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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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[*]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
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
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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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Sintetic geometry problem
ICE_CNME_4   1
N 16 minutes ago by Ianis
Source: Math Gazette Contest 2025
Let there be the triangle ABC and the points E ∈ (AC), F ∈ (AB), such that BE and CF are concurrent in O.
If {L} = AO ∩ EF and K ∈ BC, such that LK ⊥ BC, show that EKL = FKL.
1 reply
ICE_CNME_4
an hour ago
Ianis
16 minutes ago
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Sequences problem
BBNoDollar   1
N an hour ago by ICE_CNME_4
Source: Mathematical Gazette Contest
Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.

⁡Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
1 reply
BBNoDollar
5 hours ago
ICE_CNME_4
an hour ago
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Geometry with orthocenter config
thdnder   4
N an hour ago by ohhh
Source: Own
Let $ABC$ be a triangle, and let $AD, BE, CF$ be its altitudes. Let $H$ be its orthocenter, and let $O_B$ and $O_C$ be the circumcenters of triangles $AHC$ and $AHB$. Let $G$ be the second intersection of the circumcircles of triangles $FDO_B$ and $EDO_C$. Prove that the lines $DG$, $EF$, and $A$-median of $\triangle ABC$ are concurrent.
4 replies
thdnder
Apr 29, 2025
ohhh
an hour ago
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Random modulos
m4thbl3nd3r   6
N 2 hours ago by GreekIdiot
Find all pair of integers $(x,y)$ s.t $x^2+3=y^7$
6 replies
m4thbl3nd3r
Apr 7, 2025
GreekIdiot
2 hours ago
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Concurrency in Parallelogram
amuthup   89
N 2 hours ago by happypi31415
Source: 2021 ISL G1
Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.
89 replies
amuthup
Jul 12, 2022
happypi31415
2 hours ago
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deleting multiple or divisor in pairs from 2-50 on a blackboard
parmenides51   1
N 2 hours ago by TheBaiano
Source: 2023 May Olympiad L2 p3
The $49$ numbers $2,3,4,...,49,50$ are written on the blackboard . An allowed operation consists of choosing two different numbers $a$ and $b$ of the blackboard such that $a$ is a multiple of $b$ and delete exactly one of the two. María performs a sequence of permitted operations until she observes that it is no longer possible to perform any more. Determine the minimum number of numbers that can remain on the board at that moment.
1 reply
parmenides51
Mar 24, 2024
TheBaiano
2 hours ago
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at everystep a, b, c are replaced by a+\gcd(b,c), b+\gcd(a,c), c+\gcd(a,b)
NJAX   9
N 2 hours ago by atdaotlohbh
Source: 2nd Al-Khwarizmi International Junior Mathematical Olympiad 2024, Day2, Problem 8
Three positive integers are written on the board. In every minute, instead of the numbers $a, b, c$, Elbek writes $a+\gcd(b,c), b+\gcd(a,c), c+\gcd(a,b)$ . Prove that there will be two numbers on the board after some minutes, such that one is divisible by the other.
Note. $\gcd(x,y)$ - Greatest common divisor of numbers $x$ and $y$

Proposed by Sergey Berlov, Russia
9 replies
NJAX
May 31, 2024
atdaotlohbh
2 hours ago
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Easy complete system of residues problem in Taiwan TST
Fysty   6
N 3 hours ago by Primeniyazidayi
Source: 2025 Taiwan TST Round 1 Independent Study 1-N
Find all positive integers $n$ such that there exist two permutations $a_0,a_1,\ldots,a_{n-1}$ and $b_0,b_1,\ldots,b_{n-1}$ of the set $\lbrace0,1,\ldots,n-1\rbrace$, satisfying the condition
$$ia_i\equiv b_i\pmod{n}$$for all $0\le i\le n-1$.

Proposed by Fysty
6 replies
Fysty
Mar 5, 2025
Primeniyazidayi
3 hours ago
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JBMO Shortlist 2022 A2
Lukaluce   13
N 3 hours ago by Rayvhs
Source: JBMO Shortlist 2022
Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 3$. Prove that
$$\frac{x + 3}{y + z} + \frac{y + 3}{z + x} + \frac{z + 3}{x + y} + 3 \ge 27 \cdot \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^2}{(x + y + z)^3}.$$
Proposed by Petar Filipovski, Macedonia
13 replies
Lukaluce
Jun 26, 2023
Rayvhs
3 hours ago
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A very beautiful geo problem
TheMathBob   4
N 4 hours ago by ravengsd
Source: Polish MO Finals P2 2023
Given an acute triangle $ABC$ with their incenter $I$. Point $X$ lies on $BC$ on the same side as $B$ wrt $AI$. Point $Y$ lies on the shorter arc $AB$ of the circumcircle $ABC$. It is given that $$\angle AIX = \angle XYA = 120^\circ.$$Prove that $YI$ is the angle bisector of $XYA$.
4 replies
TheMathBob
Mar 29, 2023
ravengsd
4 hours ago
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Inspired by old results
sqing   6
N 4 hours ago by Jamalll
Source: Own
Let $ a,b>0 , a^2+b^2+ab+a+b=5 . $ Prove that
$$ \frac{ 1 }{a+b+ab+1}+\frac{6}{a^2+b^2+ab+1}\geq \frac{7}{4}$$$$ \frac{ 1 }{a+b+ab+1}+\frac{1}{a^2+b^2+ab+1}\geq \frac{1}{2}$$$$ \frac{41}{a+b+2}+\frac{ab}{a^3+b^3+2} \geq \frac{21}{2}$$
6 replies
sqing
Apr 29, 2025
Jamalll
4 hours ago
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A Duality Operation on Decreasing Integer Sequences
Ritangshu   0
4 hours ago
Let \( S \) be the set of all sequences \( (a_1, a_2, \ldots) \) of non-negative integers such that
(i) \( a_1 \geq a_2 \geq \cdots \); and
(ii) there exists a positive integer \( N \) such that \( a_n = 0 \) for all \( n \geq N \).

Define the dual of the sequence \( (a_1, a_2, \ldots) \in S \) to be the sequence \( (b_1, b_2, \ldots) \), where, for \( m \geq 1 \),
\( b_m \) is the number of \( a_n \)'s which are greater than or equal to \( m \).

(i) Show that the dual of a sequence in \( S \) belongs to \( S \).

(ii) Show that the dual of the dual of a sequence in \( S \) is the original sequence itself.

(iii) Show that the duals of distinct sequences in \( S \) are distinct.
0 replies
Ritangshu
4 hours ago
0 replies
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Property of a function
Ritangshu   0
4 hours ago
Let \( f(x, y) = xy \), where \( x \geq 0 \) and \( y \geq 0 \).
Prove that the function \( f \) satisfies the following property:

\[ f\left( \lambda x + (1 - \lambda)x',\; \lambda y + (1 - \lambda)y' \right) > \min\{f(x, y),\; f(x', y')\} \]
for all \( (x, y) \ne (x', y') \) and for all \( \lambda \in (0, 1) \).

0 replies
Ritangshu
4 hours ago
0 replies
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Subset of digits to express as a sum
anantmudgal09   46
N 4 hours ago by anudeep
Source: INMO 2020 P3
Let $S$ be a subset of $\{0,1,2,\dots ,9\}$. Suppose there is a positive integer $N$ such that for any integer $n>N$, one can find positive integers $a,b$ so that $n=a+b$ and all the digits in the decimal representations of $a,b$ (expressed without leading zeros) are in $S$. Find the smallest possible value of $|S|$.

Proposed by Sutanay Bhattacharya

Original Wording
46 replies
anantmudgal09
Jan 19, 2020
anudeep
4 hours ago
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interesting function equation (fe) in IR
skellyrah   2
N Apr 23, 2025 by jasperE3
Source: mine
find all function F: IR->IR such that $$ xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy) $$
2 replies
skellyrah
Apr 23, 2025
jasperE3
Apr 23, 2025
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interesting function equation (fe) in IR
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functional equation
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real numbers
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skellyrah
23 posts
skellyrah
#1 h  Apr 23, 2025, 9:51 AM
Y by
find all function F: IR->IR such that $$ xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy) $$
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CrazyInMath
457 posts
CrazyInMath
#2 h  Apr 23, 2025, 11:37 AM
Y by
solution
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jasperE3
11288 posts
jasperE3
#4 h  Apr 23, 2025, 8:33 PM
Y by
skellyrah wrote:
find all function F: IR->IR such that $$ xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy) $$

If IR means irrational numbers, and the problem is to find all $f$ such that $xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy)$ for all $x,y\in\mathbb{IR}$ then setting $x=f(y)^{-1}$ gives a contradiction
This post has been edited 1 time. Last edited by jasperE3, Apr 23, 2025, 9:43 PM
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