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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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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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
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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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Two circles tangents
Eeightqx   11
N 4 minutes ago by awesomeming327.
Source: 2024 GPO P2
Let $\triangle ABC$ be a acute triangle with altitudes $AD,BE,CF$. $H$ is the orthocenter of $\triangle ABC$. The angle bisectors of $\angle BEC,\angle BFC$ intersect $BC$ at $X,Y$, respectively. The circle $\odot(BC)$ with diameter $BC$ intersects segment $AD$ at $Z$. Show that $\odot(BC)$ is tangent to $\odot(XYZ)$.
11 replies
Eeightqx
Feb 14, 2024
awesomeming327.
4 minutes ago
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Polynomial...
Sadigly   0
15 minutes ago
Source: Azerbaijan Senior NMO 2020
Find all nonzero polynomials $P(x)$ with real coefficents, that satisfies $$P(x)^3+3P(x)^2=P(x^3)-3P(-x)$$for all real numbers $x$
0 replies
Sadigly
15 minutes ago
0 replies
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Regular 2021-gon
Sadigly   0
17 minutes ago
Source: Azerbaijan Senior NMO 2020
A regular 2021-gon is divided into 2019 triangles,such that no diagonals intersect. Prove that at least 3 of the 2019 triangles are isoscoles
0 replies
Sadigly
17 minutes ago
0 replies
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Circumcircle of ADM
v_Enhance   67
N 20 minutes ago by Ilikeminecraft
Source: USA TSTST 2012, Problem 7
Triangle $ABC$ is inscribed in circle $\Omega$. The interior angle bisector of angle $A$ intersects side $BC$ and $\Omega$ at $D$ and $L$ (other than $A$), respectively. Let $M$ be the midpoint of side $BC$. The circumcircle of triangle $ADM$ intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respectively. Let $N$ be the midpoint of segment $PQ$, and let $H$ be the foot of the perpendicular from $L$ to line $ND$. Prove that line $ML$ is tangent to the circumcircle of triangle $HMN$.
67 replies
v_Enhance
Jul 19, 2012
Ilikeminecraft
20 minutes ago
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Nice geometry...
Sadigly   0
20 minutes ago
Source: Azerbaijan Senior NMO 2020
Let $ABC$ be a scalene triangle, and let $I$ be its incenter. A point $D$ is chosen on line $BC$, such that the circumcircle of triangle $BID$ intersects $AB$ at $E\neq B$, and the circumcircle of triangle $CID$ intersects $AC$ at $F\neq C$. Circumcircle of triangle $EDF$ intersects $AB$ and $AC$ at $M$ and $N$, respectively. Lines $FD$ and $IC$ intersect at $Q$, and lines $ED$ and $BI$ intersect at $P$. Prove that $EN\parallel MF\parallel PQ$.
0 replies
Sadigly
20 minutes ago
0 replies
J
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Diophantine involving cube
Sadigly   0
25 minutes ago
Source: Azerbaijan Senior NMO 2020
$a;b;c;d\in\mathbb{Z^+}$. Solve the equation: $$2^{a!}+2^{b!}+2^{c!}=d^3$$
0 replies
Sadigly
25 minutes ago
0 replies
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Easy combinatorics
Sadigly   5
N 30 minutes ago by GreekIdiot
Source: Azerbaijan Junior NMO 2018
First $20$ positive integers are written on a board. It is known that, after you erase a number from the board, there exists a number that is equal to the arithmetic mean of the rest of the numbers left on the board. Find all the numbers that could've been erased.
5 replies
1 viewing
Sadigly
an hour ago
GreekIdiot
30 minutes ago
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Easy algebra
Sadigly   1
N 35 minutes ago by Vivaandax
Source: Azerbaijan Junior NMO 2018
$x^{11}+x^7+x^3=1$. $$x^{\alpha}=x^4+x^3-1.\hspace{4mm} \alpha=?$$
1 reply
Sadigly
an hour ago
Vivaandax
35 minutes ago
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For an advanced method, search Lagrange Interpolation (those who knows)
Sadigly   2
N 36 minutes ago by Sadigly
Source: Azerbaijan Senior NMO 2018
$P(x)$ is a fifth degree polynomial. $P(2018)=1$, $P(2019)=2$ $P(2020)=3$, $P(2021)=4$, $P(2022)=5$. $P(2017)=?$
2 replies
Sadigly
an hour ago
Sadigly
36 minutes ago
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Easy geometry...
Sadigly   1
N 39 minutes ago by Nuran2010
Source: Azerbaijan NMO 2018
A circle $\omega$ and a point $T$ outside the circle is given. Let a tangent from $T$ to $\omega$ touch $\omega$ at $A$, and take points $B,C$ lying on $\omega$ such that $T,B,C$ are colinear. The bisector of $\angle ATC$ intersects $AB$ and $AC$ at $P$ and $Q$,respectively. Prove that $PA=\sqrt{PB\cdot QC}$
1 reply
Sadigly
an hour ago
Nuran2010
39 minutes ago
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Easy inequality...
Sadigly   0
41 minutes ago
Source: Azerbaijan Senior NMO 2020
$x,y,z\in\mathbb{R^+}$. If $xyz=1$, then prove the following: $$\sum\frac{x^6+2}{x^3}\geq3(\frac{x}{y}+\frac{y}{z}+\frac{z}{x})$$
0 replies
Sadigly
41 minutes ago
0 replies
J
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Easy inequality...
Sadigly   2
N 41 minutes ago by ehuseyinyigit
Source: Azerbaijan Junior NMO 2018
$a;b\in\mathbb{R^+}$. Prove the following inequality: $$\sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{a}}\leq\sqrt[3]{2(a+b)(\frac1{a}+\frac1{b})}$$
2 replies
Sadigly
an hour ago
ehuseyinyigit
41 minutes ago
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reals associated with 1024 points
bin_sherlo   1
N an hour ago by AnSoLiN
Source: Türkiye 2025 JBMO TST P8
Pairwise distinct points $P_1,\dots,P_{1024}$, which lie on a circle, are marked by distinct reals $a_1,\dots,a_{1024}$. Let $P_i$ be $Q-$good for a $Q$ on the circle different than $P_1,\dots,P_{1024}$, if and only if $a_i$ is the greatest number on at least one of the two arcs $P_iQ$. Let the score of $Q$ be the number of $Q-$good points on the circle. Determine the greatest $k$ such that regardless of the values of $a_1,\dots,a_{1024}$, there exists a point $Q$ with score at least $k$.
1 reply
bin_sherlo
3 hours ago
AnSoLiN
an hour ago
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JBMO Combinatorics vibes
Sadigly   0
an hour ago
Source: Azerbaijan Senior NMO 2018
Numbers $1,2,3...,100$ are written on a board. $A$ and $B$ plays the following game: They take turns choosing a number from the board and deleting them. $A$ starts first. They sum all the deleted numbers. If after a player's turn (after he deletes a number on the board) the sum of the deleted numbers can't be expressed as difference of two perfect squares,then he loses, if not, then the game continues as usual. Which player got a winning strategy?
0 replies
Sadigly
an hour ago
0 replies
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J
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Hard NT problem
tiendat004   2
N Apr 1, 2025 by avinashp
Given two odd positive integers $a,b$ are coprime. Consider the sequence $(x_n)$ given by $x_0=2,x_1=a,x_{n+2}=ax_{n+1}+bx_n,$ $\forall n\geq 0$. Suppose that there exist positive integers $m,n,p$ such that $mnp$ is even and $\dfrac{x_m}{x_nx_p}$ is an integer. Prove that the numerator in its simplest form of $\dfrac{m}{np}$ is an odd integer greater than $1$.
2 replies
tiendat004
Aug 15, 2024
avinashp
Apr 1, 2025
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Hard NT problem
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Reveal topic tags
number theory
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tiendat004
4 posts
tiendat004
#1 h  Aug 15, 2024, 2:55 PM
Y by
Given two odd positive integers $a,b$ are coprime. Consider the sequence $(x_n)$ given by $x_0=2,x_1=a,x_{n+2}=ax_{n+1}+bx_n,$ $\forall n\geq 0$. Suppose that there exist positive integers $m,n,p$ such that $mnp$ is even and $\dfrac{x_m}{x_nx_p}$ is an integer. Prove that the numerator in its simplest form of $\dfrac{m}{np}$ is an odd integer greater than $1$.
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Dajinmei04288
1 post
Dajinmei04288
#2 h  Apr 1, 2025, 6:32 PM
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Very hard
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avinashp
1281 posts
avinashp
#3 h  Apr 1, 2025, 6:36 PM
Y by
tiendat004 wrote:
Given two odd positive integers $a,b$ are coprime. Consider the sequence $(x_n)$ given by $x_0=2,x_1=a,x_{n+2}=ax_{n+1}+bx_n,$ $\forall n\geq 0$. Suppose that there exist positive integers $m,n,p$ such that $mnp$ is even and $\dfrac{x_m}{x_nx_p}$ is an integer. Prove that the numerator in its simplest form of $\dfrac{m}{np}$ is an odd integer greater than $1$.

:fool:
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