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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
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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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Problem 1
SlovEcience   1
N a minute ago by KAME06
Prove that
\[ C(p-1, k-1) \equiv (-1)^{k-1} \pmod{p} \]for \( 1 \leq k \leq p-1 \), where \( C(n, m) \) is the binomial coefficient \( n \) choose \( m \).
1 reply
SlovEcience
an hour ago
KAME06
a minute ago
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a hard geometry problen
Tuguldur   0
19 minutes ago
Let $ABCD$ be a convex quadrilateral. Suppose that the circles with diameters $AB$ and $CD$ intersect at points $X$ and $Y$. Let $P=AC\cap BD$ and $Q=AD\cap BC$. Prove that the points $P$, $Q$, $X$ and $Y$ are concyclic.
( $AB$ and $CD$ are not the diagnols)
0 replies
Tuguldur
19 minutes ago
0 replies
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hard problem
Cobedangiu   0
21 minutes ago
$1\le a\le 2,1\le b \le 2:$ Find max of $A$ (and prove) $: A=(a+b^2+\frac{4}{a^2}+\frac{2}{b})(b+a^2+\frac{4}{b^2}+\frac{2}{a})$
0 replies
Cobedangiu
21 minutes ago
0 replies
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Problem 2
SlovEcience   0
24 minutes ago
Let \( a, n \) be positive integers and \( p \) be an odd prime such that:
\[ a^p \equiv 1 \pmod{p^n}. \]Prove that:
\[ a \equiv 1 \pmod{p^{n-1}}. \]
0 replies
SlovEcience
24 minutes ago
0 replies
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Regarding Maaths olympiad prepration
omega2007   1
N 29 minutes ago by GreekIdiot
<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )

Do you haves compilled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your prespective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.
1 reply
omega2007
an hour ago
GreekIdiot
29 minutes ago
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Difference between being pre-qualified and pre-approved for a mortgage
smitjohn   0
30 minutes ago
Source: Home
In the context of a Southern Home Ownership Programs, it’s essential to understand the difference between pre-qualification and pre-approval. Pre-qualification is an informal estimate of how much you might be able to borrow, based on self-reported financial information. It's a good first step, but it doesn’t carry much weight with sellers.

Pre-approval, however, is a formal process where a lender verifies your income, credit score, and debts. Once pre-approved, you’ll receive a letter showing you're a serious buyer—often giving you an edge in competitive markets. Many home ownership programs require pre-approval before offering down payment assistance or other benefits. Getting pre-approved shows you're financially ready and serious about buying. It also helps you set a realistic home budget and avoid falling for homes you can’t afford. Always aim for pre-approval to give your offer strength and move forward with confidence.
0 replies
smitjohn
30 minutes ago
0 replies
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Induction
Mathlover_1   2
N 33 minutes ago by GreekIdiot
Hello, can you share links of same interesting induction problems in algebra
2 replies
Mathlover_1
Mar 24, 2025
GreekIdiot
33 minutes ago
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n-gon function
ehsan2004   10
N 44 minutes ago by Zany9998
Source: Romanian IMO Team Selection Test TST 1996, problem 1
Let $ f: \mathbb{R}^2 \rightarrow \mathbb{R} $ be a function such that for every regular $ n $-gon $ A_1A_2 \ldots A_n $ we have $ f(A_1)+f(A_2)+\cdots +f(A_n)=0 $. Prove that $ f(x)=0 $ for all reals $ x $.
10 replies
ehsan2004
Sep 13, 2005
Zany9998
44 minutes ago
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Functional equations
hanzo.ei   13
N an hour ago by GreekIdiot
Source: Greekldiot
Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$
13 replies
1 viewing
hanzo.ei
Mar 29, 2025
GreekIdiot
an hour ago
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Congruency in sum of digits base q
buzzychaoz   3
N an hour ago by sttsmet
Source: China Team Selection Test 2016 Test 3 Day 2 Q4
Let $a,b,b',c,m,q$ be positive integers, where $m>1,q>1,|b-b'|\ge a$. It is given that there exist a positive integer $M$ such that
$$S_q(an+b)\equiv S_q(an+b')+c\pmod{m}$$
holds for all integers $n\ge M$. Prove that the above equation is true for all positive integers $n$. (Here $S_q(x)$ is the sum of digits of $x$ taken in base $q$).
3 replies
buzzychaoz
Mar 26, 2016
sttsmet
an hour ago
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Unsolved NT, 3rd time posting
GreekIdiot   11
N an hour ago by GreekIdiot
Source: own
Solve $5^x-2^y=z^3$ where $x,y,z \in \mathbb Z$
Hint
11 replies
GreekIdiot
Mar 26, 2025
GreekIdiot
an hour ago
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Bashing??
John_Mgr   2
N an hour ago by GreekIdiot
I have learned little about what bashing mean as i am planning to start geo, feels like its less effort required and doesnt need much knowledge about the synthetic solutions?
what do you guys recommend ? also state the major difference of them... especially of bashing pros and cons..
2 replies
John_Mgr
3 hours ago
GreekIdiot
an hour ago
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Inspired by JK1603JK
sqing   13
N an hour ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $ab+bc+ca=1.$ Prove that$$\frac{abc-2}{abc-1}\ge \frac{4(a^2b+b^2c+c^2a)}{a^3b+b^3c+c^3a+1} $$
13 replies
sqing
Today at 3:31 AM
sqing
an hour ago
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A simple power
Rushil   19
N an hour ago by Raj_singh1432
Source: Indian RMO 1993 Problem 2
Prove that the ten's digit of any power of 3 is even.
19 replies
Rushil
Oct 16, 2005
Raj_singh1432
an hour ago
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J
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trigonometry trivia
Raja Oktovin   1
N Nov 17, 2009 by Luis González
Source: Indonesia IMO 2007 TST, Stage 2, Test 1, Problem 1
Let $ P$ be a point in triangle $ ABC$, and define $ \alpha,\beta,\gamma$ as follows: \[ \alpha=\angle BPC-\angle BAC, \quad \beta=\angle CPA-\angle \angle CBA, \quad \gamma=\angle APB-\angle ACB.\] Prove that \[ PA\dfrac{\sin \angle BAC}{\sin \alpha}=PB\dfrac{\sin \angle CBA}{\sin \beta}=PC\dfrac{\sin \angle ACB}{\sin \gamma}.\]
1 reply
Raja Oktovin
Nov 15, 2009
Luis González
Nov 17, 2009
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trigonometry trivia
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Reveal topic tags
trigonometry
geometry
circumcircle
geometry proposed
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Source: Indonesia IMO 2007 TST, Stage 2, Test 1, Problem 1
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Raja Oktovin
277 posts
Raja Oktovin
#1 h  Nov 15, 2009, 6:26 AM • 2 Y
Y by Adventure10, Mango247
Let $ P$ be a point in triangle $ ABC$, and define $ \alpha,\beta,\gamma$ as follows: \[ \alpha=\angle BPC-\angle BAC, \quad \beta=\angle CPA-\angle \angle CBA, \quad \gamma=\angle APB-\angle ACB.\] Prove that \[ PA\dfrac{\sin \angle BAC}{\sin \alpha}=PB\dfrac{\sin \angle CBA}{\sin \beta}=PC\dfrac{\sin \angle ACB}{\sin \gamma}.\]
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The post below has been deleted. Click to close.
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Luis González
4145 posts
Luis González
#2 h  Nov 17, 2009, 5:30 PM • 1 Y
Y by Adventure10
Let $ A',B',C'$ be the second intersections of rays $ AP,BP,CP$ with the circumcircle $ (O)$ of $ \triangle ABC.$ Angle chase gives

$ \angle B'CP = \angle BPC - \angle BB'C = \angle BPC - \angle BAC = \alpha$

Similarly, $ \angle C'AP = \beta$ and $ \angle A'BP = \gamma.$

By sine law in the triangles $ \triangle PCB',\triangle PAC'$ and $ \triangle PBA'$ we obtain

$ \frac {\sin\widehat{BAC}}{\sin \alpha} = \frac {PC}{PA} \ , \ \frac {\sin\widehat{CBA}}{\sin \beta} = \frac {PA}{PC'} \ , \ \frac {\sin\widehat{ACB}}{\sin \gamma} = \frac {PB}{PA'}$

From power of $ P$ WRT $ (O) \ , $ $PA \cdot PA' = PB \cdot PB' = PC \cdot PC'$ we get

$ PA \cdot \frac {\sin\widehat{BAC}}{\sin \alpha} = PB \cdot \frac {\sin\widehat{CBA}}{\sin \beta} = PC \cdot \frac {\sin\widehat{ACB}}{\sin \gamma}$
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