab+b^2 }+\frac{b }{ 1-ab+a^2 }+\frac{1 }{1+2ab }\leq \frac{7}{3}$$$$\frac{a}{ 1+ab+b^2 }+\frac{b }{ 1+ab+a^2 } +\frac{ab }{1+ab }\leq \frac{7}{6 }$$
16 replies
sqing
Apr 25, 2025
martianrunner
an hour ago
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Reflection of D moves on a line
Ankoganit   4
N an hour ago by bin_sherlo
Source: KöMaL A. 705
Triangle $ABC$ has orthocenter $H$. Let $D$ be a point distinct from the vertices on the circumcircle of $ABC$. Suppose that circle $BHD$ meets $AB$ at $P\ne B$, and circle $CHD$ meets $AC$ at $Q\ne C$. Prove that as $D$ moves on the circumcircle, the reflection of $D$ across line $PQ$ also moves on a fixed circle.

Michael Ren
4 replies
Ankoganit
Nov 11, 2017
bin_sherlo
an hour ago
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USAMO 2003 Problem 1
MithsApprentice   67
N an hour ago by L13832
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
67 replies
MithsApprentice
Sep 27, 2005
L13832
an hour ago
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D1024 : Can you do that?
Dattier   0
an hour ago
Source: les dattes à Dattier
Let $x_{n+1}=x_n^2+1$ and $x_0=1$.

Can you calculate $\sum\limits_{i=1}^{2^{2025}} x_i \mod 10^{30}$?
0 replies
Dattier
an hour ago
0 replies
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Azer and Babek playing a game on a chessboard
Nuran2010   0
an hour ago
Source: Azerbaijan Al-Khwarizmi IJMO TST
Azer and Babek have a $8 \times 8$ chessboard. Initially, Azer colors all cells of this chessboard with some colors. Then, Babek takes $2$ rows and $2$ columns and looks at the $4$ cells in the intersection. Babek wants to have all these $4$ cells in a same color, but Azer doesn't. With at least how many colors, Azer can reach his goal?
0 replies
Nuran2010
an hour ago
0 replies
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Circumcircle of one triangle passes from another's circumcenter.
Nuran2010   0
2 hours ago
Source: Azerbaijan Al-Khwarizmi IJMO TST 2024
In a parallelogram $ABCD$,$\angle A<90^\circ$ and $AB<BC$. Interior angle bisector of $\angle BAD$ intersects $BC$ at $M$, and $DC$ at $N$.Prove that circumcircle of $BCD$ passes from circumcenter of $CMN$.
0 replies
Nuran2010
2 hours ago
0 replies
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books for olympiads
frost23   1
N 2 hours ago by M.Roueintan
can you tell me some good books for maths olympiads
1 reply
frost23
2 hours ago
M.Roueintan
2 hours ago
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Sequence
lgx57   7
N 3 hours ago by jasperE3
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.
7 replies
lgx57
Apr 27, 2025
jasperE3
3 hours ago
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Geometry Angle Chasing
Sid-darth-vater   6
N 4 hours ago by sunken rock
Is there a way to do this without drawing obscure auxiliary lines? (the auxiliary lines might not be obscure I might just be calling them obscure)

For example I tried rotating triangle MBC 80 degrees around point C (so the BC line segment would now lie on segment AC) but I couldn't get any results. Any help would be appreciated!
6 replies
Sid-darth-vater
Apr 21, 2025
sunken rock
4 hours ago
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AB = AF when <ACB = <ABE, circumcircle related (Kharkiv City IX 2013 - Ukr)
parmenides51   1
N Mar 13, 2020 by ironball
On the side $AB$ of the triangle $ABC$, the point $D$ is chosen such that $AC = CD$. On the arc $BC$ of the circumscribed circle of the triangle $BCD$ (not containing the point $D$), the point $E$ is selected such that $\angle ACB = \angle ABE$. On the extension of the segment $BC$ beyond point $C$, a point $F$ is marked such that $CE = CF$. Prove that $AB = AF$.
1 reply
parmenides51
Mar 13, 2020
ironball
Mar 13, 2020
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AB = AF when <ACB = <ABE, circumcircle related (Kharkiv City IX 2013 - Ukr)
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Reveal topic tags
geometry
circumcircle
equal angles
equal segments
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parmenides51
30650 posts
parmenides51
#1 h  Mar 13, 2020, 5:07 AM
Y by
On the side $AB$ of the triangle $ABC$, the point $D$ is chosen such that $AC = CD$. On the arc $BC$ of the circumscribed circle of the triangle $BCD$ (not containing the point $D$), the point $E$ is selected such that $\angle ACB = \angle ABE$. On the extension of the segment $BC$ beyond point $C$, a point $F$ is marked such that $CE = CF$. Prove that $AB = AF$.
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ironball
110 posts
ironball
#2 h  Mar 13, 2020, 7:50 AM • 3 Y
Y by Mango247, Mango247, Mango247
Try to prove that$\triangle CDE $ $\cong$ $\triangle CAF$,then $\angle CED=\angle CFA=\angle ABF$
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