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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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A strange NT problem
flower417477   0
14 minutes ago
Source: unknown
$p$ is a given prime number.$A=\{a_1,a_2,\cdots,a_{p-1}\}$ is a set which $\prod\limits_{i=1}^{p-1}a_i\equiv\frac{p-1}{2}\pmod p$.
Prove that there're at least $\frac{p-1}{2}$ non-empty subsets $B$ of $A$ such that $\sum\limits_{b\in B}b\equiv 1\pmod p$
0 replies
flower417477
14 minutes ago
0 replies
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combi/nt
blug   0
33 minutes ago
Prove that every positive integer $n$ can be written in the form
$$n=2^{a_1}3^{b_1}+2^{a_2}3^{b_2}+..., $$where $a_i, b_j$ are non negative integers, such that
$$2^x3^y\nmid 2^z3^t$$for every $x, y, z, t$.
0 replies
blug
33 minutes ago
0 replies
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Interesting inequalities
sqing   2
N 39 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0 , a(b+c)=k.$ Prove that
$$\frac{1}{a+1}+\frac{2}{b+1}+\frac{1}{c+1}\geq \frac{4\sqrt{k}-6}{ k-2}$$Where $5\leq k\in N^+.$
Let $ a,b,c\geq 0 , a(b+c)=9.$ Prove that
$$\frac{1}{a+1}+\frac{2}{b+1}+\frac{1}{c+1}\geq \frac{6}{7}$$
2 replies
sqing
2 hours ago
sqing
39 minutes ago
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Find smallest value of (x^2 + y^2 + z^2)/(xyz)
orl   7
N an hour ago by Bryan0224
Source: CWMO 2001, Problem 4
Let $ x, y, z$ be real numbers such that $ x + y + z \geq xyz$. Find the smallest possible value of $ \frac {x^2 + y^2 + z^2}{xyz}$.
7 replies
orl
Dec 27, 2008
Bryan0224
an hour ago
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shadow of a cylinder, shadow of a cone
vanstraelen   0
an hour ago

a) Given is a right cylinder of height $2R$ and radius $R$.
The sun shines on this solid at an angle of $45^{\circ}$.
What is the area of the shadow that this solid casts on the plane of the botom base?

b) Given is a right cone of height $2R$ and radius $R$.
The sun shines on this solid at an angle of $45^{\circ}$.
What is the area of the shadow that this solid casts on the plane of the base?
0 replies
vanstraelen
an hour ago
0 replies
J
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easy substitutions for a functional in reals
Circumcircle   9
N an hour ago by Bardia7003
Source: Kosovo Math Olympiad 2025, Grade 11, Problem 2
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ with the property that for every real numbers $x$ and $y$ it holds that
$$f(x+yf(x+y))=f(x)+f(xy)+y^2.$$
9 replies
Circumcircle
Nov 16, 2024
Bardia7003
an hour ago
J
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writing words around circle, two letters
jasperE3   1
N an hour ago by pi_quadrat_sechstel
Source: VJIMC 2000 2.2
If we write the sequence $\text{AAABABBB}$ along the perimeter of a circle, then every word of the length $3$ consisting of letters $A$ and $B$ (i.e. $\text{AAA}$, $\text{AAB}$, $\text{ABA}$, $\text{BAB}$, $\text{ABB}$, $\text{BBB}$, $\text{BBA}$, $\text{BAA}$) occurs exactly once on the perimeter. Decide whether it is possible to write a sequence of letters from a $k$-element alphabet along the perimeter of a circle in such a way that every word of the length $l$ (i.e. an ordered $l$-tuple of letters) occurs exactly once on the perimeter.
1 reply
jasperE3
Jul 27, 2021
pi_quadrat_sechstel
an hour ago
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Arithmetic Sequence of Products
GrantStar   19
N an hour ago by OronSH
Source: IMO Shortlist 2023 N4
Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products
\[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\]form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.
19 replies
GrantStar
Jul 17, 2024
OronSH
an hour ago
J
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Inequality Involving Complex Numbers with Modulus Less Than 1
tom-nowy   0
an hour ago
Let $x,y,z$ be complex numbers such that $|x|<1, |y|<1,$ and $|z|<1$.
Prove that $$ |x+y+z|^2 +3>|xy+yz+zx|^2+3|xyz|^2 .$$
0 replies
1 viewing
tom-nowy
an hour ago
0 replies
J
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Inequality
nguyentlauv   2
N an hour ago by nguyentlauv
Source: Own
Let $a,b,c$ be positive real numbers such that $ab+bc+ca=3$ and $k\ge 0$, prove that
$$\frac{\sqrt{a+1}}{b+c+k}+\frac{\sqrt{b+1}}{c+a+k}+\frac{\sqrt{c+1}}{a+b+k} \geq \frac{3\sqrt{2}}{k+2}.$$
2 replies
nguyentlauv
May 6, 2025
nguyentlauv
an hour ago
J
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japan 2021 mo
parkjungmin   0
an hour ago

The square box question

Is there anyone who can release it
0 replies
parkjungmin
an hour ago
0 replies
J
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Triangle on a tetrahedron
vanstraelen   0
an hour ago

Given a regular tetrahedron $(A,BCD)$ with edges $l$.
Construct at the apex $A$ three perpendiculars to the three lateral faces.
Take a point on each perpendicular at a distance $l$ from the apex such that these three points lie above the apex.
Calculate the lenghts of the sides of the triangle.
0 replies
vanstraelen
an hour ago
0 replies
J
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Cube Sphere
vanstraelen   4
N 2 hours ago by pieMax2713

Given the cube $\left(\begin{array}{ll} EFGH \\ ABCD \end{array}\right)$ with edge $6$ cm.
Find the volume of the sphere passing through $A,B,C,D$ and tangent to the plane $(EFGH)$.
4 replies
vanstraelen
3 hours ago
pieMax2713
2 hours ago
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Square number
linkxink0603   3
N 2 hours ago by Zok_G8D
Find m is positive interger such that m^4+3^m is square number
3 replies
linkxink0603
5 hours ago
Zok_G8D
2 hours ago
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J
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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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lgx57
41 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
41 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
41 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. $a_n=f(n)$
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steve4916
18 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
41 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
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jasperE3
11310 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
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Vivaandax
84 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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