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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
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
Apr 2, 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
J
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all functions satisfying f(x+yf(x))+y = xy + f(x+y)
falantrng   19
N a minute ago by megarnie
Source: Balkan MO 2025 P3
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[f(x+yf(x))+y = xy + f(x+y).\]
Proposed by Giannis Galamatis, Greece
19 replies
+7 w
falantrng
4 hours ago
megarnie
a minute ago
J
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Troll inequality
B1t   8
N 4 minutes ago by B1t
Source: Mongolian TST 2025 P5
For positive real numbers \( x, y > 0 \), define
\[ E(x,y) = x^{-\!y} + y. \]
1. If \( x, y \geq 1 \), prove that
\[ E(x,y) + E(y,x) \geq 4. \]
2. If \( 0 < x, y < 1 \), prove that
\[ \frac{2}{1+xy} + x + y < E(x,y) + E(y,x) < 2 + \frac{x}{y} + \frac{y}{x}. \]
8 replies
+1 w
B1t
2 hours ago
B1t
4 minutes ago
J
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Inequalities
hn111009   0
4 minutes ago
Source: Somewhere
Let $a,b,c$ be non-negative number. Prove that $$\left(a+bc\right)^2+\left(b+ca\right)^2+\left(c+ab\right)^2\ge \sqrt{2}\left(a+b\right)\left(b+c\right)\left(c+a\right)$$
0 replies
hn111009
4 minutes ago
0 replies
J
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hard problem
Rename   0
10 minutes ago
Determine the largest constant $K\geq 0$ so that:
$$\frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2$$with all real numbers $a; b; c$ satisfies $ab+bc+ca=abc$

P/s: Do you know which exam question this problem is actually in, at first I remembered but now I forgot
0 replies
Rename
10 minutes ago
0 replies
J
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Good Permutations in Modulo n
swynca   4
N 10 minutes ago by Ivan_Borsenco
Source: BMO 2025 P1
An integer $n > 1$ is called $\emph{good}$ if there exists a permutation $a_1, a_2, a_3, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$, such that:
$(i)$ $a_i$ and $a_{i+1}$ have different parities for every $1 \leq i \leq n-1$;
$(ii)$ the sum $a_1 + a_2 + \cdots + a_k$ is a quadratic residue modulo $n$ for every $1 \leq k \leq n$.
Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.
4 replies
+2 w
swynca
2 hours ago
Ivan_Borsenco
10 minutes ago
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Inspired by Mongolian 2025
sqing   1
N 16 minutes ago by sqing
Source: Own
Let \( x, y \geq 1 \). Prove that
$$ \frac{2}{1+xy} + x + y \geq \frac{x}{y} + \frac{y}{x}+1$$Let \(0< x, y \leq 1 \). Prove that
$$ \frac{2}{1+xy} + x + y \leq \frac{x}{y} + \frac{y}{x}+1$$
1 reply
sqing
17 minutes ago
sqing
16 minutes ago
J
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Arbitrary point on BC and its relation with orthocenter
falantrng   10
N 21 minutes ago by hukilau17
Source: Balkan MO 2025 P2
In an acute-angled triangle \(ABC\), \(H\) be the orthocenter of it and \(D\) be any point on the side \(BC\). The points \(E, F\) are on the segments \(AB, AC\), respectively, such that the points \(A, B, D, F\) and \(A, C, D, E\) are cyclic. The segments \(BF\) and \(CE\) intersect at \(P.\) \(L\) is a point on \(HA\) such that \(LC\) is tangent to the circumcircle of triangle \(PBC\) at \(C.\) \(BH\) and \(CP\) intersect at \(X\). Prove that the points \(D, X, \) and \(L\) lie on the same line.
10 replies
+1 w
falantrng
4 hours ago
hukilau17
21 minutes ago
J
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One more sequence with infinite prime divisors
Assassino9931   1
N 28 minutes ago by NO_SQUARES
Source: Balkan MO Shortlist 2024 N3
Prove that there are infinitely many primes such that each divides an integer of the form $2^n + 3^{n+2} + 5^{n+1}$ for some positive integer $n$.
1 reply
Assassino9931
3 hours ago
NO_SQUARES
28 minutes ago
J
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easy functional
B1t   12
N 29 minutes ago by complex2math
Source: Mongolian TST 2025 P1.
Denote the set of real numbers by $\mathbb{R}$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for all $x, y, z \in \mathbb{R}$,
\[ f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)). \]
12 replies
B1t
Yesterday at 6:45 AM
complex2math
29 minutes ago
J
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Interesting inequality
sqing   5
N 34 minutes ago by Mathzeus1024
Let $ a,b> 0 $ and $ a+b+ab=1. $ Prove that
$$\frac{1}{1+a^2} + \frac{1}{1+b^2} +a+b\leq \frac{5}{\sqrt{2}}-1 $$
5 replies
sqing
Today at 3:35 AM
Mathzeus1024
34 minutes ago
J
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Only n-digits 1234
Assassino9931   2
N an hour ago by GreekIdiot
Source: Balkan MO Shortlist 2024 N1
Let $n$ be a positive integer. Do there exist distinct $n$-digit positive integers $x$ and $y$, with each of their digits being $1,2,3$ or $4$, such that $4^n$ divides $x-y$?
2 replies
Assassino9931
3 hours ago
GreekIdiot
an hour ago
J
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BMO 2024 SL C1
GreekIdiot   5
N an hour ago by GreekIdiot
Let $n$, $k$ be positive integers. Julia and Florian play a game on a $2n \times 2n$ board. Julia
has secretly tiled the entire board with invisible dominos. Florian now chooses $k$ cells.
All dominos covering at least one of these cells then turn visible. Determine the minimal
value of $k$ such that Florian has a strategy to always deduce the entire tiling.
5 replies
GreekIdiot
3 hours ago
GreekIdiot
an hour ago
J
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K-pop sequences
L567   9
N an hour ago by fearsum_fyz
Source: India EGMO TST 2023/5
Let $k$ be a positive integer. A sequence of integers $a_1, a_2, \cdots$ is called $k$-pop if the following holds: for every $n \in \mathbb{N}$, $a_n$ is equal to the number of distinct elements in the set $\{a_1, \cdots , a_{n+k} \}$. Determine, as a function of $k$, how many $k$-pop sequences there are.

Proposed by Sutanay Bhattacharya
9 replies
L567
Dec 10, 2022
fearsum_fyz
an hour ago
J
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BMO 2024 SL A3
MuradSafarli   4
N an hour ago by GreekIdiot

A3.
Find all triples \((a, b, c)\) of positive real numbers that satisfy the system:
\[ \begin{aligned} 11bc - 36b - 15c &= abc \\ 12ca - 10c - 28a &= abc \\ 13ab - 21a - 6b &= abc. \end{aligned} \]
4 replies
MuradSafarli
3 hours ago
GreekIdiot
an hour ago
J
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J
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Find the locus of vertices A
Amir Hossein   1
N Sep 25, 2010 by oneplusone
Given a circle $K$, find the locus of vertices $A$ of parallelograms $ABCD$ with diagonals $AC \leq BD$, such that $BD$ is inside $K$.
1 reply
Amir Hossein
Sep 22, 2010
oneplusone
Sep 25, 2010
J
Find the locus of vertices A
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Reveal topic tags
geometry
parallelogram
Locus problems
Locus
circle
IMO Shortlist
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Amir Hossein
5452 posts
Amir Hossein
#1 h  Sep 22, 2010, 7:21 AM • 2 Y
Y by Adventure10, Mango247
Given a circle $K$, find the locus of vertices $A$ of parallelograms $ABCD$ with diagonals $AC \leq BD$, such that $BD$ is inside $K$.
Z K Y
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oneplusone
1459 posts
oneplusone
#2 h  Sep 25, 2010, 5:22 AM • 2 Y
Y by Adventure10, Mango247
Note that $AC\leq BD\iff \angle A\geq 90$. Clearly any point $A$ in the circle works. Now if $A$ is outside, the maximum possible $\angle A$ is when $AB,AD$ are tangents to the circle. So $A$ must be in the circle with the same center as $K$ but the radius is $\sqrt2$ times that of $K$, and the locus is all the points in that circle.
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