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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
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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Equation over a finite field
loup blanc   2
N 16 minutes ago by loup blanc
Find the set of $x\in\mathbb{F}_{5^5}$ such that the equation in the unknown $y\in \mathbb{F}_{5^5}$:
$x^3y+y^3+x=0$ admits $3$ roots: $a,a,b$ s.t. $a\not=b$.
2 replies
loup blanc
Yesterday at 6:08 PM
loup blanc
16 minutes ago
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Can a 0-1 matrix square to the matrix with all ones?
Tintarn   4
N an hour ago by loup blanc
Source: IMC 2024, Problem 3
For which positive integers $n$ does there exist an $n \times n$ matrix $A$ whose entries are all in $\{0,1\}$, such that $A^2$ is the matrix of all ones?
4 replies
+1 w
Tintarn
Aug 7, 2024
loup blanc
an hour ago
J
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Two times derivable real function
Valentin Vornicu   12
N an hour ago by Rohit-2006
Source: RMO 2008, 11th Grade, Problem 3
Let $ f: \mathbb R \to \mathbb R$ be a function, two times derivable on $ \mathbb R$ for which there exist $ c\in\mathbb R$ such that
\[ \frac { f(b)-f(a) }{b-a} \neq f'(c) ,\] for all $ a\neq b \in \mathbb R$.

Prove that $ f''(c)=0$.
12 replies
Valentin Vornicu
Apr 30, 2008
Rohit-2006
an hour ago
J
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Bounding number of solutions for floor function equation
Ciobi_   1
N an hour ago by sarjinius
Source: Romania NMO 2025 9.3
Let $n \geq 2$ be a positive integer. Consider the following equation: \[ \{x\}+\{2x\}+ \dots + \{nx\} = \lfloor x \rfloor + \lfloor 2x \rfloor + \dots + \lfloor 2nx \rfloor\]a) For $n=2$, solve the given equation in $\mathbb{R}$.
b) Prove that, for any $n \geq 2$, the equation has at most $2$ real solutions.
1 reply
Ciobi_
Apr 2, 2025
sarjinius
an hour ago
J
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nice system of equations
outback   4
N 2 hours ago by Raj_singh1432
Solve in positive numbers the system

$ x_1+\frac{1}{x_2}=4, x_2+\frac{1}{x_3}=1, x_3+\frac{1}{x_4}=4, ..., x_{99}+\frac{1}{x_{100}}=4, x_{100}+\frac{1}{x_1}=1$
4 replies
outback
Oct 8, 2008
Raj_singh1432
2 hours ago
J
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Putnam 1972 A2
sqrtX   2
N 2 hours ago by KAME06
Source: Putnam 1972
Let $S$ be a set with a binary operation $\ast$ such that
1) $a \ast(a\ast b)=b$ for all $a,b\in S$.
2) $(a\ast b)\ast b=a$ for all $a,b\in S$.
Show that $\ast$ is commutative and give an example where $\ast$ is not associative.
2 replies
sqrtX
Feb 17, 2022
KAME06
2 hours ago
J
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Two circles, a tangent line and a parallel
Valentin Vornicu   103
N 2 hours ago by zuat.e
Source: IMO 2000, Problem 1, IMO Shortlist 2000, G2
Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP = EQ$.
103 replies
Valentin Vornicu
Oct 24, 2005
zuat.e
2 hours ago
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Inequalities
idomybest   3
N 3 hours ago by damyan
Source: The Interesting Around Technical Analysis Three Variable Inequalities
The problem is in the attachment below.
3 replies
idomybest
Oct 15, 2021
damyan
3 hours ago
J
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Function on positive integers with two inputs
Assassino9931   2
N 3 hours ago by Assassino9931
Source: Bulgaria Winter Competition 2025 Problem 10.4
The function $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is such that $f(a,b) + f(b,c) = f(ac, b^2) + 1$ for any positive integers $a,b,c$. Assume there exists a positive integer $n$ such that $f(n, m) \leq f(n, m + 1)$ for all positive integers $m$. Determine all possible values of $f(2025, 2025)$.
2 replies
Assassino9931
Jan 27, 2025
Assassino9931
3 hours ago
J
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Normal but good inequality
giangtruong13   4
N 3 hours ago by IceyCold
Source: From a province
Let $a,b,c> 0$ satisfy that $a+b+c=3abc$. Prove that: $$\sum_{cyc} \frac{ab}{3c+ab+abc} \geq \frac{3}{5} $$
4 replies
giangtruong13
Mar 31, 2025
IceyCold
3 hours ago
J
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Tiling rectangle with smaller rectangles.
MarkBcc168   61
N 3 hours ago by YaoAOPS
Source: IMO Shortlist 2017 C1
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.

Proposed by Jeck Lim, Singapore
61 replies
MarkBcc168
Jul 10, 2018
YaoAOPS
3 hours ago
J
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A magician has one hundred cards numbered 1 to 100
Valentin Vornicu   49
N 3 hours ago by YaoAOPS
Source: IMO 2000, Problem 4, IMO Shortlist 2000, C1
A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn.

How many ways are there to put the cards in the three boxes so that the trick works?
49 replies
Valentin Vornicu
Oct 24, 2005
YaoAOPS
3 hours ago
J
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Nice inequality
sqing   2
N 3 hours ago by Seungjun_Lee
Source: WYX
Let $a_1,a_2,\cdots,a_n (n\ge 2)$ be real numbers . Prove that : There exist positive integer $k\in \{1,2,\cdots,n\}$ such that $$\sum_{i=1}^{n}\{kx_i\}(1-\{kx_i\})<\frac{n-1}{6}.$$Where $\{x\}=x-\left \lfloor x \right \rfloor.$
2 replies
sqing
Apr 24, 2019
Seungjun_Lee
3 hours ago
J
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Concurrency
Dadgarnia   27
N 3 hours ago by zuat.e
Source: Iranian TST 2020, second exam day 2, problem 4
Let $ABC$ be an isosceles triangle ($AB=AC$) with incenter $I$. Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$. $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$, respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$. Prove that $AD$, $MN$ and $BC$ are concurrent.

Proposed by Alireza Dadgarnia
27 replies
Dadgarnia
Mar 12, 2020
zuat.e
3 hours ago
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J
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a small generalization
pprime   0
May 28, 2018
1. Prove that $\sum\limits_{n=0}^{+\infty }{\arctan \left( \frac{x}{{{n}^{2}}+1} \right)}=\frac{\arctan x}{2}+\arctan \left( \frac{\tan \left( \pi \sqrt[4]{1+{{x}^{2}}}\sin \left( \frac{\arctan x}{2} \right) \right)}{\tanh \left( \pi \sqrt[4]{1+{{x}^{2}}}\cos \left( \frac{\arctan x}{2} \right) \right)} \right)$

2. Prove that $\sum\limits_{n=0}^{+\infty }{\arctan \left( \frac{x}{{{n}^{2}}-1} \right)}=\frac{\arctan x}{2}+\arctan \left( \frac{\tan \left( \pi \sqrt[4]{1+{{x}^{2}}}\cos \left( \frac{\arctan x}{2} \right) \right)}{\tanh \left( \pi \sqrt[4]{1+{{x}^{2}}}\sin \left( \frac{\arctan x}{2} \right) \right)} \right)$
0 replies
pprime
May 28, 2018
0 replies
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a small generalization
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pprime
600 posts
pprime
#1 h  May 28, 2018, 10:57 PM • 2 Y
Y by Adventure10, Mango247
1. Prove that $\sum\limits_{n=0}^{+\infty }{\arctan \left( \frac{x}{{{n}^{2}}+1} \right)}=\frac{\arctan x}{2}+\arctan \left( \frac{\tan \left( \pi \sqrt[4]{1+{{x}^{2}}}\sin \left( \frac{\arctan x}{2} \right) \right)}{\tanh \left( \pi \sqrt[4]{1+{{x}^{2}}}\cos \left( \frac{\arctan x}{2} \right) \right)} \right)$

2. Prove that $\sum\limits_{n=0}^{+\infty }{\arctan \left( \frac{x}{{{n}^{2}}-1} \right)}=\frac{\arctan x}{2}+\arctan \left( \frac{\tan \left( \pi \sqrt[4]{1+{{x}^{2}}}\cos \left( \frac{\arctan x}{2} \right) \right)}{\tanh \left( \pi \sqrt[4]{1+{{x}^{2}}}\sin \left( \frac{\arctan x}{2} \right) \right)} \right)$
This post has been edited 1 time. Last edited by pprime, May 28, 2018, 10:58 PM
Reason: typing error
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