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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
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[*]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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>=512 different isosceles triangles whose vertices have the same color
parmenides51   1
N 39 minutes ago by AlexCenteno2007
Source: Mathematics Regional Olympiad of Mexico West 2016 P6
The vertices of a regular polygon with $2016$ sides are colored gold or silver. Prove that there are at least $512$ different isosceles triangles whose vertices have the same color.
1 reply
+1 w
parmenides51
Sep 7, 2022
AlexCenteno2007
39 minutes ago
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Fourth power ineq
Project_Donkey_into_M4   1
N an hour ago by sqing
Source: 2018 Mock RMO tdp and kayak P1
Let $a,b,c,d \in \mathbb{R}^+$ such that $a+b+c+d \leq 1$. Prove that\[\sqrt[4]{(1-a^4)(1-b^4)(1-c^4)(1-d^4)}\geq 255\cdot abcd.\]
1 reply
Project_Donkey_into_M4
Yesterday at 6:20 PM
sqing
an hour ago
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Is this FE solvable?
ItzsleepyXD   0
an hour ago
Source: Original
Let $c_1,c_2 \in \mathbb{R^+}$. Find all $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ such that for all $x,y \in \mathbb{R^+}$ $$f(x+c_1f(y))=f(x)+c_2f(y)$$
0 replies
ItzsleepyXD
an hour ago
0 replies
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Dear Sqing: So Many Inequalities...
hashtagmath   36
N an hour ago by sqing
I have noticed thousands upon thousands of inequalities that you have posted to HSO and was wondering where you get the inspiration, imagination, and even the validation that such inequalities are true? Also, what do you find particularly appealing and important about specifically inequalities rather than other branches of mathematics? Thank you :)
36 replies
hashtagmath
Oct 30, 2024
sqing
an hour ago
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Advanced topics in Inequalities
va2010   18
N an hour ago by sqing
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!
18 replies
1 viewing
va2010
Mar 7, 2015
sqing
an hour ago
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two subsets with no fewer than four common elements.
micliva   39
N 2 hours ago by de-Kirschbaum
Source: All-Russian Olympiad 1996, Grade 9, First Day, Problem 4
In the Duma there are 1600 delegates, who have formed 16000 committees of 80 persons each. Prove that one can find two committees having no fewer than four common members.

A. Skopenkov
39 replies
micliva
Apr 18, 2013
de-Kirschbaum
2 hours ago
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3 knightlike moves is enough
sarjinius   2
N 2 hours ago by cooljoseph
Source: Philippine Mathematical Olympiad 2025 P6
An ant is on the Cartesian plane. In a single move, the ant selects a positive integer $k$, then either travels [list]
[*] $k$ units vertically (up or down) and $2k$ units horizontally (left or right); or
[*] $k$ units horizontally (left or right) and $2k$ units vertically (up or down).
[/list]
Thus, for any $k$, the ant can choose to go to one of eight possible points.
Prove that, for any integers $a$ and $b$, the ant can travel from $(0, 0)$ to $(a, b)$ using at most $3$ moves.
2 replies
sarjinius
Mar 9, 2025
cooljoseph
2 hours ago
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16th ibmo - uruguay 2001/q3.
carlosbr   21
N 2 hours ago by de-Kirschbaum
Source: Spanish Communities
Let $S$ be a set of $n$ elements and $S_1,\ S_2,\dots,\ S_k$ are subsets of $S$ ($k\geq2$), such that every one of them has at least $r$ elements.

Show that there exists $i$ and $j$, with $1\leq{i}<j\leq{k}$, such that the number of common elements of $S_i$ and $S_j$ is greater or equal to: $r-\frac{nk}{4(k-1)}$
21 replies
carlosbr
Apr 15, 2006
de-Kirschbaum
2 hours ago
J
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Weird Geo
Anto0110   1
N 2 hours ago by cooljoseph
In a trapezium $ABCD$, the sides $AB$ and $CD$ are parallel and the angles $\angle ABC$ and $\angle BAD$ are acute. Show that it is possible to divide the triangle $ABC$ into 4 disjoint triangle $X_1. . . , X_4$ and the triangle $ABD$ into 4 disjoint triangles $Y_1,. . . , Y_4$ such that the triangles $X_i$ and $Y_i$ are congruent for all $i$.
1 reply
Anto0110
Yesterday at 9:24 PM
cooljoseph
2 hours ago
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Hard FE R^+
DNCT1   5
N 3 hours ago by jasperE3
Find all functions $f:\mathbb{R^+}\to\mathbb{R^+}$ such that
$$f(3x+f(x)+y)=f(4x)+f(y)\quad\forall x,y\in\mathbb{R^+}$$
5 replies
DNCT1
Dec 30, 2020
jasperE3
3 hours ago
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Maximum of Incenter-triangle
mpcnotnpc   4
N 4 hours ago by mpcnotnpc
Triangle $\Delta ABC$ has side lengths $a$, $b$, and $c$. Select a point $P$ inside $\Delta ABC$, and construct the incenters of $\Delta PAB$, $\Delta PBC$, and $\Delta PAC$ and denote them as $I_A$, $I_B$, $I_C$. What is the maximum area of the triangle $\Delta I_A I_B I_C$?
4 replies
mpcnotnpc
Mar 25, 2025
mpcnotnpc
4 hours ago
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Something nice
KhuongTrang   26
N 4 hours ago by KhuongTrang
Source: own
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
26 replies
KhuongTrang
Nov 1, 2023
KhuongTrang
4 hours ago
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Tiling rectangle with smaller rectangles.
MarkBcc168   59
N 5 hours ago by Bonime
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
59 replies
MarkBcc168
Jul 10, 2018
Bonime
5 hours ago
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Existence of AP of interesting integers
DVDthe1st   34
N 5 hours ago by DeathIsAwe
Source: 2018 China TST Day 1 Q2
A number $n$ is interesting if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.
34 replies
DVDthe1st
Jan 2, 2018
DeathIsAwe
5 hours ago
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J
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Perhaps a classic with parameter
mihaig   1
N Apr 16, 2025 by LLriyue
Find the largest positive constant $r$ such that
$$a^2+b^2+c^2+d^2+2\left(abcd\right)^r\geq6$$for all reals $a\geq1\geq b\geq c\geq d\geq0$ satisfying $a+b+c+d=4.$
1 reply
mihaig
Jan 7, 2025
LLriyue
Apr 16, 2025
J
Perhaps a classic with parameter
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Reveal topic tags
inequalities
parameterization
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mihaig
7339 posts
mihaig
#1 h  Jan 7, 2025, 6:01 PM
Y by
Find the largest positive constant $r$ such that
$$a^2+b^2+c^2+d^2+2\left(abcd\right)^r\geq6$$for all reals $a\geq1\geq b\geq c\geq d\geq0$ satisfying $a+b+c+d=4.$
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LLriyue
2 posts
LLriyue
#2 h  Apr 16, 2025, 4:12 AM
Y by
good Question
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