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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
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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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Number Theory
MuradSafarli   1
N a minute ago by Amkan2022
Find all natural numbers \( a, b, c \) such that

\[ 2^a \cdot 3^b + 1 = 5^c. \]
1 reply
MuradSafarli
11 minutes ago
Amkan2022
a minute ago
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RMM 2019 Problem 2
math90   77
N 2 minutes ago by ihatemath123
Source: RMM 2019
Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$. Let $E$ be the midpoint of $AC$. Denote by $\omega$ and $\Omega$ the circumcircles of the triangles $ABE$ and $CDE$, respectively. Let $P$ be the crossing point of the tangent to $\omega$ at $A$ with the tangent to $\Omega$ at $D$. Prove that $PE$ is tangent to $\Omega$.

Jakob Jurij Snoj, Slovenia
77 replies
math90
Feb 23, 2019
ihatemath123
2 minutes ago
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Two circles concur on a line
math154   59
N 16 minutes ago by Mathandski
Source: ELMO Shortlist 2012, G1; also ELMO #1
In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$.

Ray Li.
59 replies
math154
Jul 2, 2012
Mathandski
16 minutes ago
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functional equation
Anni   7
N 21 minutes ago by HamstPan38825
Source: albanian TST 2008 bmo
Find all functions $f: \mathbb R \to \mathbb R$ such that
\[ f(x+f(y))=y+f(x+1),\]for all $x,y \in \mathbb R$.
7 replies
Anni
May 24, 2009
HamstPan38825
21 minutes ago
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"pseudo-Fibonnaci" sequence
pohoatza   11
N 27 minutes ago by asdf334
Source: IMO Shortlist 2006, Algebra 3
The sequence $c_{0}, c_{1}, . . . , c_{n}, . . .$ is defined by $c_{0}= 1, c_{1}= 0$, and $c_{n+2}= c_{n+1}+c_{n}$ for $n \geq 0$. Consider the set $S$ of ordered pairs $(x, y)$ for which there is a finite set $J$ of positive integers such that $x=\textstyle\sum_{j \in J}{c_{j}}$, $y=\textstyle\sum_{j \in J}{c_{j-1}}$. Prove that there exist real numbers $\alpha$, $\beta$, and $M$ with the following property: An ordered pair of nonnegative integers $(x, y)$ satisfies the inequality \[m < \alpha x+\beta y < M\] if and only if $(x, y) \in S$.

Remark: A sum over the elements of the empty set is assumed to be $0$.
11 replies
pohoatza
Jun 28, 2007
asdf334
27 minutes ago
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IMO ShortList 2001, algebra problem 4
orl   35
N 28 minutes ago by HamstPan38825
Source: IMO ShortList 2001, algebra problem 4
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, satisfying \[ f(xy)(f(x) - f(y)) = (x-y)f(x)f(y) \] for all $x,y$.
35 replies
orl
Sep 30, 2004
HamstPan38825
28 minutes ago
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Arithmetic Sequence of Products
GrantStar   18
N 28 minutes ago by hgomamogh
Source: IMO Shortlist 2023 N4
Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products
\[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\]form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.
18 replies
+1 w
GrantStar
Jul 17, 2024
hgomamogh
28 minutes ago
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sequence (.) eventually becomes constant.
N.T.TUAN   60
N an hour ago by shendrew7
Source: USAMO 2007
Let $n$ be a positive integer. Define a sequence by setting $a_{1}= n$ and, for each $k > 1$, letting $a_{k}$ be the unique integer in the range $0\leq a_{k}\leq k-1$ for which $a_{1}+a_{2}+...+a_{k}$ is divisible by $k$. For instance, when $n = 9$ the obtained sequence is $9,1,2,0,3,3,3,...$. Prove that for any $n$ the sequence $a_{1},a_{2},...$ eventually becomes constant.
60 replies
N.T.TUAN
Apr 26, 2007
shendrew7
an hour ago
J
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A line tangent to a circumcircle
ericxyzhu   6
N an hour ago by zhenghua
Source: Lusophon Mathematical Olympiad 2021 Problem 3
Let triangle $ABC$ be an acute triangle with $AB\neq AC$. The bisector of $BC$ intersects the lines $AB$ and $AC$ at points $F$ and $E$, respectively. The circumcircle of triangle $AEF$ has center $P$ and intersects the circumcircle of triangle $ABC$ at point $D$ with $D$ different to $A$.

Prove that the line $PD$ is tangent to the circumcircle of triangle $ABC$.
6 replies
ericxyzhu
Dec 19, 2021
zhenghua
an hour ago
J
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Number theory
Ecrin_eren   1
N an hour ago by Ecrin_eren
Show that there are no prime numbers satisfying the equation

(p + r)^q + (q + r)^p = (p + q)^r.

1 reply
Ecrin_eren
Mar 15, 2025
Ecrin_eren
an hour ago
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f(f(x)+y)+f(x+y)=2x+2f(y)
parmenides51   4
N an hour ago by jasperE3
Source: 2015 AGCN Competition p1 by bobthesmartypants https://artofproblemsolving.com/community/c5h1128876p5232794
Find all functions $f:\mathbb{R}_{\ge 0}\to \mathbb{R}_{\ge 0}$ satisfying$$f(f(x)+y)+f(x+y)=2x+2f(y)$$
4 replies
parmenides51
Dec 5, 2023
jasperE3
an hour ago
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Divisibility of a triple
goodar2006   52
N 2 hours ago by zuat.e
Source: Iran TST 2013-First exam-2nd day-P5
Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$?

Proposed by Mahan Malihi
52 replies
goodar2006
Apr 19, 2013
zuat.e
2 hours ago
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Rectangle EFGH in incircle, prove that QIM = 90
v_Enhance   61
N 2 hours ago by daixiahu
Source: Taiwan 2014 TST1, Problem 3
Let $ABC$ be a triangle with incenter $I$, and suppose the incircle is tangent to $CA$ and $AB$ at $E$ and $F$. Denote by $G$ and $H$ the reflections of $E$ and $F$ over $I$. Let $Q$ be the intersection of $BC$ with $GH$, and let $M$ be the midpoint of $BC$. Prove that $IQ$ and $IM$ are perpendicular.
61 replies
v_Enhance
Jul 18, 2014
daixiahu
2 hours ago
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USAMO PYQ
JetFire008   1
N 2 hours ago by v_Enhance
Source: USAMO 1993
Let $ABCD$ be a convex quadrilateral with perpendicular diagonals intersecting at $O$. Prove that the reflections of $O$ across $AB$, BC, CD, DA$ are concyclic.
1 reply
JetFire008
5 hours ago
v_Enhance
2 hours ago
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Problem 2830
sqing   0
Today at 8:15 AM
Source: SXTB (2)2025
Let $ a,b>0 $ and $ \frac{1}{a^2+1}+ \frac{1}{b^2+1}=t $ $(1<t<2). $ Find the value range of $ a+b. $
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0 replies
sqing
Today at 8:15 AM
0 replies
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Problem 2830
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Reveal topic tags
inequalities proposed
algebra
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Source: SXTB (2)2025
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sqing
41050 posts
sqing
#1 h  Today at 8:15 AM
Y by
Let $ a,b>0 $ and $ \frac{1}{a^2+1}+ \frac{1}{b^2+1}=t $ $(1<t<2). $ Find the value range of $ a+b. $
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This post has been edited 1 time. Last edited by sqing, Today at 8:24 AM
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