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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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Easy Number Theory
math_comb01   37
N 10 minutes ago by John_Mgr
Source: INMO 2024/3
Let $p$ be an odd prime and $a,b,c$ be integers so that the integers $$a^{2023}+b^{2023},\quad b^{2024}+c^{2024},\quad a^{2025}+c^{2025}$$are divisible by $p$.
Prove that $p$ divides each of $a,b,c$.
$\quad$
Proposed by Navilarekallu Tejaswi
37 replies
math_comb01
Jan 21, 2024
John_Mgr
10 minutes ago
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ALGEBRA INEQUALITY
Tony_stark0094   3
N 11 minutes ago by sqing
$a,b,c > 0$ Prove that $$\frac{a^2+bc}{b+c} + \frac{b^2+ac}{a+c} + \frac {c^2 + ab}{a+b} \geq a+b+c$$
3 replies
Tony_stark0094
5 hours ago
sqing
11 minutes ago
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Inspired by hlminh
sqing   3
N 16 minutes ago by sqing
Source: Own
Let $ a,b,c $ be real numbers such that $ a^2+b^2+c^2=1. $ Prove that $$ |a-kb|+|b-kc|+|c-ka|\leq \sqrt{3k^2+2k+3}$$Where $ k\geq 0 . $
3 replies
+1 w
sqing
Yesterday at 4:43 AM
sqing
16 minutes ago
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A Familiar Point
v4913   51
N 28 minutes ago by xeroxia
Source: EGMO 2023/6
Let $ABC$ be a triangle with circumcircle $\Omega$. Let $S_b$ and $S_c$ respectively denote the midpoints of the arcs $AC$ and $AB$ that do not contain the third vertex. Let $N_a$ denote the midpoint of arc $BAC$ (the arc $BC$ including $A$). Let $I$ be the incenter of $ABC$. Let $\omega_b$ be the circle that is tangent to $AB$ and internally tangent to $\Omega$ at $S_b$, and let $\omega_c$ be the circle that is tangent to $AC$ and internally tangent to $\Omega$ at $S_c$. Show that the line $IN_a$, and the lines through the intersections of $\omega_b$ and $\omega_c$, meet on $\Omega$.
51 replies
v4913
Apr 16, 2023
xeroxia
28 minutes ago
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Apple sharing in Iran
mojyla222   3
N an hour ago by math-helli
Source: Iran 2025 second round p6
Ali is hosting a large party. Together with his $n-1$ friends, $n$ people are seated around a circular table in a fixed order. Ali places $n$ apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction).

Find all values of $n$ such that after some number of steps, the situation reaches a point where each person has exactly one apple.
3 replies
mojyla222
Apr 20, 2025
math-helli
an hour ago
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Iran second round 2025-q1
mohsen   5
N an hour ago by math-helli
Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.
5 replies
mohsen
Apr 19, 2025
math-helli
an hour ago
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Iran Team Selection Test 2016
MRF2017   9
N an hour ago by SimplisticFormulas
Source: TST3,day1,P2
Let $ABC$ be an arbitrary triangle and $O$ is the circumcenter of $\triangle {ABC}$.Points $X,Y$ lie on $AB,AC$,respectively such that the reflection of $BC$ WRT $XY$ is tangent to circumcircle of $\triangle {AXY}$.Prove that the circumcircle of triangle $AXY$ is tangent to circumcircle of triangle $BOC$.
9 replies
MRF2017
Jul 15, 2016
SimplisticFormulas
an hour ago
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Some nice summations
amitwa.exe   30
N an hour ago by P162008
Problem 1: $\Omega=\left(\sum_{0\le i\le j\le k}^{\infty} \frac{1}{3^i\cdot4^j\cdot5^k}\right)\left(\mathop{{\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}}}_{i\neq j\neq k}\frac{1}{3^i\cdot3^j\cdot3^k}\right)=?$
30 replies
amitwa.exe
May 24, 2024
P162008
an hour ago
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Combo problem
soryn   3
N 2 hours ago by soryn
The school A has m1 boys and m2 girls, and ,the school B has n1 boys and n2 girls. Each school is represented by one team formed by p students,boys and girls. If f(k) is the number of cases for which,the twice schools has,togheter k girls, fund f(k) and the valute of k, for which f(k) is maximum.
3 replies
soryn
Yesterday at 6:33 AM
soryn
2 hours ago
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Looking for the smallest ghost
Justpassingby   5
N 3 hours ago by venhancefan777
Source: 2021 Mexico Center Zone Regional Olympiad, problem 1
Let $p$ be an odd prime number. Let $S=a_1,a_2,\dots$ be the sequence defined as follows: $a_1=1,a_2=2,\dots,a_{p-1}=p-1$, and for $n\ge p$, $a_n$ is the smallest integer greater than $a_{n-1}$ such that in $a_1,a_2,\dots,a_n$ there are no arithmetic progressions of length $p$. We say that a positive integer is a ghost if it doesn’t appear in $S$.
What is the smallest ghost that is not a multiple of $p$?

Proposed by Guerrero
5 replies
Justpassingby
Jan 17, 2022
venhancefan777
3 hours ago
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non-symmetric ineq (for girls)
easternlatincup   36
N 3 hours ago by Tony_stark0094
Source: Chinese Girl's MO 2007
For $ a,b,c\geq 0$ with $ a+b+c=1$, prove that

$ \sqrt{a+\frac{(b-c)^2}{4}}+\sqrt{b}+\sqrt{c}\leq \sqrt{3}$
36 replies
1 viewing
easternlatincup
Dec 30, 2007
Tony_stark0094
3 hours ago
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Turbo's en route to visit each cell of the board
Lukaluce   20
N 3 hours ago by Mathgloggers
Source: EGMO 2025 P5
Let $n > 1$ be an integer. In a configuration of an $n \times n$ board, each of the $n^2$ cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate $90^{\circ}$ counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of $n$, the maximum number of good cells over all possible starting configurations.

Proposed by Melek Güngör, Turkey
20 replies
Lukaluce
Apr 14, 2025
Mathgloggers
3 hours ago
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Divisibility on 101 integers
BR1F1SZ   3
N 3 hours ago by ClassyPeach
Source: Argentina Cono Sur TST 2024 P2
There are $101$ positive integers $a_1, a_2, \ldots, a_{101}$ such that for every index $i$, with $1 \leqslant i \leqslant 101$, $a_i+1$ is a multiple of $a_{i+1}$. Determine the greatest possible value of the largest of the $101$ numbers.
3 replies
BR1F1SZ
Aug 9, 2024
ClassyPeach
3 hours ago
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BMO 2021 problem 3
VicKmath7   19
N 3 hours ago by NuMBeRaToRiC
Source: Balkan MO 2021 P3
Let $a, b$ and $c$ be positive integers satisfying the equation $(a, b) + [a, b]=2021^c$. If $|a-b|$ is a prime number, prove that the number $(a+b)^2+4$ is composite.

Proposed by Serbia
19 replies
VicKmath7
Sep 8, 2021
NuMBeRaToRiC
3 hours ago
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line passing though incenter, bisects perimeter iff bisects area
parmenides51   1
N Aug 11, 2018 by BBBBMMMMLLLLE
Source: Mathematical Excalibur P179
Prove that in any triangle, a line passing through the incenter cuts the perimeter of the triangle in half if and only if it cuts the area of the triangle in half .
1 reply
parmenides51
Aug 11, 2018
BBBBMMMMLLLLE
Aug 11, 2018
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line passing though incenter, bisects perimeter iff bisects area
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Reveal topic tags
geometry
incenter
perimeter
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Source: Mathematical Excalibur P179
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parmenides51
30630 posts
parmenides51
#1 h  Aug 11, 2018, 12:54 AM • 1 Y
Y by Adventure10
Prove that in any triangle, a line passing through the incenter cuts the perimeter of the triangle in half if and only if it cuts the area of the triangle in half .
Z K Y
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BBBBMMMMLLLLE
349 posts
BBBBMMMMLLLLE
#2 h  Aug 11, 2018, 4:40 AM • 2 Y
Y by Adventure10, Mango247
Without loss of generality, let the line intersect $AB$ at $P$ and $AC$ at $Q$.
Now the perimeter is divided in half iff $AP+AQ=s$. Then note that $[APQ] = [API]+[AQI] = r \left(\frac{AP}{2} + \frac{AQ}{2}\right) = rs/2 = [ABC]/2$, as desired. All our steps were reversible, so we have proven the converse also.
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