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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Thursday at 11:16 PM
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
1 viewing
jlacosta
Thursday at 11:16 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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Calculating combinatorial numbers
lgx57   6
N 9 minutes ago by persamaankuadrat
Try to simplify this expression:

$$\sum_{i=1}^n \sum_{j=1}^i C_{n}^i C_{n}^j$$
6 replies
lgx57
Mar 30, 2025
persamaankuadrat
9 minutes ago
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Sharygin 2025 CR P18
Gengar_in_Galar   5
N an hour ago by hectorleo123
Source: Sharygin 2025
Let $ABCD$ be a quadrilateral such that the excircles $\omega_{1}$ and $\omega_{2}$ of triangles $ABC$ and $BCD$ touching their sides $AB$ and $BD$ respectively touch the extension of $BC$ at the same point $P$. The segment $AD$ meets $\omega_{2}$ at point $Q$, and the line $AD$ meets $\omega_{1}$ at $R$ and $S$. Prove that one of angles $RPQ$ and $SPQ$ is right
Proposed by: I.Kukharchuk
5 replies
Gengar_in_Galar
Mar 10, 2025
hectorleo123
an hour ago
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BMO 2025
GreekIdiot   10
N an hour ago by tranducphat
Does anyone have the problems? They should have finished by now.
10 replies
GreekIdiot
Apr 27, 2025
tranducphat
an hour ago
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Infinitely many numbers of a given form
Stefan4024   19
N 2 hours ago by cursed_tangent1434
Source: EGMO 2016 Day 2 Problem 6
Let $S$ be the set of all positive integers $n$ such that $n^4$ has a divisor in the range $n^2 +1, n^2 + 2,...,n^2 + 2n$. Prove that there are infinitely many elements of $S$ of each of the forms $7m, 7m+1, 7m+2, 7m+5, 7m+6$ and no elements of $S$ of the form $7m+3$ and $7m+4$, where $m$ is an integer.
19 replies
Stefan4024
Apr 13, 2016
cursed_tangent1434
2 hours ago
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Very easy case of a folklore polynomial equation
Assassino9931   1
N 2 hours ago by iamnotgentle
Source: Bulgaria EGMO TST 2025 P6
Determine all polynomials $P(x)$ of odd degree with real coefficients such that $P(x^2 + 2025) = P(x)^2 + 2025$.
1 reply
Assassino9931
3 hours ago
iamnotgentle
2 hours ago
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Process on scalar products and permutations
Assassino9931   2
N 2 hours ago by Assassino9931
Source: RMM Shortlist 2024 C1
Fix an integer $n\geq 2$. Consider $2n$ real numbers $a_1,\ldots,a_n$ and $b_1,\ldots, b_n$. Let $S$ be the set of all pairs $(x, y)$ of real numbers for which $M_i = a_ix + b_iy$, $i=1,2,\ldots,n$ are pairwise distinct. For every such pair sort the corresponding values $M_1, M_2, \ldots, M_n$ increasingly and let $M(i)$ be the $i$-th term in the list thus sorted. This denes an index permutation of $1,2,\ldots,n$. Let $N$ be the number of all such permutations, as the pairs run through all of $S$. In terms of $n$, determine the largest value $N$ may achieve over all possible choices of $a_1,\ldots,a_n,b_1,\ldots,b_n$.
2 replies
Assassino9931
4 hours ago
Assassino9931
2 hours ago
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Square problem
Jackson0423   2
N 2 hours ago by Jackson0423
Construct a square such that the distances from an interior point to the vertices (in clockwise order) are
1,7,8,4 respectively.
2 replies
Jackson0423
Yesterday at 4:08 PM
Jackson0423
2 hours ago
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IMO Shortlist Problems
ABCD1728   2
N 2 hours ago by ABCD1728
Source: IMO official website
Where can I get the official solution for ISL before 2005? The official website only has solutions after 2006. Thanks :)
2 replies
ABCD1728
Yesterday at 12:44 PM
ABCD1728
2 hours ago
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Estimate on number of progressions
Assassino9931   1
N 2 hours ago by BlizzardWizard
Source: RMM Shortlist 2024 C4
Let $n$ be a positive integer. For a set $S$ of $n$ real numbers, let $f(S)$ denote the number of increasing arithmetic progressions of length at least two all of whose terms are in $S$. Prove that, if $S$ is a set of $n$ real numbers, then
\[ f(S) \leq \frac{n^2}{4} + f(\{1,2,\ldots,n\})\]
1 reply
Assassino9931
4 hours ago
BlizzardWizard
2 hours ago
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2^x+3^x = yx^2
truongphatt2668   10
N 2 hours ago by MittenpunktpointX9
Prove that the following equation has infinite integer solutions:
$$2^x+3^x = yx^2$$
10 replies
truongphatt2668
Apr 22, 2025
MittenpunktpointX9
2 hours ago
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find the radius of circumcircle!
jennifreind   1
N 2 hours ago by ricarlos
In $\triangle \rm ABC$, $ \angle \rm B$ is acute, $\rm{\overline{BC}} = 8$, and $\rm{\overline{AC}} = 3\rm{\overline{AB}}$. Let point $\rm D$ be the intersection of the tangent to the circumcircle of $\triangle \rm ABC$ at point $\rm A$ and the perpendicular bisector of segment $\rm{\overline{BC}}$. Given that $\rm{\overline{AD}} = 6$, find the radius of the circumcircle of $\triangle \rm BCD$.
IMAGE
1 reply
jennifreind
Yesterday at 2:12 PM
ricarlos
2 hours ago
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Find the functions
Ecrin_eren   7
N 3 hours ago by jasperE3
"Find all differentiable functions f that satisfy the condition f(x) + f(y) = f((x + y) / (1 - xy)) for all x, y ∈ R, where xy ≠ 1."
7 replies
1 viewing
Ecrin_eren
Thursday at 8:58 PM
jasperE3
3 hours ago
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2025 CMIMC team p7, rephrased
scannose   12
N 3 hours ago by lpieleanu
In the expansion of $(x^2 + x + 1)^{2024}$, find the number of terms with coefficient divisible by $3$.
12 replies
scannose
Apr 18, 2025
lpieleanu
3 hours ago
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Floor function
Ecrin_eren   1
N 5 hours ago by alexheinis

How many different reel value of a are there which satisfies the equation floor(a) [a-floor(a)]=2024a ?
1 reply
Ecrin_eren
Yesterday at 5:38 PM
alexheinis
5 hours ago
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Recursion
Sid-darth-vater   6
N Apr 20, 2025 by vanstraelen
Help, I can't characterize ts and I dunno what to do
6 replies
Sid-darth-vater
Apr 20, 2025
vanstraelen
Apr 20, 2025
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Recursion
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Reveal topic tags
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Sid-darth-vater
42 posts
Sid-darth-vater
#1 h  Apr 20, 2025, 3:02 AM
Y by
Help, I can't characterize ts and I dunno what to do
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aidan0626
1884 posts
aidan0626
#2 h  Apr 20, 2025, 5:14 AM
Y by
well by testing small cases i got the general form to be Click to reveal hidden text, which can probably be proven by induction
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Sid-darth-vater
42 posts
Sid-darth-vater
#3 h  Apr 20, 2025, 3:40 PM
Y by
Ahh yeah, it works out. tyty. also, can you kinda explain how you thought of the numerator part? like I had figured out $a_2 = \frac{7}{24}, a_3 = \frac{13}{96},$ and $a_4 = \frac{25}{384}$ but I couldn't find the $3 \cdot 2^{n-1} + 1$ portion.
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vanstraelen
9005 posts
vanstraelen
#4 h  Apr 20, 2025, 4:22 PM
Y by
$a_{1}=\frac{4}{6},a_{2}=\frac{7}{24},a_{3}=\frac{13}{96},a_{4}=\frac{25}{384},a_{5}=\frac{49}{1536},a_{6}=\frac{97}{6144},\cdots$

$a_{n}=\frac{3 \cdot 2^{n-1} +1}{3 \cdot 2^{2n-1}}$.
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aidan0626
1884 posts
aidan0626
#5 h  Apr 20, 2025, 4:55 PM
Y by
Sid-darth-vater wrote:
Ahh yeah, it works out. tyty. also, can you kinda explain how you thought of the numerator part? like I had figured out $a_2 = \frac{7}{24}, a_3 = \frac{13}{96},$ and $a_4 = \frac{25}{384}$ but I couldn't find the $3 \cdot 2^{n-1} + 1$ portion.
well I noticed that the differences between consecutive numerators was 3, 6, 12, etc.
and so I got a geometric series which resulted in that
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Sid-darth-vater
42 posts
Sid-darth-vater
#6 h  Apr 20, 2025, 5:52 PM
Y by
mmmm i see, thanks!
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vanstraelen
9005 posts
vanstraelen
#7 h  Apr 20, 2025, 5:59 PM
Y by
$a_{k}=\frac{3 \cdot 2^{k-1} +1}{3 \cdot 2^{2k-1}}=2^{-k}+\frac{1}{3} \cdot 2^{1-2k}$.

$S=\lim_{n \to \infty} \sum_{k=1}^{n}a_{k}=\lim_{n \to \infty} \sum_{k=1}^{n}\left[2^{-k}+\frac{1}{3} \cdot 2^{1-2k}\right]$,
$S=\lim_{n \to \infty} \left[1-2^{-n}+\frac{1}{3}(\frac{2}{3}-\frac{2^{1-2n}}{2})\right]=1+\frac{2}{9}=\frac{11}{9}$.
This post has been edited 1 time. Last edited by vanstraelen, Apr 20, 2025, 6:00 PM
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