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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
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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
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Concurrency with 10 lines
oVlad   0
2 minutes ago
Source: Romania EGMO TST 2017 Day 1 P1
Consider five points on a circle. For every three of them, we draw the perpendicular from the centroid of the triangle they determine to the line through the remaining two points. Prove that the ten lines thus formed are concurrent.
0 replies
oVlad
2 minutes ago
0 replies
J
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number theory FE
pomodor_ap   1
N 6 minutes ago by rrrMath
Source: Own, PDC002-P7
Let $f : \mathbb{Z}^+ \to \mathbb{Z}^+$ be a function such that
$$f(m) + mn + n^2 \mid f(m)^2 + m^2 f(n) + f(n)^2$$for all $m, n \in \mathbb{Z}^+$. Find all such functions $f$.
1 reply
pomodor_ap
2 hours ago
rrrMath
6 minutes ago
J
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hard problem....
Cobedangiu   2
N 7 minutes ago by Cobedangiu
let $a,b,c$ be the lengths of the sides of the triangle. Prove that:
$(a+b+c)(\dfrac{3a-b}{a^2+ab}+\dfrac{3b-c}{b^2+bc}+\dfrac{3c-a}{c^2+ac})\le 9$
2 replies
Cobedangiu
3 hours ago
Cobedangiu
7 minutes ago
J
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(a+b)^k-a^k-b^k
Math-wiz   8
N 8 minutes ago by MATHS_ENTUSIAST
Source: PRMO 2019 Leg 2 P8
Let $F_k(a,b)=(a+b)^k-a^k-b^k$ and let $S={1,2,3,4,5,6,7,8,9,10}$. For how many ordered pairs $(a,b)$ with $a,b\in S$ and $a\leq b$ is $\frac{F_5(a,b)}{F_3(a,b)}$ an integer?
8 replies
Math-wiz
Aug 25, 2019
MATHS_ENTUSIAST
8 minutes ago
J
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Let \( a, b, c \) be positive real numbers satisfying \[ a^2 + c^2 = b(a + c). \
Jackson0423   1
N 31 minutes ago by Mathzeus1024
Let \( a, b, c \) be positive real numbers satisfying
\[ a^2 + c^2 = b(a + c). \]Let
\[ m = \min \left( \frac{a^2 + ab + b^2}{ab + bc + ca} \right). \]Find the value of \( 2024m \).
1 reply
Jackson0423
Apr 16, 2025
Mathzeus1024
31 minutes ago
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Collect ...
luutrongphuc   0
34 minutes ago
Find all functions $f: \mathbb{R^+} \rightarrow \mathbb{R^+}$ such that:
$$f\left(f(xy)+1\right)=xf\left(x+f(y)\right)$$
0 replies
luutrongphuc
34 minutes ago
0 replies
J
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Two very hard parallel
jayme   0
an hour ago
Source: own inspired by EGMO
Dear Mathlinkers,

1. ABC a triangle
2. D, E two point on the segment BC so that BD = DE= EC
3. M, N the midpoint of ED, AE
4. H the orthocenter of the acutangle triangle ADE
5. 1, 2 the circumcircle of the triangle DHM, EHN
6. P, Q the second point of intersection of 1 and BM, 2 and CN
7. U, V the second points of intersection of 2 and MN, PQ.

Prove : UV is parallel to PM.

Sincerely
Jean-Louis
0 replies
jayme
an hour ago
0 replies
J
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Weighted graph problem
egxa   1
N an hour ago by internationalnick123456
Source: All Russian 2025 10.4
In the plane, $106$ points are marked, no three of which are collinear. All possible segments between them are drawn. Grisha assigned to each drawn segment a real number with absolute value no greater than $1$. For every group of $6$ marked points, he calculated the sum of the numbers on all $15$ connecting segments. It turned out that the absolute value of each such sum is at least \(C\), and there are both positive and negative such sums. What is the maximum possible value of \(C\)?
1 reply
egxa
Apr 18, 2025
internationalnick123456
an hour ago
J
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Inequality
giangtruong13   2
N an hour ago by sqing
Let $a,b,c >0$ such that: $a^2+b^2+c^2=3$. Prove that: $$\frac{b^2}{a}+\frac{c^2}{b}+\frac{a^2}{c}+abc \geq 4$$
2 replies
giangtruong13
6 hours ago
sqing
an hour ago
J
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abc(a+b+c)=3, show that prod(a+b)>=8 [Indian RMO 2012(b) Q4]
Potla   27
N 2 hours ago by sqing
Let $a,b,c$ be positive real numbers such that $abc(a+b+c)=3.$ Prove that we have
\[(a+b)(b+c)(c+a)\geq 8.\]
Also determine the case of equality.
27 replies
Potla
Dec 2, 2012
sqing
2 hours ago
J
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Prove the inequality with the condition (a+1)(b+1)(c+1)=8
hlminh   1
N 2 hours ago by quacksaysduck
Let $a,b,c>0$ such that $(a+1)(b+1)(c+1)=8.$ Prove that $abc(a+b+c)\leq 3.$
1 reply
hlminh
4 hours ago
quacksaysduck
2 hours ago
J
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integer functional equation
ABCDE   147
N 2 hours ago by Adywastaken
Source: 2015 IMO Shortlist A2
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\]holds for all $x,y\in\mathbb{Z}$.
147 replies
ABCDE
Jul 7, 2016
Adywastaken
2 hours ago
J
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real+ FE
pomodor_ap   0
2 hours ago
Source: Own, PDC001-P7
Let $f : \mathbb{R}^+ \to \mathbb{R}^+$ be a function such that
$$f(x)f(x^2 + y f(y)) = f(x)f(y^2) + x^3$$for all $x, y \in \mathbb{R}^+$. Determine all such functions $f$.
0 replies
pomodor_ap
2 hours ago
0 replies
J
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Is this FE solvable?
ItzsleepyXD   2
N 2 hours ago by ItzsleepyXD
Source: Original
Let $c_1,c_2 \in \mathbb{R^+}$. Find all $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ such that for all $x,y \in \mathbb{R^+}$ $$f(x+c_1f(y))=f(x)+c_2f(y)$$
2 replies
ItzsleepyXD
Today at 3:02 AM
ItzsleepyXD
2 hours ago
J
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J
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Unsolved NT, 3rd time posting
GreekIdiot   11
N Apr 4, 2025 by GreekIdiot
Source: own
Solve $5^x-2^y=z^3$ where $x,y,z \in \mathbb Z$
Hint
11 replies
GreekIdiot
Mar 26, 2025
GreekIdiot
Apr 4, 2025
J
Unsolved NT, 3rd time posting
G H J
Reveal topic tags
number theory
modular arithmetic
Exponents
unsolved
L
G H BBookmark kLocked kLocked NReply
Source: own
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GreekIdiot
173 posts
GreekIdiot
#1 h  Mar 26, 2025, 11:40 AM • 1 Y
Y by AlexCenteno2007
Solve $5^x-2^y=z^3$ where $x,y,z \in \mathbb Z$
Hint
This post has been edited 2 times. Last edited by GreekIdiot, Mar 26, 2025, 11:41 AM
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GreekIdiot
173 posts
GreekIdiot
#2 h  Mar 28, 2025, 8:50 AM
Y by
guys bump
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whwlqkd
97 posts
whwlqkd
#3 h  Mar 29, 2025, 1:50 AM
Y by
Bump
It is so hard
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DarkDementor
45 posts
DarkDementor
#4 h  Mar 29, 2025, 2:19 AM
Y by
Wrong mbmbmbmb
This post has been edited 1 time. Last edited by DarkDementor, Apr 4, 2025, 11:54 PM
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whwlqkd
97 posts
whwlqkd
#5 h  Mar 29, 2025, 3:36 AM
Y by
z^3 can be 3(mod 4)
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GreekIdiot
173 posts
GreekIdiot
#6 h  Apr 1, 2025, 5:48 PM
Y by
whwlqkd wrote:
Bump
It is so hard

Yeah lol. I took a known BMO SL problem and tried changing some things. The problem was $5^x-3^y=z^2$ or something similar. I changed the $3$ so that modulo $4$ would not solve it immediately and then the power of $z$ to make it a lot harder. Also just noticed that $8$ has a nice property $d=1(mod \: 2) \implies d^3=d(mod \: 8)$.
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GreekIdiot
173 posts
GreekIdiot
#8 h  Apr 3, 2025, 8:37 PM
Y by
Bumping...
DarkDementor wrote:
Mod 4 then LTE should work.
Mind explaining??
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ektorasmiliotis
102 posts
ektorasmiliotis
#9 h  Apr 3, 2025, 8:43 PM
Y by
I don't think mod 4 and lte works... maybe we should try something different, simple mods might not help enough
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ektorasmiliotis
102 posts
ektorasmiliotis
#10 h  Apr 3, 2025, 8:48 PM
Y by
if mod 4 works with a perfect cube...i'll be surprised
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iniffur
536 posts
iniffur
#11 h  Apr 4, 2025, 12:12 AM
Y by
Setting $5^k=X, ~2^t=Y$ wherein $~~k,~t =0.1,2,3....$

It would appear that the equation with x=odd, y= odd of formula:


$5X^2-2Y^2=Z^3$

comprises a non trivial solution $~(X,Y,Z)= (1, 4, -3)\rightarrow (x,y,z)=(1,5,-3)$

@ below: amendments have been made to meet your objection, Thx. I found the solution merely by inspection. What i am suggesting is merely a possible approach other than modulos (but which could turn out to be tougher than the original).
This post has been edited 2 times. Last edited by iniffur, Apr 4, 2025, 7:03 AM
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mathprodigy2011
313 posts
mathprodigy2011
#12 h  Apr 4, 2025, 2:05 AM
Y by
iniffur wrote:
Setting $~5^x=X,~~2^y=Y,~~$the given problem would boil down to solving:

$X^2-Y^2=Z^3$

$5X^2-Y^2=Z^3$

$X^2-2Y^2=Z^3$

$5X^2-2Y^2=Z^3$

A non trivial solution can be found in the last one $~(X,Y,Z)= (1, 4, -3)\rightarrow (x,y,z)=(1,5,-3)$

i mean you definitely got a solution but you wrote X^2 and Y^2 at the start of your proof but isn't is supposed to be X-Y cuz theres no squares in the original? maybe im a bit confused
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GreekIdiot
173 posts
GreekIdiot
#13 h  Apr 4, 2025, 3:19 PM
Y by
Cool idea... Other solutions are pretty easy to come up with. I wonder how we ll prove uniqueness though.
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