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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 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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Expected number of flips
Bread10   7
N 20 minutes ago by Pengu14
An unfair coin has a $\frac{4}{7}$ probability of coming up heads and $\frac{3}{7}$ probability of coming up tails. The expected number of flips necessary to first see the sequence $HHTHTHHT$ in that consecutive order can be written as $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. Find the number of factors of $n$.

$\textbf{(A)}~40\qquad\textbf{(B)}~42\qquad\textbf{(C)}~44\qquad\textbf{(D)}~45\qquad\textbf{(E)}~48$
7 replies
Bread10
2 hours ago
Pengu14
20 minutes ago
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stat moment
fruitmonster97   1
N 25 minutes ago by aidan0626
Define xiooix deviation of a set as the average positive difference from each element and the average of said set. Prove that the xiooix deviation of a set is less than or equal to standard deviation.
1 reply
fruitmonster97
an hour ago
aidan0626
25 minutes ago
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Help with Competitive Geometry?
REACHAW   3
N 33 minutes ago by REACHAW
Hi everyone,
I'm struggling a lot with geometry. I've found algebra, number theory, and even calculus to be relatively intuitive. However, when I took geometry, I found it very challenging. I stumbled my way through the class and can do the basic 'textbook' geometry problems, but still struggle a lot with geometry in competitive math. I find myself consistently skipping the geometry problems during contests (even the easier/first ones).

It's difficult for me to see the solution path. I can do the simpler textbook tasks (eg. find congruent triangles) but not more complex ones (eg. draw these two lines to form similar triangles).

Do you have any advice, resources, or techniques I should try?
3 replies
REACHAW
Yesterday at 11:51 PM
REACHAW
33 minutes ago
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Find all real functions withf(x^2 + yf(z)) = xf(x) + zf(y)
Rushil   32
N an hour ago by InterLoop
Source: INMO 2005 Problem 6
Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that \[ f(x^2 + yf(z)) = xf(x) + zf(y) , \] for all $x, y, z \in \mathbb{R}$.
32 replies
Rushil
Aug 23, 2005
InterLoop
an hour ago
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fractional part
Ecrin_eren   3
N an hour ago by rchokler
{x^2}+{x}=0.64

How many positive real values of x satisfy this equation?
3 replies
Ecrin_eren
Apr 13, 2025
rchokler
an hour ago
J
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IMO 2018 Problem 5
orthocentre   78
N 2 hours ago by chenghaohu
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
78 replies
orthocentre
Jul 10, 2018
chenghaohu
2 hours ago
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3-variable inequality with min(ab,bc,ca)>=1
mathwizard888   71
N 3 hours ago by Burak0609
Source: 2016 IMO Shortlist A1
Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$
Proposed by Tigran Margaryan, Armenia
71 replies
mathwizard888
Jul 19, 2017
Burak0609
3 hours ago
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IMO Shortlist 2011, Algebra 7
orl   23
N 3 hours ago by bin_sherlo
Source: IMO Shortlist 2011, Algebra 7
Let $a,b$ and $c$ be positive real numbers satisfying $\min(a+b,b+c,c+a) > \sqrt{2}$ and $a^2+b^2+c^2=3.$ Prove that

\[\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.\]

Proposed by Titu Andreescu, Saudi Arabia
23 replies
orl
Jul 11, 2012
bin_sherlo
3 hours ago
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Ant wanna come to A
Rohit-2006   1
N 4 hours ago by Rohit-2006
An insect starts from $A$ and in $10$ steps and has to reach $A$ again. But in between one of the s steps and can't go $A$. Find probability. For example $ABCDCDEDEA$ is valid but $ABCDEDEDEA$ is not valid.
1 reply
Rohit-2006
4 hours ago
Rohit-2006
4 hours ago
J
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one cyclic formed by two cyclic
CrazyInMath   31
N 4 hours ago by juckter
Source: EGMO 2025/3
Let $ABC$ be an acute triangle. Points $B, D, E$, and $C$ lie on a line in this order and satisfy $BD = DE = EC$. Let $M$ and $N$ be the midpoints of $AD$ and $AE$, respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$, respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic.
31 replies
CrazyInMath
Apr 13, 2025
juckter
4 hours ago
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Divisibility NT FE
CHESSR1DER   10
N 4 hours ago by CHESSR1DER
Source: Own
Find all functions $f$ $N \rightarrow N$ such for any $a,b$:
$(a+b)|a^{f(b)} + b^{f(a)}$.
10 replies
CHESSR1DER
Yesterday at 7:07 PM
CHESSR1DER
4 hours ago
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Prove that x1=x2=....=x2025
Rohit-2006   8
N 4 hours ago by Project_Donkey_into_M4
Source: A mock
The real numbers $x_1,x_2,\cdots,x_{2025}$ satisfy,
$$x_1+x_2=2\bar{x_1}, x_2+x_3=2\bar{x_2},\cdots, x_{2025}+x_1=2\bar{x_{2025}}$$Where {$\bar{x_1},\cdots,\bar{x_{2025}}$} is a permutation of $x_1,x_2,\cdots,x_{2025}$. Prove that $x_1=x_2=\cdots=x_{2025}$
8 replies
Rohit-2006
Apr 9, 2025
Project_Donkey_into_M4
4 hours ago
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IMO 2011 Problem 4
Amir Hossein   92
N 5 hours ago by LobsterJuice
Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \cdots, 2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.
Determine the number of ways in which this can be done.

Proposed by Morteza Saghafian, Iran
92 replies
Amir Hossein
Jul 19, 2011
LobsterJuice
5 hours ago
J
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Squares in an Octagon
kred9   1
N 5 hours ago by crazydog
Source: 2025 Utah Math Olympiad #1
A regular octagon and all of its diagonals are drawn. Find, with proof, the number of squares that appear in the resulting diagram. (The side of each square must lie along one of the edges or diagonals of the octagon.)
1 reply
kred9
Apr 5, 2025
crazydog
5 hours ago
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computational in an convex pentagon (Formula of Unity 2019 gr8,9,10)
parmenides51   1
N May 1, 2020 by vanstraelen
In a convex pentagon $ABCDE, \angle A = 60^o$, and all other angles are equal. It is known that $AB = 6$,$CD = 4$,$EA = 7$. Find the distance from $A$ to the line $CD$.
1 reply
parmenides51
Apr 29, 2020
vanstraelen
May 1, 2020
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computational in an convex pentagon (Formula of Unity 2019 gr8,9,10)
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Reveal topic tags
pentagon
distance
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equal angles
Formula of Unity
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parmenides51
30630 posts
parmenides51
#1 h  Apr 29, 2020, 12:35 AM • 1 Y
Y by Mango247
In a convex pentagon $ABCDE, \angle A = 60^o$, and all other angles are equal. It is known that $AB = 6$,$CD = 4$,$EA = 7$. Find the distance from $A$ to the line $CD$.
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vanstraelen
8963 posts
vanstraelen
#2 h  May 1, 2020, 4:44 PM
Y by
Let $A(0,0),B(6,0)$.
The line $BC$ has equation $y=\sqrt{3}(x-6)$, then point $C(\lambda,\sqrt{3}(\lambda-6))$.
The line $CD$ has equation $y-\sqrt{3}(\lambda-6)=-\sqrt{3}(x-\lambda) \quad\quad (1)$.

The line $AE$ has equation $y=\sqrt{3}x$, then the point $E(\frac{7}{2},\frac{7\sqrt{3}}{2})$.
The line $ED$ has equation $y=\frac{7\sqrt{3}}{2}$.

The lines $CD$ and $ED$ cut in the point $D(2\lambda-\frac{19}{2},\frac{7\sqrt{3}}{2})$.
The condition $CD=4$ gives us $\lambda=\frac{15}{2}$.

Distance from point $A$ to the line $CD$, see (1), equals $\frac{9\sqrt{3}}{2}$.
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