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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 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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Vasc = 1?
Li4   0
2 minutes ago
Source: 2025 Taiwan TST Round 3 Independent Study 1-N
Find all integer tuples $(a, b, c)$ such that
\[(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1. \]
Proposed by Li4, Untro368, usjl and YaWNeeT.
0 replies
Li4
2 minutes ago
0 replies
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Symmetric Homogeneous Polynomial of Degree 3
USJL   3
N 15 minutes ago by USJL
Source: 2025 Taiwan TST Round 3 Independent Study 1-A
Find all symmetric homogeneous polynomials $P(x,y,z)$ with real coefficients of degree $3$ such that $P(1,x,x^2)$ divides $P(-(x+1)^3,x,x^2)$.

Proposed by usjl
3 replies
USJL
an hour ago
USJL
15 minutes ago
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Collect ...
luutrongphuc   4
N an hour ago by Rayanelba
Find all functions $f: \mathbb{R^+} \rightarrow \mathbb{R^+}$ such that:
$$f\left(f(xy)+1\right)=xf\left(x+f(y)\right)$$
4 replies
luutrongphuc
Apr 21, 2025
Rayanelba
an hour ago
J
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Algebra problem
kjhgyuio   0
an hour ago
........
0 replies
kjhgyuio
an hour ago
0 replies
J
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No three collinear
USJL   0
an hour ago
Source: 2025 Taiwan TST Round 3 Mock P6
Given a positive integer $n\geq 3$. A convex polygon is said to be $n$-good if it contains $n$ lattice points where any three of them are not collinear.

(a) Show that there exists an $n$-good convex polygon with area at most $4n^2$.
(b) Show that there exists a constant $c>0$ so that any $n$-good convex polygon has area at least $cn^2$.

Proposed by usjl
0 replies
USJL
an hour ago
0 replies
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Polynomials with even coefficient
USJL   0
an hour ago
Source: 2025 Taiwan TST Round 3 Mock P3
Find the number of polynomials $F(x,y)$ satisfying the following:
1. $\deg F \leq 2048$;
2. the coefficients of $F(x,y)$ are $0$ and $1$;
3. $F(x,0)=F(0,y)\equiv 0$;
4. $F(x+z,y)+F(z,y)+F(x,y+z)+F(x,z)$ has all of its coefficients being even.

Proposed by usjl
0 replies
USJL
an hour ago
0 replies
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Inspired by old results
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b,c>0 $ and $ a+b+c=3. $ Prove that
$$ \frac{2}{a}+\frac {2}{ab}+\frac{1}{abc}\geq 4$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{2}{abc}\geq 2+\sqrt 3$$$$ \frac{3}{a}+\frac {3}{ab}+\frac{1}{abc}\geq\frac {7+\sqrt {13}}{2}$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{3}{abc}\geq\frac {5+\sqrt {21}}{2}$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{4}{abc}\geq 3+2\sqrt 2$$
1 reply
sqing
an hour ago
sqing
an hour ago
J
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Distant and Difference
USJL   0
an hour ago
Source: 2025 Taiwan TST Round 3 Independent Study 2-C
There are $N$ points on the plane with diameter $D$.
Show that there exist two distinct points $X,Y$ and two not necessarily distinct points $A,B$ not equal to $X$ or $Y$ satisfying that
\[|AX-XY|+|BY-XY|\leq \frac{2D}{N-2}.\]
Proposed by usjl
0 replies
USJL
an hour ago
0 replies
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easy functional
B1t   6
N an hour ago by cazanova19921
Source: Mongolian TST 2025 P1.
Denote the set of real numbers by $\mathbb{R}$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for all $x, y, z \in \mathbb{R}$,
\[ f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)). \]
6 replies
B1t
Today at 6:45 AM
cazanova19921
an hour ago
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gcd (a^n+b,b^n+a) is constant
EthanWYX2009   80
N an hour ago by santhoshn
Source: 2024 IMO P2
Determine all pairs $(a,b)$ of positive integers for which there exist positive integers $g$ and $N$ such that
$$\gcd (a^n+b,b^n+a)=g$$holds for all integers $n\geqslant N.$ (Note that $\gcd(x, y)$ denotes the greatest common divisor of integers $x$ and $y.$)

Proposed by Valentio Iverson, Indonesia
80 replies
EthanWYX2009
Jul 16, 2024
santhoshn
an hour ago
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Benelux fe
ErTeeEs06   7
N an hour ago by Rayanelba
Source: BxMO 2025 P1
Does there exist a function $f:\mathbb{R}\to \mathbb{R}$ such that $$f(x^2+f(y))=f(x)^2-y$$for all $x, y\in \mathbb{R}$?
7 replies
ErTeeEs06
2 hours ago
Rayanelba
an hour ago
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AZE JBMO TST
IstekOlympiadTeam   6
N an hour ago by Namisgood
Source: AZE JBMO TST
Find all non-negative solutions to the equation $2013^x+2014^y=2015^z$
6 replies
IstekOlympiadTeam
May 2, 2015
Namisgood
an hour ago
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$5^t + 3^x4^y = z^2$
Namisgood   1
N 2 hours ago by skellyrah
Source: JBMO shortlist 2017
Solve in nonnegative integers the equation $5^t + 3^x4^y = z^2$
1 reply
Namisgood
3 hours ago
skellyrah
2 hours ago
J
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4 var inequality
sqing   0
2 hours ago
Source: Own
Let $ a,b,c,d\geq -1 $ and $ a+b+c+d=2. $ Prove that$$ab+bc+cd\leq \frac{13}{4}$$$$ab+bc+cd-d\leq \frac{17}{4}$$$$ ab+bc+cd+2d \leq \frac{37}{4}$$$$ab+bc+cd+2da \leq 5$$$$ab+bc+cd-da \leq 6$$$$a +ab-bc+cd+ d \leq 8$$
0 replies
sqing
2 hours ago
0 replies
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Find the minimum
sqing   5
N Apr 14, 2025 by sqing
Source: China
Let $ABC$ be a triangle with $ BC=2AB$ and the rea is $2 . $ Find the minimum of $AC. $
5 replies
sqing
Apr 14, 2025
sqing
Apr 14, 2025
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Find the minimum
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Reveal topic tags
inequalities proposed
geometry
algebra
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Source: China
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sqing
41839 posts
sqing
#1 h  Apr 14, 2025, 8:58 AM
Y by
Let $ABC$ be a triangle with $ BC=2AB$ and the rea is $2 . $ Find the minimum of $AC. $
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kjhgyuio
52 posts
kjhgyuio
#2 h  Apr 14, 2025, 9:06 AM
Y by
sqing wrote:
Let $ABC$ be a triangle with $ BC=2AB$ and the rea is $2 . $ Find the minimum of $AC. $

what does rea mean
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lbh_qys
552 posts
lbh_qys
#3 h  Apr 14, 2025, 9:16 AM
Y by
sqing wrote:
Let $ABC$ be a triangle with $ BC=2AB$ and the rea is $2 . $ Find the minimum of $AC. $

Let $ABC$ be a triangle with $ BC = k AB$ and $k > 1$, prove that $AC^2 \geq 2(k - \frac{1}{k})\mathrm{Area}_{ABC} $.
This post has been edited 1 time. Last edited by lbh_qys, Apr 14, 2025, 9:18 AM
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sqing
41839 posts
sqing
#4 h  Apr 14, 2025, 12:37 PM
Y by
Very nice.Thank lbh_qys
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pooh123
27 posts
pooh123
#5 h  Apr 14, 2025, 12:44 PM
Y by
sqing wrote:
Let $ABC$ be a triangle with $BC=2AB$ and the area is $2$. Find the minimum of $AC$.
Let \( [XYZ] \) denote the area of triangle \( XYZ \).

We have:
\[ 4 = 2[ABC] = AB \cdot BC \cdot \sin B = 2AB^2 \sin B, \]so
\[ AB^2 = \frac{2}{\sin B}. \]
Then:
\[ AC^2 = AB^2 + BC^2 - 2AB \cdot BC \cdot \cos B = AB^2(5 - 4\cos B) = \frac{10 - 8\cos B}{\sin B}. \]
Using the Cauchy-Schwarz inequality, we have:
\[ 8\cos B + 6\sin B \leq \sqrt{(8^2 + 6^2)(\cos^2 B + \sin^2 B)} = \sqrt{100} = 10, \]so
\[ 10 - 8\cos B \geq 6\sin B, \]which implies
\[ \frac{10 - 8\cos B}{\sin B} \geq 6. \]
Hence, the minimum of \( AC \) is \( \sqrt{6} \).
This post has been edited 1 time. Last edited by pooh123, Apr 14, 2025, 12:45 PM
Reason: typo
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sqing
41839 posts
sqing
#6 h  Apr 14, 2025, 12:56 PM
Y by
Good.Thanks.
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