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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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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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Beautiful geometry
m4thbl3nd3r   0
a minute ago
Let $\omega$ be the circumcircle of triangle $ABC$, $M$ is the midpoint of $BC$ and $E$ be the second intersection of $AM$ and $\omega$. Tangent line of $\omega$ at $E$ intersects $BC$ at $P$, let $PKL$ be a transversal of $\omega$ and $X,Y$ be intersections of $AK,AL$ with $BC$. Let $PF$ be a tangent line of $\omega$. Prove that $LYFP$ is cyclic
0 replies
m4thbl3nd3r
a minute ago
0 replies
J
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powers sums and triangular numbers
gaussious   2
N 5 minutes ago by Quidditch
prove 1^k+2^k+3^k + \cdots + n^k \text{is divisible by } \frac{n(n+1)}{2} \text{when} k \text{is odd}
2 replies
gaussious
4 hours ago
Quidditch
5 minutes ago
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prove |a-b| is a square, given a-b=a^2c-b^2d
Alpha314159   4
N 23 minutes ago by Leman_Nabiyeva
Source: Macau Inter High School Competition
Let $a, b$ be integers such that there are consecutive integers $c,d$ satisfy $$a-b=a^2 c-b^2 d$$.
Prove : $|a-b|$ is a perfect square.
4 replies
1 viewing
Alpha314159
Mar 7, 2020
Leman_Nabiyeva
23 minutes ago
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Inspired by pennypc123456789
sqing   0
32 minutes ago
Source: Own
Let $ a,b,c\geq 0 $and $ab+bc+ca\neq 0.$ Prove that
$$ \frac{9-8\sqrt 2+5\sqrt 5}{6}\leq\frac{a + b}{a + 2b + c} + \dfrac{b + c}{b + 2c + a}+\dfrac{c + a}{c + 2a + b}\leq \frac{9+8\sqrt 2-5\sqrt 5}{6}$$
0 replies
sqing
32 minutes ago
0 replies
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The Bank of Bath
TelMarin   100
N 35 minutes ago by Ihatecombin
Source: IMO 2019, problem 5
The Bank of Bath issues coins with an $H$ on one side and a $T$ on the other. Harry has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly $k>0$ coins showing $H$, then he turns over the $k$th coin from the left; otherwise, all coins show $T$ and he stops. For example, if $n=3$ the process starting with the configuration $THT$ would be $THT \to HHT \to HTT \to TTT$, which stops after three operations.

(a) Show that, for each initial configuration, Harry stops after a finite number of operations.

(b) For each initial configuration $C$, let $L(C)$ be the number of operations before Harry stops. For example, $L(THT) = 3$ and $L(TTT) = 0$. Determine the average value of $L(C)$ over all $2^n$ possible initial configurations $C$.

Proposed by David Altizio, USA
100 replies
TelMarin
Jul 17, 2019
Ihatecombin
35 minutes ago
J
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Permutations of Integers from 1 to n
Twoisntawholenumber   74
N 37 minutes ago by Maximilian113
Source: 2020 ISL C1
Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the
sequence $1$, $2$, $\dots$ , $n$ satisfying
$$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$.

Proposed by United Kingdom
74 replies
Twoisntawholenumber
Jul 20, 2021
Maximilian113
37 minutes ago
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11^n+2^n+6=m^3
jungle_wang   5
N 40 minutes ago by Rayanelba
Source: 2024IMOC
Find all positive integers $(m,n)$ such that
$$11^n+2^n+6=m^3$$
5 replies
jungle_wang
Aug 1, 2024
Rayanelba
40 minutes ago
J
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Inequality from my inequality training.
Orkhan-Ashraf_2002   2
N an hour ago by sqing
Let $a,b,c$ non-negative real numbers,but $ab+bc+ca\not=$0.Prove that
\[1\leq \frac{a+b}{a+4b+c}+\frac{b+c}{b+4c+a}+\frac{c+a}{c+4a+b}\leq \frac{4}{3}\]
2 replies
Orkhan-Ashraf_2002
Aug 21, 2016
sqing
an hour ago
J
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inequalities
pennypc123456789   3
N an hour ago by arqady
If $a,b,c$ are positive real numbers, then
$$ \frac{a + b}{a + 7b + c} + \dfrac{b + c}{b + 7c + a}+\dfrac{c + a}{c + 7a + b} \geq \dfrac{2}{3}$$
we can generalize this problem
3 replies
pennypc123456789
3 hours ago
arqady
an hour ago
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Using Humpty point on Trapezoid ??
FireBreathers   1
N an hour ago by aidenkim119
Given a trapezoid $ABCD$ with $AD//BC$. Let point $H$ be orthocenter $ABD$ and $M$ midpoint $AD$. It is also known that $HC$ perpendicular to $BM$. Let $X$ be a point on the segment $AB$ such that $XH=BH$ and point $Y$ be the intersection of $CX$ and $BD$. Prove that $AXYD$ concyclic
1 reply
FireBreathers
4 hours ago
aidenkim119
an hour ago
J
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IMO 2014 Problem 2
v_Enhance   60
N an hour ago by math-olympiad-clown
Source: 0
Let $n \ge 2$ be an integer. Consider an $n \times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that, for each peaceful configuration of $n$ rooks, there is a $k \times k$ square which does not contain a rook on any of its $k^2$ unit squares.
60 replies
v_Enhance
Jul 8, 2014
math-olympiad-clown
an hour ago
J
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Counting friends in two ways
joybangla   18
N an hour ago by Mathworld314
Source: ISI Entrance 2014, P1
Suppose a class contains $100$ students. Let, for $1\le i\le 100$, the $i^{\text{th}}$ student have $a_i$ many friends. For $0\le j\le 99$ let us define $c_j$ to be the number of students who have strictly more than $j$ friends. Show that \begin{align*} & \sum_{i=1}^{100}a_i=\sum_{j=0}^{99}c_j \end{align*}
18 replies
joybangla
May 11, 2014
Mathworld314
an hour ago
J
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Function equation
luci1337   0
2 hours ago
find all function $f:R \rightarrow R$ such that:
$2f(x)f(x+y)-f(x^2)=\frac{x}{2}(f(2x)+f(f(y)))$ with all $x,y$ is real number
0 replies
luci1337
2 hours ago
0 replies
J
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Coaxal Circles
fattypiggy123   29
N 2 hours ago by sttsmet
Source: China TSTST Test 2 Day 1 Q3
Let $ABCD$ be a quadrilateral and let $l$ be a line. Let $l$ intersect the lines $AB,CD,BC,DA,AC,BD$ at points $X,X',Y,Y',Z,Z'$ respectively. Given that these six points on $l$ are in the order $X,Y,Z,X',Y',Z'$, show that the circles with diameter $XX',YY',ZZ'$ are coaxal.
29 replies
fattypiggy123
Mar 13, 2017
sttsmet
2 hours ago
J
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J
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The greatest number of balls
Amir Hossein   4
N Mar 25, 2011 by math_explorer
What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \times 10 \times 1 \ ?$
4 replies
Amir Hossein
Sep 29, 2010
math_explorer
Mar 25, 2011
J
The greatest number of balls
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Reveal topic tags
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Amir Hossein
5452 posts
Amir Hossein
#1 h  Sep 29, 2010, 11:55 AM • 2 Y
Y by Adventure10, Mango247
What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \times 10 \times 1 \ ?$
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chen_alex
12 posts
chen_alex
#2 h  Mar 24, 2011, 11:50 PM • 1 Y
Y by Adventure10
Well, wouldn't it be $100$ balls? If the radius is $1/2$, then the ball's diameter would be equal to $1$. Thus, the number of balls that can fit into the box with the given dimensions is equal to $ (10/1)\times\ (10/1)\times\ (1/1) = 100$
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math_explorer
583 posts
math_explorer
#3 h  Mar 25, 2011, 12:15 AM • 1 Y
Y by Adventure10
That's actually not the optimal packing. You can fit $106$ balls by packing rows of 10,9,10,9,10,9,10,9,10,10,10.

I think that's optimal, although I don't have a proof.
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chen_alex
12 posts
chen_alex
#4 h  Mar 25, 2011, 12:17 AM • 1 Y
Y by Adventure10
Yeah, i just actually realized that there was a better way of packing the balls. I think I can prove that i can fit $106$ circles of radius 1/2 into a 10 by 10 rectangle. I'm trying to use the same logic though now to apply it to 3-D...might take some time.
This post has been edited 1 time. Last edited by chen_alex, Mar 25, 2011, 12:35 AM
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math_explorer
583 posts
math_explorer
#5 h  Mar 25, 2011, 12:30 AM • 1 Y
Y by Adventure10
You don't have to do any work to extend it to three dimensions, at least for this problem. Since the box has a thickness of 1 unit, all the balls are fixed along that dimension, and taking the cross-section with a plane 1/2 unit along this dimension yields circles of radius 1/2 in a 10x10 box. So the problem is exactly the same as the 2-D 10x10 problem.
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