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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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Functional equation wrapped in f's
62861   35
N 16 minutes ago by ihatemath123
Source: RMM 2019 Problem 5
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying
\[f(x + yf(x)) + f(xy) = f(x) + f(2019y),\]for all real numbers $x$ and $y$.
35 replies
62861
Feb 24, 2019
ihatemath123
16 minutes ago
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JBMO Shortlist 2023 C1
Orestis_Lignos   6
N 21 minutes ago by zhenghua
Source: JBMO Shortlist 2023, C1
Given is a square board with dimensions $2023 \times 2023$, in which each unit cell is colored blue or red. There are exactly $1012$ rows in which the majority of cells are blue, and exactly $1012$ columns in which the majority of cells are red.

What is the maximal possible side length of the largest monochromatic square?
6 replies
Orestis_Lignos
Jun 28, 2024
zhenghua
21 minutes ago
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Number Theory
MuradSafarli   6
N 27 minutes ago by krish6_9
Find all natural numbers \( a, b, c \) such that

\[ 2^a \cdot 3^b + 1 = 5^c. \]
6 replies
MuradSafarli
4 hours ago
krish6_9
27 minutes ago
J
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Equilateral triangle geo
MathSaiyan   1
N 44 minutes ago by ricarlos
Source: PErA 2025/3
Let \( ABC \) be an equilateral triangle with circumcenter \( O \). Let \( X \) and \( Y \) be two points on segments \( AB \) and \( AC \), respectively, such that \( \angle XOY = 60^\circ \). If \( T \) is the reflection of \( O \) with respect to line \( XY \), prove that lines \( BT \) and \( OY \) are parallel.
1 reply
MathSaiyan
Yesterday at 1:47 PM
ricarlos
44 minutes ago
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IMO 2009, Problem 5
orl   86
N an hour ago by Ilikeminecraft
Source: IMO 2009, Problem 5
Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths
\[ a, f(b) \text{ and } f(b + f(a) - 1).\]
(A triangle is non-degenerate if its vertices are not collinear.)

Proposed by Bruno Le Floch, France
86 replies
orl
Jul 16, 2009
Ilikeminecraft
an hour ago
J
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IMO 2023 P2
799786   88
N an hour ago by Frd_19_Hsnzde
Source: IMO 2023 P2
Let $ABC$ be an acute-angled triangle with $AB < AC$. Let $\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ again at $E \neq A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P \neq B$. Prove that the line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\angle BAC$.
88 replies
799786
Jul 8, 2023
Frd_19_Hsnzde
an hour ago
J
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Diagonals BD,CE concurrent with diameter AO in cyclic ABCDE
WakeUp   10
N an hour ago by zhenghua
Source: Romanian TST 2002
Let $ABCDE$ be a cyclic pentagon inscribed in a circle of centre $O$ which has angles $\angle B=120^{\circ},\angle C=120^{\circ},$ $\angle D=130^{\circ},\angle E=100^{\circ}$. Show that the diagonals $BD$ and $CE$ meet at a point belonging to the diameter $AO$.

Dinu Șerbănescu
10 replies
WakeUp
Feb 5, 2011
zhenghua
an hour ago
J
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Parallel lines in two-circle configuration
Tintarn   3
N 2 hours ago by zhenghua
Source: Francophone 2024, Senior P3
Let $ABC$ be an acute triangle, $\omega$ its circumcircle and $O$ its circumcenter. The altitude from $A$ intersects $\omega$ in a point $D \ne A$ and the segment $AC$ intersects the circumcircle of $OCD$ in a point $E \ne C$. Finally, let $M$ be the midpoint of $BE$. Show that $DE$ is parallel to $OM$.
3 replies
Tintarn
Apr 4, 2024
zhenghua
2 hours ago
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IMO Shortlist 2013, Algebra #5
lyukhson   33
N 2 hours ago by HamstPan38825
Source: IMO Shortlist 2013, Algebra #5
Let $\mathbb{Z}_{\ge 0}$ be the set of all nonnegative integers. Find all the functions $f: \mathbb{Z}_{\ge 0} \rightarrow \mathbb{Z}_{\ge 0} $ satisfying the relation
\[ f(f(f(n))) = f(n+1 ) +1 \]
for all $ n\in \mathbb{Z}_{\ge 0}$.
33 replies
lyukhson
Jul 9, 2014
HamstPan38825
2 hours ago
J
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f(2) = 7, find all integer functions [Taiwan 2014 Quizzes]
v_Enhance   56
N 2 hours ago by Maximilian113
Find all increasing functions $f$ from the nonnegative integers to the integers satisfying $f(2)=7$ and \[ f(mn) = f(m) + f(n) + f(m)f(n) \] for all nonnegative integers $m$ and $n$.
56 replies
v_Enhance
Jul 18, 2014
Maximilian113
2 hours ago
J
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China MO 2021 P6
NTssu   22
N 2 hours ago by HamstPan38825
Source: CMO 2021 P6
Find $f: \mathbb{Z}_+ \rightarrow \mathbb{Z}_+$, such that for any $x,y \in \mathbb{Z}_+$, $$f(f(x)+y)\mid x+f(y).$$
22 replies
NTssu
Nov 25, 2020
HamstPan38825
2 hours ago
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Very concex function
lomos_lupin   48
N 2 hours ago by Ilikeminecraft
Source: USAM0 2000 #1 (billzhao)
Call a real-valued function $ f$ very convex if
\[ \frac {f(x) + f(y)}{2} \ge f\left(\frac {x + y}{2}\right) + |x - y| \]
holds for all real numbers $ x$ and $ y$. Prove that no very convex function exists.
48 replies
lomos_lupin
Aug 8, 2005
Ilikeminecraft
2 hours ago
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USAMO 2001 Problem 6
MithsApprentice   20
N 3 hours ago by Ritwin
Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.
20 replies
MithsApprentice
Sep 30, 2005
Ritwin
3 hours ago
J
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Bijection on the set of integers
talkon   18
N 3 hours ago by HamstPan38825
Source: InfinityDots MO 2 Problem 2
Determine all bijections $f:\mathbb Z\to\mathbb Z$ satisfying
$$f^{f(m+n)}(mn) = f(m)f(n)$$for all integers $m,n$.

Note: $f^0(n)=n$, and for any positive integer $k$, $f^k(n)$ means $f$ applied $k$ times to $n$, and $f^{-k}(n)$ means $f^{-1}$ applied $k$ times to $n$.

Proposed by talkon
18 replies
talkon
Apr 9, 2018
HamstPan38825
3 hours ago
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\frac{a}{b+c}\sqrt{ \frac{b^2}{c^2+a^2}+\frac{c^2}{a^2+b^2}} \geq\sqrt{\frac{2}{
sqing   2
N Jul 26, 2024 by sqing
Source: Own
Let $ a,b,c>0 $ and $ a\geq b+c.$ Prove that
$$\frac{a}{b+c}\sqrt{ \frac{b^2}{c^2+a^2}+\frac{c^2}{a^2+b^2}} \geq\sqrt{\frac{2}{5}}$$$$\frac{a}{b+c}\sqrt{ \frac{b^2}{c^2+2a^2}+\frac{c^2}{2a^2+b^2}} \geq\frac{\sqrt{2}}{3}$$$$\frac{a}{b+c}\sqrt{ \frac{b^2}{2c^2+a^2}+\frac{c^2}{a^2+2b^2}} \geq\frac{1}{\sqrt{3}}$$
2 replies
sqing
Jul 25, 2024
sqing
Jul 26, 2024
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\frac{a}{b+c}\sqrt{ \frac{b^2}{c^2+a^2}+\frac{c^2}{a^2+b^2}} \geq\sqrt{\frac{2}{
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Reveal topic tags
inequalities proposed
algebra
inequalities
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Source: Own
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sqing
41050 posts
sqing
#1 h  Jul 25, 2024, 4:31 AM • 1 Y
Y by GeoKing
Let $ a,b,c>0 $ and $ a\geq b+c.$ Prove that
$$\frac{a}{b+c}\sqrt{ \frac{b^2}{c^2+a^2}+\frac{c^2}{a^2+b^2}} \geq\sqrt{\frac{2}{5}}$$$$\frac{a}{b+c}\sqrt{ \frac{b^2}{c^2+2a^2}+\frac{c^2}{2a^2+b^2}} \geq\frac{\sqrt{2}}{3}$$$$\frac{a}{b+c}\sqrt{ \frac{b^2}{2c^2+a^2}+\frac{c^2}{a^2+2b^2}} \geq\frac{1}{\sqrt{3}}$$
This post has been edited 1 time. Last edited by sqing, Jul 25, 2024, 8:57 AM
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frc
37 posts
frc
#2 h  Jul 26, 2024, 3:11 AM
Y by
https://artofproblemsolving.com/community/c6h3365546_fraca2_b2c2__leftfracb2c2a2fracc2a2b2_rightgeq
Notice$\frac{a^2}{(b+c)^2}\geq \frac{a^2}{2(b^2+c^2)}$
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sqing
41050 posts
sqing
#3 h  Jul 26, 2024, 3:38 AM
Y by
Very good.Thanks.
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