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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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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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JBMO TST Bosnia and Herzegovina 2023 P4
FishkoBiH   0
9 minutes ago
Source: JBMO TST Bosnia and Herzegovina 2023 P4
Let $n$ be a positive integer. A board with a format $n*n$ is divided in $n*n$ equal squares.Determine all integers $n$≥3 such that the board can be covered in $2*1$ (or $1*2$) pieces so that there is exactly one empty square in each row and each column.
0 replies
FishkoBiH
9 minutes ago
0 replies
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JBMO TST Bosnia and Herzegovina 2023 P3
FishkoBiH   0
14 minutes ago
Source: JBMO TST Bosnia and Herzegovina 2023 P3
Let ABC be an acute triangle with an incenter $I$.The Incircle touches sides $AC$ and $AB$ in $E$ and $F$ ,respectivly. Lines CI and EF intersect at $S$. The point $T$≠$I$ is on the line AI so that $EI$=$ET$.If $K$ is the foot of the altitude from $C$ in triangle $ABC$,prove that points $K$,$S$ and $T$ are colinear.
0 replies
FishkoBiH
14 minutes ago
0 replies
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Locus of points P in triangle ABC
v_Enhance   25
N 21 minutes ago by alexanderchew
Source: USA January TST for IMO 2016, Problem 3
Let $ABC$ be an acute scalene triangle and let $P$ be a point in its interior. Let $A_1$, $B_1$, $C_1$ be projections of $P$ onto triangle sides $BC$, $CA$, $AB$, respectively. Find the locus of points $P$ such that $AA_1$, $BB_1$, $CC_1$ are concurrent and $\angle PAB + \angle PBC + \angle PCA = 90^{\circ}$.
25 replies
v_Enhance
May 17, 2016
alexanderchew
21 minutes ago
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JBMO TST Bosnia and Herzegovina 2023 P2
FishkoBiH   0
22 minutes ago
Source: JBMO TST Bosnia and Herzegovina 2023 P2
Determine all non negative integers $x$ and $y$ such that $6^x$ + $2^y$ + 2 is a perfect square.
0 replies
FishkoBiH
22 minutes ago
0 replies
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JBMO TST Bosnia and Herzegovina 2023 P1
FishkoBiH   0
27 minutes ago
Source: JBMO TST Bosnia and Herzegovina 2023 P1
Determine all real numbers $a, b, c, d$ for which

$ab+cd=6$
$ac+bd=3$
$ad+bc=2$
$a+b+c+d=6$
0 replies
FishkoBiH
27 minutes ago
0 replies
J
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number theory diophantic with factorials and primes
skellyrah   4
N an hour ago by skellyrah
Source: by me
find all triplets of non negative integers (a,b,p) where p is prime such that $$ a! + b! + 7ab = p^2 $$
4 replies
skellyrah
Feb 16, 2025
skellyrah
an hour ago
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Inequality em981
oldbeginner   17
N an hour ago by sqing
Source: Own
Let $a, b, c>0, a+b+c=3$. Prove that
\[\sqrt{a+\frac{9}{b+2c}}+\sqrt{b+\frac{9}{c+2a}}+\sqrt{c+\frac{9}{a+2b}}+\frac{2(ab+bc+ca)}{9}\ge\frac{20}{3}\]
17 replies
oldbeginner
Sep 22, 2016
sqing
an hour ago
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primes,exponentials,factorials
skellyrah   6
N an hour ago by skellyrah
find all primes p,q such that $$ \frac{p^q+q^p-p-q}{p!-q!} $$is a prime number
6 replies
skellyrah
Apr 30, 2025
skellyrah
an hour ago
J
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Serbian selection contest for the IMO 2025 - P5
OgnjenTesic   2
N an hour ago by GreenTea2593
Source: Serbian selection contest for the IMO 2025
Determine the smallest positive real number $\alpha$ such that there exists a sequence of positive real numbers $(a_n)$, $n \in \mathbb{N}$, with the property that for every $n \in \mathbb{N}$ it holds that:
\[ a_1 + \cdots + a_{n+1} < \alpha \cdot a_n. \]Proposed by Pavle Martinović
2 replies
OgnjenTesic
May 22, 2025
GreenTea2593
an hour ago
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centroid wanted, point that minimizes sum of squares of distances from sides
parmenides51   1
N an hour ago by SuperBarsh
Source: Oliforum Contest V 2017 p9 https://artofproblemsolving.com/community/c2487525_oliforum_contes
Given a triangle $ABC$, let $ P$ be the point which minimizes the sum of squares of distances from the sides of the triangle. Let $D, E, F$ the projections of $ P$ on the sides of the triangle $ABC$. Show that $P$ is the barycenter of $DEF$.

(Jack D’Aurizio)
1 reply
parmenides51
Sep 29, 2021
SuperBarsh
an hour ago
J
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Strictly monotone polynomial with an extra condition
Popescu   11
N 2 hours ago by Iveela
Source: IMSC 2024 Day 2 Problem 2
Let $\mathbb{R}_{>0}$ be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions $f:\mathbb{R}_{>0} \to \mathbb{R}$ such that there exists a two-variable polynomial $P(x, y)$ with real coefficients satisfying
$$ f(xy)=P(f(x), f(y)) $$for all $x, y\in\mathbb{R}_{>0}$.

Proposed by Navid Safaei, Iran
11 replies
Popescu
Jun 29, 2024
Iveela
2 hours ago
J
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Hard geometry
Lukariman   1
N 2 hours ago by ricarlos
Given triangle ABC, any line d intersects AB at D, intersects AC at E, intersects BC at F. Let O1,O2,O3 be the centers of the circles circumscribing triangles ADE, BDF, CFE. Prove that the orthocenter of triangle O1O2O3 lies on line d.
1 reply
Lukariman
May 12, 2025
ricarlos
2 hours ago
J
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Russian Diophantine Equation
LeYohan   1
N 2 hours ago by Natrium
Source: Moscow, 1963
Find all integer solutions to

$\frac{xy}{z} + \frac{xz}{y} + \frac{yz}{x} = 3$.
1 reply
LeYohan
Yesterday at 2:59 PM
Natrium
2 hours ago
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Simple Geometry
AbdulWaheed   6
N 2 hours ago by Gggvds1
Source: EGMO
Try to avoid Directed angles
Let ABC be an acute triangle inscribed in circle $\Omega$. Let $X$ be the midpoint of the arc $\overarc{BC}$ not containing $A$ and define $Y, Z$ similarly. Show that the orthocenter of $XYZ$ is the incenter $I$ of $ABC$.
6 replies
AbdulWaheed
May 23, 2025
Gggvds1
2 hours ago
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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
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Unsolved NT, 3rd time posting
G H J
Reveal topic tags
number theory
modular arithmetic
Exponents
unsolved
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G H BBookmark kLocked kLocked NReply
Source: own
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GreekIdiot
256 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
256 posts
GreekIdiot
#2 h  Mar 28, 2025, 8:50 AM
Y by
guys bump
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whwlqkd
128 posts
whwlqkd
#3 h  Mar 29, 2025, 1:50 AM
Y by
Bump
It is so hard
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DarkDementor
46 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
128 posts
whwlqkd
#5 h  Mar 29, 2025, 3:36 AM
Y by
z^3 can be 3(mod 4)
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GreekIdiot
256 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
256 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
109 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
109 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
538 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
342 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
256 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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