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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 3:18 PM
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0 replies
jlacosta
Yesterday at 3:18 PM
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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Inequality
Ecrin_eren   1
N 15 minutes ago by sqing
For non-negative real numbers x and y , if the inequality

x^2(y^2 + 9) + x^4y + 3y^2 \geq 27

holds, what is the minimum possible value of

x^2 + y

1 reply
Ecrin_eren
30 minutes ago
sqing
15 minutes ago
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Inequality
Ecrin_eren   1
N 19 minutes ago by sqing
If the real numbers x,y,z satisfy the equation

x^4 + y^4 + z^4 = 5

then what is the maximum possible value of the expression

3x^4 + 8xyz^2
1 reply
+1 w
Ecrin_eren
32 minutes ago
sqing
19 minutes ago
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Inequality
SunnyEvan   3
N 23 minutes ago by DKI
Let $a$, $b$, $c$ be non-negative real numbers, no two of which are zero. Prove that :
$$ \sum \frac{3ab-2bc+3ca}{3b^2+bc+3c^2} \geq \frac{12}{7}$$
3 replies
SunnyEvan
Tuesday at 9:54 AM
DKI
23 minutes ago
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Mmo 9-10 graders P5
Bet667   3
N 41 minutes ago by Quantum-Phantom
Let $a,b,c,d$ be real numbers less than 2.Then prove that $\frac{a^3}{b^2+4}+\frac{b^3}{c^2+4}+\frac{c^3}{d^2+4}+\frac{d^3}{a^2+4}\le4$
3 replies
Bet667
2 hours ago
Quantum-Phantom
41 minutes ago
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A number theory problem from the British Math Olympiad
Rainbow1971   13
N an hour ago by ektorasmiliotis
Source: British Math Olympiad, 2006/2007, round 1, problem 6
I am a little surprised to find that I am (so far) unable to solve this little problem:

[quote]Let $n$ be an integer. Show that, if $2 + 2 \sqrt{1+12n^2}$ is an integer, then it is a perfect square.[/quote]

I set $k := \sqrt{1+12n^2}$. If $2 + 2 \sqrt{1+12n^2}$ is an integer, then $k (=\sqrt{1+12n^2})$ is at least rational, so that $1 + 12n^2$ must be a perfect square then. Using Conway's topograph method, I have found out that the smallest non-negative pairs $(n, k)$ for which this happens are $(0,1), (2,7), (28,97)$ and $(390, 1351)$, and that, for every such pair $(n,k)$, the "next" such pair can be calculated as
$$ \begin{bmatrix} 7 & 2 \\ 24 & 7 \end{bmatrix} \begin{bmatrix} n \\ k \end{bmatrix} .$$The eigenvalues of that matrix are irrational, however, so that any calculation which uses powers of that matrix is a little cumbersome. There must be an easier way, but I cannot find it. Can you?

Thank you.




13 replies
Rainbow1971
Mar 28, 2025
ektorasmiliotis
an hour ago
J
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inequalities
Cobedangiu   9
N an hour ago by DKI
problem
9 replies
Cobedangiu
Mar 31, 2025
DKI
an hour ago
J
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All Black Cells
David-Vieta   8
N an hour ago by sato2718
Source: 2023 China TST Problem 24
Let $n$ be a positive integer. Initially, a $2n \times 2n$ grid has $k$ black cells and the rest white cells. The following two operations are allowed :
(1) If a $2\times 2$ square has exactly three black cells, the fourth is changed to a black cell;
(2) If there are exactly two black cells in a $2 \times 2$ square, the black cells are changed to white and white to black.
Find the smallest positive integer $k$ such that for any configuration of the $2n \times 2n$ grid with $k$ black cells, all cells can be black after a finite number of operations.
8 replies
David-Vieta
Apr 1, 2023
sato2718
an hour ago
J
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Similar triangles and parallelism
KAME06   2
N 2 hours ago by jordiejoh
Source: OMEC Ecuador National Olympiad Final Round 2022 N3 P5 day 2
Let $ABC$ be a 90-degree triangle with hypotenuse $BC$. Let $D$ and $E$ distinct points on segment $BC$ and $P, Q$ be the foot of the perpendicular from $D$ to $AB$ and $E$ to $AC$, respectively. $DP$ and $EQ$ intersect at $R$.
Lines $CR$ and $AB$ intersect at $M$ and lines $BR$ and $AC$ intersect at $N$.
Prove that $MN \parallel BC$ if and only if $BD=CE$.
2 replies
KAME06
Nov 4, 2024
jordiejoh
2 hours ago
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Easy FE; source unknown
NamelyOrange   5
N 2 hours ago by Maxificial
Find (with proof) all $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x)\ge x$ and $f(f(x)) = x$.
5 replies
NamelyOrange
Yesterday at 8:48 PM
Maxificial
2 hours ago
J
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Modular Matching Pairs
steven_zhang123   2
N 2 hours ago by CHN_Lucas
Source: China TST 2025 P20
Let \( n \) be an odd integer, \( m = \frac{n+1}{2} \). Consider \( 2m \) integers \( a_1, a_2, \ldots, a_m, b_1, b_2, \ldots, b_m \) such that for any \( 1 \leq i < j \leq m \), \( a_i \not\equiv a_j \pmod{n} \) and \( b_i \not\equiv b_j \pmod{n} \). Prove that the number of \( k \in \{0, 1, \ldots, n-1\} \) for which satisfy \( a_i + b_j \equiv k \pmod{n} \) for some \( i \neq j \), $i, j \in \left \{ 1,2,\cdots,m \right \} $ is greater than \( n - \sqrt{n} - \frac{1}{2} \).
2 replies
steven_zhang123
Mar 29, 2025
CHN_Lucas
2 hours ago
J
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Turkish MO 1994 P5
xeroxia   9
N 2 hours ago by Primeniyazidayi
Source: Turkish Mathematical Olympiad 2nd Round 1994
Find the set of all ordered pairs $(s,t)$ of positive integers such that \[t^{2}+1=s(s+1).\]
9 replies
1 viewing
xeroxia
Sep 27, 2006
Primeniyazidayi
2 hours ago
J
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Another FE
Ankoganit   43
N 2 hours ago by jasperE3
Source: India IMO Training Camp 2016, Practice test 2, Problem 2
Find all functions $f:\mathbb R\to\mathbb R$ such that $$f\left( x^2+xf(y)\right)=xf(x+y)$$for all reals $x,y$.
43 replies
Ankoganit
Jul 22, 2016
jasperE3
2 hours ago
J
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Geometry problem
Raul_S_Baz   1
N 3 hours ago by sunken rock
IMAGE
1 reply
Raul_S_Baz
Yesterday at 8:49 PM
sunken rock
3 hours ago
J
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Sequence of numbers in form of a^2+b^2
TheOverlord   12
N 3 hours ago by ihategeo_1969
Source: Iran TST 2015, exam 1, day 1 problem 3
Let $ b_1<b_2<b_3<\dots $ be the sequence of all natural numbers which are sum of squares of two natural numbers.
Prove that there exists infinite natural numbers like $m$ which $b_{m+1}-b_m=2015$ .
12 replies
TheOverlord
May 11, 2015
ihategeo_1969
3 hours ago
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J
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This a better problem than that FE: Geometry
First   6
N Sep 3, 2016 by henry24816
Let ABCD be a quadrilateral. If a circle can be inscribed in it, prove that AB +CD= BC+DA. (The Pitot theorem)
6 replies
First
Sep 2, 2016
henry24816
Sep 3, 2016
J
This a better problem than that FE: Geometry
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Reveal topic tags
geometry
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First
2352 posts
First
#1 h  Sep 2, 2016, 9:51 PM • 2 Y
Y by Adventure10, Mango247
Let ABCD be a quadrilateral. If a circle can be inscribed in it, prove that AB +CD= BC+DA. (The Pitot theorem)
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checkmatetang
3454 posts
checkmatetang
#2 h  Sep 2, 2016, 9:52 PM • 2 Y
Y by Adventure10, Mango247
You put in the easy version. The better statement is quadrilateral $ABCD$ is circumscriptable if and only if $AB+CD=AD+BC.$

First, we prove the "if". Let the incircle meet $AB, BC, CD, DA$ at $E,F,G,H$ respectively. Then, $AE=AH, BE=BF, CF=CG, DG=DH.$ Summing then combining gives $AB+CD=AD+BC.$

Now, we assume $AB+CD=AD+BC$ and prove $ABCD$ is crcumscriptable. Construct circle $O$ tangent to $AD, DC, BC.$ We generate point $B'$ on $BC$ such that $AB'$ is tangent to circle $O$ as well. Thus, by the "if" statement, $AB'+CD=AD+B'C.$ By hypothesis, $AB+CD=AD+BC.$ Thus, $AB-AB'=BC-B'C.$ Note that since $BC-B'C=BB',$ we have $AB-AB'=BB'.$ By the triangle inequality, this means $B'$ lies on $AB.$ However, $B'$ also lies on $BC,$ so $B'=B$ as we wished.
This post has been edited 1 time. Last edited by checkmatetang, Sep 2, 2016, 9:53 PM
Reason: more info!
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First
2352 posts
First
#3 h  Sep 2, 2016, 9:56 PM • 2 Y
Y by Adventure10, Mango247
What does circumscriptable mean? Do you mean circumscribable? I knew, how to prove it, it was mainly redemption for my ridiculous problem I posted earlier.
This post has been edited 1 time. Last edited by First, Sep 2, 2016, 9:58 PM
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checkmatetang
3454 posts
checkmatetang
#4 h  Sep 2, 2016, 9:58 PM • 2 Y
Y by Adventure10, Mango247
No, circumscriptable is the same as tangential, which means a circle can be inscribed in the figure.
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tree3
641 posts
tree3
#5 h  Sep 3, 2016, 1:05 AM • 2 Y
Y by Adventure10, Mango247
Consider the intersections of the quadrilateral and circle. For distinct intersections each by two tangents implies that there are four sets of tangent lines. Let them be w,x,y, and z. Now w and x, say are on one side, x and y on another, y and z on another, and z and w on the last. Now w+x+y+z, summing up two opposite sides equals itself. Q.E.D.
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jayme
9775 posts
jayme
#6 h  Sep 3, 2016, 6:54 AM • 1 Y
Y by Adventure10
Dear Mathlinkers,

see
http://jl.ayme.pagesperso-orange.fr/Docs/Urquhart.pdf p. 6...
and also

http://jl.ayme.pagesperso-orange.fr/Docs/Demir.pdf p. 7-11.

Sincerely
Jean-Louis
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henry24816
268 posts
henry24816
#7 h  Sep 3, 2016, 1:30 PM • 1 Y
Y by Adventure10
I just used a bunch of congruent tangents, added up their measures and matched up numbers.
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