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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 3:18 PM
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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0 replies
jlacosta
Yesterday at 3:18 PM
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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Basis vectors type question
RenheMiResembleRice   1
N 8 minutes ago by RenheMiResembleRice
Source: Yurou Ju, Tuo Guan
Solve the following attached with explanation.
1 reply
RenheMiResembleRice
2 hours ago
RenheMiResembleRice
8 minutes ago
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Strange limit
Snoop76   5
N 41 minutes ago by Alphaamss
Find: $\lim_{n \to \infty} n\cdot\sum_{k=1}^n \frac 1 {k(n-k)!}$
5 replies
Snoop76
Mar 29, 2025
Alphaamss
41 minutes ago
J
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Game About Passing Pencils
WilliamSChen   0
an hour ago
A group of $n$ children sit in a circle facing inward with $n > 2$, and each child starts with an arbitrary even number of pencils. Each minute, each child simultaneously passes exactly half of all of their pencils to the child to their right. Then, all children that have an odd number of pencils receive one more pencil.
Prove that after a finite amount of time, the children will all have the same number of pencils.

I do not know the source.
0 replies
WilliamSChen
an hour ago
0 replies
J
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An nxn Checkboard
MithsApprentice   26
N an hour ago by NicoN9
Source: USAMO 1999 Problem 1
Some checkers placed on an $n \times n$ checkerboard satisfy the following conditions:

(a) every square that does not contain a checker shares a side with one that does;

(b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side.

Prove that at least $(n^{2}-2)/3$ checkers have been placed on the board.
26 replies
MithsApprentice
Oct 3, 2005
NicoN9
an hour ago
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Is this FE solvable?
Mathdreams   4
N 2 hours ago by Mathdreams
Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ such that \[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]for all reals $x$ and $y$.
4 replies
Mathdreams
Tuesday at 6:58 PM
Mathdreams
2 hours ago
J
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Coaxial circles related to Gergon point
Headhunter   0
2 hours ago
Source: I tried but can't find the source...
Hi, everyone.

In $\triangle$$ABC$, $Ge$ is the Gergon point and the incircle $(I)$ touch $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively.
Let the circumcircles of $\triangle IDGe$, $\triangle IEGe$, $\triangle IFGe$ be $O_{1}$ , $O_{2}$ , $O_{3}$ respectively.

Reflect $O_{1}$ in $ID$ and then we get the circle $O'_{1}$
Reflect $O_{2}$ in $IE$ and then the circle $O'_{2}$
Reflect $O_{3}$ in $IF$ and then the circle $O'_{3}$

Prove that $O'_{1}$ , $O'_{2}$ , $O'_{3}$ are coaxial.
0 replies
Headhunter
2 hours ago
0 replies
J
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Equation with powers
a_507_bc   6
N 2 hours ago by EVKV
Source: Serbia JBMO TST 2024 P1
Find all non-negative integers $x, y$ and primes $p$ such that $$3^x+p^2=7 \cdot 2^y.$$
6 replies
a_507_bc
May 25, 2024
EVKV
2 hours ago
J
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no numbers of the form 80...01 are squares
Marius_Avion_De_Vanatoare   2
N 2 hours ago by EVKV
Source: Moldova JTST 2024 P5
Prove that a number of the form $80\dots01$ (there is at least 1 zero) can't be a perfect square.
2 replies
1 viewing
Marius_Avion_De_Vanatoare
Jun 10, 2024
EVKV
2 hours ago
J
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Equivalent definition for C^1 functions
Ciobi_   1
N 2 hours ago by KAME06
Source: Romania NMO 2025 11.3
Prove that, for a function $f \colon \mathbb{R} \to \mathbb{R}$, the following $2$ statements are equivalent:
a) $f$ is differentiable, with continuous first derivative.
b) For any $a\in\mathbb{R}$ and for any two sequences $(x_n)_{n\geq 1},(y_n)_{n\geq 1}$, convergent to $a$, such that $x_n \neq y_n$ for any positive integer $n$, the sequence $\left(\frac{f(x_n)-f(y_n)}{x_n-y_n}\right)_{n\geq 1}$ is convergent.
1 reply
Ciobi_
Yesterday at 1:54 PM
KAME06
2 hours ago
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f((x XOR f(y)) + y) = (f(x) XOR y) + y
the_universe6626   3
N 3 hours ago by jasperE3
Source: Janson MO 5 P4
Find all functions $f:\mathbb{Z}_{\ge0}\rightarrow\mathbb{Z}_{\ge0}$ such that
\[f((x\oplus f(y))+y)=(f(x)\oplus y)+y\]Note: $\oplus$ denotes the bitwise XOR operation. For example, $1001_2 \oplus 101_2 = 1100_2$.

(Proposed by ja.)
3 replies
the_universe6626
Feb 21, 2025
jasperE3
3 hours ago
J
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2024 8's
Marius_Avion_De_Vanatoare   3
N 3 hours ago by EVKV
Source: Moldova JTST 2024 P2
Prove that the number $ \underbrace{88\dots8}_\text{2024\; \textrm{times}}$ is divisible by 2024.
3 replies
Marius_Avion_De_Vanatoare
Jun 10, 2024
EVKV
3 hours ago
J
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pretty well known
dotscom26   0
3 hours ago
Let $\triangle ABC$ be a scalene triangle such that $\Omega$ is its incircle. $AB$ is tangent to $\Omega$ at $D$. A point $E$ ($E \notin \Omega$) is located on $BC$.

Let $\omega_1$, $\omega_2$, and $\omega_3$ be the incircles of the triangles $BED$, $ADE$, and $AEC$, respectively.

Show that the common tangent to $\omega_1$ and $\omega_3$ is also tangent to $\omega_2$.

0 replies
dotscom26
3 hours ago
0 replies
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Thanks u!
Ruji2018252   6
N 3 hours ago by jasperE3
Find all $f:\mathbb{R}\to\mathbb{R}$ and
\[ f(x+y)+f(x^2+f(y))=f(f(x))^2+f(x)+f(y)+y,\forall x,y\in\mathbb{R}\]
6 replies
Ruji2018252
Mar 26, 2025
jasperE3
3 hours ago
J
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Range of solutions to the log equation
obihs   1
N 4 hours ago by rchokler
Source: Own
Let $n$ be a positive integer, and consider the equation:
$$(\log x)^n - x + 1 = 0\quad\cdots(\heartsuit)$$Answer the following questions. You may assume that $2.7<e<2.72$ is known.

$(1)\quad$ Determine the number of real solutions of equation $(\heartsuit)$ for each $n$.

$(2)\quad$ For $n\ge 3$ , let $r_n$ be the segond largest real solution of $(\heartsuit)$.

$(\i)\quad$ Find $\alpha$ such that $\lim_{n\to\infty} r_n =\alpha.$

$(\i\i)\quad$ Find $\lfloor\beta\rfloor$, where $\beta$ is defined as

$$\lim_{n\to\infty}n(r_n-\alpha)=\beta.$$
1 reply
obihs
Tuesday at 6:18 PM
rchokler
4 hours ago
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Double derivative coated integral
RenheMiResembleRice   1
N Mar 28, 2025 by HacheB2031
Source: Lufang Yue, Lianru Meng
Show that J>0. For this to be true, I guess we also need the condition for c and d to be positive, isn't that right?
1 reply
RenheMiResembleRice
Mar 28, 2025
HacheB2031
Mar 28, 2025
J
Double derivative coated integral
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Reveal topic tags
calculus
derivative
integration
L
G H BBookmark kLocked kLocked NReply
Source: Lufang Yue, Lianru Meng
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RenheMiResembleRice
257 posts
RenheMiResembleRice
#1 h  Mar 28, 2025, 3:13 AM
Y by
Show that J>0. For this to be true, I guess we also need the condition for c and d to be positive, isn't that right?
Attachments:
This post has been edited 1 time. Last edited by RenheMiResembleRice, Mar 28, 2025, 3:14 AM
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HacheB2031
330 posts
HacheB2031
#2 h  Mar 28, 2025, 5:07 AM
Y by
there's infinite functions where $J<0$
like $H(x)=-e^x,$ $H'(x)=-e^x,$ $\displaystyle J=\int_c^dH''(x)\,\text dx=H'(d)-H'(c)=e^c-e^d,$ and since $c<d$ then $e^c<e^d$ and so $e^c-e^d=J<0$
there would need to be some restriction on $H$
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