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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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Symmetric System's Cyclic Sum
worthawholebean   6
N 32 minutes ago by kilobyte144
If $ a+1=b+2=c+3=d+4=a+b+c+d+5$, then $ a+b+c+d$ is

$ \text{(A)}\ -5 \qquad \text{(B)}\ -10/3 \qquad \text{(C)}\ -7/3 \qquad \text{(D)}\ 5/3 \qquad \text{(E)}\ 5$
6 replies
worthawholebean
Feb 11, 2008
kilobyte144
32 minutes ago
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Weird expression being integer.
MarkBcc168   24
N 42 minutes ago by AR17296174
Source: IMO Shortlist 2017 N5
Find all pairs $(p,q)$ of prime numbers which $p>q$ and
$$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$is an integer.
24 replies
MarkBcc168
Jul 10, 2018
AR17296174
42 minutes ago
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Find the value of n - ILL 1990 MEX1
Amir Hossein   3
N an hour ago by maromex
During the class interval, $n$ children sit in a circle and play the game described below. The teacher goes around the children clockwisely and hands out candies to them according to the following regulations: Select a child, give him a candy; and give the child next to the first child a candy too; then skip over one child and give next child a candy; then skip over two children; give the next child a candy; then skip over three children; give the next child a candy;...

Find the value of $n$ for which the teacher can ensure that every child get at least one candy eventually (maybe after many circles).
3 replies
Amir Hossein
Sep 18, 2010
maromex
an hour ago
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Self-evident inequality trick
Lukaluce   14
N an hour ago by sqing
Source: 2025 Junior Macedonian Mathematical Olympiad P4
Let $x, y$, and $z$ be positive real numbers, such that $x^2 + y^2 + z^2 = 3$. Prove the inequality
\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]When does the equality hold?
14 replies
+1 w
Lukaluce
May 18, 2025
sqing
an hour ago
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maximum value
moldovan   4
N an hour ago by MathIQ.
Source: Austria 1989
Natural numbers $ a \le b \le c \le d$ satisfy $ a+b+c+d=30$. Find the maximum value of the product $ P=abcd.$
4 replies
moldovan
Jul 10, 2009
MathIQ.
an hour ago
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R to R, with x+f(xy)=f(1+f(y))x
NicoN9   5
N an hour ago by jasperE3
Source: Own.
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that\[ x+f(xy)=f(1+f(y))x \]for all $x, y\in \mathbb{R}$.
5 replies
NicoN9
May 11, 2025
jasperE3
an hour ago
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Chain of floors
Assassino9931   1
N 2 hours ago by pi_quadrat_sechstel
Source: Vojtech Jarnik IMC 2025, Category I, P2
Determine all real numbers $x>1$ such that
\[ \left\lfloor\frac{n+1}{x}\right\rfloor = n - \left\lfloor \frac{n}{x} \right\rfloor + \left \lfloor \frac{\left \lfloor \frac{n}{x} \right\rfloor}{x}\right \rfloor - \left \lfloor \frac{\left \lfloor \frac{\left\lfloor \frac{n}{x} \right\rfloor}{x} \right\rfloor}{x}\right \rfloor + \cdots \]for any positive integer $n$.
1 reply
Assassino9931
May 2, 2025
pi_quadrat_sechstel
2 hours ago
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2015 Taiwan TST Round 2 Quiz 1 Problem 2
wanwan4343   8
N 2 hours ago by Want-to-study-in-NTU-MATH
Source: 2015 Taiwan TST Round 2 Quiz 1 Problem 2
Let $\omega$ be the incircle of triangle $ABC$ and $\omega$ touches $BC$ at $D$. $AD$ meets $\omega$ again at $L$. Let $K$ be $A$-excenter, and $M,N$ be the midpoint of $BC,KM$, respectively. Prove that $B,C,N,L$ are concyclic.
8 replies
wanwan4343
Jul 12, 2015
Want-to-study-in-NTU-MATH
2 hours ago
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Inspired by Butterfly
sqing   2
N 2 hours ago by sqing
Source: Own
Let $ a,b,c>0. $ Prove that
$$a^2+b^2+c^2+ab+bc+ca+abc-3(a+b+c) \geq 34-14\sqrt 7$$$$a^2+b^2+c^2+ab+bc+ca+abc-\frac{433}{125}(a+b+c) \geq \frac{2(57475-933\sqrt{4665})}{3125} $$
2 replies
sqing
Today at 8:52 AM
sqing
2 hours ago
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Triangle form by perpendicular bisector
psi241   52
N 2 hours ago by ihatemath123
Source: IMO Shortlist 2018 G5
Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. A line $\ell$ intersects the lines $AI$, $BI$, and $CI$ at points $D$, $E$, and $F$, respectively, distinct from the points $A$, $B$, $C$, and $I$. The perpendicular bisectors $x$, $y$, and $z$ of the segments $AD$, $BE$, and $CF$, respectively determine a triangle $\Theta$. Show that the circumcircle of the triangle $\Theta$ is tangent to $\Omega$.
52 replies
psi241
Jul 17, 2019
ihatemath123
2 hours ago
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Inequalities
sqing   5
N 2 hours ago by sqing
Let $ a,b,c>0. $ Prove that$$a^2+b^2+c^2+abc-k(a+b+c)\geq 3k+2-2(k+1)\sqrt{k+1}$$Where $7\geq k \in N^+.$
$$a^2+b^2+c^2+abc-3(a+b+c)\geq-5$$
5 replies
sqing
Yesterday at 2:23 PM
sqing
2 hours ago
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Regional Olympiad - FBH 2018 Grade 9 Problem 3
gobathegreat   5
N 2 hours ago by justaguy_69
Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018
Let $p$ and $q$ be prime numbers such that $p^2+pq+q^2$ is perfect square. Prove that $p^2-pq+q^2$ is prime
5 replies
gobathegreat
Sep 18, 2018
justaguy_69
2 hours ago
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Find the value of k/m
zolfmark   4
N 2 hours ago by Shan3t
If we have
K=1/1+1/3+1/4+......1/2019

m=2018+2017/2+2016/3+......2/2017+1/2018
4 replies
zolfmark
Mar 1, 2024
Shan3t
2 hours ago
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Inequalities
sqing   1
N 3 hours ago by sqing
Let $ a,b,c\geq 0 ,a+b+c\leq 3. $ Prove that
$$a^2+b^2+c^2+ab +2ca+2bc + abc \leq \frac{251}{27}$$$$ a^2+b^2+c^2+ab+2ca+2bc + \frac{2}{5}abc \leq \frac{4861}{540}$$$$ a^2+b^2+c^2+ab+2ca+2bc + \frac{7}{20}abc \leq \frac{2381411}{26460}$$
1 reply
sqing
4 hours ago
sqing
3 hours ago
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Sequence
lgx57   8
N Apr 30, 2025 by Vivaandax
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.
8 replies
lgx57
Apr 27, 2025
Vivaandax
Apr 30, 2025
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Sequence
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Reveal topic tags
algebra
Sequences
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lgx57
43 posts
lgx57
#1 h  Apr 27, 2025, 12:56 PM
Y by
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.
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lgx57
43 posts
lgx57
#2 h  Apr 27, 2025, 12:57 PM
Y by
I can only find that $a_n \sim \sqrt{2n}$.
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aops-g5-gethsemanea2
3456 posts
aops-g5-gethsemanea2
#3 h  Apr 27, 2025, 2:05 PM
Y by
lgx57 wrote:
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.

do you mean closed form or explicit formula of $a_n$?
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lgx57
43 posts
lgx57
#4 h  Apr 27, 2025, 2:19 PM
Y by
aops-g5-gethsemanea2 wrote:
lgx57 wrote:
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.

do you mean closed form or explicit formula of $a_n$?

Just find a function $f$ ,s.t. $a_n=f(n)$
Z K Y
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steve4916
62 posts
steve4916
#5 h  Apr 27, 2025, 2:24 PM
Y by
now prove me if im wrong but there is no simple closed form for this
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lgx57
43 posts
lgx57
#6 h  Apr 27, 2025, 2:48 PM
Y by
steve4916 wrote:
now prove me if im wrong but there is no simple closed form for this

Why?
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johnnie.walker
2 posts
johnnie.walker
#7 h  Apr 29, 2025, 1:42 PM
Y by
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jasperE3
11366 posts
jasperE3
#8 h  Apr 29, 2025, 3:38 PM
Y by
lgx57 wrote:
steve4916 wrote:
now prove me if im wrong but there is no simple closed form for this

Why?

why would there be a closed form
Z K Y
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Vivaandax
87 posts
Vivaandax
#9 h  Apr 30, 2025, 5:08 AM
Y by
You can bound the value of a_n quite well (consider IMO Shortlist 1975 Problem 14), but there is not an explicit formula to calculate a_n.
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