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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.

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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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Crazy number theory
MTA_2024   5
N 24 minutes ago by bjump
Find all couple $(p;q)$ of primes (greater than 5) such that : $$pq \mid (5^q-3^q)(5^p-3^p)$$
5 replies
MTA_2024
3 hours ago
bjump
24 minutes ago
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hard number theory problem
Zavyk09   0
33 minutes ago
Source: forgotten
Find all couple $(x, y)$ of positive integers such that:
$$2^n + 3^n \mid x^n + y^n, \forall n \in \mathbb{N}^*$$
0 replies
Zavyk09
33 minutes ago
0 replies
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The return of a legend inequality
giangtruong13   2
N 43 minutes ago by polishedhardwoodtable
Source: Legacy
Given that $0<a,b,c,d<1$.Prove that: $$ (1-a)(1-b)(1-c)(1-d) > 1-a-b-c-d $$
2 replies
giangtruong13
2 hours ago
polishedhardwoodtable
43 minutes ago
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Slightly weird points which are not so weird
Pranav1056   9
N an hour ago by Retemoeg
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 $\angle IML = \angle DVM$
9 replies
Pranav1056
Jul 9, 2023
Retemoeg
an hour ago
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2023 factors and perfect cube
proxima1681   4
N an hour ago by anudeep
Source: Indian Statistical Institute (ISI) UGB 2023 P4
Let $n_1, n_2, \cdots , n_{51}$ be distinct natural numbers each of which has exactly $2023$ positive integer factors. For instance, $2^{2022}$ has exactly $2023$ positive integer factors $1,2, 2^{2}, 2^{3}, \cdots 2^{2021}, 2^{2022}$. Assume that no prime larger than $11$ divides any of the $n_{i}$'s. Show that there must be some perfect cube among the $n_{i}$'s.
4 replies
proxima1681
May 14, 2023
anudeep
an hour ago
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circle geometry showing perpendicularity
Kyj9981   1
N an hour ago by Retemoeg
Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A line through $B$ intersects $\omega_1$ and $\omega_2$ at points $C$ and $D$, respectively. Line $AD$ intersects $\omega_1$ at point $E \neq A$, and line $AC$ intersects $\omega_2$ at point $F \neq A$. If $O$ is the circumcenter of $\triangle AEF$, prove that $OB \perp CD$.
1 reply
Kyj9981
6 hours ago
Retemoeg
an hour ago
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Hard problem
Tendo_Jakarta   5
N an hour ago by Tendo_Jakarta
Let the sequence \(x_{n}\) be such that
\[u_{1} = 1; \quad u_{n+1} = \dfrac{u_{1} + u_{2} +...+u_{n}}{n}+n-1 \quad \forall n \in \mathbb{N^{*}}\]and \(y_{n} =\dfrac{1}{u_{1}u_{2}} + \dfrac{1}{u_{3}u_{4}} + ... + \dfrac{1}{u_{2n-1}u_{2n}} \quad \forall n \geq 1\). Find \(\lim_{n\rightarrow\infty}{y_{n}}\).
5 replies
Tendo_Jakarta
2 hours ago
Tendo_Jakarta
an hour ago
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Oh no! Inequality again?
mathisreaI   108
N an hour ago by Maximilian113
Source: IMO 2022 Problem 2
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for each $x \in \mathbb{R}^+$, there is exactly one $y \in \mathbb{R}^+$ satisfying $$xf(y)+yf(x) \leq 2$$
108 replies
mathisreaI
Jul 13, 2022
Maximilian113
an hour ago
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Equation with powers
a_507_bc   5
N an hour ago by ali123456
Source: Serbia JBMO TST 2024 P1
Find all non-negative integers $x, y$ and primes $p$ such that $$3^x+p^2=7 \cdot 2^y.$$
5 replies
a_507_bc
May 25, 2024
ali123456
an hour ago
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hermoso, raiz primitiva, orden??
holaquehace707070   0
an hour ago
Sea n un numero natural con mas de 2021 dıgitos donde ninguno de el-
los es 8 o 9. Suponga que n no tiene factores comunes con 2021. Demuestre que es posible
aumentar uno de los dıgitos de n en a lo mas 2 de modo que el numero resultante sea multiplo
de 2021.
0 replies
holaquehace707070
an hour ago
0 replies
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Discrete problem
ThanhNg   0
an hour ago
Prove that : From any 161 natural numbers, we can always find 81 numbers whose sum is divisible by 81.
0 replies
ThanhNg
an hour ago
0 replies
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[PMO23 Qualifying I.1] Sides and Diagonals
kae_3   4
N 2 hours ago by JetFire008
In a convex polygon, the number of diagonals is $23$ times the number of its sides. How many sides does it have?
$\text{(a) }46 \qquad\text{(b) }49 \qquad\text{(c) }66 \qquad\text{(d) }69$

Answer Confirmation
4 replies
kae_3
Feb 9, 2025
JetFire008
2 hours ago
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The return of an inequality
giangtruong13   2
N 2 hours ago by sqing
Let $a,b,c$ be real positive number satisfy that: $a+b+c=1$. Prove that: $$\sum_{cyc} \frac{a}{b^2+c^2} \geq \frac{3}{2}$$
2 replies
giangtruong13
2 hours ago
sqing
2 hours ago
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Questions problem 10 and 11 in a Test
Math2030   2
N 2 hours ago by vanstraelen
Exercise P10. (4 points) Given triangle ABC. Let D and E be the midpoints of AB and CD respectively. Suppose angle ACD = 2 angle DEB. Prove that 2 .angle AED = angle DCB +180°.


Exercise P11. Given rectangle ABCD and a point P in the plane. Let X, Y, Z, W, S and T be the orthogonal projections of P onto the lines AB, BC, CD, DA, AC and BD respectively. The perpendicular bisectors of XY and W Z intersect at Q. The perpendicular bisectors of Y Z and XW intersect at R. Prove that QR is parallel to ST.
2 replies
Math2030
Mar 12, 2025
vanstraelen
2 hours ago
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a+b+c=3 ine
jokehim   1
N 3 hours ago by Cool12345678
Problem. Given non-negative real numbers $a,b,c$ satisfying $a+b+c=3.$ Prove that $$\color{black}{\frac{a\left(b+c\right)}{bc+3}+\frac{b\left(c+a\right)}{ca+3}+\frac{c\left(a+b\right)}{ab+3}\le \frac{3}{2}.}$$Proposed by Phan Ngoc Chau
1 reply
jokehim
Today at 9:48 AM
Cool12345678
3 hours ago
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a+b+c=3 ine
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Reveal topic tags
inequalities
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jokehim
1014 posts
jokehim
#1 h  Today at 9:48 AM
Y by
Problem. Given non-negative real numbers $a,b,c$ satisfying $a+b+c=3.$ Prove that $$\color{black}{\frac{a\left(b+c\right)}{bc+3}+\frac{b\left(c+a\right)}{ca+3}+\frac{c\left(a+b\right)}{ab+3}\le \frac{3}{2}.}$$Proposed by Phan Ngoc Chau
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Cool12345678
14 posts
Cool12345678
#2 h  3 hours ago
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Since $(2,0,0)$ majorizes $(1,1,0)$ and given the condition $a+b+c=3$, by Muirhead, we rewrite $$\sum_{cyc} \frac{a\left(b+c\right)}{bc+3}\leq\sum_{cyc} \frac{a\left(3-a\right)}{a^2+3}$$
Now, let $f(x)=\frac{x\left(3-x\right)}{x^2+3}$. Then, setting $y=\frac{x\left(3-x\right)}{x^2+3}$, we get $$ x^2y + 3y = 3x-x^2 \implies x^2(y + 1) + 3y-3x = 0.$$Using the discriminant of a quadratic, we get $$9-4(y+1)(3y)\geq0 \implies y\leq \frac{1}{2}$$Therefore, the maximum value of $f(x)$ is $\frac{1}{2}$ so for all $a,b,c$ we have $$\frac{a\left(3-a\right)}{a^2+3}\leq \frac{1}{2}$$Hence, $$\sum_{cyc} \frac{a\left(3-a\right)}{a^2+3}\leq \frac{3}{2}$$as desired.
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