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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 11:16 PM
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
Yesterday at 11:16 PM
0 replies
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
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
Functional equations in IMO TST
sheripqr   49
N 21 minutes ago by clarkculus
Source: Iran TST 1996
Find all functions $f: \mathbb R \to \mathbb R$ such that $$ f(f(x)+y)=f(x^2-y)+4f(x)y $$ for all $x,y \in \mathbb R$
49 replies
sheripqr
Sep 14, 2015
clarkculus
21 minutes ago
Good Permutations in Modulo n
swynca   8
N 22 minutes ago by Thapakazi
Source: BMO 2025 P1
An integer $n > 1$ is called $\emph{good}$ if there exists a permutation $a_1, a_2, a_3, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$, such that:
$(i)$ $a_i$ and $a_{i+1}$ have different parities for every $1 \leq i \leq n-1$;
$(ii)$ the sum $a_1 + a_2 + \cdots + a_k$ is a quadratic residue modulo $n$ for every $1 \leq k \leq n$.
Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.
8 replies
swynca
Apr 27, 2025
Thapakazi
22 minutes ago
Interesting inequalities
sqing   0
25 minutes ago
Source: Own
Let $ a,b,c>0 $ and $ (a+b)^2 (a+c)^2=16abc. $ Prove that
$$ 2a+b+c\leq \frac{128}{27}$$$$ \frac{9}{2}a+b+c\leq \frac{864}{125}$$$$3a+b+c\leq 24\sqrt{3}-36$$$$5a+b+c\leq \frac{4(8\sqrt{6}-3)}{9}$$
0 replies
sqing
25 minutes ago
0 replies
Geometry..Pls
Jackson0423   3
N 30 minutes ago by Jackson0423
In equilateral triangle \( ABC \), let \( AB = 10 \). Point \( D \) lies on segment \( BC \) such that \( BC = 4 \cdot DC \). Let \( O \) and \( I \) be the circumcenter and incenter of triangle \( ABD \), respectively. Let \( O' \) and \( I' \) be the circumcenter and incenter of triangle \( ACD \), respectively. Suppose that lines \( OI \) and \( O'I' \) intersect at point \( X \). Find the length of \( XD \).
3 replies
Jackson0423
Yesterday at 2:43 PM
Jackson0423
30 minutes ago
Functional Equation f(x(1+y)) = f(x)(1+f(y))
minimario   3
N 40 minutes ago by jasperE3
Solve in $\mathbb{R}: f(x(1+y)) = f(x)(1+f(y))$
3 replies
minimario
Aug 16, 2015
jasperE3
40 minutes ago
How many variables?
Lukaluce   1
N 42 minutes ago by a_507_bc
Source: BMO SL 2024 A1
Let $u, v, w$ be positive reals. Prove that there is a cyclic permutation $(x, y, z)$ of $(u, v, w)$ such that the inequality:
$$\frac{a}{xa + yb + zc} + \frac{b}{xb + yc  + za} + \frac{c}{xc + ya + zb} \ge 
\frac{3}{x + y + z}$$holds for all positive real numbers $a,  b$ and $c$.
1 reply
Lukaluce
an hour ago
a_507_bc
42 minutes ago
find the radius of circumcircle!
jennifreind   0
an hour ago
In $\triangle \rm ABC$, $  \angle \rm B$ is acute, $\rm{\overline{BC}} = 8$, and $\rm{\overline{AC}} = 3\rm{\overline{AB}}$. Let point $\rm D$ be the intersection of the tangent to the circumcircle of $\triangle \rm ABC$ at point $\rm A$ and the perpendicular bisector of segment $\rm{\overline{BC}}$. Given that $\rm{\overline{AD}} = 6$, find the radius of the circumcircle of $\triangle \rm BCD$.
IMAGE
0 replies
jennifreind
an hour ago
0 replies
Nice one
imnotgoodatmathsorry   0
an hour ago
Source: Own
With $x,y,z >0$.Prove that: $\frac{xy}{4y+4z+x} + \frac{yz}{4z+4x+y} +\frac{zx}{4x+4y+z} \le \frac{x+y+z}{9}$
0 replies
imnotgoodatmathsorry
an hour ago
0 replies
Geometric inequality in quadrilateral
BBNoDollar   1
N an hour ago by nabodorbuco2
Source: Romanian Mathematical Gazette 2025
Let ABCD be a convex quadrilateral with angles BAD and BCD obtuse, and let the points E, F ∈ BD, such that AE ⊥ BD and CF ⊥ BD.
Prove that 1/(AE*CF) ≥ 1/(AB*BC) + 1/(AD*CD) .
1 reply
BBNoDollar
2 hours ago
nabodorbuco2
an hour ago
How many numbers ?
brokendiamond   0
an hour ago
How many 9-digit numbers can be formed using digits 1 to 9, with the condition that no even digits are adjacent to each other ?
0 replies
brokendiamond
an hour ago
0 replies
-2 belongs to S
WakeUp   4
N an hour ago by jasperE3
Source: Baltic Way 1996 Q12
Let $S$ be a set of integers containing the numbers $0$ and $1996$. Suppose further that any integer root of any non-zero polynomial with coefficients in $S$ also belongs to $S$. Prove that $-2$ belongs to $S$.
4 replies
WakeUp
Mar 19, 2011
jasperE3
an hour ago
Inspired by Kunihiko_Chikaya
sqing   0
an hour ago
Source: Own
Let $a,\ b,\ c$ be real numbers such that $ a+b+c=1  $ and $ a^2+b^2+c^2=23 . $ Prove that
$$ -47\leq a^3+b^3-3c\leq 78$$$$\frac{509-278\sqrt{34}}{27}\leq a^3+b^3-3abc\leq \frac{509+278\sqrt{34}}{27}$$
0 replies
sqing
an hour ago
0 replies
Distributing coins in a circle
quacksaysduck   2
N an hour ago by genius_007
Source: JOM 2025 Mock 1 P4
There are $n$ people arranged in a circle, and $n^{n^n}$ coins are distributed among them, where each person has at least $n^n$ coins. Each person is then assigned a random index number in $\{1,2,...n\}$ such that no two people have the same number. Then every minute, if $i$ is the number of minutes passed, the person with index number congruent to $i$ mod $n$ will give a coin to the person on his left or right. After some time, everyone has the same number of coins.

For what $n$ is this always possible, regardless of the original distribution of coins and index numbers?

(Proposed by Ho Janson)
2 replies
quacksaysduck
Jan 26, 2025
genius_007
an hour ago
Familiar cyclic quad config
Rijul saini   11
N an hour ago by ihategeo_1969
Source: India IMOTC Practice Test 1 Problem 2
Let $ABCD$ be a convex cyclic quadrilateral with circumcircle $\omega$. Let $BA$ produced beyond $A$ meet $CD$ produced beyond $D$, at $L$. Let $\ell$ be a line through $L$ meeting $AD$ and $BC$ at $M$ and $N$ respectively, so that $M,D$ (respectively $N,C$) are on opposite sides of $A$ (resp. $B$). Suppose $K$ and $J$ are points on the arc $AB$ of $\omega$ not containing $C,D$ so that $MK, NJ$ are tangent to $\omega$. Prove that $K,J,L$ are collinear.

Proposed by Rijul Saini
11 replies
Rijul saini
May 31, 2024
ihategeo_1969
an hour ago
function
CarlFriedrichGauss-1777   5
N Apr 28, 2025 by lksb
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that:
$f(2021+xf(y))=yf(x+y+2021)$
5 replies
CarlFriedrichGauss-1777
Jun 4, 2021
lksb
Apr 28, 2025
function
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G H BBookmark kLocked kLocked NReply
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CarlFriedrichGauss-1777
51 posts
#1
Y by
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that:
$f(2021+xf(y))=yf(x+y+2021)$
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Hopeooooo
819 posts
#2 • 1 Y
Y by Mango247
$x,y$ are real numbers or positive real?????
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jasperE3
11287 posts
#3
Y by
Assuming it holds for all $x,y\in\mathbb R^+$.
If $\exists u:f(u)>1$ then
$P\left(\frac u{f(u)-1},u\right)\Rightarrow\exists k:f(k)=0$, contradiction.
So $f(x)\le1$ for all $x$. Then $f(2021+xf(y))\le y$, and taking $x=\frac1{f\left(\frac12f(2022)\right)}$ and $y=\frac12f(2022)$ is a contradiction.
This post has been edited 1 time. Last edited by jasperE3, Jun 5, 2021, 2:40 AM
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Blackbeam999
16 posts
#4
Y by
jasperE3 wrote:
Assuming it holds for all $x,y\in\mathbb R^+$.
If $\exists u:f(u)>1$ then
$P\left(\frac u{f(u)-1},u\right)\Rightarrow\exists k:f(k)=0$, contradiction.
So $f(x)\le1$ for all $x$. Then $f(2021+xf(y))\le y$, and taking $x=\frac1{f\left(\frac12f(2022)\right)}$ and $y=\frac12f(2022)$ is a contradiction.

I don’t get it why f(k)=0 for some k?
Z K Y
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jasperE3
11287 posts
#5
Y by
Blackbeam999 wrote:
I don’t get it why f(k)=0 for some k?

what a terrible solution who wrote that

lemme rewrite it and see if it's more clear:
CarlFriedrichGauss-1777 wrote:
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that:
$f(2021+xf(y))=yf(x+y+2021)$

Claim: $f(x)\le1$ for all $x>2021$
Suppose otherwise, then for some $u>2021$ we have $f(u)>1$.
Plugging in $x=\frac u{f(u)-1}$ and $y=u$ we get:
$$(u-1)f\left(\frac{uf(u)}{f(u)-1}\right)=0$$and since $u>2021$ we have $u-1\ne0$ and so $f\left(\frac{uf(u)}{f(u)-1}\right)=0$, which contradicts that the codomain of $f$ is $\mathbb R^+$.
So the claim is proven.

Then since $x+y+2021>2021$, $P(x,y)$ implies:
$$f(2021+xf(y))\le y,$$so $f(x+2021)\le y$ for all $x,y>0$, but taking $y=\frac12f(x+2021)$ gives an immediate contradiction.
Hence no solutions.
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lksb
169 posts
#6
Y by
$x\to \frac{y}{f(y)-1}\implies f\left(2021+\frac{yf(y)}{f(y)-1}\right)=yf\left(2021+\frac{yf(y)}{f(y)-1}\right)\implies y=1 \text{, Abs!}$
Therefore, $f(x)=1$(obviously doesn't work) or $f(y)-1\leq0\implies f(y)\leq1\ \forall y\in\mathbb{R}$
$\implies f(2021+xf(y))\leq y\implies\ x\to\frac{1}{f(\frac{1}{2}f(2022))}, y\to \frac{1}{2}f(2022)\implies f(2022)\leq \frac{1}{2}f(2022)<f(2022),\ \text{Abs!}$
Therefore, no such function exists
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