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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
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.

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0 replies
jlacosta
May 1, 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
J
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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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Own made functional equation
Primeniyazidayi   5
N 7 minutes ago by JARP091
Source: own(probably)
Find all functions $f:R \rightarrow R$ such that $xf(x^2+2f(y)-yf(x))=f(x)^3-f(y)(f(x^2)-2f(x))$ for all $x,y \in \mathbb{R}$
5 replies
Primeniyazidayi
May 26, 2025
JARP091
7 minutes ago
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Iran TST P6
luutrongphuc   4
N an hour ago by AblonJ
Find all function $f: \mathbb{R^+} \to \mathbb{R^+}$ such that:
$$f \left( f(f(xy))+x^2\right)=f(y)f(x)-f(y)f(x+y)$$
4 replies
luutrongphuc
May 26, 2025
AblonJ
an hour ago
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Product of opposite sides
nabodorbuco2   0
an hour ago
Source: Original
Let $ABCDEF$ a regular hexagon inscribed in a circle $\Omega$. Let $P_i$ be a point inside $\Omega$ and $P_e$ its polar reflection wrt $\Omega$. The rays $AP_i,BP_i,CP_i,DP_i,EP_i,FP_i$ meet $\Omega$ again at $A_i,B_i,C_i,D_i,E_i,F_i$. Call $Q_I$ the polygon formed by the vertices $A_i,B_i,C_i,D_i,E_i,F_i$. Similarly construct the polygon $Q_E$ using $P_e$ instead.

Show that $Q_I$ and $Q_E$ are congruent.
0 replies
nabodorbuco2
an hour ago
0 replies
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Brilliant Problem
M11100111001Y1R   7
N an hour ago by flower417477
Source: Iran TST 2025 Test 3 Problem 3
Find all sequences \( (a_n) \) of natural numbers such that for every pair of natural numbers \( r \) and \( s \), the following inequality holds:
\[ \frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2 \]
7 replies
M11100111001Y1R
May 27, 2025
flower417477
an hour ago
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In Cyclic Quadrilateral ABCD, find AB^2+BC^2-CD^2-AD^2
Darealzolt   1
N an hour ago by Beelzebub
Source: KTOM April 2025 P8
Given Cyclic Quadrilateral \(ABCD\) with an area of \(2025\), with \(\angle ABC = 45^{\circ}\). If \( 2AC^2 = AB^2+BC^2+CD^2+DA^2\), Hence find the value of \(AB^2+BC^2-CD^2-DA^2\).
1 reply
Darealzolt
Today at 4:10 AM
Beelzebub
an hour ago
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Another FE
M11100111001Y1R   3
N an hour ago by AndreiVila
Source: Iran TST 2025 Test 2 Problem 3
Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for all $x,y>0$ we have:
$$f(f(f(xy))+x^2)=f(y)(f(x)-f(x+y))$$
3 replies
M11100111001Y1R
Yesterday at 8:03 AM
AndreiVila
an hour ago
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Iran TST Starter
M11100111001Y1R   4
N an hour ago by flower417477
Source: Iran TST 2025 Test 1 Problem 1
Let \( a_n \) be a sequence of positive real numbers such that for every \( n > 2025 \), we have:
\[ a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i} \]Prove that there exists a natural number \( M \) such that for all \( n > M \), the following holds:
\[ a_n < \frac{1}{1404} \]
4 replies
M11100111001Y1R
May 27, 2025
flower417477
an hour ago
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Inequality with abc=1
tenplusten   10
N an hour ago by Adywastaken
Source: JBMO 2011 Shortlist A7
$\boxed{\text{A7}}$ Let $a,b,c$ be positive reals such that $abc=1$.Prove the inequality $\sum\frac{2a^2+\frac{1}{a}}{b+\frac{1}{a}+1}\geq 3$
10 replies
tenplusten
May 15, 2016
Adywastaken
an hour ago
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A game of cutting
k.vasilev   11
N 2 hours ago by NicoN9
Source: All-Russian Olympiad 2019 grade 10 problem 2
Pasha and Vova play the following game, making moves in turn; Pasha moves first. Initially, they have a large piece of plasticine. By a move, Pasha cuts one of the existing pieces into three(of arbitrary sizes), and Vova merges two existing pieces into one. Pasha wins if at some point there appear to be $100$ pieces of equal weights. Can Vova prevent Pasha's win?
11 replies
k.vasilev
Apr 23, 2019
NicoN9
2 hours ago
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Problem 10
SlovEcience   1
N 2 hours ago by lbh_qys
Let \( x, y, z \) be positive real numbers satisfying
\[ xy + yz + zx = 3xyz. \]Prove that
\[ \sqrt{\frac{x}{3y^2z^2 + xyz}} + \sqrt{\frac{y}{3x^2z^2 + xyz}} + \sqrt{\frac{z}{3x^2y^2 + xyz}} \le \frac{3}{2}. \]
1 reply
SlovEcience
3 hours ago
lbh_qys
2 hours ago
J
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Cup of Combinatorics
M11100111001Y1R   8
N 2 hours ago by sansgankrsngupta
Source: Iran TST 2025 Test 4 Problem 2
There are \( n \) cups labeled \( 1, 2, \dots, n \), where the \( i \)-th cup has capacity \( i \) liters. In total, there are \( n \) liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.

$a)$ Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most \( \frac{4n}{3} \) steps.

$b)$ Prove that in at most \( \frac{5n}{3} \) steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
8 replies
M11100111001Y1R
May 27, 2025
sansgankrsngupta
2 hours ago
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2-var inequality
sqing   2
N 3 hours ago by sqing
Source: Own
Let $ a,b> 0 , ab(a+b+1) =3.$ Prove that$$\frac{1}{a^2}+\frac{1}{b^2}+\frac{24}{(a+b)^2} \geq 8$$$$ \frac{a}{b^2}+\frac{b}{a^2}+\frac{49}{(a+ b)^2} \geq \frac{57}{4}$$Let $ a,b> 0 , (a+b)(ab+1) =4.$ Prove that$$\frac{1}{a^2}+\frac{1}{b^2}+\frac{40}{(a+b)^2} \geq 12$$$$\frac{a}{b^2}+\frac{b}{a^2}+\frac{76}{(a+ b)^2} \geq 21$$
2 replies
sqing
May 25, 2025
sqing
3 hours ago
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2-var inequality
sqing   10
N 3 hours ago by sqing
Source: Own
Let $ a,b>0 , a^2+b^2-ab\leq 1 . $ Prove that
$$a^3+b^3 -\frac{a^4}{b+1} -\frac{b^4}{a+1} \leq 1 $$
10 replies
sqing
May 27, 2025
sqing
3 hours ago
J
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Complex number
ronitdeb   1
N 4 hours ago by alexheinis
Let $z_1, ... ,z_5$ be vertices of regular pentagon inscribed in a circle whose radius is $2$ and center is at $6+i8$. Find all possible values of $z_1^2+z_2^2+...+z_5^2$
1 reply
ronitdeb
Yesterday at 6:13 PM
alexheinis
4 hours ago
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r /R <= cos (180^o/n), 2 circles and a convex n-gon
parmenides51   1
N Nov 19, 2021 by rafaello
Source: 2010 Baltic Way Shortlist SWE4 BW https://artofproblemsolving.com/community/c2641329_baltic_way_shortlist__geometry
A convex $n$-gon encloses a circle of radius $ r$, and is itself enclosed within a circle of radius $R$. Prove that
$$\frac{r}{R} \le \cos \frac{180^o}{n}.$$
1 reply
parmenides51
Nov 18, 2021
rafaello
Nov 19, 2021
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r /R <= cos (180^o/n), 2 circles and a convex n-gon
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Reveal topic tags
convex polygon
geometric inequality
geometry
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Source: 2010 Baltic Way Shortlist SWE4 BW https://artofproblemsolving.com/community/c2641329_baltic_way_shortlist__geometry
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parmenides51
30653 posts
parmenides51
#1 h  Nov 18, 2021, 4:10 PM
Y by
A convex $n$-gon encloses a circle of radius $ r$, and is itself enclosed within a circle of radius $R$. Prove that
$$\frac{r}{R} \le \cos \frac{180^o}{n}.$$
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rafaello
1079 posts
rafaello
#2 h  Nov 19, 2021, 10:32 PM
Y by
Let $A_1A_2\ldots A_n$ be the $n$-gon with $O$ as the center of circle $\omega$, which encloses the $n$-gon. Let $C_i=\overline{OA_i}\cap \omega$. Let $B_1,B_2,\ldots, B_n$ be the feet from $O$ to $\overline{C_1C_2},\overline{C_2C_3},\ldots,\overline{C_nC_1}$, respectively.
\begin{align*} R\cos{\frac{180^\circ}{n}}&\geq R\cdot \frac{\cos{\angle C_1OB_1}+\cos{\angle B_1OC_2}+\ldots +\cos{\angle B_nOC_1}}{2n}\\&=\frac{OB_1+OB_2+\ldots+OB_n}{n}\\&\geq \frac{2S_{C_1C_2\ldots C_n}}{C_1C_2+\ldots + C_nC_1}\\&\geq r, \end{align*}where the the first inequality comes from Jensen's inequality and the second from Chebyshev's inequality.
This post has been edited 2 times. Last edited by rafaello, Dec 28, 2021, 12:36 AM
Reason: fixed
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