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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
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0 replies
jlacosta
Mar 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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Slightly weird points which are not so weird
Pranav1056   10
N 4 minutes ago by kes0716
Source: India TST 2023 Day 4 P1
Suppose an acute scalene triangle $ABC$ has incentre $I$ and incircle touching $BC$ at $D$. Let $Z$ be the antipode of $A$ in the circumcircle of $ABC$. Point $L$ is chosen on the internal angle bisector of $\angle BZC$ such that $AL = LI$. Let $M$ be the midpoint of arc $BZC$, and let $V$ be the midpoint of $ID$. Prove that $\angle IML = \angle DVM$
10 replies
Pranav1056
Jul 9, 2023
kes0716
4 minutes ago
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classical R+ FE
jasperE3   1
N 9 minutes ago by pco
Source: kent2207, based on 2019 Slovenia TST
wanted to post this problem in its own thread: https://artofproblemsolving.com/community/c6h1784825p34307772
Find all functions $f:\mathbb R^+\to\mathbb R^+$ for which:
$$f(f(x)+f(y))=yf(1+yf(x))$$for all $x,y\in\mathbb R^+$.
1 reply
jasperE3
Yesterday at 3:55 PM
pco
9 minutes ago
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Physics disguised as math
everythingpi3141592   7
N 12 minutes ago by kes0716
Source: India IMOTC 2024 Day 4 Problem 2
There are $n\ge 3$ particles on a circle situated at the vertices of a regular $n$-gon. All these particles move on the circle with the same constant speed. One of the particles moves in the clockwise direction while all others move in the anti-clockwise direction. When particles collide, that is, they are all at the same point, they all reverse the direction of their motion and continue with the same speed as before.

Let $s$ be the smallest number of collisions after which all particles return to their original positions. Find $s$.

Proposed by N.V. Tejaswi
7 replies
everythingpi3141592
May 31, 2024
kes0716
12 minutes ago
J
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D1016 : A strange result about the palindrom polynomials
Dattier   3
N 34 minutes ago by Dattier
Source: les dattes à Dattier
Let $Q\in \{-1,1,0\}[x]$ with $Q(1)=0$.

Is it true that $\exists P \in \mathbb Z[x]$ palindrom with $P | Q$ ?
3 replies
Dattier
Monday at 12:13 PM
Dattier
34 minutes ago
J
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Equation with integer part
soruz   2
N an hour ago by lbh_qys
Determine $ x \in [1, \infty) $ for which $x^2+(2^x-x)^2=2^ {\left\lfloor { 2^x-x}\right\rfloor}$, where $ \left\lfloor {.}\right\rfloor$ denote the integer part.
2 replies
soruz
Feb 26, 2025
lbh_qys
an hour ago
J
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2015 Taiwan TST Round 1 Quiz 1 Problem 1
wanwan4343   10
N an hour ago by PrinceChen
Source: 2015 Taiwan TST Round 1 Quiz 1 Problem 1
Find all primes $p,q,r$ such that $qr-1$ is divisible by $p$, $pr-1$ is divisible by $q$, $pq-1$ is divisible by $r$.
10 replies
wanwan4343
Jul 12, 2015
PrinceChen
an hour ago
J
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Inspired by BaCaPhe
sqing   8
N 2 hours ago by sqing
Source: Own
Let $ a,b,c \ge 0 $ and $ ab + bc + ca \ge 4 + abc. $ Prove that
$$ a^2 + b^2 + c^2 \ge 8$$
8 replies
sqing
Today at 3:39 AM
sqing
2 hours ago
J
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USAMO 2000 Problem 4
MithsApprentice   33
N 2 hours ago by MaxSze
Find the smallest positive integer $n$ such that if $n$ squares of a $1000 \times 1000$ chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.
33 replies
MithsApprentice
Oct 1, 2005
MaxSze
2 hours ago
J
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Cool NT with Sets and Mod
pear333   0
2 hours ago
Find all integers $a$ such that there exists a set $X$ of $6$ integers satisfying the following condition: for each $k = 1,2,...,36$, there exist $x, y \in X$ such that $ax+y-k$ is divisible by $37$.
0 replies
pear333
2 hours ago
0 replies
J
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postaffteff
JetFire008   14
N 2 hours ago by Korean_fish_Kaohsiung
Source: Internet
Let $P$ be the Fermat point of a $\triangle ABC$. Prove that the Euler line of the triangles $PAB$, $PBC$, $PCA$ are concurrent and the point of concurrence is $G$, the centroid of $\triangle ABC$.
14 replies
JetFire008
Mar 15, 2025
Korean_fish_Kaohsiung
2 hours ago
J
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Family of functions
Davdav1232   4
N 2 hours ago by Davdav1232
Source: Israel TST 2025 test 4 p1
Let \(\mathcal{F}\) be a family of functions from \(\mathbb{R}^+ \to \mathbb{R}^+\). It is known that for all \( f, g \in \mathcal{F} \), there exists \( h \in \mathcal{F} \) such that for all \( x, y \in \mathbb{R}^+ \), the following equation holds:

\[ y^2 \cdot f\left(\frac{g(x)}{y}\right) = h(xy) \]
Prove that for all \( f \in \mathcal{F} \) and all \( x \in \mathbb{R}^+ \), the following identity is satisfied:

\[ f\left(\frac{x}{f(x)}\right) = 1. \]
4 replies
Davdav1232
Feb 3, 2025
Davdav1232
2 hours ago
J
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Twin Prime Diophantine
awesomeming327.   18
N 3 hours ago by EthanWYX2009
Source: CMO 2025
Determine all positive integers $a$, $b$, $c$, $p$, where $p$ and $p+2$ are odd primes and
\[2^ap^b=(p+2)^c-1.\]
18 replies
awesomeming327.
Mar 7, 2025
EthanWYX2009
3 hours ago
J
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Sharygin 2025 CR P12
Gengar_in_Galar   7
N 3 hours ago by maths_enthusiast_0001
Source: Sharygin 2025
Circles $\omega_{1}$ and $\omega_{2}$ are given. Let $M$ be the midpoint of the segment joining their centers, $X$, $Y$ be arbitrary points on $\omega_{1}$, $\omega_{2}$ respectively such that $MX=MY$. Find the locus of the midpoints of segments $XY$.
Proposed by: L Shatunov
7 replies
Gengar_in_Galar
Mar 10, 2025
maths_enthusiast_0001
3 hours ago
J
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Problem 1
randomusername   71
N 4 hours ago by MihaiT
Source: IMO 2015, Problem 1
We say that a finite set $\mathcal{S}$ of points in the plane is balanced if, for any two different points $A$ and $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say that $\mathcal{S}$ is centre-free if for any three different points $A$, $B$ and $C$ in $\mathcal{S}$, there is no points $P$ in $\mathcal{S}$ such that $PA=PB=PC$.

(a) Show that for all integers $n\ge 3$, there exists a balanced set consisting of $n$ points.

(b) Determine all integers $n\ge 3$ for which there exists a balanced centre-free set consisting of $n$ points.

Proposed by Netherlands
71 replies
randomusername
Jul 10, 2015
MihaiT
4 hours ago
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Two linear forms are simultaneously divisible by 83
djb86   3
N Nov 29, 2020 by mrdriller
Let $m$ and $n$ be positive integers. Show that $25m+ 3n$ is divisible by $83$ if and only if so is $3m+ 7n$.
3 replies
djb86
Apr 19, 2013
mrdriller
Nov 29, 2020
J
Two linear forms are simultaneously divisible by 83
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Reveal topic tags
algebra
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djb86
445 posts
djb86
#1 h  Apr 19, 2013, 6:21 PM • 2 Y
Y by Adventure10, Mango247
Let $m$ and $n$ be positive integers. Show that $25m+ 3n$ is divisible by $83$ if and only if so is $3m+ 7n$.
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fractals
3028 posts
fractals
#2 h  Apr 19, 2013, 6:23 PM • 2 Y
Y by Adventure10, Mango247
Solution
EDIT: sure
This post has been edited 1 time. Last edited by fractals, Apr 19, 2013, 10:20 PM
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maplestory
1458 posts
maplestory
#3 h  Apr 19, 2013, 9:05 PM • 1 Y
Y by Adventure10
please stop labeling your outlines as solutions.
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mrdriller
32 posts
mrdriller
#4 h  Nov 29, 2020, 5:52 AM • 2 Y
Y by Mango247, Mango247
Suppose that 83|25m+3n then 83|25m-80n then 83|5m-16n then 83|88m-16n then 83|11m-2n then 83|-72m-2n then 83|72m+2n then 83|36m+n then 83|36m+84n then 83|3m+7n \leftrightarrow then 83|3m+7n then 83|3m+83*9m+7n+83n then 83|750m+90m then 83|25m+3n.
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