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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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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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Reducibility of 2x^2 cyclotomic
vincentwant   1
N 7 minutes ago by rchokler
Let $S$ denote the set of all positive integers less than $1020$ that are relatively prime to $1020$. Let $\omega=\cos\frac{\pi}{510}+i\sin\frac{\pi}{510}$. Is the polynomial $$\prod_{n\in S}(2x^2-\omega^n)$$reducible over the rational numbers, given that it has integer coefficients?
1 reply
vincentwant
an hour ago
rchokler
7 minutes ago
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Fibonacci...?
Jackson0423   1
N 18 minutes ago by KAME06
The sequence \( F \) is defined by \( F_0 = F_1 = 2025 \) and for all positive integers \( n \geq 2 \), \( F_n = F_{n-1} + F_{n-2} \). Show that for every positive integer \( k \), there exists a suitable positive integer \( j \) such that \( F_j \) is a multiple of \( k \).
1 reply
Jackson0423
44 minutes ago
KAME06
18 minutes ago
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4 concyclic points
buzzychaoz   18
N 43 minutes ago by bjump
Source: Japan Mathematical Olympiad Finals 2015 Q4
Scalene triangle $ABC$ has circumcircle $\Gamma$ and incenter $I$. The incircle of triangle $ABC$ touches side $AB,AC$ at $D,E$ respectively. Circumcircle of triangle $BEI$ intersects $\Gamma$ again at $P$ distinct from $B$, circumcircle of triangle $CDI$ intersects $\Gamma$ again at $Q$ distinct from $C$. Prove that the $4$ points $D,E,P,Q$ are concyclic.
18 replies
buzzychaoz
Apr 1, 2016
bjump
43 minutes ago
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angles in triangle
AndrewTom   33
N an hour ago by zuat.e
Source: BrMO 2012/13 Round 2
The point $P$ lies inside triangle $ABC$ so that $\angle ABP = \angle PCA$. The point $Q$ is such that $PBQC$ is a parallelogram. Prove that $\angle QAB = \angle CAP$.
33 replies
1 viewing
AndrewTom
Feb 1, 2013
zuat.e
an hour ago
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Geometry Proof
Jackson0423   0
an hour ago
In triangle \( \triangle ABC \), point \( P \) on \( AB \) satisfies \( DB = BC \) and \( \angle DCA = 30^\circ \).
Let \( X \) be the point where the perpendicular from \( B \) to line \( DC \) meets the angle bisector of \( \angle BCA \).
Then, the relation \( AD \cdot DC = BD \cdot AX \) holds.

Prove that \( \triangle ABC \) is an isosceles triangle.
0 replies
Jackson0423
an hour ago
0 replies
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a nice prob for number theory
Jackson0423   0
an hour ago
Source: number theory
Let \( n \) be a positive integer, and let its positive divisors be
\[ d_1 < d_2 < \cdots < d_k. \]Define \( f(n) \) to be the number of ordered pairs \( (i, j) \) with \( 1 \le i, j \le k \) such that \( \gcd(d_i, d_j) = 1 \).

Find \( f(3431 \times 2999) \).

Also, find a general formula for \( f(n) \) when
\[ n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}, \]where the \( p_i \) are distinct primes and the \( e_i \) are positive integers.
0 replies
1 viewing
Jackson0423
an hour ago
0 replies
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Very easy NT
GreekIdiot   6
N an hour ago by ektorasmiliotis
Prove that there exists no natural number $n>1$ such that $n \mid 2^n-1$.
6 replies
GreekIdiot
2 hours ago
ektorasmiliotis
an hour ago
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Functionnal equation
Rayanelba   0
an hour ago
Source: Own
Find all functions $f:\mathbb{R}_{>0}\to \mathbb{R}_{>0}$ that verify the following equation for all $x,y\in \mathbb{R}_{>0}$:
$f(x+yf(f(x)))+f(\frac{x}{y})=\frac{x}{y}+f(x+xy)$
0 replies
Rayanelba
an hour ago
0 replies
J
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Great sequence problem
Assassino9931   1
N an hour ago by internationalnick123456
Source: Balkan MO Shortlist 2024 N4
Let $k$ be a positive integer. Determine all sequences $(a_n)_{n\geq 1}$ of positive integers such that
$$ a_{n+2}(a_{n+1} - k) = a_n(a_{n+1} + k) $$for all positive integers $n$.
1 reply
Assassino9931
Apr 27, 2025
internationalnick123456
an hour ago
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INMO 2018 -- Problem #3
integrated_JRC   44
N an hour ago by bjump
Source: INMO 2018
Let $\Gamma_1$ and $\Gamma_2$ be two circles with respective centres $O_1$ and $O_2$ intersecting in two distinct points $A$ and $B$ such that $\angle{O_1AO_2}$ is an obtuse angle. Let the circumcircle of $\Delta{O_1AO_2}$ intersect $\Gamma_1$ and $\Gamma_2$ respectively in points $C (\neq A)$ and $D (\neq A)$. Let the line $CB$ intersect $\Gamma_2$ in $E$ ; let the line $DB$ intersect $\Gamma_1$ in $F$. Prove that, the points $C, D, E, F$ are concyclic.
44 replies
integrated_JRC
Jan 21, 2018
bjump
an hour ago
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Queue geo
vincentwant   0
an hour ago
Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $Y, Z$ be the feet of the altitudes from $B, C$ to $AC, AB$ respectively. Let $D$ be the midpoint of $BC$. Let $\omega_1$ be the circle with diameter $AD$. Let $Q\neq A$ be the intersection of $(ABC)$ and $\omega$. Let $H$ be the orthocenter of $ABC$. Let $K$ be the intersection of $AQ$ and $BC$. Let $l_1,l_2$ be the lines through $Q$ tangent to $\omega,(AYZ)$ respectively. Let $I$ be the intersection of $l_1$ and $KH$. Let $P$ be the intersection of $l_2$ and $YZ$. Let $l$ be the line through $I$ parallel to $HD$ and let $O'$ be the reflection of $O$ across $l$. Prove that $O'P$ is tangent to $(KPQ)$.
0 replies
vincentwant
an hour ago
0 replies
J
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Linear colorings mod 2^n
vincentwant   0
an hour ago
Let $n$ be a positive integer. The ordered pairs $(x,y)$ where $x,y$ are integers in $[0,2^n)$ are each labeled with a positive integer less than or equal to $2^n$ such that every label is used exactly $2^n$ times and there exist integers $a_1,a_2,\dots,a_{2^n}$ and $b_1,b_2,\dots,b_{2^n}$ such that the following property holds: For any two lattice points $(x_1,y_1)$ and $(x_2,y_2)$ that are both labeled $t$, there exists an integer $k$ such that $x_2-x_1-ka_t$ and $y_2-y_1-kb_t$ are both divisible by $2^n$. How many such labelings exist?
0 replies
vincentwant
an hour ago
0 replies
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sqrt(n) or n+p (Generalized 2017 IMO/1)
vincentwant   0
an hour ago
Let $p$ be an odd prime. Define $f(n)$ over the positive integers as follows:
$$f(n)=\begin{cases} \sqrt{n}&\text{ if n is a perfect square} \\ n+p&\text{ otherwise} \end{cases}$$
Let $p$ be chosen such that there exists an ordered pair of positive integers $(n,k)$ where $n>1,p\nmid n$ such that $f^k(n)=n$. Prove that there exists at least three integers $i$ such that $1\leq i\leq k$ and $f^i(n)$ is a perfect square.
0 replies
vincentwant
an hour ago
0 replies
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thanks u!
Ruji2018252   0
an hour ago
Can you guys tell me if there is any link to look up articles on aops?
0 replies
Ruji2018252
an hour ago
0 replies
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R> = 2ρ, where R , ρ are circumradius , inradius resp.
parmenides51   1
N Jan 16, 2023 by Arslan
Source: Spanish Mathematical Olympiad 1993 P3
Prove that in every triangle the diameter of the incircle is not greater than the radius of the circumcircle.
1 reply
parmenides51
Aug 1, 2018
Arslan
Jan 16, 2023
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R> = 2ρ, where R , ρ are circumradius , inradius resp.
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Reveal topic tags
geometry
inradius
circumradius
geometric inequality
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G H BBookmark kLocked kLocked NReply
Source: Spanish Mathematical Olympiad 1993 P3
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parmenides51
30650 posts
parmenides51
#1 h  Aug 1, 2018, 3:31 PM • 2 Y
Y by Adventure10, Mango247
Prove that in every triangle the diameter of the incircle is not greater than the radius of the circumcircle.
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Arslan
269 posts
Arslan
#2 h  Jan 16, 2023, 6:55 PM
Y by
It is a famous Euler's theorem: Given the triangle $ABC$, where $O$ is the circumcenter, $I$ the incenter, $R$ the circumradius and $r$ the inradius, then $$OI=\sqrt{R^2-2Rr}.$$Since the distance between two points is non-negative: $$0 \le OI \implies 0 \le \sqrt{R^2-2rR}=\sqrt{R(R-2r)} \implies 0 \le R-2r \implies 2r \le R.$$Equality occurs iff $O=I$, i.e., $ABC$ is an equilateral triangle.
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