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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)!

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[*]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
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
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
Number Theory
fasttrust_12-mn   14
N 32 minutes ago by Namisgood
Source: Pan African Mathematics Olympiad P1
Find all positive intgers $a,b$ and $c$ such that $\frac{a+b}{a+c}=\frac{b+c}{b+a}$ and $ab+bc+ca$ is a prime number
14 replies
fasttrust_12-mn
Aug 15, 2024
Namisgood
32 minutes ago
find question
mathematical-forest   5
N an hour ago by Jupiterballs
Are there any contest questions that seem simple but are actually difficult? :-D
5 replies
mathematical-forest
Thursday at 10:19 AM
Jupiterballs
an hour ago
Find the value
sqing   17
N an hour ago by jkim0656
Source: 2024 China Fujian High School Mathematics Competition
Let $f(x)=a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0,$ $a_i\in\{-1,1\} ,i=0,1,2,\cdots,6 $ and $f(2)=-53 .$ Find the value of $f(1).$
17 replies
sqing
Jun 22, 2024
jkim0656
an hour ago
Own made functional equation
Primeniyazidayi   10
N an hour ago by Phat_23000245
Source: own(probably)
Find all functions $f:R \rightarrow R$ such that $xf(x^2+2f(y)-yf(x))=f(x)^3-f(y)(f(x^2)-2f(x))$ for all $x,y \in \mathbb{R}$
10 replies
Primeniyazidayi
May 26, 2025
Phat_23000245
an hour ago
Tough inequality
TUAN2k8   4
N an hour ago by Phat_23000245
Source: Own
Let $n \ge 2$ be an even integer and let $x_1,x_2,...,x_n$ be real numbers satisfying $x_1^2+x_2^2+...+x_n^2=n$.
Prove that
$\sum_{1 \le i < j \le n} \frac{x_ix_j}{x_i^2+x_j^2+1} \ge \frac{-n}{6}$
4 replies
TUAN2k8
May 28, 2025
Phat_23000245
an hour ago
Guess period of function
a1267ab   9
N 2 hours ago by HamstPan38825
Source: USA TST 2025
Let $n$ be a positive integer. Ana and Banana play a game. Banana thinks of a function $f\colon\mathbb{Z}\to\mathbb{Z}$ and a prime number $p$. He tells Ana that $f$ is nonconstant, $p<100$, and $f(x+p)=f(x)$ for all integers $x$. Ana's goal is to determine the value of $p$. She writes down $n$ integers $x_1,\dots,x_n$. After seeing this list, Banana writes down $f(x_1),\dots,f(x_n)$ in order. Ana wins if she can determine the value of $p$ from this information. Find the smallest value of $n$ for which Ana has a winning strategy.

Anthony Wang
9 replies
a1267ab
Dec 14, 2024
HamstPan38825
2 hours ago
Inequality with abc=1
tenplusten   11
N 2 hours ago by sqing
Source: JBMO 2011 Shortlist A7
$\boxed{\text{A7}}$ Let $a,b,c$ be positive reals such that $abc=1$.Prove the inequality $\sum\frac{2a^2+\frac{1}{a}}{b+\frac{1}{a}+1}\geq 3$
11 replies
tenplusten
May 15, 2016
sqing
2 hours ago
Central sequences
EeEeRUT   13
N 2 hours ago by v_Enhance
Source: EGMO 2025 P2
An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ of positive integers is called central if for every positive integer $n$ , the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$.

Show that there exists an infinite sequence $b_1, b_2, b_3, \dots$ of positive integers such that for every central sequence $a_1, a_2, a_3, \dots, $ there are infinitely many positive integers $n$ with $a_n = b_n$.
13 replies
EeEeRUT
Apr 16, 2025
v_Enhance
2 hours ago
Interesting inequality
sqing   0
2 hours ago
Source: Own
Let $ a,b,c\geq  0 , a^2+b^2+c^2 =3.$ Prove that
$$ a^4+ b^4+c^4+6abc\leq9$$$$ a^3+ b^3+  c^3+3( \sqrt{3}-1)abc\leq 3\sqrt 3$$
0 replies
sqing
2 hours ago
0 replies
IMO Shortlist 2014 C7
hajimbrak   19
N 3 hours ago by quantam13
Let $M$ be a set of $n \ge 4$ points in the plane, no three of which are collinear. Initially these points are connected with $n$ segments so that each point in $M$ is the endpoint of exactly two segments. Then, at each step, one may choose two segments $AB$ and $CD$ sharing a common interior point and replace them by the segments $AC$ and $BD$ if none of them is present at this moment. Prove that it is impossible to perform $n^3 /4$ or more such moves.

Proposed by Vladislav Volkov, Russia
19 replies
hajimbrak
Jul 11, 2015
quantam13
3 hours ago
<BAC = 2 <ABC wanted, AC + AI = BC given , incenter I
parmenides51   3
N 3 hours ago by LeYohan
Source: 2020 Dutch IMO TST 1.1
In acute-angled triangle $ABC, I$ is the center of the inscribed circle and holds $| AC | + | AI | = | BC |$. Prove that $\angle BAC = 2 \angle ABC$.
3 replies
parmenides51
Nov 21, 2020
LeYohan
3 hours ago
China South East Mathematical Olympiad 2014 Q3B
sqing   5
N 3 hours ago by MathLuis
Source: China Zhejiang Fuyang , 27 Jul 2014
Let $p$ be a primes ,$x,y,z $ be positive integers such that $x<y<z<p$ and $\{\frac{x^3}{p}\}=\{\frac{y^3}{p}\}=\{\frac{z^3}{p}\}$.
Prove that $(x+y+z)|(x^5+y^5+z^5).$
5 replies
sqing
Aug 17, 2014
MathLuis
3 hours ago
Parallelograms and concyclicity
Lukaluce   32
N 4 hours ago by v_Enhance
Source: EGMO 2025 P4
Let $ABC$ be an acute triangle with incentre $I$ and $AB \neq AC$. Let lines $BI$ and $CI$ intersect the circumcircle of $ABC$ at $P \neq B$ and $Q \neq C$, respectively. Consider points $R$ and $S$ such that $AQRB$ and $ACSP$ are parallelograms (with $AQ \parallel RB, AB \parallel QR, AC \parallel SP$, and $AP \parallel CS$). Let $T$ be the point of intersection of lines $RB$ and $SC$. Prove that points $R, S, T$, and $I$ are concyclic.
32 replies
Lukaluce
Apr 14, 2025
v_Enhance
4 hours ago
Gcd of N and its coprime pair sum
EeEeRUT   18
N 4 hours ago by lksb
Source: EGMO 2025 P1
For a positive integer $N$, let $c_1 < c_2 < \cdots < c_m$ be all positive integers smaller than $N$ that are coprime to $N$. Find all $N \geqslant 3$ such that $$\gcd( N, c_i + c_{i+1}) \neq 1$$for all $1 \leqslant i \leqslant m-1$

Here $\gcd(a, b)$ is the largest positive integer that divides both $a$ and $b$. Integers $a$ and $b$ are coprime if $\gcd(a, b) = 1$.

Proposed by Paulius Aleknavičius, Lithuania
18 replies
EeEeRUT
Apr 16, 2025
lksb
4 hours ago
PAMO Problem 4: Perpendicular lines
DylanN   11
N Apr 23, 2025 by ATM_
Source: 2019 Pan-African Mathematics Olympiad, Problem 4
The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ meet at $D$. The circumcircle of $\triangle BCD$ meets sides $AC$ and $AB$ again at $E$ and $F$ respectively. Let $O$ be the circumcentre of $\triangle ABC$. Show that $AO$ is perpendicular to $EF$.
11 replies
DylanN
Apr 9, 2019
ATM_
Apr 23, 2025
PAMO Problem 4: Perpendicular lines
G H J
G H BBookmark kLocked kLocked NReply
Source: 2019 Pan-African Mathematics Olympiad, Problem 4
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DylanN
194 posts
#1 • 2 Y
Y by Adventure10, Mango247
The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ meet at $D$. The circumcircle of $\triangle BCD$ meets sides $AC$ and $AB$ again at $E$ and $F$ respectively. Let $O$ be the circumcentre of $\triangle ABC$. Show that $AO$ is perpendicular to $EF$.
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pad
1671 posts
#3 • 2 Y
Y by Adventure10, Mango247
We know $AD$ is the $A$-symmedian of $\triangle ABC$, so it is well known that $B,C,D,O$ are concyclic. So $(BCD)=(BCO)$. Then $\angle EAO = \angle CAO=90-B$, and $\angle AEF=180-\angle CEF=180-\angle CBF=B$ since $E,C,F,B$ are concyclic. Therefore, $AO\perp EF$.
Z K Y
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DylanN
194 posts
#4 • 2 Y
Y by Adventure10, Mango247
It's not important that $D$ is defined as the intersection of the tangents. We can replace the circumcircle of $\triangle BCD$ with any circle that passes through $B$ and $C$. Indeed, we know that $\angle CAO = 90^\circ - \angle B$, and since $BCEF$ is cyclic, we have that $\angle B = \angle CEF$, from which the result follows.
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khanhnx
1618 posts
#5 • 1 Y
Y by Adventure10
This problem has been posted in here
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AshAuktober
1013 posts
#6
Y by
Observe that we can obtain $\Delta AEF$ by reflecting $\Delta ABC$ about its $A-$angle bisector and taking a homothety at $A$. Now since the circumcentre and orthocentre are isogonal conjugates, $O$ has to lie on the $A-$altitude of $\Delta AEF$, and therefore we are done. $\square$
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Aiden-1089
302 posts
#7
Y by
Let $H$ be the orthocenter of $ABC$.
$\measuredangle BEC = \measuredangle BDC = \measuredangle BFC \implies BCEF$ is concyclic $\implies \Delta ABC \sim \Delta AEF$.
Consider taking a suitable homothety about A then reflecting across the angle bisector of $\angle BAC$, such that $BC$ goes to $EF$. Under this transformation, line $AH$ goes to line $AO$ because they are isogonal in $\angle BAC$. Since $AH \perp BC$, we have $AO \perp EF$.
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Jishnu4414l
155 posts
#8
Y by
We know that $\measuredangle OAC=90^{\circ}-\measuredangle CBA=\measuredangle KAE$
Also, $\measuredangle CBA=\measuredangle CEF=\measuredangle AEK$
Now, $\measuredangle KAE+\measuredangle AEK+\measuredangle EKA=0$
$\implies 90^{\circ}-\measuredangle CBA+\measuredangle CBA +\measuredangle EKA=0$
$\implies \measuredangle EKA=90^{\circ}$
And we are done!
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fasttrust_12-mn
118 posts
#9
Y by
Claim: $O$ lies on the circumcircle of $\triangle BDC$
proof:
$$\measuredangle BAC =\measuredangle BCD \implies \measuredangle BDC= 180^{\circ}- 2\times \measuredangle BAC$$, we know that
$$\measuredangle BOC = 2\times \measuredangle BAC$$so
$$OBDC  \text{ is cyclic quad }$$
$$\measuredangle FBC=\measuredangle AEF $$$$\measuredangle OAC=\measuredangle OCA$$$$\measuredangle OCB=90^{\circ}$$we know that
$$\measuredangle AEG=90^{\circ}-\measuredangle BAO-\measuredangle OAE + \measuredangle BAO\implies 90^{\circ}-\measuredangle OAE$$$$\measuredangle AGE =180^{\circ}-(90^{\circ}-\measuredangle OAE+\measuredangle  OAE)=90^{\circ}$$hence we are done $\blacksquare$
Attachments:
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Tony_stark0094
69 posts
#10
Y by
$$\angle EBF=\angle BAF+\angle AFB=\angle A + \angle BDC=\angle A + 180 - 2\angle A = 180 - \angle A$$$$\implies FA=FB$$$$by\ symmetry\ we \ can\ get \ that\ EA=EC$$hence $F$ lies on perpendicular bisector of $AB$ and $E$ lies on perpendicular bisector of $AC$ moreover $G$ also lies on these lines
$\implies G$ is the orthocentre of $\Delta AEF$ hence $AG$ is perpendicular to $EF$
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Rayanelba
20 posts
#11 • 1 Y
Y by ATM_
To prove that $AO$ perpendicular to $EF$ , it suffies to prove that: $OAC=\frac{\pi}{2}-\measuredangle CFE$
We know that : $\measuredangle OAC=\frac{\pi}{2}-\measuredangle CBA$
And : $\measuredangle CBA=\pi-\measuredangle CBE=\measuredangle CFE$ (because CFEB is cyclic)
$\implies \measuredangle OAC=\frac{\pi}{2}-\measuredangle CFE$
Q.E.D
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ATM_
23 posts
#12
Y by
nice solution*
This post has been edited 1 time. Last edited by ATM_, Apr 23, 2025, 11:08 PM
Reason: .
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ATM_
23 posts
#13
Y by
here is a cute sketch
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