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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!
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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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A number theory about divisors which no one fully solved at the contest
nAalniaOMliO   4
N 25 minutes ago by AL1296
Source: Belarusian national olympiad 2024
Let's call a pair of positive integers $(k,n)$ interesting if $n$ is composite and for every divisor $d<n$ of $n$ at least one of $d-k$ and $d+k$ is also a divisor of $n$
Find the number of interesting pairs $(k,n)$ with $k \leq 100$
M. Karpuk
4 replies
nAalniaOMliO
Jul 24, 2024
AL1296
25 minutes ago
J
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Inspired by A Romanian competition question
sqing   8
N an hour ago by Davut1102
Source: Own
Let $ a,b,c $ be reals such that $ a^2+b^2 +ab+bc+ca=1. $ Prove that
$$ (a+ b) c- a b \leq1$$Let $ a,b,c $ be reals such that $ a^2+b^2+c^2+ab+bc+ca =1. $ Prove that
$$ 29(a+ b) c - 10a b \leq 10$$Let $ a,b,c $ be reals such that $ a^2+b^2+c^2+bc+ca=1. $ Prove that
$$ 149(a+ b) c- 100a b \leq50$$
8 replies
1 viewing
sqing
Mar 18, 2025
Davut1102
an hour ago
J
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Conditional maximum
giangtruong13   0
an hour ago
Source: Specialized Math
Let $a,b$ satisfy that: $1 \leq a \leq2$ and $1 \leq b \leq 2$. Find the maximum: $$A=(a+b^2+\frac{4}{a^2}+\frac{2}{b})(b+a^2+\frac{4}{b^2}+\frac{2}{a})$$
0 replies
giangtruong13
an hour ago
0 replies
J
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Inequality with real numbers
JK1603JK   3
N an hour ago by arqady
Source: unknown
Let a,b,c are real numbers. Prove that (a^3+b^3+c^3+3abc)^4+(a+b+c)^3(a+b-c)^3(-a+b+c)^3(a-b+c)^3>=0
3 replies
JK1603JK
Today at 6:48 AM
arqady
an hour ago
J
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Inequalities
sqing   3
N an hour ago by sqing
Let $ a,b> 0$ and $ a+b=1 . $ Prove that
$$ \frac{1}{a}+\frac{1}{b}\geq \frac{4+\frac{k}{4096}}{1+ ka^7b^7}$$Where $\frac{8192}{3}\geq k>0 .$
$$ \frac{1}{a}+\frac{1}{b}\geq \frac{\frac{14}{3}}{1+ \frac{8192}{3}a^7b^7}$$
3 replies
sqing
2 hours ago
sqing
an hour ago
J
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Number theory question with many (confusing) variables
urfinalopp   0
an hour ago
Given m,n,p,q \in \mathbb{N+}, find all solutions to 2^{m}3^{n}+5^{p}=7^{q}$

One of the paths I've found is to boil it down to solving two non-simultaneous equations 2^{m_1}+5^{n_1}=7^{q_1} and
7^{m_1}+5^{n_1}=2^{q_1} but its too hard. Any other approaches/solutions or a continuation of this path?
0 replies
urfinalopp
an hour ago
0 replies
J
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Fractional Inequality
sqing   31
N an hour ago by Marcus_Zhang
Source: Chinese Girls Mathematical Olympiad 2012, Problem 1
Let $ a_1, a_2,\ldots, a_n$ be non-negative real numbers. Prove that
$\frac{1}{1+ a_1}+\frac{ a_1}{(1+ a_1)(1+ a_2)}+\frac{ a_1 a_2}{(1+ a_1)(1+ a_2)(1+ a_3)}+$ $\cdots+\frac{ a_1 a_2\cdots a_{n-1}}{(1+ a_1)(1+ a_2)\cdots (1+ a_n)} \le 1.$
31 replies
sqing
Aug 10, 2012
Marcus_Zhang
an hour ago
J
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Dealing with Multiple Circles
Wildabandon   3
N 2 hours ago by Wildabandon
Source: PEMNAS Brawijaya University Senior High School Semifinal 2023 P4
A non-isosceles triangle $ABC$ and $\ell$ is tangent to the circumcircle of triangle $ABC$ through point $C$. Points $D$ and $E$ are the midpoints of segments $BC$ and $CA$ respectively, then line $AD$ and line $BE$ intersect $\ell$ at points $A_1$ and $B_1$ respectively. Line $AB_1$ and line $BA_1$ intersect the circumcircle of triangle $ABC$ at points $X$ and $Y$ respectively. Prove that $X$, $Y$, $D$ and $E$ concyclic.
3 replies
Wildabandon
Dec 1, 2024
Wildabandon
2 hours ago
J
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Inequalities
sqing   0
2 hours ago
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that$$a^3b+b^3c+c^3a+\frac{473}{256}abc\le\frac{27}{256}$$Equality holds when $ a=b=c=\frac{1}{3} $ or $ a=0,b=\frac{3}{4},c=\frac{1}{4} $ or $ a=\frac{1}{4} ,b=0,c=\frac{3}{4} $
or $ a=\frac{3}{4} ,b=\frac{1}{4},c=0. $
0 replies
sqing
2 hours ago
0 replies
J
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help title
nguyenvana   0
2 hours ago
Source: no from book
An and Binh play a game on a square board of size (2n+1)x(2n+1) with An going first. Initially, all the squares on the board are white. In each turn, An colors a white square blue and Binh colors a white square red. The game ends after both players have colored all the squares on the board. An wins if, for any two blue squares, there exists at least one chain of neighboring blue squares connecting them (two squares are called neighboring if they have at least one vertex in common). Otherwise, Binh wins. Determine the player with the winning strategy in the following cases:
a) with n=1
b) with n>=2
0 replies
nguyenvana
2 hours ago
0 replies
J
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funny title
nguyenvana   0
2 hours ago
Source: no from book
Find all the functions f: R+ to R+ which satisfy the functional equation:
f(2f(x)+f(y)+xy)=xy+2x+y (x,y R+)
0 replies
nguyenvana
2 hours ago
0 replies
J
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Inequalities from SXTX
sqing   11
N 2 hours ago by byron-aj-tom
T702. Let $ a,b,c>0 $ and $ a+2b+3c=\sqrt{13}. $ Prove that $$ \sqrt{a^2+1} +2\sqrt{b^2+1} +3\sqrt{c^2+1} \geq 7$$S
T703. Let $ a,b $ be real numbers such that $ a+b\neq 0. $. Find the minimum of $ a^2+b^2+(\frac{1-ab}{a+b} )^2.$
T704. Let $ a,b,c>0 $ and $ a+b+c=3. $ Prove that $$ \frac{a^2+7}{(c+a)(a+b)} + \frac{b^2+7}{(a+b)(b+c)} +\frac{c^2+7}{(b+c)(c+a)} \geq 6$$S
11 replies
sqing
Feb 18, 2025
byron-aj-tom
2 hours ago
J
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2018 VNTST Problem 1
gausskarl   6
N 2 hours ago by cursed_tangent1434
Source: 2018 Vietnam Team Selection Test
Let $ABC$ be a acute, non-isosceles triangle. $D,\ E,\ F$ are the midpoints of sides $AB,\ BC,\ AC$, resp. Denote by $(O),\ (O')$ the circumcircle and Euler circle of $ABC$. An arbitrary point $P$ lies inside triangle $DEF$ and $DP,\ EP,\ FP$ intersect $(O')$ at $D',\ E',\ F'$, resp. Point $A'$ is the point such that $D'$ is the midpoint of $AA'$. Points $B',\ C'$ are defined similarly.
a. Prove that if $PO=PO'$ then $O\in(A'B'C')$;
b. Point $A'$ is mirrored by $OD$, its image is $X$. $Y,\ Z$ are created in the same manner. $H$ is the orthocenter of $ABC$ and $XH,\ YH,\ ZH$ intersect $BC, AC, AB$ at $M,\ N,\ L$ resp. Prove that $M,\ N,\ L$ are collinear.
6 replies
gausskarl
Mar 30, 2018
cursed_tangent1434
2 hours ago
J
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Inspired by m4thbl3nd3r
sqing   0
2 hours ago
Source: Own
Let $ a,b,c\geq 0 $ and $a+b+c=3$. Prove that$$a^3b+b^3c+c^3a+\frac{1419}{256}abc\le\frac{2187}{256}$$Equality holds when $ a=b=c=1 $ or $ a=0,b=\frac{9}{4},c=\frac{3}{4} $ or $ a=\frac{3}{4} ,b=0,c=\frac{9}{4} $
or $ a=\frac{9}{4} ,b=\frac{3}{4},c=0. $
0 replies
sqing
2 hours ago
0 replies
J
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J
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PIE Help
Rice_Farmer   5
N Mar 19, 2025 by Rice_Farmer
How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and reflection.

Non ai way btw pls
5 replies
Rice_Farmer
Mar 15, 2025
Rice_Farmer
Mar 19, 2025
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PIE Help
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Reveal topic tags
reflection
rotation
counting
pie
Inclusion Exclusion Principle
L
G H BBookmark kLocked kLocked NReply
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Rice_Farmer
891 posts
Rice_Farmer
#1 h  Mar 15, 2025, 11:47 PM
Y by
How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and reflection.

Non ai way btw pls
This post has been edited 2 times. Last edited by Rice_Farmer, Mar 16, 2025, 3:35 AM
Z K Y
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Rice_Farmer
891 posts
Rice_Farmer
#3 h  Mar 16, 2025, 3:25 AM
Y by
Is that AI?
Z K Y
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EaZ_Shadow
1110 posts
EaZ_Shadow
#4 h  Mar 16, 2025, 3:33 AM
Y by
Rice_Farmer wrote:
Is that AI?

Yep ofc
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Rice_Farmer
891 posts
Rice_Farmer
#6 h  Mar 19, 2025, 1:26 AM
Y by
Help pls?
Z K Y
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Awesomeness_in_a_bun
473 posts
Awesomeness_in_a_bun
#7 h  Mar 19, 2025, 1:31 AM
Y by
burnside's lemma first, and then use PIE to eliminate all 2 color and 1 color solutions
Z K Y
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Rice_Farmer
891 posts
Rice_Farmer
#8 h  Mar 19, 2025, 1:35 AM
Y by
Wait what’s burnsides lemma I haven’t learned that yet
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