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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. 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 upcoming events:
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[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 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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Every popular person is the best friend of a popular person?
yunxiu   8
N 22 minutes ago by HHGB
Source: 2012 European Girls’ Mathematical Olympiad P6
There are infinitely many people registered on the social network Mugbook. Some pairs of (different) users are registered as friends, but each person has only finitely many friends. Every user has at least one friend. (Friendship is symmetric; that is, if $A$ is a friend of $B$, then $B$ is a friend of $A$.)
Each person is required to designate one of their friends as their best friend. If $A$ designates $B$ as her best friend, then (unfortunately) it does not follow that $B$ necessarily designates $A$ as her best friend. Someone designated as a best friend is called a $1$-best friend. More generally, if $n> 1$ is a positive integer, then a user is an $n$-best friend provided that they have been designated the best friend of someone who is an $(n-1)$-best friend. Someone who is a $k$-best friend for every positive integer $k$ is called popular.
(a) Prove that every popular person is the best friend of a popular person.
(b) Show that if people can have infinitely many friends, then it is possible that a popular person is not the best friend of a popular person.

Romania (Dan Schwarz)
8 replies
yunxiu
Apr 13, 2012
HHGB
22 minutes ago
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2021 EGMO P2: f(xf(x)+y) = f(y) + x^2 for rational x, y
anser   80
N 30 minutes ago by math-olympiad-clown
Source: 2021 EGMO P2
Find all functions $f:\mathbb{Q}\to\mathbb{Q}$ such that the equation
\[f(xf(x)+y) = f(y) + x^2\]holds for all rational numbers $x$ and $y$.

Here, $\mathbb{Q}$ denotes the set of rational numbers.
80 replies
anser
Apr 13, 2021
math-olympiad-clown
30 minutes ago
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D1033 : A problem of probability for dominoes 3*1
Dattier   1
N 30 minutes ago by Dattier
Source: les dattes à Dattier
Let $G$ a grid of 9*9, we choose a little square in $G$ of this grid three times, we can choose three times the same.

What the probability of cover with 3*1 dominoes this grid removed by theses little squares (one, two or three) ?
1 reply
Dattier
May 15, 2025
Dattier
30 minutes ago
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2010 Japan MO Finals
parkjungmin   4
N 38 minutes ago by parkjungmin
Is there anyone who can solve question problem 5?
4 replies
parkjungmin
May 15, 2025
parkjungmin
38 minutes ago
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something...
SunnyEvan   3
N an hour ago by SunnyEvan
Source: unknown
Try to prove : $$ \sum csc^{20} \frac{2^{i} \pi}{7} csc^{23} \frac{2^{j}\pi }{7} csc^{2023} \frac{2^{k} \pi}{7} $$is a rational number.
Where $ (i,j,k)=(1,2,3) $ and other permutations.
3 replies
SunnyEvan
May 5, 2025
SunnyEvan
an hour ago
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IMO 2010 Problem 2
orl   89
N an hour ago by fearsum_fyz
Given a triangle $ABC$, with $I$ as its incenter and $\Gamma$ as its circumcircle, $AI$ intersects $\Gamma$ again at $D$. Let $E$ be a point on the arc $BDC$, and $F$ a point on the segment $BC$, such that $\angle BAF=\angle CAE < \dfrac12\angle BAC$. If $G$ is the midpoint of $IF$, prove that the meeting point of the lines $EI$ and $DG$ lies on $\Gamma$.

Proposed by Tai Wai Ming and Wang Chongli, Hong Kong
89 replies
orl
Jul 7, 2010
fearsum_fyz
an hour ago
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vieta jumping
kailash_mk   2
N an hour ago by Thapakazi
Find all pairs of positive integers $(a,b)$ such that

Hint 1 : Try to show that both a and b have to be odd numbers
Hint 2 : Co-prime

\[a+1|b^2+1 \ , \ b+1|a^2+1.\]
2 replies
kailash_mk
Mar 14, 2020
Thapakazi
an hour ago
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Find tha maximum value
sqing   2
N 2 hours ago by sqing
Source: China Zhejiang High School Mathematics Competition 2025 Q7
Let $ x,y,z $ be reals such that $ 5x^2+6y^2+6z^2-8yz\leq 1. $ Find tha maximum value of $ x+y+z. $
2 replies
sqing
2 hours ago
sqing
2 hours ago
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Find tha minimum value
sqing   3
N 2 hours ago by sqing
Source: China Zhejiang High School Mathematics Competition 2025 Q6
Let $ x,y $ be reals such that $ x^2+y^2=1 $ and $ x\neq 1. $ Find tha minimum value of $ \frac{\max\{2x-3,3y-1\}}{\min\{x-1,\frac{3}{2}y\}}. $
3 replies
sqing
2 hours ago
sqing
2 hours ago
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CSMGO P6: Incenter lies on radax of two interesting circles
amar_04   13
N 2 hours ago by WLOGQED1729
Source: https://artofproblemsolving.com/community/c594864h2372843p19407517
Let $\triangle ABC$ be a triangle with the incenter $I$, and let the incircle $\omega$ of $\triangle ABC$ touch $\overline{BC},\overline{CA},\overline{AB}$ at points $D,E,F$ respectively. Let $\overline{AD}$ intersect $\omega$ again at a point $X\ne D$. Let the lines through $E$ and $F$ parallel to $AD$ intersect $\omega$ again at points $P\ne E, Q\ne F$ respectively. Prove that $I$ lies on the common chord of the circumcircle of $\triangle XBQ$ and the circumcircle of $\triangle XCP$.
13 replies
amar_04
Feb 16, 2021
WLOGQED1729
2 hours ago
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Beijing High School Mathematics Competition 2025 Q1
SunnyEvan   2
N 2 hours ago by SunnyEvan
Let $ a,b,c,d \in R^+ $. Prove that:
$$ \frac{1}{a^4+b^4+c^4+abcd}+\frac{1}{b^4+c^4+d^4+abcd}+\frac{1}{c^4+d^4+a^4+abcd}+\frac{1}{d^4+a^4+b^4+abcd} \leq \frac{1}{abcd} $$
2 replies
SunnyEvan
3 hours ago
SunnyEvan
2 hours ago
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Inspired by Zhejiang 2025
sqing   0
2 hours ago
Source: Own
Let $ x,y,z $ be reals such that $ 5x^2+6y^2+6z^2-8yz\leq 5. $ Prove that$$ x+y+z\leq \sqrt{6}$$
0 replies
sqing
2 hours ago
0 replies
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Lord Evan the Reflector
whatshisbucket   24
N 2 hours ago by Trasher_Cheeser12321
Source: ELMO 2018 #3, 2018 ELMO SL G3
Let $A$ be a point in the plane, and $\ell$ a line not passing through $A$. Evan does not have a straightedge, but instead has a special compass which has the ability to draw a circle through three distinct noncollinear points. (The center of the circle is not marked in this process.) Additionally, Evan can mark the intersections between two objects drawn, and can mark an arbitrary point on a given object or on the plane.

(i) Can Evan construct* the reflection of $A$ over $\ell$?

(ii) Can Evan construct the foot of the altitude from $A$ to $\ell$?

*To construct a point, Evan must have an algorithm which marks the point in finitely many steps.

Proposed by Zack Chroman
24 replies
whatshisbucket
Jun 28, 2018
Trasher_Cheeser12321
2 hours ago
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Iran second round 2025-q1
mohsen   7
N 2 hours ago by Mathgloggers
Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.
7 replies
mohsen
Apr 19, 2025
Mathgloggers
2 hours ago
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BC = BD + DA if AB=AC, <A = 100^o, BD angle bisector
parmenides51   3
N Aug 25, 2021 by zerononnatural
Source: Indonesia INAMO Shortlist 2008 G10
Given a triangle $ABC$ with $AB = AC$, angle $\angle A = 100^o$ and $BD$ bisector of angle $\angle B$. Prove that $$BC = BD + DA.$$
3 replies
parmenides51
Aug 25, 2021
zerononnatural
Aug 25, 2021
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BC = BD + DA if AB=AC, <A = 100^o, BD angle bisector
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Reveal topic tags
geometry
equal segments
isosceles
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G H BBookmark kLocked kLocked NReply
Source: Indonesia INAMO Shortlist 2008 G10
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parmenides51
30652 posts
parmenides51
#1 h  Aug 25, 2021, 9:09 PM
Y by
Given a triangle $ABC$ with $AB = AC$, angle $\angle A = 100^o$ and $BD$ bisector of angle $\angle B$. Prove that $$BC = BD + DA.$$
This post has been edited 2 times. Last edited by parmenides51, Aug 25, 2021, 9:10 PM
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OlympusHero
17020 posts
OlympusHero
#2 h  Aug 25, 2021, 10:07 PM
Y by
There's probably a super slick solution but we'll just take the obvious route.

Solution
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i3435
1350 posts
i3435
#3 h  Aug 25, 2021, 10:24 PM
Y by
Sol
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zerononnatural
5 posts
zerononnatural
#4 h  Aug 25, 2021, 11:19 PM
Y by
Let $T$ be on $BC$ such that $BD = BT$. With a simple angle chasing we found that $TD = TC$, so we just need to prove that $TC = TD = DA$, but this is simple since $BADT$ is cyclic, then $\angle TAD = \angle ATD = 20º$. And we are done.
This post has been edited 1 time. Last edited by zerononnatural, Aug 25, 2021, 11:19 PM
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