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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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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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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
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AZE JBMO TST
IstekOlympiadTeam   4
N 6 minutes ago by Namisgood
Source: AZE JBMO TST
Find all non-negative solutions to the equation $2013^x+2014^y=2015^z$
4 replies
1 viewing
IstekOlympiadTeam
May 2, 2015
Namisgood
6 minutes ago
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Mount Inequality erupts on a sequence :o
GrantStar   89
N 7 minutes ago by sangsidhya
Source: 2023 IMO P4
Let $x_1,x_2,\dots,x_{2023}$ be pairwise different positive real numbers such that
\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]is an integer for every $n=1,2,\dots,2023.$ Prove that $a_{2023} \geqslant 3034.$
89 replies
GrantStar
Jul 9, 2023
sangsidhya
7 minutes ago
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Showing that is not a square
Kyj9981   2
N 11 minutes ago by DTforever
Find all $n$ such that $(2^{n}-1)(5^{n}-1)$ is a perfect square.
2 replies
Kyj9981
19 minutes ago
DTforever
11 minutes ago
J
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2013 China girls' Mathematical Olympiad problem 7
s372102   5
N 11 minutes ago by Nari_Tom
As shown in the figure, $\odot O_1$ and $\odot O_2$ touches each other externally at a point $T$, quadrilateral $ABCD$ is inscribed in $\odot O_1$, and the lines $DA$, $CB$ are tangent to $\odot O_2$ at points $E$ and $F$ respectively. Line $BN$ bisects $\angle ABF$ and meets segment $EF$ at $N$. Line $FT$ meets the arc $\widehat{AT}$ (not passing through the point $B$) at another point $M$ different from $A$. Prove that $M$ is the circumcenter of $\triangle BCN$.
5 replies
s372102
Aug 13, 2013
Nari_Tom
11 minutes ago
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Max and min of Sum of d_k^2
Kunihiko_Chikaya   1
N 19 minutes ago by Mathzeus1024
Source: 2012 Yokohama National University entrance exam/Economic, #1
Given $n$ points $P_k(x_k,\ y_k)\ (k=1,\ 2,\ 3,\ \cdots,\ n)$ on the $xy$-plane.
Let $a=\sum_{k=1}^n x_k^2,\ b=\sum_{k=1}^n y_k^2,\ c=\sum_{k=1}^{n} x_ky_k$. Denote by $d_k$ the distance between $P_k$ and the line $l : x\cos \theta +y\sin \theta =0$. Let $L=\sum_{k=1}^n d_k^2$.

Answer the following questions:

(1) Express $L$ in terms of $a,\ b,\ c,\ \theta$.

(2) When $\theta$ moves in the range of $0\leq \theta <\pi$, express the maximum and minimum value of $L$ in terms of $a,\ b,\ c$.
1 reply
Kunihiko_Chikaya
Feb 27, 2012
Mathzeus1024
19 minutes ago
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1:1 correspondance + Graph Theory
jaydenkaka   1
N 22 minutes ago by jaydenkaka
Source: Own
Lets define Graph G as a graph with "n" vortexes and no edges. Define C(G) as a number of cycles that starts from a point, visit all points exactly once, and comes back to the point that they started (The paths made can't cross each other). Define R(G) as a number of routes that starts from a point, visit all points exactly once, and finishes at another point. (The paths made cannot cross each other.) Show that R(G)≥n*C(G). Also, show the reason why is it inequality instead of an equation, and show the equrilibrum conditions.

(See attachment for example)
1 reply
jaydenkaka
Oct 24, 2024
jaydenkaka
22 minutes ago
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Domain of (a, b) satisfying inequality with fraction
Kunihiko_Chikaya   1
N an hour ago by Mathzeus1024
Source: 2014 Kyoto University entrance exam/Science, Problem 4
For real constants $a,\ b$, define a function $f(x)=\frac{ax+b}{x^2+x+1}.$

Draw the domain of the points $(a,\ b)$ such that the inequality :

\[f(x) \leq f(x)^3-2f(x)^2+2\]

holds for all real numbers $x$.
1 reply
Kunihiko_Chikaya
Feb 26, 2014
Mathzeus1024
an hour ago
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ISL 2023 C2
OronSH   11
N an hour ago by zRevenant
Source: ISL 2023 C2
Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties:
[list=disc]
[*]every term in the sequence is less than or equal to $2^{2023}$, and
[*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\][/list]
11 replies
OronSH
Jul 17, 2024
zRevenant
an hour ago
J
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2024 IMO P1
EthanWYX2009   101
N an hour ago by santhoshn
Source: 2024 IMO P1
Determine all real numbers $\alpha$ such that, for every positive integer $n,$ the integer
$$\lfloor\alpha\rfloor +\lfloor 2\alpha\rfloor +\cdots +\lfloor n\alpha\rfloor$$is a multiple of $n.$ (Note that $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z.$ For example, $\lfloor -\pi\rfloor =-4$ and $\lfloor 2\rfloor= \lfloor 2.9\rfloor =2.$)

Proposed by Santiago Rodríguez, Colombia
101 replies
EthanWYX2009
Jul 16, 2024
santhoshn
an hour ago
J
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Equal angles with midpoint of $AH$
Stuttgarden   2
N 2 hours ago by HormigaCebolla
Source: Spain MO 2025 P4
Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$, satisfying $AB<AC$. The tangent line at $A$ to the circumcicle of $ABC$ intersects $BC$ in $T$. Let $X$ be the midpoint of $AH$. Prove that $\angle ATX=\angle OTB$.
2 replies
Stuttgarden
Mar 31, 2025
HormigaCebolla
2 hours ago
J
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3 var inequalities
sqing   2
N 2 hours ago by sqing
Source: Own
Let $ a,b> 0 $ and $ a+b\leq 2ab . $ Prove that
$$ \frac{ a + b }{ a^2(1+ b^2)} \leq\frac{1 }{\sqrt 2}-\frac{1 }{2}$$$$ \frac{ a +ab+ b }{ a^2(1+ b^2)} \leq \sqrt 2-1$$$$ \frac{ a +a^2b^2+ b }{ a^2(1+ b^2)} \leq\frac{\sqrt5 }{2}$$
2 replies
1 viewing
sqing
Yesterday at 1:13 PM
sqing
2 hours ago
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Flipping L's
MarkBcc168   12
N 2 hours ago by zRevenant
Source: IMO Shortlist 2023 C1
Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A move consists of the following steps.
[list=1]
[*]select a $2\times 2$ square in the grid;
[*]flip the coins in the top-left and bottom-right unit squares;
[*]flip the coin in either the top-right or bottom-left unit square.
[/list]
Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves.

Thanasin Nampaisarn, Thailand
12 replies
MarkBcc168
Jul 17, 2024
zRevenant
2 hours ago
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\frac{1}{5-2a}
Havu   1
N 4 hours ago by Havu
Let $a,b,c \ge \frac{1}{2}$ and $a^2+b^2+c^2=3$. Find minimum:
\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]
1 reply
Havu
Yesterday at 9:56 AM
Havu
4 hours ago
J
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<DPA+ <AQD =< QIP wanted, incircle circumcircle related
parmenides51   41
N 5 hours ago by Ilikeminecraft
Source: IMo 2019 SL G6
Let $I$ be the incentre of acute-angled triangle $ABC$. Let the incircle meet $BC, CA$, and $AB$ at $D, E$, and $F,$ respectively. Let line $EF$ intersect the circumcircle of the triangle at $P$ and $Q$, such that $F$ lies between $E$ and $P$. Prove that $\angle DPA + \angle AQD =\angle QIP$.

(Slovakia)
41 replies
parmenides51
Sep 22, 2020
Ilikeminecraft
5 hours ago
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J
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Dividing Pairs
Jackson0423   1
N Apr 13, 2025 by ND_
Source: Own
Let \( a \) and \( b \) be positive integers.
Suppose that \( a \) is a divisor of \( b^2 + 1 \) and \( b \) is a divisor of \( a^2 + 1 \).
Find all such pairs \( (a, b) \).
1 reply
Jackson0423
Apr 13, 2025
ND_
Apr 13, 2025
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Dividing Pairs
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Reveal topic tags
number theory
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Jackson0423
42 posts
Jackson0423
#1 h  Apr 13, 2025, 8:39 AM
Y by
Let \( a \) and \( b \) be positive integers.
Suppose that \( a \) is a divisor of \( b^2 + 1 \) and \( b \) is a divisor of \( a^2 + 1 \).
Find all such pairs \( (a, b) \).
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ND_
44 posts
ND_
#2 h  Apr 13, 2025, 9:20 AM
Y by
$(F_{2n-1}, F_{2n+1})$, where $F_n$ is the Fibonacci series
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