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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
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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Find the number of ordered triples (p,q,r)
Darealzolt   2
N 5 minutes ago by elizhang101412
Let \(p,q,r\) be prime numbers such that
\[ \frac{1}{pq}+\frac{1}{qr}+\frac{1}{pr}=\frac{1}{839} \]Hence find the numbers of ordered triples \(\{p,q,r\}\)
2 replies
Darealzolt
39 minutes ago
elizhang101412
5 minutes ago
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Find the sum of all the products a_i a_j
Darealzolt   1
N 18 minutes ago by alexheinis
Among the 100 constants \( \{ a_1,a_2,a_3,\dots,a_{100} \} \),there are \(39\) equal to \( -1\), and \(61\) equal to \(1\). Find the sum of all the products \(a_i a_j\) , where \(a \leq i < j \leq 100\).
1 reply
Darealzolt
3 hours ago
alexheinis
18 minutes ago
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Max and Min
Butterfly   0
39 minutes ago

Let $a_1,a_2,\cdots,a_n$ be an arrangement of $\{1,2,3,\cdots,n\}$. Find the maximum and minimum values of $$\frac{a_1}{a_2}+\frac{a_2}{a_3}+\cdots+\frac{a_{n-1}}{a_n}+\frac{a_n}{a_1}.$$
0 replies
+1 w
Butterfly
39 minutes ago
0 replies
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Cup of Combinatorics
M11100111001Y1R   4
N 43 minutes ago by sami1618
Source: Iran TST 2025 Test 4 Problem 2
There are \( n \) cups labeled \( 1, 2, \dots, n \), where the \( i \)-th cup has capacity \( i \) liters. In total, there are \( n \) liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.

$a)$ Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most \( \frac{4n}{3} \) steps.

$b)$ Prove that in at most \( \frac{5n}{3} \) steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
4 replies
M11100111001Y1R
May 27, 2025
sami1618
43 minutes ago
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(c^n+1)/(2^na+b) is an integer for all n
parmenides51   2
N an hour ago by Assassino9931
Source: Ukraine TST 2010 p6
Find all pairs of odd integers $a$ and $b$ for which there exists a natural number$ c$ such that the number $\frac{c^n+1}{2^na+b}$ is integer for all natural $n$.
2 replies
parmenides51
May 4, 2020
Assassino9931
an hour ago
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Nine point circle + Perpendicularities
YaoAOPS   18
N an hour ago by AndreiVila
Source: 2025 CTST P2
Suppose $\triangle ABC$ has $D$ as the midpoint of $BC$ and orthocenter $H$. Let $P$ be an arbitrary point on the nine point circle of $ABC$. The line through $P$ perpendicular to $AP$ intersects $BC$ at $Q$. The line through $A$ perpendicular to $AQ$ intersects $PQ$ at $X$. If $M$ is the midpoint of $AQ$, show that $HX \perp DM$.
18 replies
YaoAOPS
Mar 5, 2025
AndreiVila
an hour ago
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IOQM P26 2024
SomeonecoolLovesMaths   5
N an hour ago by SomeonecoolLovesMaths
The sum of $\lfloor x \rfloor$ for all real numbers $x$ satisfying the equation $16 + 15x + 15x^2 = \lfloor x \rfloor ^3$ is:
5 replies
SomeonecoolLovesMaths
Sep 8, 2024
SomeonecoolLovesMaths
an hour ago
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Inequality conjecture
RainbowNeos   0
an hour ago
Show (or deny) that there exists an absolute constant $C>0$ that, for all $n$ and $n$ positive real numbers $x_i ,1\leq i \leq n$, there is
\[\sum_{i=1}^n \frac{x_i^2}{\sum_{j=1}^i x_j}\geq C \ln n\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}\]
0 replies
RainbowNeos
an hour ago
0 replies
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inequality 2905
pennypc123456789   0
an hour ago
Consider positive real numbers \( x, y, z \) that satisfy the condition
\[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 3. \]Find the maximum value of the expression
\[ P = \dfrac{yz}{\sqrt[3]{3y^2z^2+ 3x^2y^2z^2+ x^2z^2 + x^2y^2}} + \frac{xz}{\sqrt[3]{3x^2z^2 + 3x^2y^2z^2 + x^2y^2 + y^2z^2}} + \frac{xy}{\sqrt[3]{3x^2y^2 + 3x^2y^2z^2 +y^2z^2 + x^2z^2}}. \]
0 replies
pennypc123456789
an hour ago
0 replies
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IOQM 2023-24 P-4
lifeismathematics   8
N an hour ago by SomeonecoolLovesMaths
Let $x, y$ be positive integers such that
$$ x^4=(x-1)\left(y^3-23\right)-1 . $$Find the maximum possible value of $x+y$.
8 replies
lifeismathematics
Sep 3, 2023
SomeonecoolLovesMaths
an hour ago
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Inspired by m4thbl3nd3r
sqing   3
N 2 hours ago by sqing
Source: Own
Let $ a, b,c>0,b+c>a$. Prove that$$\sqrt{\frac{a}{b+c-a}}-\frac{2a^2-b^2-c^2}{(a+b)(a+c)}\geq 1$$$$\frac{a}{b+c-a}-\frac{2a^2-b^2-c^2}{(a+b)(a+c)} \geq \frac{4\sqrt 2}{3}-1$$
3 replies
sqing
Today at 3:43 AM
sqing
2 hours ago
J
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Inspired by qrxz17
sqing   7
N 2 hours ago by sqing
Source: Own
Let $a, b,c>0 ,(a^2+b^2+c^2)^2 - 2(a^4+b^4+c^4) = 27 $. Prove that $$a+b+c\geq 3\sqrt {3}$$
7 replies
sqing
6 hours ago
sqing
2 hours ago
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Geometry problem
Whatisthepurposeoflife   2
N 2 hours ago by Whatisthepurposeoflife
Source: Derived from MEMO 2024 I3
Triangle ∆ABC is scalene the circle w that goes through the points A and B intersects AC at E BC at D let the Lines BE and AD intersect at point F. And let the tangents A and B of circle w Intersect at point G.
Prove that C F and G are collinear
2 replies
Whatisthepurposeoflife
Yesterday at 1:45 PM
Whatisthepurposeoflife
2 hours ago
J
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A Sequence of +1's and -1's
ike.chen   36
N 2 hours ago by maromex
Source: ISL 2022/C1
A $\pm 1$-sequence is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
36 replies
ike.chen
Jul 9, 2023
maromex
2 hours ago
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Help me please
Hahahsafdk3l   2
N Mar 29, 2025 by Hahahsafdk3l
Find all functions $f: \mathbb{R^+} \to \mathbb{R^+}$ such that $f(2f(x) + f(y) + xyy) = xy + 2x + y, \forall x, y > 0$
2 replies
Hahahsafdk3l
Mar 29, 2025
Hahahsafdk3l
Mar 29, 2025
J
Help me please
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Reveal topic tags
functional equation
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Hahahsafdk3l
33 posts
Hahahsafdk3l
#1 h  Mar 29, 2025, 2:21 AM
Y by
Find all functions $f: \mathbb{R^+} \to \mathbb{R^+}$ such that $f(2f(x) + f(y) + xyy) = xy + 2x + y, \forall x, y > 0$
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jasperE3
11386 posts
jasperE3
#2 h  Mar 29, 2025, 2:40 AM
Y by
Hahahsafdk3l wrote:
Find all functions $f: \mathbb{R^+} \to \mathbb{R^+}$ such that $f(2f(x) + f(y) + xyy) = xy + 2x + y, \forall x, y > 0$

Almost certainly a typo of this question.
https://artofproblemsolving.com/community/c4h3536833p34392946
BTW there is a search function on AoPS that you can use, most problems you've posted have already been posted elsewhere on AoPS so there is no need for the post.
This post has been edited 2 times. Last edited by jasperE3, Mar 29, 2025, 2:53 AM
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Hahahsafdk3l
33 posts
Hahahsafdk3l
#3 h  Mar 29, 2025, 11:43 PM
Y by
thank you
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