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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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Imtersecting two regular pentagons
Miquel-point   2
N 24 minutes ago by ohiorizzler1434
Source: KoMaL B. 5093
The intersection of two congruent regular pentagons is a decagon with sides of $a_1,a_2,\ldots ,a_{10}$ in this order. Prove that
\[a_1a_3+a_3a_5+a_5a_7+a_7a_9+a_9a_1=a_2a_4+a_4a_6+a_6a_8+a_8a_{10}+a_{10}a_2.\]
2 replies
Miquel-point
4 hours ago
ohiorizzler1434
24 minutes ago
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monving balls in 2018 boxes
parmenides51   1
N an hour ago by venhancefan777
Source: 1st Mathematics Regional Olympiad of Mexico Northwest 2018 P1
There are $2018$ boxes $C_1$, $C_2$, $C_3$,..,$C_{2018}$. The $n$-th box $C_n$ contains $n$ balls.
A move consists of the following steps:
a) Choose an integer $k$ greater than $1$ and choose $m$ a multiple of $k$.
b) Take a ball from each of the consecutive boxes $C_{m-1}$, $C_m$, $C_{m+1}$ and move the $3$ balls to the box $C_{m+k}$.
With these movements, what is the largest number of balls we can get in the box $2018$?
1 reply
1 viewing
parmenides51
Sep 6, 2022
venhancefan777
an hour ago
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inequality
danilorj   0
an hour ago
Let $a, b, c$ be nonnegative real numbers such that $a + b + c = 3$. Prove that
\[ \frac{a}{4 - b} + \frac{b}{4 - c} + \frac{c}{4 - a} + \frac{1}{16}(1 - a)^2(1 - b)^2(1 - c)^2 \leq 1, \]and determine all such triples $(a, b, c)$ where the equality holds.
0 replies
danilorj
an hour ago
0 replies
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P,Q,B are collinear
MNJ2357   28
N 2 hours ago by Ilikeminecraft
Source: 2020 Korea National Olympiad P2
$H$ is the orthocenter of an acute triangle $ABC$, and let $M$ be the midpoint of $BC$. Suppose $(AH)$ meets $AB$ and $AC$ at $D,E$ respectively. $AH$ meets $DE$ at $P$, and the line through $H$ perpendicular to $AH$ meets $DM$ at $Q$. Prove that $P,Q,B$ are collinear.
28 replies
MNJ2357
Nov 21, 2020
Ilikeminecraft
2 hours ago
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Chinese Girls Mathematical Olympiad 2017, Problem 7
Hermitianism   45
N 2 hours ago by Ilikeminecraft
Source: Chinese Girls Mathematical Olympiad 2017, Problem 7
This is a very classical problem.
Let the $ABCD$ be a cyclic quadrilateral with circumcircle $\omega_1$.Lines $AC$ and $BD$ intersect at point $E$,and lines $AD$,$BC$ intersect at point $F$.Circle $\omega_2$ is tangent to segments $EB,EC$ at points $M,N$ respectively,and intersects with circle $\omega_1$ at points $Q,R$.Lines $BC,AD$ intersect line $MN$ at $S,T$ respectively.Show that $Q,R,S,T$ are concyclic.
45 replies
Hermitianism
Aug 16, 2017
Ilikeminecraft
2 hours ago
J
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D1031 : A general result on polynomial 1
Dattier   1
N 2 hours ago by Dattier
Source: les dattes à Dattier
Let $P(x,y) \in \mathbb Q(x,y)$ with $\forall (a,b) \in \mathbb Z^2, P(a,b) \in \mathbb Z $.

Is it true that $P(x,y) \in \mathbb Q[x,y]$?
1 reply
Dattier
5 hours ago
Dattier
2 hours ago
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Asymmetric FE
sman96   18
N 2 hours ago by jasperE3
Source: BdMO 2025 Higher Secondary P8
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that$$f(xf(y)-y) + f(xy-x) + f(x+y) = 2xy$$for all $x, y \in \mathbb{R}$.
18 replies
sman96
Feb 8, 2025
jasperE3
2 hours ago
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Easy Geometry
pokmui9909   6
N 2 hours ago by reni_wee
Source: FKMO 2025 P4
Triangle $ABC$ satisfies $\overline{CA} > \overline{AB}$. Let the incenter of triangle $ABC$ be $\omega$, which touches $BC, CA, AB$ at $D, E, F$, respectively. Let $M$ be the midpoint of $BC$. Let the circle centered at $M$ passing through $D$ intersect $DE, DF$ at $P(\neq D), Q(\neq D)$, respecively. Let line $AP$ meet $BC$ at $N$, line $BP$ meet $CA$ at $L$. Prove that the three lines $EQ, FP, NL$ are concurrent.
6 replies
pokmui9909
Mar 30, 2025
reni_wee
2 hours ago
J
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Old hard problem
ItzsleepyXD   3
N 3 hours ago by Funcshun840
Source: IDK
Let $ABC$ be a triangle and let $O$ be its circumcenter and $I$ its incenter.
Let $P$ be the radical center of its three mixtilinears and let $Q$ be the isogonal conjugate of $P$.
Let $G$ be the Gergonne point of the triangle $ABC$.
Prove that line $QG$ is parallel with line $OI$ .
3 replies
ItzsleepyXD
Apr 25, 2025
Funcshun840
3 hours ago
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\frac{2^{n!}-1}{2^n-1} be a square
AlperenINAN   10
N 3 hours ago by Nuran2010
Source: Turkey JBMO TST 2024 P5
Find all positive integer values of $n$ such that the value of the
$$\frac{2^{n!}-1}{2^n-1}$$is a square of an integer.
10 replies
AlperenINAN
May 13, 2024
Nuran2010
3 hours ago
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Beautiful Angle Sum Property in Hexagon with Incenter
Raufrahim68   0
4 hours ago
Hello everyone! I discovered an interesting geometric property and would like to share it with the community. I'm curious if this is a known result and whether it can be generalized.

Problem Statement:
Let
A
B
C
D
E
K
ABCDEK be a convex hexagon with an incircle centered at
O
O. Prove that:

∠
A
O
B
+
∠
C
O
D
+
∠
E
O
K
=
180
∘
∠AOB+∠COD+∠EOK=180
∘
0 replies
Raufrahim68
4 hours ago
0 replies
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Anything real in this system must be integer
Assassino9931   7
N 4 hours ago by Leman_Nabiyeva
Source: Al-Khwarizmi International Junior Olympiad 2025 P1
Determine the largest integer $c$ for which the following statement holds: there exists at least one triple $(x,y,z)$ of integers such that
\begin{align*} x^2 + 4(y + z) = y^2 + 4(z + x) = z^2 + 4(x + y) = c \end{align*}and all triples $(x,y,z)$ of real numbers, satisfying the equations, are such that $x,y,z$ are integers.

Marek Maruin, Slovakia
7 replies
Assassino9931
May 9, 2025
Leman_Nabiyeva
4 hours ago
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CIIM 2011 First day problem 3
Ozc   2
N 4 hours ago by pi_quadrat_sechstel
Source: CIIM 2011
Let $f(x)$ be a rational function with complex coefficients whose denominator does not have multiple roots. Let $u_0, u_1,... , u_n$ be the complex roots of $f$ and $w_1, w_2,..., w_m$ be the roots of $f'$. Suppose that $u_0$ is a simple root of $f$. Prove that
\[ \sum_{k=1}^m \frac{1}{w_k - u_0} = 2\sum_{k = 1}^n\frac{1}{u_k - u_0}.\]
2 replies
Ozc
Oct 3, 2014
pi_quadrat_sechstel
4 hours ago
J
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IMO 2009 P2, but in space
Miquel-point   1
N 4 hours ago by Miquel-point
Source: KoMaL A. 485
Let $ABCD$ be a tetrahedron with circumcenter $O$. Suppose that the points $P, Q$ and $R$ are interior points of the edges $AB, AC$ and $AD$, respectively. Let $K, L, M$ and $N$ be the centroids of the triangles $PQD$, $PRC,$ $QRB$ and $PQR$, respectively. Prove that if the plane $PQR$ is tangent to the sphere $KLMN$ then $OP=OQ=OR.$

1 reply
Miquel-point
4 hours ago
Miquel-point
4 hours ago
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J
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Very easy NT
GreekIdiot   4
N May 1, 2025 by GreekIdiot
Prove that there exists no natural number $n>1$ such that $n \mid 2^n-1$.
4 replies
GreekIdiot
Apr 30, 2025
GreekIdiot
May 1, 2025
J
Very easy NT
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Reveal topic tags
number theory
bezout
Elementary
Fermat s Little Theorem
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G H BBookmark kLocked kLocked NReply
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GreekIdiot
234 posts
GreekIdiot
#1 h  Apr 30, 2025, 2:49 PM
Y by
Prove that there exists no natural number $n>1$ such that $n \mid 2^n-1$.
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NO_SQUARES
1132 posts
NO_SQUARES
#2 h  Apr 30, 2025, 3:22 PM
Y by
GreekIdiot wrote:
Prove that there exists no natural number $n>1$ such that $n \mid 2^n-1$.

https://artofproblemsolving.com/community/q2h150382p32726736
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ektorasmiliotis
107 posts
ektorasmiliotis
#7 h  Apr 30, 2025, 4:14 PM
Y by
GreekIdiot wrote:
Prove that there exists no natural number $n>1$ such that $n \mid 2^n-1$.

Consider L(n)=the smallest prime divisor of n for n>1
We use the lemma L((x^n -1)/(x-1))>= L(n) (note that there is no equality case for x=2)
Z K Y
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Primeniyazidayi
106 posts
Primeniyazidayi
#8 h  Apr 30, 2025, 5:46 PM
Y by
Quick proof:
Let $p$ be the smallest prime divisor of $n$.Then note that $ord_p(2) \mid n$ because $2^n \equiv 1 (mod p)$.By Euler we also have $ord_p(2) \mid p-1$.Then $ord_p(2) \mid (n,p-1)=1$ and $ord_p(2)=1$.But then $2 \equiv 1 (mod p)$,absurd.Done.
This post has been edited 1 time. Last edited by Primeniyazidayi, Apr 30, 2025, 5:47 PM
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GreekIdiot
234 posts
GreekIdiot
#10 h  May 1, 2025, 12:40 PM
Y by
Let $n \in \mathbb N$ such that $n \mid 2^n-1$ and $p$ be its minimal prime divisor. Obviously $p>2$.
Then by condition $p \mid n \implies p\mid 2^n-1 \implies 2^n \equiv 1 \: mod \: p$
By Fermat's little theorem $2^{p-1} \equiv 1 \: mod \: p$
$p \mid n \implies p-1 \nmid n$ meaning $gcd(n,p-1)=1$
By bezout lemma $\exists \: x,y \in \mathbb Z$ such that $xn+y(p-1)=1$
Then $2^{xn+y(p-1)} \equiv 2^1 \: mod \: p$
But $2^{xn} \cdot 2^{y(p-1)}\equiv 1 \cdot 1 \equiv 1 \: mod \: p$ Contradiction qed
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