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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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[*]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
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
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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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Interesting inequalities
sqing   4
N 32 minutes ago by SunnyEvan
Source: Own
Let $ a,b,c,d\geq 0 , a+b+c+d \leq 4.$ Prove that
$$a(kbc+bd+cd) \leq \frac{64k}{27}$$$$a (b+c) (kb c+ b d+ c d) \leq \frac{27k}{4}$$Where $ k\geq 2. $
4 replies
sqing
Yesterday at 12:44 PM
SunnyEvan
32 minutes ago
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Long and wacky inequality
Royal_mhyasd   5
N 44 minutes ago by Royal_mhyasd
Source: Me
Let $x, y, z$ be positive real numbers such that $x^2 + y^2 + z^2 = 12$. Find the minimum value of the following sum :
$$\sum_{cyc}\frac{(x^3+2y)^3}{3x^2yz - 16z - 8yz + 6x^2z}$$knowing that the denominators are positive real numbers.
5 replies
Royal_mhyasd
May 12, 2025
Royal_mhyasd
44 minutes ago
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Polynomials algebra
Foxellar   2
N an hour ago by elizhang101412
\textbf{9.} The real root of the polynomial \( p(x) = 8x^3 - 3x^2 - 3x - 1 \) can be written in the form
\[ \frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c}, \]where \( a, b, \) and \( c \) are positive integers. Find the value of \( a + b + c \).
2 replies
Foxellar
2 hours ago
elizhang101412
an hour ago
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Collinear points
tenplusten   2
N an hour ago by Blackbeam999
Let $A,B,C$ be three collinear points and $D,E,F$ three other collinear points. Let $G,H,I$ be the intersection of the lines $BE,CF$ $AD,CF$ and $AD,CE$,respectively. If $AI=HD$ and $CH=GF$.Prove that $BI=GE$



I hope you will use Pappus theorem in your solutions.
2 replies
tenplusten
Jun 20, 2016
Blackbeam999
an hour ago
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Some number theory
EeEeRUT   4
N 2 hours ago by juckter
Source: Thailand MO 2025 P9
Let $p$ be an odd prime and $S = \{1,2,3,\dots, p\}$
Assume that $U: S \rightarrow S$ is a bijection and $B$ is an integer such that $$B\cdot U(U(a)) - a \: \text{ is a multiple of} \: p \: \text{for all} \: a \in S$$Show that $B^{\frac{p-1}{2}} -1$ is a multiple of $p$.
4 replies
EeEeRUT
May 14, 2025
juckter
2 hours ago
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Simple Geometry
AbdulWaheed   0
2 hours ago
Source: EGMO
Try to avoid Directed angles
Let ABC be an acute triangle inscribed in circle $\Omega$. Let $X$ be the midpoint of the arc $\overarc{BC}$ not containing $A$ and define $Y, Z$ similarly. Show that the orthocenter of $XYZ$ is the incenter $I$ of $ABC$.
0 replies
AbdulWaheed
2 hours ago
0 replies
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Graph Theory
ABCD1728   0
2 hours ago
Can anyone provide the PDF version of "Graphs: an introduction" by Radu Bumbacea (XYZ press), thanks!
0 replies
ABCD1728
2 hours ago
0 replies
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Number theory for people who love theory
Assassino9931   3
N 2 hours ago by NamelyOrange
Source: Bulgaria RMM TST 2019
Prove that there is no positive integer $n$ such that $2^n + 1$ divides $5^n-1$.
3 replies
Assassino9931
Jul 31, 2024
NamelyOrange
2 hours ago
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Interesting inequalities
sqing   0
3 hours ago
Source: Own
Let $ a,b> 0 , a+b+a^2+b^2=2.$ Prove that
$$ab+ \frac{k}{a+b+ab} \geq \frac{3-k+(k-1)\sqrt{5}}{2}$$Where $ k\geq 2. $
$$ab+ \frac{2}{a+b+ab} \geq \frac{1+\sqrt{5}}{2}$$$$ab+ \frac{3}{a+b+ab} \geq \sqrt{5} $$
0 replies
sqing
3 hours ago
0 replies
J
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Stability of Additive Cauchy Equation
doanquangdang   1
N 4 hours ago by jasperE3
Show that if $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies
$$ |f(x+y)-f(x)-f(y)-x y| \leq \varepsilon\left(|x|^p+|y|^p\right) $$for some $\varepsilon>0,$ $p \in[0,1)$ and for all $x, y \in \mathbb{R}$, then there exists a unique solution $a: \mathbb{R} \rightarrow \mathbb{R}$ of the functional equation $a(x+y)=$ $a(x)+a(y)$ for all $x, y \in \mathbb{R}$ such that
$$ \left|f(x)-a(x)-\frac{1}{2} x^2\right| \leq \frac{2}{2-2^p} \varepsilon|x|^p $$for all $x \in \mathbb{R}$.
1 reply
doanquangdang
Aug 16, 2024
jasperE3
4 hours ago
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Polynomials with common roots and coefficients
VicKmath7   10
N 4 hours ago by math-olympiad-clown
Source: Balkan MO SL 2020 A3
Let $P(x), Q(x)$ be distinct polynomials of degree $2020$ with non-zero coefficients. Suppose that they have $r$ common real roots counting multiplicity and $s$ common coefficients. Determine the maximum possible value of $r + s$.

Demetres Christofides, Cyprus
10 replies
VicKmath7
Sep 9, 2021
math-olympiad-clown
4 hours ago
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Serbian selection contest for the IMO 2025 - P3
OgnjenTesic   1
N 4 hours ago by korncrazy
Source: Serbian selection contest for the IMO 2025
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that:
- $f$ is strictly increasing,
- there exists $M \in \mathbb{N}$ such that $f(x+1) - f(x) < M$ for all $x \in \mathbb{N}$,
- for every $x \in \mathbb{Z}$, there exists $y \in \mathbb{Z}$ such that
\[ f(y) = \frac{f(x) + f(x + 2024)}{2}. \]Proposed by Pavle Martinović
1 reply
OgnjenTesic
Yesterday at 4:06 PM
korncrazy
4 hours ago
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Interesting inequalities
sqing   0
4 hours ago
Source: Own
Let $ a,b,c \geq 0 , a+b+c+abc = 4.$ Prove that
$$ a+ab^2+\frac{15}{4}ab^2c^3 \leq \frac{2(100+13\sqrt{13})}{27}$$$$ 2a+ab^2+ 4ab^2c^3\leq \frac{4(68+5\sqrt{10})}{27}$$$$ 3a+ab^2+ \frac{9}{2}ab^2c^3\leq \frac{2(172+7\sqrt{7})}{27}$$$$a+ab^2+ 3.75982ab^2c^3 \leq \frac{2(100+13\sqrt{13})}{27}$$$$ 2a+ab^2+ 4.21981ab^2c^3\leq \frac{4(68+5\sqrt{10})}{27}$$$$ 3a+ab^2+4.73626ab^2c^3\leq \frac{2(172+7\sqrt{7})}{27}$$
0 replies
sqing
4 hours ago
0 replies
J
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Inspired by old results
sqing   0
5 hours ago
Source: Own
Let $ a,b,c \geq 0 , a+b+c+abc = 4.$ Prove that
$$ a+ab+2ab^2c^3 \leq \frac{25}{4}$$$$ 2a+ab+\frac{29}{10}ab^2c^3 \leq 9$$$$3a+ab+4ab^2c^3 \leq \frac{49}{4}$$$$ 2a+ab+2.9746371ab^2c^3 \leq 9$$$$3a+ab+4.062494ab^2c^3 \leq \frac{49}{4}$$
0 replies
sqing
5 hours ago
0 replies
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Number theory
MuradSafarli   1
N Apr 29, 2025 by Sadigly
Prove that for any natural number \( n \) :

\[ 1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3). \]
1 reply
MuradSafarli
Apr 29, 2025
Sadigly
Apr 29, 2025
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Number theory
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number theory
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MuradSafarli
110 posts
MuradSafarli
#1 h  Apr 29, 2025, 7:39 PM
Y by
Prove that for any natural number \( n \) :

\[ 1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3). \]
This post has been edited 1 time. Last edited by MuradSafarli, Apr 29, 2025, 7:41 PM
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Sadigly
227 posts
Sadigly
#3 h  Apr 29, 2025, 8:28 PM
Y by
Just try to write it like $\text{something something}\binom{8n+3}{4n+1}$
This post has been edited 1 time. Last edited by Sadigly, Apr 29, 2025, 8:32 PM
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