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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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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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real functional equation
DottedCaculator   20
N 39 minutes ago by megahertz13
Source: OMMC POTM February 2022
Find all functions $f:\mathbb R \to \mathbb R$ (from the set of real numbers to itself) where$$f(x-y)+xf(x-1)+f(y)=x^2$$for all reals $x,y.$

Proposed by cj13609517288
20 replies
DottedCaculator
Nov 2, 2023
megahertz13
39 minutes ago
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Simson lines on OH circle
DVDTSB   3
N an hour ago by Funcshun840
Source: Romania TST 2025 Day 2 P4
Let \( ABC \) and \( DEF \) be two triangles inscribed in the same circle, centered at \( O \), and sharing the same orthocenter \( H \ne O \). The Simson lines of the points \( D, E, F \) with respect to triangle \( ABC \) form a non-degenerate triangle \( \Delta \).
Prove that the orthocenter of \( \Delta \) lies on the circle with diameter \( OH \).

Note. Assume that the points \( A, F, B, D, C, E \) lie in this order on the circle and form a convex, non-degenerate hexagon.

Proposed by Andrei Chiriță
3 replies
DVDTSB
May 13, 2025
Funcshun840
an hour ago
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Twisted easy Number Theory
reni_wee   1
N an hour ago by reni_wee
Source: ONTCP Example 2.1.1.
Let $m$ and $n$ be positive integers posessing the following property: the equation
$$\gcd(11k-1, m) = \gcd(11k-1 , n)$$holds for all positive integers $k$. Prove that $m = 11^rn$ for some integer $r$.
1 reply
1 viewing
reni_wee
an hour ago
reni_wee
an hour ago
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Minimize
lgx57   1
N an hour ago by Math-lover1
Minimize $\sqrt{\cos^2 x+(2-\sin x)^2}+\dfrac{1}{2}\sqrt{(\sqrt 3-\cos x)^2+(\sin x+1)^2}$
1 reply
lgx57
5 hours ago
Math-lover1
an hour ago
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graph thory
o.k.oo   0
an hour ago
There are 10 people at a party. None of the 3 friends of each person are friends with each other. What is the maximum number of friends at this party?
0 replies
o.k.oo
an hour ago
0 replies
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IMO Shortlist 2012, Geometry 8
lyukhson   33
N 2 hours ago by awesomeming327.
Source: IMO Shortlist 2012, Geometry 8
Let $ABC$ be a triangle with circumcircle $\omega$ and $\ell$ a line without common points with $\omega$. Denote by $P$ the foot of the perpendicular from the center of $\omega$ to $\ell$. The side-lines $BC,CA,AB$ intersect $\ell$ at the points $X,Y,Z$ different from $P$. Prove that the circumcircles of the triangles $AXP$, $BYP$ and $CZP$ have a common point different from $P$ or are mutually tangent at $P$.

Proposed by Cosmin Pohoata, Romania
33 replies
lyukhson
Jul 29, 2013
awesomeming327.
2 hours ago
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Consecutive squares are floors
ICE_CNME_4   11
N 2 hours ago by ICE_CNME_4

Determine how many positive integers \( n \) have the property that both
\[ \left\lfloor \sqrt{2n - 1} \right\rfloor \quad \text{and} \quad \left\lfloor \sqrt{3n + 2} \right\rfloor \]are consecutive perfect squares.
11 replies
ICE_CNME_4
Yesterday at 1:50 PM
ICE_CNME_4
2 hours ago
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Finding all possible $n$ on a strange division condition!!
MathLuis   10
N 2 hours ago by justaguy_69
Source: Bolivian Cono Sur Pre-TST 2021 P1
Find the sum of all positive integers $n$ such that
$$\frac{n+11}{\sqrt{n-1}}$$is an integer.
10 replies
MathLuis
Nov 12, 2021
justaguy_69
2 hours ago
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IMO 2012 P5
mathmdmb   123
N 3 hours ago by SimplisticFormulas
Source: IMO 2012 P5
Let $ABC$ be a triangle with $\angle BCA=90^{\circ}$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK=BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL=AC$. Let $M$ be the point of intersection of $AL$ and $BK$.

Show that $MK=ML$.

Proposed by Josef Tkadlec, Czech Republic
123 replies
mathmdmb
Jul 11, 2012
SimplisticFormulas
3 hours ago
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Fixed line
TheUltimate123   14
N 3 hours ago by amirhsz
Source: ELMO Shortlist 2023 G4
Let \(D\) be a point on segment \(PQ\). Let \(\omega\) be a fixed circle passing through \(D\), and let \(A\) be a variable point on \(\omega\). Let \(X\) be the intersection of the tangent to the circumcircle of \(\triangle ADP\) at \(P\) and the tangent to the circumcircle of \(\triangle ADQ\) at \(Q\). Show that as \(A\) varies, \(X\) lies on a fixed line.

Proposed by Elliott Liu and Anthony Wang
14 replies
TheUltimate123
Jun 29, 2023
amirhsz
3 hours ago
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Geometry
MathsII-enjoy   0
3 hours ago
Given triangle $ABC$ inscribed in $(O)$, $M, N$ are respectively the midpoints of the major and minor arcs $BC$. Let $R$ be $A-mix$, $D$ is the intersection point of the line through $A$ parallel to $BC$ with $(O)$. $DI$ intersects $AR$ at $K$, take $L$ on $AK$ so that $LI//BC$. $NL$ intersects $(O)$ at $G$, $AG$ intersects $LI$ at $Z$. Prove that: $ZM$ is perpendicular to $KN$.
0 replies
MathsII-enjoy
3 hours ago
0 replies
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Computing functions
BBNoDollar   7
N 3 hours ago by ICE_CNME_4
Let $f : [0, \infty) \to [0, \infty)$, $f(x) = \dfrac{ax + b}{cx + d}$, with $a, d \in (0, \infty)$, $b, c \in [0, \infty)$. Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
\[ f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0. \](For $n \in \mathbb{N}^*$ and $x \geq 0$, the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )
7 replies
BBNoDollar
May 18, 2025
ICE_CNME_4
3 hours ago
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Combinatorics
P162008   2
N 4 hours ago by ostriches88
$4$ girls and $4$ boys are to be seated in a line. Find the total number of ways such that boys and girls are alternate and a particular boy and girl are never adjacent to each other in any arrangement.
2 replies
P162008
6 hours ago
ostriches88
4 hours ago
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Combinatorics
P162008   1
N 4 hours ago by alexheinis
A cricket team comprising of $11$ players named $A,B,C,\cdots,J,K$ is to be sent for batting. If $A$ wants to bat before $J$ and $J$ wants to bat after $G.$ Then find the total number of batting orders if other players could go in any order.
1 reply
1 viewing
P162008
6 hours ago
alexheinis
4 hours ago
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Theory of Equations
P162008   4
N Apr 25, 2025 by vanstraelen
Let $a,b,c,d$ and $e\in [-2,2]$ such that $\sum_{cyc} a = 0, \sum_{cyc} a^3 = 0, \sum_{cyc} a^5 = 10.$ Find the value of $\sum_{cyc} a^2.$
4 replies
P162008
Apr 23, 2025
vanstraelen
Apr 25, 2025
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Theory of Equations
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P162008
210 posts
P162008
#1 h  Apr 23, 2025, 11:27 AM
Y by
Let $a,b,c,d$ and $e\in [-2,2]$ such that $\sum_{cyc} a = 0, \sum_{cyc} a^3 = 0, \sum_{cyc} a^5 = 10.$ Find the value of $\sum_{cyc} a^2.$
This post has been edited 2 times. Last edited by P162008, Apr 23, 2025, 11:28 AM
Reason: Typo
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vanstraelen
9059 posts
vanstraelen
#2 h  Apr 23, 2025, 7:48 PM • 1 Y
Y by P162008
Polynomial $P(x)=x^{5}-5x^{3}+5x-2$.
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P162008
210 posts
P162008
#3 h  Apr 23, 2025, 10:18 PM
Y by
vanstraelen wrote:
Polynomial $P(x)=x^{5}-5x^{3}+5x-2$.

Yeah correct
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P162008
210 posts
P162008
#4 h  Apr 25, 2025, 7:46 AM
Y by
vanstraelen wrote:
Polynomial $P(x)=x^{5}-5x^{3}+5x-2$.

What was your approach?
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vanstraelen
9059 posts
vanstraelen
#5 h  Apr 25, 2025, 3:11 PM
Y by
$P(x)=x^{5}+p_{4}x^{4}+p_{3}x^{3}+p_{2}x^{2}+p_{1}x+p_{0}$.

$a$ is a root $\Rightarrow a^{5}+p_{4}a^{4}+p_{3}a^{3}+p_{2}a^{2}+p_{1}a+p_{0}=0$, etc.
$\sum_{cyc} a^5+p_{4} \cdot \sum_{cyc} a^4+p_{3} \cdot \sum_{cyc} a^3+p_{2} \cdot \sum_{cyc} a^2+p_{1} \sum_{cyc} a+5p_{0}=0$
or $10+p_{4} \cdot \sum_{cyc} a^4+p_{2} \cdot \sum_{cyc} a^2+5p_{0}=0$.
Let $p_{4}=p_{2}=0$, then $p_{0}=-2$.

New equation: $P(x)=x^{5}+px^{3}+qx-2$.
We want $x=2$ is a root, then $q=-15-4p$ and $P(x)=x^{5}+px^{3}-(15+4p)x-2$.
$P(x)=(x-2)[x^{4}+2x^{3}+(p+4)x^{2}+(2p+8)x+1]$.
We try $P(x)=(x-2)[x^{4}+2x^{3}+(p+4)x^{2}+(2p+8)x+1]=(x-2)(x^{2}+tx-1)^{2}$
and we find $t=1,p=-5$.
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