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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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Geometry Problem
Itoz   3
N 19 minutes ago by Itoz
Source: Own
Given $\triangle ABC$. Let the perpendicular line from $A$ to $BC$ meets $BC,\odot(ABC)$ at points $S,K$, respectively, and the foot from $B$ to $AC$ is $L$. $\odot (AKL)$ intersects line $AB$ at $T(\neq A)$, $\odot(AST)$ intersects line $AC$ at $M(\neq A)$, and lines $TM,CK$ intersect at $N$.

Prove that $\odot(CNM)$ is tangent to $\odot (BST)$.
3 replies
Itoz
Apr 18, 2025
Itoz
19 minutes ago
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Nasty Floor Sum with Omega Function
Kezer   9
N 32 minutes ago by lpieleanu
Source: Bulgaria 1989, Evan Chen's Summation Handout
Let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicity. Evaluate \[ \sum_{n=1}^{1989} (-1)^{\Omega(n)}\left\lfloor \frac{1989}{n} \right \rfloor. \]
9 replies
Kezer
Jul 15, 2017
lpieleanu
32 minutes ago
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combinatorial geo question
SAAAAAAA_B   2
N an hour ago by R8kt
Kuba has two finite families $\mathcal{A}, \mathcal{B}$ of convex polygons (in the plane). It turns out that every point of the plane lies in the same number of elements of $\mathcal{A}$ as elements of $\mathcal{B}$. Prove that $|\mathcal{A}| = |\mathcal{B}|$.

\textit{Note:} We treat segments and points as degenerate convex polygons, and they can be elements of $\mathcal{A}$ or $\mathcal{B}$.
2 replies
SAAAAAAA_B
Apr 14, 2025
R8kt
an hour ago
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Two very hard parallel
jayme   6
N an hour ago by jayme
Source: own inspired by EGMO
Dear Mathlinkers,

1. ABC a triangle
2. D, E two point on the segment BC so that BD = DE= EC
3. M, N the midpoint of ED, AE
4. H the orthocenter of the acutangle triangle ADE
5. 1, 2 the circumcircle of the triangle DHM, EHN
6. P, Q the second point of intersection of 1 and BM, 2 and CN
7. U, V the second points of intersection of 2 and MN, PQ.

Prove : UV is parallel to PM.

Sincerely
Jean-Louis
6 replies
jayme
Yesterday at 12:46 PM
jayme
an hour ago
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Integer representation
RL_parkgong_0106   0
an hour ago
Source: Own
Show that for any positive integer $n$, there exists some positive integer $k$ that makes the following equation have no integer root $(x_1, x_2, x_3, \dots, x_n)$.

$$x_1^{2^1}+x_2^{2^2}+x_3^{2^3}+\dots+x_n^{2^n}=k$$
0 replies
RL_parkgong_0106
an hour ago
0 replies
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A cyclic inequality
KhuongTrang   1
N an hour ago by jokehim
Source: own-CRUX
IMAGE
https://cms.math.ca/.../uploads/2025/04/Wholeissue_51_4.pdf
1 reply
KhuongTrang
Yesterday at 4:18 PM
jokehim
an hour ago
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Diophantine equation !
ComplexPhi   6
N 2 hours ago by Mr.Sharkman
Source: Romania JBMO TST 2015 Day 1 Problem 4
Solve in nonnegative integers the following equation :
$$21^x+4^y=z^2$$
6 replies
ComplexPhi
May 14, 2015
Mr.Sharkman
2 hours ago
J
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Cool sequence problem
AlephG_64   1
N 2 hours ago by FarrukhKhayitboyev
Source: 2025 Finals Portuguese Mathematical Olympiad P3
A computer science teacher has asked his students to write a program that, given a list of $n$ numbers $a_1, a_2, ..., a_n$, calculates the list $b_1, b_2, ..., b_n$ where $b_k$ is the number of times the number $a_k$ appears in the list. So, for example, for the list $1,2,3,1$, the program returns the list $2,1,1,2$.

Next, the teacher asked Alexandre to run the program for a list of $2025$ numbers. Then he asked him to apply the program to the resulting list, and so on, until a number greater than or equal to $k$ appears in the list. Find the largest value of $k$ for which, whatever the initial list of $2025$ positive integers $a_1, a_2, ..., a_{2025}$, it is possible for Alexander to do what the teacher asked him to do.
1 reply
AlephG_64
Apr 5, 2025
FarrukhKhayitboyev
2 hours ago
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Inequality
hlminh   1
N 2 hours ago by arqady
Let $a,b,c>0$ such that $a^2+b^2+c^2=3.$ Prove that $\sum \frac a{\sqrt{b^2+b+c}}\leq \sqrt 3.$
1 reply
hlminh
Yesterday at 9:36 AM
arqady
2 hours ago
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IMO Shortlist 2014 N6
hajimbrak   27
N 2 hours ago by cj13609517288
Let $a_1 < a_2 < \cdots <a_n$ be pairwise coprime positive integers with $a_1$ being prime and $a_1 \ge n + 2$. On the segment $I = [0, a_1 a_2 \cdots a_n ]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_1 , \ldots , a_n$ . These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_1$.

Proposed by Serbia
27 replies
hajimbrak
Jul 11, 2015
cj13609517288
2 hours ago
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FE inequality from Iran
mojyla222   3
N 2 hours ago by amir_ali
Source: Iran 2025 second round P5
Find all functions $f:\mathbb{R}^+ \to \mathbb{R}$ such that for all $x,y,z>0$
$$ 3(x^3+y^3+z^3)\geq f(x+y+z)\cdot f(xy+yz+xz) \geq (x+y+z)(xy+yz+xz). $$
3 replies
mojyla222
Apr 19, 2025
amir_ali
2 hours ago
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binomial sum ratio
thewayofthe_dragon   2
N 2 hours ago by P162008
Source: YT
Someone please evaluate this ratio inside the log for any given n(I feel the sum doesn't have any nice closed form).
2 replies
thewayofthe_dragon
Jun 16, 2024
P162008
2 hours ago
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Maximum Area of a triangle formed by 3 Lines
Kunihiko_Chikaya   1
N 2 hours ago by Mathzeus1024
Let $a>1.$ In the $xy-$ plane with the origin $O$, the line $y=2-ax$ intersects the lines $y=x$, and $y=ax$ at the points $A,\ B$, respectively. Find the maximum value of the area of $\triangle{OAB}.$
1 reply
Kunihiko_Chikaya
Sep 28, 2020
Mathzeus1024
2 hours ago
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Ring out the Old Year and ring in the New.
Kunihiko_Chikaya   1
N 3 hours ago by Mathzeus1024
Let $a,\ b,\ c$ be positive real numbers.

Prove that

$$\sqrt[3]{\left(\frac{a^{2022}-a}{b}+\frac{2021}{a^{\frac{a}{b}}}+1\right)\left(\frac{b^{2022}-b}{c}+\frac{2021}{b^{\frac{b}{c}}}+1\right)\left(\frac{c^{2022}-c}{a}+\frac{2021}{c^{\frac{c}{a}}}+1\right)}$$
$$\geq 2022.$$
Proposed by Kunihiko Chikaya/December 31, 2021
1 reply
Kunihiko_Chikaya
Dec 31, 2021
Mathzeus1024
3 hours ago
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Circle geometry
nAalniaOMliO   2
N Apr 7, 2025 by Captainscrubz
Source: Belarusian MO 2021
A convex quadrilateral $ABCD$ is given. $\omega_1$ is a circle with diameter $BC$, $\omega_2$ is a circle with diameter $AD$. $AC$ meets $\omega_1$ and $\omega_2$ for the second time at $B_1$ and $D_1$. $BD$ meets $\omega_1$ and $\omega_2$ for the second time at $C_1$ and $A_1$. $AA_1$ meets $DD_1$ at $X$, $BB_1$ meets $CC_1$ at $Y$. $\omega_1$ intersects $\omega_2$ at $P$ and $Q$. $XY$ meets $PQ$ at $N$.
Prove that $XN=NY$.
2 replies
nAalniaOMliO
Apr 16, 2024
Captainscrubz
Apr 7, 2025
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Circle geometry
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geometry
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Source: Belarusian MO 2021
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nAalniaOMliO
295 posts
nAalniaOMliO
#1 h  Apr 16, 2024, 11:18 AM
Y by
A convex quadrilateral $ABCD$ is given. $\omega_1$ is a circle with diameter $BC$, $\omega_2$ is a circle with diameter $AD$. $AC$ meets $\omega_1$ and $\omega_2$ for the second time at $B_1$ and $D_1$. $BD$ meets $\omega_1$ and $\omega_2$ for the second time at $C_1$ and $A_1$. $AA_1$ meets $DD_1$ at $X$, $BB_1$ meets $CC_1$ at $Y$. $\omega_1$ intersects $\omega_2$ at $P$ and $Q$. $XY$ meets $PQ$ at $N$.
Prove that $XN=NY$.
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RaeT
20 posts
RaeT
#2 h  Apr 7, 2025, 8:33 AM
Y by
is thee any solution for this?
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Captainscrubz
57 posts
Captainscrubz
#3 h  Apr 7, 2025, 10:09 AM
Y by
If suppose $X_1$ is mentioned I used it as $X'$ forgive me :P
Let $H=BB'\cap AA'$ and $G=CC'\cap DD'$
See that $GYHX$ is a parallelogram
Claim 1: $G$ and $H$ lies on line $PQ$
Proof:$GD'\cdot GD=GC\cap GC'\implies $ G has the same power wrt to $\omega_1$ and $\omega_2$
$\therefore G$ lies on $PQ$ same for $H$
$\implies$ that $N$ the midpoint of $GH$ will also lie on $PQ$
$\therefore NX=NY$
Attachments:
This post has been edited 1 time. Last edited by Captainscrubz, Apr 7, 2025, 10:11 AM
Reason: file
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