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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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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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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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Diophantine eq.
User335559   13
N 4 minutes ago by Ianis
Source: European Mathematical Cup 2017
Solve in integers the equation :
$x^2y+y^2=x^3$
13 replies
+1 w
User335559
Jan 3, 2018
Ianis
4 minutes ago
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Balkan MO SL A1 easy
tenplusten   12
N an hour ago by Primeniyazidayi
Source: Balkan MO SL 2014 A1
$\boxed{\text{A1}}$Let $a,b,c$ be positive reals numbers such that $a+b+c=1$.Prove that $2(a^2+b^2+c^2)\ge \frac{1}{9}+15abc$
12 replies
tenplusten
Sep 27, 2016
Primeniyazidayi
an hour ago
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$f(x+y^2f(y))=f(1+yf(x)).f(x),\forall x,y>0$
Zahy2106   0
an hour ago
Source: Collections
Determine all functions $f:\mathbb{R^+} \to \mathbb{R^+}$ satisfying: $f(x+y^2f(y))=f(1+yf(x)).f(x),\forall x,y>0$
0 replies
Zahy2106
an hour ago
0 replies
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floor of (an+b)/(cn+d) is surjective
Miquel-point   3
N 2 hours ago by Rohit-2006
Source: Romanian NMO 2021 grade 10 P2
Let $a,b,c,d\in\mathbb{Z}_{\ge 0}$, $d\ne 0$ and the function $f:\mathbb{Z}_{\ge 0}\to\mathbb Z_{\ge 0}$ defined by
\[f(n)=\left\lfloor \frac{an+b}{cn+d}\right\rfloor\text{ for all } n\in\mathbb{Z}_{\ge 0}.\]Prove that the following are equivalent:
[list=1]
[*] $f$ is surjective;
[*] $c=0$, $b<d$ and $0<a\le d$.
[/list]

Tiberiu Trif
3 replies
Miquel-point
Apr 15, 2023
Rohit-2006
2 hours ago
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evan chen??
Captainscrubz   1
N 2 hours ago by Captainscrubz
Let point $D$ and $E$ be on sides $AB$ and $AC$ respectively in $\triangle ABC$ such that $BD=BC=CE$. Let $O_1$ be the circumcenter of $\triangle ADE$ and let $S=DC\cap EB$. Prove that $O_1S \perp BC$
1 reply
Captainscrubz
Today at 3:50 AM
Captainscrubz
2 hours ago
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SMO 2015 open q3
dominicleejun   13
N 2 hours ago by jasperE3
Source: SMO 2015 open
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$, where $\mathbb{R}$ is the set of real numbers, such that
$f(x)f(yf(x) - 1) = x^2 f(y) - f(x) \quad\forall x,y \in \mathbb{R}$
13 replies
1 viewing
dominicleejun
Mar 31, 2018
jasperE3
2 hours ago
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thank you
Piwbo   1
N 2 hours ago by CHESSR1DER
Let $p_n$ be the n-th prime number in increasing order for $n\geq 1$. Prove that there exists a sequence of distinct prime numbers $q_n$ satisfying $q_1+q_2+...+q_n=p_n$ for all $n\geq 1 $
1 reply
Piwbo
Yesterday at 11:22 AM
CHESSR1DER
2 hours ago
J
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Difficult lattice point coloring problem
CBMaster   0
2 hours ago
Source: Korean math olympiad practice problem
Is it possible to color all lattice points in plane into 3 colors such that

1. every line passing through lattice points and parallel to x axis has these three colors infinitely many(that is, every color appears infinitely many times in those lines).

2. every line passing through lattice points and not parallel to x axis cannot have three different color lattice points on it.

I think the answer is yes, but I couldn't find an example...
0 replies
CBMaster
2 hours ago
0 replies
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Stop Projecting your insecurities
naman12   52
N 2 hours ago by ihategeo_1969
Source: 2022 USA TST #2
Let $ABC$ be an acute triangle. Let $M$ be the midpoint of side $BC$, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively. Suppose that the common external tangents to the circumcircles of triangles $BME$ and $CMF$ intersect at a point $K$, and that $K$ lies on the circumcircle of $ABC$. Prove that line $AK$ is perpendicular to line $BC$.

Kevin Cong
52 replies
naman12
Dec 12, 2022
ihategeo_1969
2 hours ago
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Digit sum
Disjeje   3
N 2 hours ago by jasperE3
Let’s say S(n) is digit sum of n does n exists thatS(n)>S(n^2)?
3 replies
Disjeje
5 hours ago
jasperE3
2 hours ago
J
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R+ Functional Equation
Mathdreams   3
N 2 hours ago by jasperE3
Source: Nepal TST 2025, Problem 3
Find all functions $f : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that \[f(f(x)) + xf(xy) = x + f(y)\]for all positive real numbers $x$ and $y$.

(Andrew Brahms, USA)
3 replies
Mathdreams
6 hours ago
jasperE3
2 hours ago
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TST Junior Romania 2025
ant_   0
2 hours ago
Source: ssmr
Consider the isosceles triangle $ABC$, with $\angle BAC > 90^\circ$, and the circle $\omega$ with center $A$ and radius $AC$. Denote by $M$ the midpoint of side $AC$. The line $BM$ intersects the circle $\omega$ for the second time in $D$. Let $E$ be a point on the circle $\omega$ such that $BE \perp AC$ and $DE \cap AC = {N}$. Show that $AN = 2AB$.
0 replies
ant_
2 hours ago
0 replies
J
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Functional equation
Pmshw   16
N 2 hours ago by jasperE3
Source: Iran 2nd round 2022 P2
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any real value of $x,y$ we have:
$$f(xf(y)+f(x)+y)=xy+f(x)+f(y)$$
16 replies
Pmshw
May 8, 2022
jasperE3
2 hours ago
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Line passes through the fix point
moony_   1
N 2 hours ago by GioOrnikapa
Source: idk
Let $ABC$ be a triangle. $P$ and $Q$ are points, such that $PA = PB$, $QA$ = $QC$ and $\angle{PBC} =\angle{QCB}$ ($P$ - inside $\triangle{ABC}$ and $Q$ - oitside). Proove that line $PQ$ passes through the fix point.
1 reply
moony_
5 hours ago
GioOrnikapa
2 hours ago
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inequalities
pennypc123456789   1
N Apr 7, 2025 by Double07
Let \( x,y \) be non-negative real numbers.Prove that :
\[ \sqrt{x^4+y^4 } +(2+\sqrt{2})xy \geq x^2+y^2 \]
1 reply
pennypc123456789
Apr 7, 2025
Double07
Apr 7, 2025
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inequalities
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inequalities
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pennypc123456789
29 posts
pennypc123456789
#1 h  Apr 7, 2025, 3:28 AM
Y by
Let \( x,y \) be non-negative real numbers.Prove that :
\[ \sqrt{x^4+y^4 } +(2+\sqrt{2})xy \geq x^2+y^2 \]
Z K Y
The post below has been deleted. Click to close.
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Double07
74 posts
Double07
#2 h  Apr 7, 2025, 12:06 PM • 1 Y
Y by pennypc123456789
$\sqrt{x^4+y^4}+(2+\sqrt{2})xy\geq x^2+y^2\iff \sqrt{x^4+y^4}\geq x^2+y^2-(2+\sqrt{2})xy$.
If $x^2+y^2-(2+\sqrt{2})xy\leq 0$, we're done.
Else, we want to prove that $x^4+y^4\geq x^4+y^4+(2+\sqrt{2})^2x^2y^2+2x^2y^2-2(2+\sqrt{2})(x^3y+xy^3)\iff 2xy(2+\sqrt{2})(x^2+y^2)\geq (8+4\sqrt{2})x^2y^2\iff $
$\iff (2+\sqrt{2})(x^2+y^2)\geq (4+2\sqrt{2})xy$.
But we know that $x^2+y^2>(2+\sqrt{2})xy$, so $(2+\sqrt{2})(x^2+y^2)>(2+\sqrt{2})^2xy=(6+4\sqrt{2})xy>(4+2\sqrt{2})xy$ so we're done.
This post has been edited 2 times. Last edited by Double07, Apr 7, 2025, 12:06 PM
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