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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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Cyclic Quads and Parallel Lines
gracemoon124   16
N 7 minutes ago by ohiorizzler1434
Source: 2015 British Mathematical Olympiad?
Let $ABCD$ be a cyclic quadrilateral. Let $F$ be the midpoint of the arc $AB$ of its circumcircle which does not contain $C$ or $D$. Let the lines $DF$ and $AC$ meet at $P$ and the lines $CF$ and $BD$ meet at $Q$. Prove that the lines $PQ$ and $AB$ are parallel.
16 replies
1 viewing
gracemoon124
Aug 16, 2023
ohiorizzler1434
7 minutes ago
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Radical Center on the Euler Line (USEMO 2020/3)
franzliszt   37
N 27 minutes ago by Ilikeminecraft
Source: USEMO 2020/3
Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $\Gamma$ denote the circumcircle of triangle $ABC$, and $N$ the midpoint of $OH$. The tangents to $\Gamma$ at $B$ and $C$, and the line through $H$ perpendicular to line $AN$, determine a triangle whose circumcircle we denote by $\omega_A$. Define $\omega_B$ and $\omega_C$ similarly.
Prove that the common chords of $\omega_A$,$\omega_B$ and $\omega_C$ are concurrent on line $OH$.

Proposed by Anant Mudgal
37 replies
franzliszt
Oct 24, 2020
Ilikeminecraft
27 minutes ago
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Functional equation with powers
tapir1729   13
N 28 minutes ago by ihategeo_1969
Source: TSTST 2024, problem 6
Determine whether there exists a function $f: \mathbb{Z}_{> 0} \rightarrow \mathbb{Z}_{> 0}$ such that for all positive integers $m$ and $n$,
\[f(m+nf(m))=f(n)^m+2024! \cdot m.\]Jaedon Whyte
13 replies
tapir1729
Jun 24, 2024
ihategeo_1969
28 minutes ago
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Powers of a Prime
numbertheorist17   34
N an hour ago by KevinYang2.71
Source: USA TSTST 2014, Problem 6
Suppose we have distinct positive integers $a, b, c, d$, and an odd prime $p$ not dividing any of them, and an integer $M$ such that if one considers the infinite sequence \begin{align*} ca &- db \\ ca^2 &- db^2 \\ ca^3 &- db^3 \\ ca^4 &- db^4 \\ &\vdots \end{align*} and looks at the highest power of $p$ that divides each of them, these powers are not all zero, and are all at most $M$. Prove that there exists some $T$ (which may depend on $a,b,c,d,p,M$) such that whenever $p$ divides an element of this sequence, the maximum power of $p$ that divides that element is exactly $p^T$.
34 replies
+1 w
numbertheorist17
Jul 16, 2014
KevinYang2.71
an hour ago
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q(x) to be the product of all primes less than p(x)
orl   18
N an hour ago by happypi31415
Source: IMO Shortlist 1995, S3
For an integer $x \geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and \[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \] for $n \geq 0$. Find all $n$ such that $x_n = 1995$.
18 replies
orl
Aug 10, 2008
happypi31415
an hour ago
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IMO 2018 Problem 5
orthocentre   80
N an hour ago by OronSH
Source: IMO 2018
Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number
$$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$.

Proposed by Bayarmagnai Gombodorj, Mongolia
80 replies
orthocentre
Jul 10, 2018
OronSH
an hour ago
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Line passes through fixed point, as point varies
Jalil_Huseynov   60
N 2 hours ago by Rayvhs
Source: APMO 2022 P2
Let $ABC$ be a right triangle with $\angle B=90^{\circ}$. Point $D$ lies on the line $CB$ such that $B$ is between $D$ and $C$. Let $E$ be the midpoint of $AD$ and let $F$ be the seconf intersection point of the circumcircle of $\triangle ACD$ and the circumcircle of $\triangle BDE$. Prove that as $D$ varies, the line $EF$ passes through a fixed point.
60 replies
Jalil_Huseynov
May 17, 2022
Rayvhs
2 hours ago
J
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Tangent to two circles
Mamadi   2
N 2 hours ago by A22-
Source: Own
Two circles \( w_1 \) and \( w_2 \) intersect each other at \( M \) and \( N \). The common tangent to two circles nearer to \( M \) touch \( w_1 \) and \( w_2 \) at \( A \) and \( B \) respectively. Let \( C \) and \( D \) be the reflection of \( A \) and \( B \) respectively with respect to \( M \). The circumcircle of the triangle \( DCM \) intersect circles \( w_1 \) and \( w_2 \) respectively at points \( E \) and \( F \) (both distinct from \( M \)). Show that the line \( EF \) is the second tangent to \( w_1 \) and \( w_2 \).
2 replies
Mamadi
May 2, 2025
A22-
2 hours ago
J
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Deduction card battle
anantmudgal09   55
N 3 hours ago by deduck
Source: INMO 2021 Problem 4
A Magician and a Detective play a game. The Magician lays down cards numbered from $1$ to $52$ face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.

Prove that the Detective can guarantee a win if and only if she is allowed to ask at least $50$ questions.

Proposed by Anant Mudgal
55 replies
1 viewing
anantmudgal09
Mar 7, 2021
deduck
3 hours ago
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Geometry
Lukariman   7
N 3 hours ago by vanstraelen
Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that $\angle HDM$ = 2∠AMP.
7 replies
Lukariman
Yesterday at 12:43 PM
vanstraelen
3 hours ago
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perpendicularity involving ex and incenter
Erken   20
N 4 hours ago by Baimukh
Source: Kazakhstan NO 2008 problem 2
Suppose that $ B_1$ is the midpoint of the arc $ AC$, containing $ B$, in the circumcircle of $ \triangle ABC$, and let $ I_b$ be the $ B$-excircle's center. Assume that the external angle bisector of $ \angle ABC$ intersects $ AC$ at $ B_2$. Prove that $ B_2I$ is perpendicular to $ B_1I_B$, where $ I$ is the incenter of $ \triangle ABC$.
20 replies
Erken
Dec 24, 2008
Baimukh
4 hours ago
J
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Isosceles Triangle Geo
oVlad   4
N 4 hours ago by Double07
Source: Romania Junior TST 2025 Day 1 P2
Consider the isosceles triangle $ABC$ with $\angle A>90^\circ$ and the circle $\omega$ of radius $AC$ centered at $A.$ Let $M$ be the midpoint of $AC.$ The line $BM$ intersects $\omega$ a second time at $D.$ Let $E$ be a point on $\omega$ such that $BE\perp AC.$ Let $N$ be the intersection of $DE$ and $AC.$ Prove that $AN=2\cdot AB.$
4 replies
oVlad
Apr 12, 2025
Double07
4 hours ago
J
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Geometry
Lukariman   1
N 4 hours ago by Primeniyazidayi
Given acute triangle ABC ,AB=b,AC=c . M is a variable point on side AB. The circle circumscribing triangle BCM intersects AC at N.

a)Let I be the center of the circle circumscribing triangle AMN. Prove that I always lies on a fixed line.

b)Let J be the center of the circle circumscribing triangle MBC. Prove that line segment IJ has a constant length.
1 reply
Lukariman
Today at 4:02 PM
Primeniyazidayi
4 hours ago
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Kingdom of Anisotropy
v_Enhance   24
N 4 hours ago by deduck
Source: IMO Shortlist 2021 C4
The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a path from $X$ to $Y$ is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called diverse if no road belongs to two or more paths in the collection.

Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$.

Proposed by Warut Suksompong, Thailand
24 replies
v_Enhance
Jul 12, 2022
deduck
4 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
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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
219 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
219 posts
GreekIdiot
#2 h  Mar 28, 2025, 8:50 AM
Y by
guys bump
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whwlqkd
99 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
99 posts
whwlqkd
#5 h  Mar 29, 2025, 3:36 AM
Y by
z^3 can be 3(mod 4)
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GreekIdiot
219 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
219 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
107 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
107 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
325 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
219 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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