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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
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
J
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all functions satisfying f(x+yf(x))+y = xy + f(x+y)
falantrng   29
N a minute ago by Uzb_Math2010
Source: Balkan MO 2025 P3
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[f(x+yf(x))+y = xy + f(x+y).\]
Proposed by Giannis Galamatis, Greece
29 replies
+1 w
falantrng
Apr 27, 2025
Uzb_Math2010
a minute ago
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2020 EGMO P4: n times the previous number of fresh permutations
alifenix-   22
N a minute ago by math-olympiad-clown
Source: 2020 EGMO P4
A permutation of the integers $1, 2, \ldots, m$ is called fresh if there exists no positive integer $k < m$ such that the first $k$ numbers in the permutation are $1, 2, \ldots, k$ in some order. Let $f_m$ be the number of fresh permutations of the integers $1, 2, \ldots, m$.

Prove that $f_n \ge n \cdot f_{n - 1}$ for all $n \ge 3$.

For example, if $m = 4$, then the permutation $(3, 1, 4, 2)$ is fresh, whereas the permutation $(2, 3, 1, 4)$ is not.
22 replies
alifenix-
Apr 18, 2020
math-olympiad-clown
a minute ago
J
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Factorial Equation
Alidq   1
N 4 minutes ago by CHESSR1DER
Solve in $\mathbb{N}$ $$\frac{x!}{(x-y)!} = 10x+2y-29$$
1 reply
Alidq
an hour ago
CHESSR1DER
4 minutes ago
J
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amazing balkan combi
egxa   6
N 17 minutes ago by DeathIsAwe
Source: BMO 2025 P4
There are $n$ cities in a country, where $n \geq 100$ is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities $A$ and $B$ we define:

$(i)$ A $\emph{path}$ between $A$ and $B$ as a sequence of distinct cities $A = C_0, C_1, \dots, C_k, C_{k+1} = B$, $k \geq 0$, such that there are direct flights between $C_i$ and $C_{i+1}$ for every $0 \leq i \leq k$;
$(ii)$ A $\emph{long path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has more cities;
$(iii)$ A $\emph{short path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has fewer cities.
Assume that for any pair of cities $A$ and $B$ in the country, there exist a long path and a short path between them that have no cities in common (except $A$ and $B$). Let $F$ be the total number of pairs of cities in the country that are connected by direct flights. In terms of $n$, find all possible values $F$

Proposed by David-Andrei Anghel, Romania.
6 replies
egxa
Apr 27, 2025
DeathIsAwe
17 minutes ago
J
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Arbitrary point on BC and its relation with orthocenter
falantrng   23
N 35 minutes ago by EVKV
Source: Balkan MO 2025 P2
In an acute-angled triangle \(ABC\), \(H\) be the orthocenter of it and \(D\) be any point on the side \(BC\). The points \(E, F\) are on the segments \(AB, AC\), respectively, such that the points \(A, B, D, F\) and \(A, C, D, E\) are cyclic. The segments \(BF\) and \(CE\) intersect at \(P.\) \(L\) is a point on \(HA\) such that \(LC\) is tangent to the circumcircle of triangle \(PBC\) at \(C.\) \(BH\) and \(CP\) intersect at \(X\). Prove that the points \(D, X, \) and \(L\) lie on the same line.

Proposed by Theoklitos Parayiou, Cyprus
23 replies
falantrng
Apr 27, 2025
EVKV
35 minutes ago
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Show that f is bijective on T
Tsukuyomi   34
N 40 minutes ago by lelouchvigeo
Source: IMO Shortlist 2017 A3
Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.
34 replies
Tsukuyomi
Jul 10, 2018
lelouchvigeo
40 minutes ago
J
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Isogonal lines at the intersection of two circles
anantmudgal09   12
N 44 minutes ago by Tony_stark0094
Source: RMO Delhi 2016, P3
Two circles $C_1$ and $C_2$ intersect each other at points $A$ and $B$. Their external common tangent (closer to $B$) touches $C_1$ at $P$ and $C_2$ at $Q$. Let $C$ be the reflection of $B$ in line $PQ$. Prove that $\angle CAP=\angle BAQ$.
12 replies
anantmudgal09
Oct 11, 2016
Tony_stark0094
44 minutes ago
J
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Diophantine equation !
ComplexPhi   10
N an hour ago by Namisgood
Determine all triples $(m , n , p)$ satisfying :
\[n^{2p}=m^2+n^2+p+1\]
where $m$ and $n$ are integers and $p$ is a prime number.
10 replies
ComplexPhi
Feb 4, 2015
Namisgood
an hour ago
J
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Triangular function and quadrilateral
Kunihiko_Chikaya   1
N an hour ago by Mathzeus1024
Source: National Defence Medical college Entrance exam November 2005
Given a convex quadrilateral such that $AB=AD=\sqrt{5},\ CB=CD=2, \tan \angle{DAB=-2,\ \angle{BCD}=2\alpha\ (0<2\alpha <\pi).}$ Find all the integers $n$ satisfying $\cos n\alpha=\cos \alpha.$
1 reply
Kunihiko_Chikaya
Jan 17, 2006
Mathzeus1024
an hour ago
J
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IMO Shortlist 2011, G5
WakeUp   71
N an hour ago by InterLoop
Source: IMO Shortlist 2011, G5
Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. Let $D$ and $E$ be the second intersection points of $\omega$ with $AI$ and $BI$, respectively. The chord $DE$ meets $AC$ at a point $F$, and $BC$ at a point $G$. Let $P$ be the intersection point of the line through $F$ parallel to $AD$ and the line through $G$ parallel to $BE$. Suppose that the tangents to $\omega$ at $A$ and $B$ meet at a point $K$. Prove that the three lines $AE,BD$ and $KP$ are either parallel or concurrent.

Proposed by Irena Majcen and Kris Stopar, Slovenia
71 replies
WakeUp
Jul 13, 2012
InterLoop
an hour ago
J
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Inspired by old results
sqing   2
N an hour ago by sqing
Source: Own
Let $ a,b>0 , a^2+b^2+ab+a+b=5 . $ Prove that
$$ \frac{ 1 }{a+b+ab+1}+\frac{6}{a^2+b^2+ab+1}\geq \frac{7}{4}$$$$ \frac{ 1 }{a+b+ab+1}+\frac{1}{a^2+b^2+ab+1}\geq \frac{1}{2}$$$$ \frac{41}{a+b+2}+\frac{ab}{a^3+b^3+2} \geq \frac{21}{2}$$
2 replies
sqing
2 hours ago
sqing
an hour ago
J
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If $b^n|a^n-1$ then $a^b >\frac {3^n}{n}$ (China TST 2009)
Fang-jh   15
N an hour ago by ihategeo_1969
Source: Chinese TST 2009 6th P1
Let $ a > b > 1, b$ is an odd number, let $ n$ be a positive integer. If $ b^n|a^n-1,$ then $ a^b > \frac {3^n}{n}.$
15 replies
Fang-jh
Apr 4, 2009
ihategeo_1969
an hour ago
J
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inequality (another entrance exam)
nai0610   1
N an hour ago by sqing
Given positive real numbers $a,b,c$ satisfying
$(a+2)b^2+(b+2)c^2+(c+2)a^2\geq 8+abc$
prove that $2(ab+bc+ca)\leq a^2(a+b)+b^2(b+c)+c^2(c+a)$
1 reply
nai0610
Jun 2, 2024
sqing
an hour ago
J
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Inspired by old results
sqing   6
N 2 hours ago by sqing
Source: Own
Let $ a,b,c>0 $ and $ a+b+c=3. $ Prove that
$$ \frac{2}{a}+\frac {2}{ab}+\frac{1}{abc}\geq 4$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{2}{abc}\geq 2+\sqrt 3$$$$ \frac{3}{a}+\frac {3}{ab}+\frac{1}{abc}\geq\frac {7+\sqrt {13}}{2}$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{3}{abc}\geq\frac {5+\sqrt {21}}{2}$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{4}{abc}\geq 3+2\sqrt 2$$
6 replies
sqing
Apr 26, 2025
sqing
2 hours ago
J
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J
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2010 PUMaC Geometry A8: covering light source by balls
azjps   9
N Sep 7, 2011 by AIME15
There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball?
9 replies
azjps
Aug 21, 2011
AIME15
Sep 7, 2011
J
2010 PUMaC Geometry A8: covering light source by balls
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Reveal topic tags
geometry
3D geometry
sphere
tetrahedron
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azjps
1405 posts
azjps
#1 h  Aug 21, 2011, 9:45 PM • 1 Y
Y by Adventure10
There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball?
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g.c.boxd
181 posts
g.c.boxd
#2 h  Aug 22, 2011, 9:15 PM • 3 Y
Y by Adventure10, Mango247, and 1 other user
probably wrong
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BOGTRO
5818 posts
BOGTRO
#3 h  Aug 22, 2011, 9:19 PM • 2 Y
Y by Adventure10, Mango247
g.c.boxd wrote:
probably wrong

That's what I did during the actual test :P

Presumably that's not the question's intention though.
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superpi83
1416 posts
superpi83
#4 h  Aug 24, 2011, 8:00 AM • 3 Y
Y by Adventure10, Mango247, and 1 other user
upper bound
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1=2
4594 posts
1=2
#5 h  Aug 24, 2011, 10:45 AM • 3 Y
Y by superpi83, Adventure10, Mango247
hint
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SCP
1502 posts
SCP
#6 h  Aug 24, 2011, 11:16 AM • 2 Y
Y by Adventure10, Mango247
But wasn't the question unclear and do they now just mean we have to put balls out the light (so no one) and the minimum we need?

I think it can't be $4$ if the solid balls can be "put together",
because otherwise with the $5$ there can be yet an other ray?
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xjava
182 posts
xjava
#7 h  Aug 24, 2011, 2:09 PM • 4 Y
Y by superpi83, Adventure10, Mango247, and 1 other user
Think of an arrangement of spheres in a tetrahedron.
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superpi83
1416 posts
superpi83
#8 h  Aug 24, 2011, 3:26 PM • 3 Y
Y by Adventure10, Mango247, and 1 other user
Ah! I think I got it! Thanks.

Solution
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1=2
4594 posts
1=2
#9 h  Sep 7, 2011, 9:59 PM • 1 Y
Y by Adventure10
I'd like to know if we can generalize this problem to n dimensions. Does anybody have any ideas for this?
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AIME15
7892 posts
AIME15
#10 h  Sep 7, 2011, 11:00 PM • 2 Y
Y by Adventure10, Mango247
I think the answer for $n$ dimensions is $n+1$. Take the generalization of a tetrahedron into multiple dimensions - here is the Wikipedia link for the 4-dimensional analog of tetrahedra.

(this is just a guess)
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