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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
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Sequence
lgx57   8
N 15 minutes ago by Vivaandax
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.
8 replies
1 viewing
lgx57
Apr 27, 2025
Vivaandax
15 minutes ago
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Inequlities
sqing   28
N 23 minutes ago by DAVROS
Let $ a,b,c\geq 0 $ and $ a^2+ab+bc+ca=3 .$ Prove that$$\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2} \geq \frac{3}{2}$$$$\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2}-bc \geq -\frac{3}{2}$$
28 replies
sqing
Jul 19, 2024
DAVROS
23 minutes ago
J
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Geometric inequality
ReticulatedPython   3
N an hour ago by ItalianZebra
Let $A$ and $B$ be points on a plane such that $AB=n$, where $n$ is a positive integer. Let $S$ be the set of all points $P$ such that $\frac{AP^2+BP^2}{(AP)(BP)}=c$, where $c$ is a real number. The path that $S$ traces is continuous, and the value of $c$ is minimized. Prove that $c$ is rational for all positive integers $n.$
3 replies
ReticulatedPython
Apr 22, 2025
ItalianZebra
an hour ago
J
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Easy Geometry Problem in Taiwan TST
chengbilly   7
N an hour ago by L13832
Source: 2025 Taiwan TST Round 1 Independent Study 2-G
Suppose $I$ and $I_A$ are the incenter and the $A$-excenter of triangle $ABC$, respectively.
Let $M$ be the midpoint of arc $BAC$ on the circumcircle, and $D$ be the foot of the
perpendicular from $I_A$ to $BC$. The line $MI$ intersects the circumcircle again at $T$ . For
any point $X$ on the circumcircle of triangle $ABC$, let $XT$ intersect $BC$ at $Y$ . Prove
that $A, D, X, Y$ are concyclic.
7 replies
chengbilly
Mar 6, 2025
L13832
an hour ago
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Overlapping game
Kei0923   3
N an hour ago by CrazyInMath
Source: 2023 Japan MO Finals 1
On $5\times 5$ squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.
3 replies
Kei0923
Feb 11, 2023
CrazyInMath
an hour ago
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Interesting Function
Kei0923   4
N 2 hours ago by CrazyInMath
Source: 2024 JMO preliminary p8
Function $f:\mathbb{Z}_{\geq 0}\rightarrow\mathbb{Z}$ satisfies
$$f(m+n)^2=f(m|f(n)|)+f(n^2)$$for any non-negative integers $m$ and $n$. Determine the number of possible sets of integers $\{f(0), f(1), \dots, f(2024)\}$.
4 replies
Kei0923
Jan 9, 2024
CrazyInMath
2 hours ago
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Functional Geometry
GreekIdiot   1
N 2 hours ago by ItzsleepyXD
Source: BMO 2024 SL G7
Let $f: \pi \to \mathbb R$ be a function from the Euclidean plane to the real numbers such that $f(A)+f(B)+f(C)=f(O)+f(G)+f(H)$ for any acute triangle $\Delta ABC$ with circumcenter $O$, centroid $G$ and orthocenter $H$. Prove that $f$ is constant.
1 reply
GreekIdiot
Apr 27, 2025
ItzsleepyXD
2 hours ago
J
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hard inequalities
pennypc123456789   1
N 2 hours ago by 1475393141xj
Given $x,y,z$ be the positive real number. Prove that

$\frac{2xy}{\sqrt{2xy(x^2+y^2)}} + \frac{2yz}{\sqrt{2yz(y^2+z^2)}} + \frac{2xz}{\sqrt{2xz(x^2+z^2)}} \le \frac{2(x^2+y^2+z^2) + xy+yz+xz}{x^2+y^2+z^2}$
1 reply
pennypc123456789
5 hours ago
1475393141xj
2 hours ago
J
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Cute R+ fe
Aryan-23   6
N 2 hours ago by jasperE3
Source: IISc Pravega, Enumeration 2023-24 Finals P1
Find all functions $f\colon \mathbb R^+ \mapsto \mathbb R^+$, such that for all positive reals $x,y$, the following is true:

$$xf(1+xf(y))= f\left(f(x) + \frac 1y\right)$$
Kazi Aryan Amin
6 replies
Aryan-23
Jan 27, 2024
jasperE3
2 hours ago
J
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Easy Combinatorial Game Problem in Taiwan TST
chengbilly   8
N 2 hours ago by CrazyInMath
Source: 2025 Taiwan TST Round 1 Independent Study 1-C
Alice and Bob are playing game on an $n \times n$ grid. Alice goes first, and they take turns drawing a black point from the coordinate set
\[\{(i, j) \mid i, j \in \mathbb{N}, 1 \leq i, j \leq n\}\]There is a constraint that the distance between any two black points cannot be an integer. The player who cannot draw a black point loses. Find all integers $n$ such that Alice has a winning strategy.

Proposed by chengbilly
8 replies
1 viewing
chengbilly
Mar 5, 2025
CrazyInMath
2 hours ago
J
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Tiling problem (Combinatorics or Number Theory?)
Rukevwe   4
N 2 hours ago by CrazyInMath
Source: 2022 Nigerian MO Round 3/Problem 3
A unit square is removed from the corner of an $n \times n$ grid, where $n \geq 2$. Prove that the remainder can be covered by copies of the figures of $3$ or $5$ unit squares depicted in the drawing below.
IMAGE

Note: Every square must be covered once and figures must not go over the bounds of the grid.
4 replies
Rukevwe
May 2, 2022
CrazyInMath
2 hours ago
J
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Finding all integers with a divisibility condition
Tintarn   15
N 2 hours ago by CrazyInMath
Source: Germany 2020, Problem 4
Determine all positive integers $n$ for which there exists a positive integer $d$ with the property that $n$ is divisible by $d$ and $n^2+d^2$ is divisible by $d^2n+1$.
15 replies
Tintarn
Jun 22, 2020
CrazyInMath
2 hours ago
J
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Find all functions
WakeUp   21
N 2 hours ago by CrazyInMath
Source: Baltic Way 2010
Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)\]
for all $x,y\in\mathbb{R}$.
21 replies
WakeUp
Nov 19, 2010
CrazyInMath
2 hours ago
J
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Generating Functions
greenplanet2050   5
N 2 hours ago by Shan3t
So im learning generating functions and i dont really understand why $1+2x+3x^2+4x^3+5x^4+…=\dfrac{1}{(1-x)^2}$

can someone help

thank you :)
5 replies
greenplanet2050
Yesterday at 10:42 PM
Shan3t
2 hours ago
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how many quadrilaterals ?
Ecrin_eren   8
N Apr 16, 2025 by mathprodigy2011
"All the diagonals of an 11-gon are drawn. How many quadrilaterals can be formed using these diagonals as sides? (The vertices of the quadrilaterals are selected from the vertices of the 11-gon.)"
8 replies
Ecrin_eren
Apr 13, 2025
mathprodigy2011
Apr 16, 2025
J
how many quadrilaterals ?
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Ecrin_eren
56 posts
Ecrin_eren
#1 h  Apr 13, 2025, 4:01 PM • 1 Y
Y by Kizaruno
"All the diagonals of an 11-gon are drawn. How many quadrilaterals can be formed using these diagonals as sides? (The vertices of the quadrilaterals are selected from the vertices of the 11-gon.)"
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mathprodigy2011
324 posts
mathprodigy2011
#2 h  Apr 14, 2025, 2:54 AM • 2 Y
Y by yodaboss1110, Kizaruno
Ecrin_eren wrote:
"All the diagonals of an 11-gon are drawn. How many quadrilaterals can be formed using these diagonals as sides? (The vertices of the quadrilaterals are selected from the vertices of the 11-gon.)"

wait so basically your asking for the number of quadrilaterals that share all vertices with the 11-gon and all sides are diagonals?
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Ecrin_eren
56 posts
Ecrin_eren
#3 h  Apr 14, 2025, 5:15 AM
Y by
Yes exactly
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Ecrin_eren
56 posts
Ecrin_eren
#4 h  Apr 14, 2025, 2:00 PM • 1 Y
Y by Kizaruno
Any ideas ?
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mathprodigy2011
324 posts
mathprodigy2011
#5 h  Apr 14, 2025, 7:59 PM • 2 Y
Y by Ecrin_eren, Kizaruno
Ecrin_eren wrote:
Any ideas ?
PIE could work nicely. 11 choose 4 total ways. Then count the number of quadrilaterals with at least 1 side shared and count with PIE. I'll write a full solution later since im at school rn. Stars and bars could also be good lol
This post has been edited 1 time. Last edited by mathprodigy2011, Apr 15, 2025, 5:32 PM
Reason: .
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Ecrin_eren
56 posts
Ecrin_eren
#6 h  Apr 15, 2025, 5:18 PM
Y by
@mathprodigy could you please give your solution
This post has been edited 1 time. Last edited by Ecrin_eren, Apr 15, 2025, 5:18 PM
Reason: Mb
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mathprodigy2011
324 posts
mathprodigy2011
#7 h  Apr 15, 2025, 5:31 PM
Y by
Ecrin_eren wrote:
@mathprodigy could you please give your solution

Click to reveal hidden text Don't think im right after doing this :skull:, No latex cuz im still at school
This post has been edited 1 time. Last edited by mathprodigy2011, Apr 15, 2025, 5:32 PM
Reason: school
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alcumus_grinder_bad
1 post
alcumus_grinder_bad
#9 h  Apr 16, 2025, 3:51 PM • 1 Y
Y by Kizaruno
i could be super duper wrong becuase i am not very good at math but isnt this just the number of ways to choose 4 vertices out of a set of 11 such that each vertex isn't adjacent to another? in which case u could do a sort of stars and bars type approach I think.

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mathprodigy2011
324 posts
mathprodigy2011
#10 h  Apr 16, 2025, 5:27 PM • 1 Y
Y by Kizaruno
alcumus_grinder_bad wrote:
i could be super duper wrong becuase i am not very good at math but isnt this just the number of ways to choose 4 vertices out of a set of 11 such that each vertex isn't adjacent to another? in which case u could do a sort of stars and bars type approach I think.

Click to reveal hidden text

well you have to adjust for each vertex so i don't think this is right but im unsure
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