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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
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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 marginal profit..
ArmiAldi   1
N 27 minutes ago by Juno_34
Source: can someone help me
The total profit selling x units of books is P(x) = (6x - 7)(9x - 8) .
Find the marginal average profit function?
1 reply
ArmiAldi
Mar 2, 2008
Juno_34
27 minutes ago
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Marginal Profit
NC4723   1
N 39 minutes ago by Juno_34
Please help me solve this
1 reply
NC4723
Dec 11, 2015
Juno_34
39 minutes ago
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Sequence
Titibuuu   2
N an hour ago by Tkn
Let \( a_1 = a \), and for all \( n \geq 1 \), define the sequence \( \{a_n\} \) by the recurrence
\[ a_{n+1} = a_n^2 + 1 \]Prove that there is no natural number \( n \) such that
\[ \prod_{k=1}^{n} \left( a_k^2 + a_k + 1 \right) \]is a perfect square.
2 replies
Titibuuu
Today at 2:22 AM
Tkn
an hour ago
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Combinatorics
imnotgoodatmathsorry   0
an hour ago
Source: By @irregular22104
Given two positive integers $a,b$ written on the board. We apply the following rule: At each step, we will add all the numbers that are the sum of the two numbers on the board so that the sum does not appear on the board. For example, if the two initial numbers are $2,5$; then the numbers on the board after step 1 are $2,5,7$; after step 2 are $2,5,7,9,12;...$
1) With $a = 3$; $b = 12$, prove that the number 2024 cannot appear on the board.
2) With $a = 2$; $b = 34$, prove that the number 2024 can appear on the board.
0 replies
imnotgoodatmathsorry
an hour ago
0 replies
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Iranian geometry configuration
Assassino9931   0
an hour ago
Source: Al-Khwarizmi Junior International Olympiad 2025 P7
Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$, such that $CD$ is not a diameter of its circumcircle. The lines $AD$ and $BC$ intersect at point $P$, so that $A$ lies between $D$ and $P$, and $B$ lies between $C$ and $P$. Suppose triangle $PCD$ is acute and let $H$ be its orthocenter. The points $E$ and $F$ on the lines $BC$ and $AD$, respectively, are such that $BD \parallel HE$ and $AC\parallel HF$. The line through $E$, perpendicular to $BC$, intersects $AD$ at $L$, and the line through $F$, perpendicular to $AD$, intersects $BC$ at $K$. Prove that the points $K$, $L$, $O$ are collinear.

Amir Parsa Hosseini Nayeri, Iran
0 replies
Assassino9931
an hour ago
0 replies
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Inequality, inequality, inequality...
Assassino9931   0
an hour ago
Source: Al-Khwarizmi Junior International Olympiad 2025 P6
Let $a,b,c$ be real numbers such that \[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]Find the smallest possible value of $a^2 + b^2 + c^2$.

Binh Luan and Nhan Xet, Vietnam
0 replies
Assassino9931
an hour ago
0 replies
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Prime sums of pairs
Assassino9931   0
an hour ago
Source: Al-Khwarizmi Junior International Olympiad 2025 P5
Sevara writes in red $8$ distinct positive integers and then writes in blue the $28$ sums of each two red numbers. At most how many of the blue numbers can be prime?

Marin Hristov, Bulgaria
0 replies
Assassino9931
an hour ago
0 replies
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help me solve this problem. Thanks
tnhan.129   0
an hour ago
Find f:R+ -> R such that:
(x+1/x).f(y) = f(xy) + f(y/x)
0 replies
tnhan.129
an hour ago
0 replies
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Line passing through orthocenter
pokmui9909   4
N an hour ago by Tkn
Source: KMO 2024 P7
In an acute triangle $ABC$, let a line $\ell$ pass through the orthocenter and not through point $A$. The line $\ell$ intersects line $BC$ at $P(\neq B, C)$. A line passing through $A$ and perpendicular to $\ell$ meets the circumcircle of triangle $ABC$ at $R(\neq A)$. Let the feet of the perpendiculars from $A, B$ to $\ell$ be $A', B'$, respectively. Define line $\ell_1$ as the line passing through $A'$ and perpendicular to $BC$, and line $\ell_2$ as the line passing through $B'$ and perpendicular to $CA$. Prove that if $Q$ is the reflection of the intersection of $\ell_1$ and $\ell_2$ across $\ell$, then $\angle PQR = 90^{\circ}$.
4 replies
pokmui9909
Nov 9, 2024
Tkn
an hour ago
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Inequality with number of divisors
MathLuis   12
N an hour ago by Grasshopper-
Source: Iberoamerican MO 2024 Day 1 P1
For each positive integer $n$, let $d(n)$ be the number of positive integer divisors of $n$.
Prove that for all pairs of positive integers $(a,b)$ we have that:
\[ d(a)+d(b) \le d(\gcd(a,b))+d(\text{lcm}(a,b)) \]and determine all pairs of positive integers $(a,b)$ where we have equality case.
12 replies
MathLuis
Sep 21, 2024
Grasshopper-
an hour ago
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No three collinear
USJL   1
N 2 hours ago by Photaesthesia
Source: 2025 Taiwan TST Round 3 Mock P6
Given a positive integer $n\geq 3$. A convex polygon is said to be $n$-good if it contains $n$ lattice points where any three of them are not collinear.

(a) Show that there exists an $n$-good convex polygon with area at most $4n^2$.
(b) Show that there exists a constant $c>0$ so that any $n$-good convex polygon has area at least $cn^2$.

Proposed by usjl
1 reply
USJL
Apr 26, 2025
Photaesthesia
2 hours ago
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Concentric Circles
MithsApprentice   61
N 2 hours ago by endless_abyss
Source: USAMO 1998
Let ${\cal C}_1$ and ${\cal C}_2$ be concentric circles, with ${\cal C}_2$ in the interior of ${\cal C}_1$. From a point $A$ on ${\cal C}_1$ one draws the tangent $AB$ to ${\cal C}_2$ ($B\in {\cal C}_2$). Let $C$ be the second point of intersection of $AB$ and ${\cal C}_1$, and let $D$ be the midpoint of $AB$. A line passing through $A$ intersects ${\cal C}_2$ at $E$ and $F$ in such a way that the perpendicular bisectors of $DE$ and $CF$ intersect at a point $M$ on $AB$. Find, with proof, the ratio $AM/MC$.
61 replies
MithsApprentice
Oct 9, 2005
endless_abyss
2 hours ago
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Romania NMO 2023 Grade 11 P1
DanDumitrescu   15
N 5 hours ago by anudeep
Source: Romania National Olympiad 2023
Determine twice differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which verify relation

\[ \left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}. \]
15 replies
DanDumitrescu
Apr 14, 2023
anudeep
5 hours ago
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Subset Ordered Pairs of {1, 2, ..., 10}
ahaanomegas   11
N 5 hours ago by cappucher
Source: Putnam 1990 A6
If $X$ is a finite set, let $X$ denote the number of elements in $X$. Call an ordered pair $(S,T)$ of subsets of $ \{ 1, 2, \cdots, n \} $ $ \emph {admissible} $ if $ s > |T| $ for each $ s \in S $, and $ t > |S| $ for each $ t \in T $. How many admissible ordered pairs of subsets $ \{ 1, 2, \cdots, 10 \} $ are there? Prove your answer.
11 replies
ahaanomegas
Jul 12, 2013
cappucher
5 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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Reveal topic tags
differential equation
calculus
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Riptide1901
2705 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
865 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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