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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.

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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
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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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(2^{4n+2}+1)/65 (i) integer, (ii) prime
parmenides51   5
N a minute ago by MR.1
Source: JBMO 2016 Shortlist N3
Find all positive integers $n$ such that the number $A_n =\frac{ 2^{4n+2}+1}{65}$ is
a) an integer,
b) a prime.
5 replies
parmenides51
Oct 14, 2017
MR.1
a minute ago
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Divisibility..
Sadigly   0
11 minutes ago
Source: Azerbaijan Junior MO 2025 P2
Find all $4$ consecutive even numbers, such that the square of their product is divisible by the sum of their squares.
0 replies
Sadigly
11 minutes ago
0 replies
J
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This question just asks if you can factorise 12 factorial or not
Sadigly   0
15 minutes ago
Source: Azerbaijan Junior MO 2025 P1
A teacher creates a fraction using numbers from $1$ to $12$ (including $12$). He writes some of the numbers on the numerator, and writes $\times$ (product) between each number. Then he writes the rest of the numbers in the denominator and also writes $\times$ between each number. There is at least one number both in numerator and denominator. The teacher ensures that the fraction is equal to the smallest possible integer possible.

What is this positive integer, which is also the value of the fraction?
0 replies
Sadigly
15 minutes ago
0 replies
J
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harmonic quadrilateral
Lukariman   1
N 16 minutes ago by Lukariman
Given quadrilateral ABCD inscribed in a circle with center O. CA:CB= DA:DB are satisfied. M is any point and d is a line parallel to MC. Radial projection M transforms A,B,D onto line d into A',B',D'. Prove that B' is the midpoint of A'D'.
1 reply
Lukariman
an hour ago
Lukariman
16 minutes ago
J
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Factorising and prime numbers...
Sadigly   4
N 23 minutes ago by Nuran2010
Source: Azerbaijan Senior MO 2025 P4
Prove that for any $p>2$ prime number, there exists only one positive number $n$ that makes the equation $n^2-np$ a perfect square of a positive integer
4 replies
Sadigly
Yesterday at 4:19 PM
Nuran2010
23 minutes ago
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find positive n so that exists prime p with p^n-(p-1)^n$ a power of 3
parmenides51   15
N 24 minutes ago by MR.1
Source: JBMO Shortlist 2017 NT5
Find all positive integers $n$ such that there exists a prime number $p$, such that $p^n-(p-1)^n$ is a power of $3$.

Note. A power of $3$ is a number of the form $3^a$ where $a$ is a positive integer.
15 replies
parmenides51
Jul 25, 2018
MR.1
24 minutes ago
J
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Cute Inequality
EthanWYX2009   1
N 25 minutes ago by lbh_qys
Let $a_1,\ldots ,a_n\in\mathbb R\backslash\{0\},$ determine the minimum and maximum value of
\[\frac{\sum_{i,j=1}^n|a_i+a_j|}{\sum_{i=1}^n|a_i|}.\]
1 reply
EthanWYX2009
6 hours ago
lbh_qys
25 minutes ago
J
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butterfly theorem in finnish high school math competition
parmenides51   4
N an hour ago by arzhang2001
Source: Finland 2018, p3
The chords $AB$ and $CD$ of a circle intersect at $M$, which is the midpoint of the chord $PQ$. The points $X$ and $Y$ are the intersections of the segments $AD$ and $PQ$, respectively, and $BC$ and $PQ$, respectively. Show that $M$ is the midpoint of $XY$.
4 replies
parmenides51
Sep 8, 2019
arzhang2001
an hour ago
J
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2^x + 3^y a perfect square, find positive integers x,y
parmenides51   12
N an hour ago by MR.1
Source: JBMO Shortlist 2017 NT3
Find all pairs of positive integers $(x,y)$ such that $2^x + 3^y$ is a perfect square.
12 replies
parmenides51
Jul 25, 2018
MR.1
an hour ago
J
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Geo metry
TUAN2k8   4
N an hour ago by TUAN2k8
Help me plss!
Given an acute triangle $ABC$. Points $D$ and $E$ lie on segments $AB$ and $AC$, respectively. Lines $BD$ and $CE$ intersect at point $F$. The circumcircles of triangles $BDF$ and $CEF$ intersect at a second point $P$. The circumcircles of triangles $ABC$ and $ADE$ intersect at a second point $Q$. Point $K$ lies on segment $AP$ such that $KQ \perp AQ$. Prove that triangles $\triangle BKD$ and $\triangle CKE$ are similar.
4 replies
TUAN2k8
May 6, 2025
TUAN2k8
an hour ago
J
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Functional equation
Nima Ahmadi Pour   99
N an hour ago by youochange
Source: ISl 2005, A2, Iran prepration exam
We denote by $\mathbb{R}^+$ the set of all positive real numbers.

Find all functions $f: \mathbb R^ + \rightarrow\mathbb R^ +$ which have the property:
\[f(x)f(y)=2f(x+yf(x))\]
for all positive real numbers $x$ and $y$.

Proposed by Nikolai Nikolov, Bulgaria
99 replies
Nima Ahmadi Pour
Apr 24, 2006
youochange
an hour ago
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JBMO 2018. Shortlist NT
Steve12345   14
N 2 hours ago by MR.1
Find all ordered pairs of positive integers $(m,n)$ such that :
$125*2^n-3^m=271$
14 replies
Steve12345
Jul 7, 2019
MR.1
2 hours ago
J
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2025 HMIC-5
EthanWYX2009   1
N 2 hours ago by EthanWYX2009
Source: 2025 HMIC-5
Compute the smallest positive integer $k > 45$ for which there exists a sequence $a_1, a_2, a_3, \ldots ,a_{k-1}$ of positive integers satisfying the following conditions:[list]
[*]$a_i = i$ for all integers $1 \le i \le 45;$
[*] $a_{k-i} = i$ for all integers $1 \le i \le 45;$
[*] for any odd integer $1 \le n \le k -45,$ the sequence $a_n, a_{n+1}, \ldots , a_{n+44}$ is a permutation of
$\{1, 2, \ldots , 45\}.$[/list]
Proposed by: Derek Liu
1 reply
EthanWYX2009
Wednesday at 3:16 PM
EthanWYX2009
2 hours ago
J
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JBMO 2018. Shortlist NT
Steve12345   14
N 2 hours ago by MR.1
Prove that there exist infinitely many positive integers $n$ such that $\frac{4^n+2^n+1}{n^2+n+1}$ is a positive integer.
14 replies
Steve12345
Jul 7, 2019
MR.1
2 hours ago
J
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J
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The locus of P with supplementary angles condition
WakeUp   3
N Apr 4, 2025 by Nari_Tom
Source: Baltic Way 2001
Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.
3 replies
WakeUp
Nov 17, 2010
Nari_Tom
Apr 4, 2025
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The locus of P with supplementary angles condition
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Reveal topic tags
geometry
rhombus
parallelogram
circumcircle
conics
hyperbola
cyclic quadrilateral
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Source: Baltic Way 2001
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WakeUp
1347 posts
WakeUp
#1 h  Nov 17, 2010, 7:19 PM • 2 Y
Y by Adventure10, Mango247
Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.
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yetti
2643 posts
yetti
#2 h  Nov 17, 2010, 8:29 PM • 2 Y
Y by Adventure10, Mango247
Let $ABCD$ be just a parallelogram and $\angle APD + \angle CPB = 180^\circ$ $\Longrightarrow$ circumcircles $(O_1), (O_3)$ of $\triangle APD, \triangle CPB$ are congruent. Translating $\triangle APD$ by $\overrightarrow{AB}$ into a $\triangle CP'B$ creates a cyclic quadrilateral $PBP'C$ and parallelogram $PABP'$ $\Longrightarrow$ circumcircles $(O_2), (O_3)$ of $\triangle BPA, \triangle CPB$ are also congruent. It follows that $\angle PCD = \angle DAP,$ which means that isogonal conjugate $P^*$ of $P$ WRT $\triangle ACD$ is on perpendicular bisector of $AC.$ As a result, the locus of $P$ is a rectangular circum-hyperbola of the parallelogram $ABCD,$ centered at its diagonal intersection $E.$ If $ABCD$ is a rhombus, this hyperbola degenerates to its diagonals $AC, BD.$
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isaacmeng
113 posts
isaacmeng
#3 h  May 5, 2020, 10:45 AM
Y by
What if P is not necessarily in the plane ABCD?
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Nari_Tom
117 posts
Nari_Tom
#4 h  Apr 4, 2025, 10:01 AM
Y by
There is a theorem states that if $P$ is a point lies in the plane of $ABCD$ and $\angle APD+\angle CPD=180^{\circ}$, then there exists a isogonal conjugate of $P$. Since $ABCD$ is a rhombus we can easily conclude that $P$ lies on $AC$ or $BD$.
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