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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
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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Find the value
sqing   12
N 25 minutes ago by Phat_23000245
Source: 2024 China Fujian High School Mathematics Competition
Let $f(x)=a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0,$ $a_i\in\{-1,1\} ,i=0,1,2,\cdots,6 $ and $f(2)=-53 .$ Find the value of $f(1).$
12 replies
sqing
Jun 22, 2024
Phat_23000245
25 minutes ago
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Midpoints of arcs form a similar triangle
v_Enhance   19
N 25 minutes ago by Adywastaken
Source: USA TSTST 2013, Problem 1
Let $ABC$ be a triangle and $D$, $E$, $F$ be the midpoints of arcs $BC$, $CA$, $AB$ on the circumcircle. Line $\ell_a$ passes through the feet of the perpendiculars from $A$ to $DB$ and $DC$. Line $m_a$ passes through the feet of the perpendiculars from $D$ to $AB$ and $AC$. Let $A_1$ denote the intersection of lines $\ell_a$ and $m_a$. Define points $B_1$ and $C_1$ similarly. Prove that triangle $DEF$ and $A_1B_1C_1$ are similar to each other.
19 replies
v_Enhance
Aug 13, 2013
Adywastaken
25 minutes ago
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find question
mathematical-forest   4
N 26 minutes ago by GreekIdiot
Are there any contest questions that seem simple but are actually difficult? :-D
4 replies
mathematical-forest
2 hours ago
GreekIdiot
26 minutes ago
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4 var inequality
SunnyEvan   1
N 28 minutes ago by SunnyEvan
Source: Own
Let $ x,y,z,t \in R^+ ,$ such that : $ (x+y+z+t)^2 = x+y+z+t + (x+z)(y+t) $ and $ x \geq y \geq z \geq t .$
Try to prove or disprove : $$ \frac{2 \sqrt{x+y+z+t +(x+t)(y+z)}}{x^2+y^2+z^2+t^2 +3xz+3yt+xt+yz} \geq \frac{11(x+y)(z+t)-(x+y+z+t)}{x+y+z+t +(x+z)(y+t)} $$
1 reply
SunnyEvan
5 hours ago
SunnyEvan
28 minutes ago
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Serbian selection contest for the IMO 2025 - P6
OgnjenTesic   14
N an hour ago by math90
Source: Serbian selection contest for the IMO 2025
For an $n \times n$ table filled with natural numbers, we say it is a divisor table if:
- the numbers in the $i$-th row are exactly all the divisors of some natural number $r_i$,
- the numbers in the $j$-th column are exactly all the divisors of some natural number $c_j$,
- $r_i \ne r_j$ for every $i \ne j$.

A prime number $p$ is given. Determine the smallest natural number $n$, divisible by $p$, such that there exists an $n \times n$ divisor table, or prove that such $n$ does not exist.

Proposed by Pavle Martinović
14 replies
OgnjenTesic
May 22, 2025
math90
an hour ago
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Three Distinct Divisors Sum to 2022
ike.chen   31
N an hour ago by Jupiterballs
Source: ISL 2022/N1
A number is called Norwegian if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number.
(Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)
31 replies
ike.chen
Jul 9, 2023
Jupiterballs
an hour ago
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MOP 2012 Inequality
holdmyquadrilateral   3
N an hour ago by Adywastaken
Source: MOP 2012
For $a,b,c>0$, prove that
\[\frac{a^3}{(b-c)^2+bc}+\frac{b^3}{(c-a)^2+ca}+\frac{c^3}{(a-b)^2+ab}\ge a+b+c.\]
3 replies
holdmyquadrilateral
Mar 11, 2023
Adywastaken
an hour ago
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Prove the statement
Butterfly   12
N 2 hours ago by oty
Given an infinite sequence $\{x_n\} \subseteq [0,1]$, there exists some constant $C$, for any $r>0$, among the sequence $x_n$ and $x_m$ could be chosen to satisfy $|n-m|\ge r $ and $|x_n-x_m|<\frac{C}{|n-m|}$.
12 replies
Butterfly
May 7, 2025
oty
2 hours ago
J
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strange geometry problem
Zavyk09   2
N 2 hours ago by Zavyk09
Source: own
Let $ABC$ be a triangle with circumcenter $O$ and internal bisector $AD$. Let $AD$ cuts $(O)$ again at $M$ and $MO$ cuts $(O)$ again at $N$. Point $L$ lie on $AD$ such that $(AD, LM) = -1$. The line pass through $L$ and perpendicular to $AD$ intersects $NC, NB$ at $P, Q$ respectively. Let circumcircle of $\triangle NPQ$ cuts $(O)$ at $G \ne N$. Prove that $\angle AGD = 90^{\circ}$.
2 replies
Zavyk09
Yesterday at 4:32 PM
Zavyk09
2 hours ago
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exponential diophantine with factorials
skellyrah   5
N 2 hours ago by skellyrah
find all non negative integers (x,y) such that $$ x! + y! = 2025^x + xy$$
5 replies
skellyrah
Feb 24, 2025
skellyrah
2 hours ago
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A second final attempt to make a combinatorics problem
JARP091   2
N 2 hours ago by JARP091
Source: At the time of writing this problem I do not know the source if any
Arthur Morgan is playing a game.

He has $n$ eggs, each with a hardness value $k_1, k_2, \dots, k_n$, where $\{k_1, k_2, \dots, k_n\}$ is a permutation of the set $\{1, 2, \dots, n\}$. He is throwing the eggs from an $m$-floor building.

When the $i$-th egg is dropped from the $j$-th floor, its new hardness becomes
\[ \left\lfloor \frac{k_i}{j+1} \right\rfloor. \]If $\left\lfloor \frac{k_i}{j+1} \right\rfloor = 0$, then the egg breaks and cannot be used again.

Arthur can drop each egg from a particular floor at most once.
For which values of $n$ and $m$ can Arthur always determine the correct ordering of the eggs according to their initial hardness values?
Note: The problem might be wrong or too easy
2 replies
JARP091
May 25, 2025
JARP091
2 hours ago
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Putnam 1981 A3
sqrtX   1
N 2 hours ago by Mathzeus1024
Source: Putnam 1981
Find
$$ \lim_{t\to \infty} e^{-t} \int_{0}^{t} \int_{0}^{t} \frac{e^x -e^y }{x-y} \,dx\,dy,$$or show that the limit does not exist.
1 reply
sqrtX
Mar 31, 2022
Mathzeus1024
2 hours ago
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Recurrence trouble
SomeonecoolLovesMaths   2
N 4 hours ago by SomeonecoolLovesMaths
Let $0 < x_0 < y_0$ be real numbers. Define $x_{n+1} = \frac{x_n + y_n}{2}$ and $y_{n+1} = \sqrt{x_{n+1}y_n}$.
Prove that $\lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n$ and hence find the limit.
2 replies
SomeonecoolLovesMaths
Yesterday at 11:27 AM
SomeonecoolLovesMaths
4 hours ago
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Trigo or Complex no.?
hzbrl   3
N 4 hours ago by GreenKeeper
(a) Let $y=\cos \phi+\cos 2 \phi$, where $\phi=\frac{2 \pi}{5}$. Verify by direct substitution that $y$ satisfies the quadratic equation $2 y^2=3 y+2$ and deduce that the value of $y$ is $-\frac{1}{2}$.
(b) Let $\theta=\frac{2 \pi}{17}$. Show that $\sum_{k=0}^{16} \cos k \theta=0$
(c) If $z=\cos \theta+\cos 2 \theta+\cos 4 \theta+\cos 8 \theta$, show that the value of $z$ is $-(1-\sqrt{17}) / 4$.



I could solve (a) and (b). Can anyone help me with the 3rd part please?
3 replies
hzbrl
May 27, 2025
GreenKeeper
4 hours ago
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Question
Riptide1901   1
N Apr 19, 2025 by Mathzeus1024
Isn't their a left hand side of the graph that they're leaving out? I'm confused why they only consider $x>\sqrt{15}/2$ and not $x<-\sqrt{15}/2.$
1 reply
Riptide1901
Jan 29, 2025
Mathzeus1024
Apr 19, 2025
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Question
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Reveal topic tags
differential equation
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Riptide1901
2731 posts
Riptide1901
#1 h  Jan 29, 2025, 1:35 PM
Y by
Isn't their a left hand side of the graph that they're leaving out? I'm confused why they only consider $x>\sqrt{15}/2$ and not $x<-\sqrt{15}/2.$
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Mathzeus1024
938 posts
Mathzeus1024
#2 h  Apr 19, 2025, 9:23 AM
Y by
I think you are correct since the solution is the hyperbola $x^2- \left(y+\frac{1}{2}\right)^2 = \frac{15}{4}$ whose domain is:

$x \in \left(-\infty,-\frac{\sqrt{15}}{2}\right] \cup \left[\frac{\sqrt{15}}{2},\infty\right)$

and we are not given any explicit restriction for the domain to be $\mathbb{R}^{+}$.
This post has been edited 3 times. Last edited by Mathzeus1024, Apr 19, 2025, 9:29 AM
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