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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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[*]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
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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Iranian geometry configuration
Assassino9931   2
N 36 minutes ago by Captainscrubz
Source: Al-Khwarizmi Junior International Olympiad 2025 P7
Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$, such that $CD$ is not a diameter of its circumcircle. The lines $AD$ and $BC$ intersect at point $P$, so that $A$ lies between $D$ and $P$, and $B$ lies between $C$ and $P$. Suppose triangle $PCD$ is acute and let $H$ be its orthocenter. The points $E$ and $F$ on the lines $BC$ and $AD$, respectively, are such that $BD \parallel HE$ and $AC\parallel HF$. The line through $E$, perpendicular to $BC$, intersects $AD$ at $L$, and the line through $F$, perpendicular to $AD$, intersects $BC$ at $K$. Prove that the points $K$, $L$, $O$ are collinear.

Amir Parsa Hosseini Nayeri, Iran
2 replies
1 viewing
Assassino9931
Today at 9:39 AM
Captainscrubz
36 minutes ago
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Range if \omega for No Inscribed Right Triangle y = \sin(\omega x)
ThisIsJoe   1
N an hour ago by Lankou
For a positive number \omega , determine the range of \omega for which the curve y = \sin(\omega x) has no inscribed right triangle.
Could someone help me figure out how to approach this?
1 reply
ThisIsJoe
May 8, 2025
Lankou
an hour ago
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f(m + n) >= f(m) + f(f(n)) - 1
orl   30
N 2 hours ago by ezpotd
Source: IMO Shortlist 2007, A2, AIMO 2008, TST 2, P1, Ukrainian TST 2008 Problem 8
Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition
\[ f(m + n) \geq f(m) + f(f(n)) - 1 \]
for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$

Author: Nikolai Nikolov, Bulgaria
30 replies
orl
Jul 13, 2008
ezpotd
2 hours ago
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Classic Diophantine
Adywastaken   3
N 2 hours ago by Adywastaken
Source: NMTC 2024/6
Find all natural number solutions to $3^x-5^y=z^2$.
3 replies
Adywastaken
4 hours ago
Adywastaken
2 hours ago
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Add d or Divide by a
MarkBcc168   25
N 2 hours ago by Entei
Source: ISL 2022 N3
Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define
$$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.
25 replies
MarkBcc168
Jul 9, 2023
Entei
2 hours ago
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Alice and Bob play, 8x8 table, white red black, minimum n for victory
parmenides51   14
N 2 hours ago by Ilikeminecraft
Source: JBMO Shortlist 2018 C3
The cells of a $8 \times 8$ table are initially white. Alice and Bob play a game. First Alice paints $n$ of the fields in red. Then Bob chooses $4$ rows and $4$ columns from the table and paints all fields in them in black. Alice wins if there is at least one red field left. Find the least value of $n$ such that Alice can win the game no matter how Bob plays.
14 replies
parmenides51
Jul 22, 2019
Ilikeminecraft
2 hours ago
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GEOMETRY GEOMETRY GEOMETRY
Kagebaka   71
N 2 hours ago by bin_sherlo
Source: IMO 2021/3
Let $D$ be an interior point of the acute triangle $ABC$ with $AB > AC$ so that $\angle DAB = \angle CAD.$ The point $E$ on the segment $AC$ satisfies $\angle ADE =\angle BCD,$ the point $F$ on the segment $AB$ satisfies $\angle FDA =\angle DBC,$ and the point $X$ on the line $AC$ satisfies $CX = BX.$ Let $O_1$ and $O_2$ be the circumcenters of the triangles $ADC$ and $EXD,$ respectively. Prove that the lines $BC, EF,$ and $O_1O_2$ are concurrent.
71 replies
Kagebaka
Jul 20, 2021
bin_sherlo
2 hours ago
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Equation of integers
jgnr   3
N 3 hours ago by KTYC
Source: Indonesia Mathematics Olympiad 2005 Day 1 Problem 2
For an arbitrary positive integer $ n$, define $ p(n)$ as the product of the digits of $ n$ (in decimal). Find all positive integers $ n$ such that $ 11p(n)=n^2-2005$.
3 replies
jgnr
Jun 2, 2008
KTYC
3 hours ago
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Divisibility..
Sadigly   4
N 3 hours ago by Solar Plexsus
Source: another version of azerbaijan nmo 2025
Just ignore this
4 replies
Sadigly
Yesterday at 7:37 AM
Solar Plexsus
3 hours ago
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Surjective number theoretic functional equation
snap7822   3
N 3 hours ago by internationalnick123456
Source: 2025 Taiwan TST Round 3 Independent Study 2-N
Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions:
[list=i]
[*] For all $m, n \in \mathbb{N}$, if $m > n$ and $f(m) > f(n)$, then $f(m-n) = f(n)$;
[*] $f$ is surjective.
[/list]
Find the maximum possible value of $f(2025)$.

Proposed by snap7822
3 replies
snap7822
May 1, 2025
internationalnick123456
3 hours ago
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Many Equal Sides
mathisreal   3
N 3 hours ago by QueenArwen
Source: Brazil EGMO TST 2023 #1
Let $ABC$ be a triangle with $BA=BC$ and $\angle ABC=90^{\circ}$. Let $D$ and $E$ be the midpoints of $CA$ and $BA$ respectively. The point $F$ is inside of $\triangle ABC$ such that $\triangle DEF$ is equilateral. Let $X=BF\cap AC$ and $Y=AF\cap DB$. Prove that $DX=YD$.
3 replies
mathisreal
Nov 10, 2022
QueenArwen
3 hours ago
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Calculate the distance AD
MTA_2024   3
N 3 hours ago by MTA_2024
A semi-circle is inscribed in a quadrilateral $ABCD$. The center $O$ of the semi-circle is the midpoint of segment $AD$. We have $AB=9$ and $CD=16$.
Calculate the distance $AD$.
3 replies
MTA_2024
Yesterday at 3:50 PM
MTA_2024
3 hours ago
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2012 preRMO p17, roots of equation x^3 + 3x + 5 = 0
parmenides51   11
N 4 hours ago by Pengu14
Let $x_1,x_2,x_3$ be the roots of the equation $x^3 + 3x + 5 = 0$. What is the value of the expression
$\left( x_1+\frac{1}{x_1} \right)\left( x_2+\frac{1}{x_2} \right)\left( x_3+\frac{1}{x_3} \right)$ ?
11 replies
parmenides51
Jun 17, 2019
Pengu14
4 hours ago
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Interesting question from Al-Khwarezmi olympiad 2024 P3, day1
Adventure1000   3
N 5 hours ago by sqing
Find all $x, y, z \in \left (0, \frac{1}{2}\right )$ such that
$$ \begin{cases} (3 x^{2}+y^{2}) \sqrt{1-4 z^{2}} \geq z; \\ (3 y^{2}+z^{2}) \sqrt{1-4 x^{2}} \geq x; \\ (3 z^{2}+x^{2}) \sqrt{1-4 y^{2}} \geq y. \end{cases} $$Proposed by Ngo Van Trang, Vietnam
3 replies
Adventure1000
May 7, 2025
sqing
5 hours ago
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Coin Probability
girishpimoli   1
N Apr 28, 2025 by lpieleanu
A fair coin is repeatedly tossed. If the probability that the first time head is tossed , twice in a row , is on the $9$ th and $10$ th toss, is
1 reply
girishpimoli
Apr 28, 2025
lpieleanu
Apr 28, 2025
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Coin Probability
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Reveal topic tags
probability
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girishpimoli
216 posts
girishpimoli
#1 h  Apr 28, 2025, 3:20 AM
Y by
A fair coin is repeatedly tossed. If the probability that the first time head is tossed , twice in a row , is on the $9$ th and $10$ th toss, is
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lpieleanu
3001 posts
lpieleanu
#2 h  Apr 28, 2025, 3:26 AM • 1 Y
Y by girishpimoli
For $1 \le i \le 8,$ let $p_{H, i}$ denote the probability that this occurs if the $i$th flip is heads, and let $p_{T, i}$ denote the probability that this occurs if the $i$th flip is tails. Using the fact that $$p_{H,8}=0, p_{T,8}=\frac{1}{4},$$we can work our way backwards to get $$p_{H, 1} = \frac{13}{512}, p_{T,1}=\frac{21}{512},$$giving an answer of
\begin{align*} \frac{1}{2} \cdot \frac{13}{512} + \frac{1}{2} \cdot \frac{21}{512} &= \frac{13}{1024} + \frac{21}{1024} \\ &= \frac{34}{1024} \\ &= \boxed{\frac{17}{512}}. \end{align*}$\square$
This post has been edited 1 time. Last edited by lpieleanu, Apr 28, 2025, 3:27 AM
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