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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
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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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BMO 2024 SL A1
MuradSafarli   7
N 7 minutes ago by sqing
A1.

Let \( u, v, w \) be positive reals. Prove that there is a cyclic permutation \( (x, y, z) \) of \( (u, v, w) \) such that the inequality:

\[ \frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z} \]
holds for all positive real numbers \( a, b \) and \( c \).
7 replies
MuradSafarli
Apr 27, 2025
sqing
7 minutes ago
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BMO 2024 SL A4
MuradSafarli   1
N 10 minutes ago by sqing
A4.
Let \(a \geq b \geq c \geq 0\) be real numbers such that \(ab + bc + ca = 3\).
Prove that:
\[ 3 + (2 - \sqrt{3}) \cdot \frac{(b-c)^2}{b+(\sqrt{3}-1)c} \leq a+b+c \]and determine all the cases when the equality occurs.
1 reply
1 viewing
MuradSafarli
Apr 27, 2025
sqing
10 minutes ago
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A coincidence about triangles with common incenter
flower417477   1
N 31 minutes ago by flower417477
$\triangle ABC,\triangle ADE$ have the same incenter $I$.Prove that $BCDE$ is concyclic iff $BC,DE,AI$ is concurrent
1 reply
flower417477
35 minutes ago
flower417477
31 minutes ago
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Difficult combinatorics problem about distinct sums under shifts
CBMaster   1
N 35 minutes ago by CBMaster
Source: Korea
Problem. Let $a_1, ..., a_n$ be the nonnegative integers in $\{0, 1, ..., m\}$ where $m=\left\lceil \frac{n^{2/3}}{4} \right\rceil $. Define $A=\{a_i+a_j+(j-i)|1\leq i<j\leq n\}$. Prove that $|A|\geq m$.

Bonus problem (Open). Can we prove a tighter result than the one above? That is, is there a function $f(n)$ such that $f(n)=O(n^\alpha)$ where $\alpha>\frac{2}{3}$, and the statement is still true when $m=f(n)$?
Or, is there a function $f(n)$ such that $f(n)\geq C \cdot n^{2/3}$ where $C>\frac{1}{4}$, and the statement is still true when $m=f(n)$?.
1 reply
CBMaster
Apr 27, 2025
CBMaster
35 minutes ago
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Enjoy with this one
Halykov   0
43 minutes ago
Source: own
Let \( ABC \) be a scalene triangle with circumcircle \( \Gamma \) and circumcircle \( O \). Denote \( M \) and \( N \) as the midpoints of \( AC \) and \( BC \), respectively. The altitude from \( A \) to \( BC \) intersects \( \Gamma \) again at \( D \). The line \( DN \) meets \( \Gamma \) a second time at \( T \), and \( AT \) intersects \( BC \) at \( X \). The perpendicular bisector of \( AC \) intersects \( AD \) at \( S \) and \( AB \) at \( Z \). Let the circumcircle of \( \triangle XMZ \) intersect \( BC \) again at \( Y \), and let the line \( ZY \) intersect \( AD \) at \( K \).
If \( H \) is the reflection of \( S \) over \( K \), prove that the intersection of \( CH \) and \( BM \) lies on the circumcircle of \( BOC \).
0 replies
Halykov
43 minutes ago
0 replies
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Conjecture: Intersection of diagonals of cyclic quadrilateral lies on a fixed li
kieusuong   1
N an hour ago by Aaronjudgeisgoat
Let (O) be a fixed circle and let P be a point outside the circle.
From P, draw a line that intersects the circle at points A and B.
Now, from P, draw another line that intersects the circle at points N and M so that quadrilateral ANMB is cyclic (i.e., lies on the circle).

Let AM and BN intersect at point G. Let AN and BM intersect at point T.
Let PJ be the tangent from P to circle (O), and let J be the point of tangency.

Claim (Conjecture):
As the quadrilateral ANMB varies (still inscribed in the circle), the points T, G, and J always lie on a straight line.
Moreover, this line TJ is perpendicular to the fixed chord AB.

I believe this might be a new result and would appreciate any insights or proof ideas.
Attached is a diagram for reference.
1 reply
kieusuong
2 hours ago
Aaronjudgeisgoat
an hour ago
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Inequality with condition a+b+c = ab+bc+ca (and special equality case)
DoThinh2001   69
N an hour ago by mihaig
Source: BMO 2019, problem 2
Let $a,b,c$ be real numbers such that $0 \leq a \leq b \leq c$ and $a+b+c=ab+bc+ca >0.$
Prove that $\sqrt{bc}(a+1) \geq 2$ and determine the equality cases.

(Edit: Proposed by sir Leonard Giugiuc, Romania)
69 replies
1 viewing
DoThinh2001
May 2, 2019
mihaig
an hour ago
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1 line solution to Inequality
ItzsleepyXD   1
N an hour ago by mihaig
Source: Own , Mock Thailand Mathematic Olympiad P8
Let $x_1,x_2,\dots,x_n$ be positive real integer such that $x_1^2+x_2^2+\cdots+x_n^2=2$ Prove that
$$\sum_{i=1}^{n}\frac{1}{x_i^3(x_{i-1}+x_{i+1})}\geqslant \left(\sum_{i=1}^{n}\frac{x_i}{x_{i-1}+x_{i+1}}\right)^3$$such that $x_{n+1}=x_1$ and $x_0=x_n$
1 reply
ItzsleepyXD
5 hours ago
mihaig
an hour ago
J
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a nice problem of nt from PUMaC
Namisgood   1
N an hour ago by Goutamioqmtopper
Source: PUMaC
Problem is attached
1 reply
Namisgood
Yesterday at 9:47 AM
Goutamioqmtopper
an hour ago
J
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Diophantine equation !
ComplexPhi   11
N an hour ago by Goutamioqmtopper
Determine all triples $(m , n , p)$ satisfying :
\[n^{2p}=m^2+n^2+p+1\]
where $m$ and $n$ are integers and $p$ is a prime number.
11 replies
ComplexPhi
Feb 4, 2015
Goutamioqmtopper
an hour ago
J
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Solve $\sin(17x)+\sin(13x)=\sin(7x)$
Speed2001   1
N an hour ago by rchokler
How to solve the equation:
$$ \sin(17x)+\sin(13x)=\sin(7x),\;0<x<24^{\circ} $$Approach: I'm trying to factor $\sin(18x)$ to get $x=10^{\circ}$.

Any hint would be appreciated.
1 reply
Speed2001
Yesterday at 1:06 AM
rchokler
an hour ago
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7^a - 3^b divides a^4 + b^2 (from IMO Shortlist 2007)
Dida Drogbier   39
N an hour ago by ND_
Source: ISL 2007, N1, VAIMO 2008, P4
Find all pairs of natural numbers $ (a, b)$ such that $ 7^a - 3^b$ divides $ a^4 + b^2$.

Author: Stephan Wagner, Austria
39 replies
Dida Drogbier
Apr 21, 2008
ND_
an hour ago
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Inspired by tom-nowy
sqing   0
an hour ago
Source: Own
Let $a,b,c \in [-\frac{1}{2}, \frac{1}{2}]$. Prove that
$$ \frac{1}{16}\geq (ab+bc+ca)^2+ a^2b^2c^2-(a+b+c)^2\geq -\frac{107}{64} $$$$ \frac{1}{16}\geq (ab+bc+ca)^2+3a^2b^2c^2-(a+b+c)^2\geq -\frac{105}{64} $$Let $a,b,c \in [-2,2]$. Prove that
$$172\geq (ab+bc+ca)^2+ a^2b^2c^2-(a+b+c)^2\geq -\frac{64}{11} $$$$300\geq (ab+bc+ca)^2+3a^2b^2c^2-(a+b+c)^2\geq -\frac{192}{35} $$
0 replies
sqing
an hour ago
0 replies
J
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An amazing inequality
Kei0923   6
N an hour ago by Kei0923
Source: Own (2021 Mock Japan MO Finals 5)
Let $n$ be a positive integer and $a_{1},a_{2},\dots, a_{n+1}$ be positive real numbers such that $$\displaystyle {\sum_{i=1}^{n+1}}\frac{1}{a_{i}+n^2}=\frac{1}{n}.$$Prove that $$\sum_{i=1}^{n+1} \frac{1}{a_{i}^{2021}+n^{2020}} \geq \frac{1}{n^{2020}}.$$
6 replies
Kei0923
Mar 14, 2021
Kei0923
an hour ago
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J
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n=y^2+108
Havu   6
N Apr 3, 2025 by ektorasmiliotis
Given the positive integer $n = y^2 + 108$ where $y \in \mathbb{N}$.
Prove that $n$ cannot be a perfect cube of a positive integer.
6 replies
Havu
Apr 3, 2025
ektorasmiliotis
Apr 3, 2025
J
n=y^2+108
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Reveal topic tags
number theory
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Havu
41 posts
Havu
#1 h  Apr 3, 2025, 4:30 PM • 1 Y
Y by MR.1
Given the positive integer $n = y^2 + 108$ where $y \in \mathbb{N}$.
Prove that $n$ cannot be a perfect cube of a positive integer.
Z K Y
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MuradSafarli
106 posts
MuradSafarli
#2 h  Apr 3, 2025, 4:52 PM
Y by
this problem is greece 2025 p4
This post has been edited 1 time. Last edited by MuradSafarli, Apr 3, 2025, 4:54 PM
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MuradSafarli
106 posts
MuradSafarli
#3 h  Apr 3, 2025, 4:53 PM
Y by
https://artofproblemsolving.com/community/c4242999_2025_greece_national_olympiad
Z K Y
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MR.1
110 posts
MR.1
#4 h  Apr 3, 2025, 5:28 PM
Y by
wow what a great time to shine. use mordell's equation which has no solution for $k=108$ :D
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MathPerson12321
3738 posts
MathPerson12321
#5 h  Apr 3, 2025, 5:30 PM
Y by
does mod 9 work
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GreekIdiot
209 posts
GreekIdiot
#6 h  Apr 3, 2025, 8:33 PM • 1 Y
Y by InftyByond
yeah mod 9 works. also FCT works too, so does legendre's symbol.
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ektorasmiliotis
103 posts
ektorasmiliotis
#7 h  Apr 3, 2025, 8:45 PM
Y by
can you post your solution with mod 9?FCT works,you can express 100 as 10^2 +2^3
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