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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 11:16 PM
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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0 replies
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jlacosta
Yesterday at 11:16 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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Fermat points of Pentagon
Jackson0423   0
9 minutes ago
It is known that, in general, a pentagon has three Fermat points. But I'm curious—if there are exactly two Fermat points inside the pentagon, under what conditions does the distance sum reach a minimum? Can you help me?
0 replies
Jackson0423
9 minutes ago
0 replies
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Two equal angles
jayme   1
N 11 minutes ago by jayme
Dear Mathlinkers,

1. ABCD a square
2. I the midpoint of AB
3. 1 the circle center at A passing through B
4. Q the point of intersection of 1 with the segment IC
5. X the foot of the perpendicular to BC from Q
6. Y the point of intersection of 1 with the segment AX
7. M the point of intersection of CY and AB.

Prove : <ACI = <IYM.

Sincerely
Jean-Louis
1 reply
jayme
Today at 6:52 AM
jayme
11 minutes ago
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Perfect square
Ecrin_eren   0
12 minutes ago
(13^n+2)/ 3 is a perfect square.How many n natural numbers satisfy?
0 replies
Ecrin_eren
12 minutes ago
0 replies
J
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Symetric inequality
Nguyenhuyen_AG   0
18 minutes ago
Let $a, \, b, \, c$ are non-negative real numbers and $k \geqslant 0.$
(i) Prove that
\[\frac{a(a^2-bc)}{\sqrt{ka+b+c}} + \frac{b(b^2-ca)}{\sqrt{kb+c+a}} + \frac{c(c^2-ab)}{\sqrt{kc+a+b}} \geqslant 0.\](ii) Prove that
\[(a^2-bc)\sqrt{ka+b+c}+(b^2-ca)\sqrt{kb+c+a}+(c^2-ab)\sqrt{kc+a+b} \geqslant 0.\]
0 replies
Nguyenhuyen_AG
18 minutes ago
0 replies
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Lord Evan the Reflector
whatshisbucket   23
N 19 minutes ago by bjump
Source: ELMO 2018 #3, 2018 ELMO SL G3
Let $A$ be a point in the plane, and $\ell$ a line not passing through $A$. Evan does not have a straightedge, but instead has a special compass which has the ability to draw a circle through three distinct noncollinear points. (The center of the circle is not marked in this process.) Additionally, Evan can mark the intersections between two objects drawn, and can mark an arbitrary point on a given object or on the plane.

(i) Can Evan construct* the reflection of $A$ over $\ell$?

(ii) Can Evan construct the foot of the altitude from $A$ to $\ell$?

*To construct a point, Evan must have an algorithm which marks the point in finitely many steps.

Proposed by Zack Chroman
23 replies
whatshisbucket
Jun 28, 2018
bjump
19 minutes ago
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4 variables with quadrilateral sides 2
mihaig   6
N 26 minutes ago by mihaig
Source: Own
Let $a,b,c,d\geq0$ satisfying
$$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$$Prove
$$\left(a+b+c+d-2\right)^2+8\geq3\left(abc+abd+acd+bcd\right).$$
6 replies
mihaig
Apr 29, 2025
mihaig
26 minutes ago
J
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Hard inequality
ys33   6
N 26 minutes ago by mihaig
Let $a, b, c, d>0$. Prove that
$\sqrt[3]{ab}+ \sqrt[3]{cd} < \sqrt[3]{(a+b+c)(b+c+d)}$.
6 replies
ys33
Today at 9:36 AM
mihaig
26 minutes ago
J
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Nice one
imnotgoodatmathsorry   2
N 26 minutes ago by imnotgoodatmathsorry
Source: Own
With $x,y,z >0$.Prove that: $\frac{xy}{4y+4z+x} + \frac{yz}{4z+4x+y} +\frac{zx}{4x+4y+z} \le \frac{x+y+z}{9}$
2 replies
1 viewing
imnotgoodatmathsorry
2 hours ago
imnotgoodatmathsorry
26 minutes ago
J
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Permutation with Matrices
SomeonecoolLovesMaths   0
33 minutes ago
Consider all $n \times n$ matrix such that $\forall$ $k \leq n$, $( a_{1k}, a_{2k}, \cdots, a_{nk} )$ is a permutation of $(1,2, \cdots, n)$, call such matrices $\textit{rowgood}$. Consider all $n \times n$ matrix such that $\forall$ $k \leq n$, $( a_{k1}, a_{k2}, \cdots, a_{kn} )$ is a permutation of $(1,2, \cdots, n)$, call such matrices $\textit{columngood}$. How many $n \times n$ matrices exist that are both $\textit{rowgood}$ and $\textit{columngood}$?
0 replies
SomeonecoolLovesMaths
33 minutes ago
0 replies
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Points U,V,F,E are concyclic (GAMO P5)
Aritra12   4
N 34 minutes ago by ihategeo_1969
Source: GAMO day 2 P5
Let $ABC$ be an acute, non-isosceles triangle, $AD,BE,CF$ be its heights and $(c)$ its circumcircle. $FE$ cuts the circumcircle at points $S,T$, with point $F$ being between points $S,E$. In addition, let $P,Q$ be the midpoints of the major and the minor arc $BC$, respectively. Line $DQ$ cuts $(c)$ at $R$. The circumcircles of triangles $RSF,TER,SFP$ and $TEP$ cut again $PR$ at points $X,Y,Z$ and $W$, respectively. Suppose $(\ell)$ is the line passing through the circumcenters of triangles $AXW,AYZ$ and $(\ell_B ),(\ell_C)$ the parallel lines through $B,C$ to $(\ell)$. If $(\ell_B)$ meets $CF$ at $U$ and $(\ell_C )$ meets $BE$ at $V$, then prove that points $U,V,F,E$ are concyclic.

$\textit{Proposed by Orestis Lignos}$
4 replies
Aritra12
Apr 12, 2021
ihategeo_1969
34 minutes ago
J
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Min and Max
giangtruong13   1
N 37 minutes ago by imnotgoodatmathsorry
Source: PTNK-HCM Specialized School's Practical Math Test (Round 1)
Let $a,b \geq 0$ such that: $a^3+b^3=2$.Prove that: $$ \sqrt[3]{4} \geq a^2-ab+b^2 \geq 1$$
1 reply
giangtruong13
an hour ago
imnotgoodatmathsorry
37 minutes ago
J
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Value of expression
Ecrin_eren   2
N 37 minutes ago by vanstraelen
Let a be a root of the equation x^3-x-1=0 , with a>1
What is the value of the expression:
∛(3a^2 - 4a) + ∛(3a^2 + 4a + 2)?
2 replies
Ecrin_eren
Today at 7:59 AM
vanstraelen
37 minutes ago
J
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Coprime sequence
Ecrin_eren   2
N 41 minutes ago by Ecrin_eren
"Let N be a natural number. Show that any two numbers from the following sequence are coprime:

2^1 + 1, 2^2 + 1, 2^4+ 1,2^8+1 ..., 2^(2^N )+ 1."
2 replies
Ecrin_eren
Yesterday at 8:53 PM
Ecrin_eren
41 minutes ago
J
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Find the functions
Ecrin_eren   5
N 42 minutes ago by Ecrin_eren
"Find all differentiable functions f that satisfy the condition f(x) + f(y) = f((x + y) / (1 - xy)) for all x, y ∈ R, where xy ≠ 1."
5 replies
Ecrin_eren
Yesterday at 8:58 PM
Ecrin_eren
42 minutes ago
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J
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A problem with a rectangle
Raul_S_Baz   14
N Today at 9:08 AM by george_54
On the sides AB and AD of the rectangle ABCD, points M and N are taken such that MB = ND. Let P be the intersection of BN and CD, and Q be the intersection of DM and CB. How can we prove that PQ || MN?
IMAGE
14 replies
Raul_S_Baz
Apr 26, 2025
george_54
Today at 9:08 AM
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A problem with a rectangle
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Reveal topic tags
geometry
rectangle
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Raul_S_Baz
17 posts
Raul_S_Baz
#1 h  Apr 26, 2025, 11:13 AM
Y by
On the sides AB and AD of the rectangle ABCD, points M and N are taken such that MB = ND. Let P be the intersection of BN and CD, and Q be the intersection of DM and CB. How can we prove that PQ || MN?
https://i.imgur.com/rV7I1qQ.png
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undefined-NaN
11 posts
undefined-NaN
#2 h  Apr 26, 2025, 1:17 PM
Y by
if PQ || MN
>
intersection of MD and BN = K
QM/MK = PN/NK

F a point in BC line and KF 90DEGREE with BC
E a point in DC line and KE 90DEGREE with DC
QB/BF=QM/MK SO MB/KF = QB/QB+BF =QM/QM+MK
PN/NK=PD/DE SO DN/KE= PN/NK+PN = QM/MK+QM
also MB=DN so KF=KE so KFCE = a square so CK is a half angle line of 90 degree
so BK/KP = BC/CP
so QK/KD = QC/DC

I will continue after from here...
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Kizaruno
2 posts
Kizaruno
#3 h  Apr 26, 2025, 2:28 PM
Y by
I really like how you tackled this problem!
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Raul_S_Baz
17 posts
Raul_S_Baz
#4 h  Apr 26, 2025, 3:08 PM
Y by
Thanx a lot! Do you have a solution?
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lpieleanu
2982 posts
lpieleanu
#5 h  Apr 26, 2025, 11:21 PM
Y by
We proceed with coordinate bashing.

Without loss of generality, let us have $$A=(0, 1), B=(a,1), C=(a,0), D=(0,0), N=(0,c), M=(a-c, 1).$$Then, it follows that $$P=\left(\frac{ac}{c-1}, 0\right), Q=\left(a, \frac{a}{a-c}\right).$$Then, the slope of line $PQ$ is
\begin{align*} \frac{\tfrac{a}{a-c}-0}{a-\tfrac{ac}{c-1}} &= \frac{ac-a}{ac-a^2} \\ &= \frac{1-c}{a-c}, \end{align*}concluding the proof. $\square$
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Raul_S_Baz
17 posts
Raul_S_Baz
#6 h  Apr 27, 2025, 4:15 AM
Y by
Thank you for this elegant solution. However, I am looking for a synthetic one.
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undefined-NaN
11 posts
undefined-NaN
#7 h  Apr 27, 2025, 11:41 AM
Y by
undefined-NaN wrote:
if PQ || MN
>
intersection of MD and BN = K
QM/MK = PN/NK

F a point in BC line and KF 90DEGREE with BC
E a point in DC line and KE 90DEGREE with DC
QB/BF=QM/MK SO MB/KF = QB/QB+BF =QM/QM+MK
PN/NK=PD/DE SO DN/KE= PN/NK+PN = QM/MK+QM
also MB=DN so KF=KE so KFCE = a square so CK is a half angle line of 90 degree
so BK/KP = BC/CP
so QK/KD = QC/DC

I will continue after from here...



ı SOLVED

let draw a line for BD
BD must be parellel to PQ || MN
because AB || DC
CB/CQ = BD/PQ
also CD/CP=AB/CP
SO ABN=~CPB
T, intersection of AB PQ
so BP/BN = BM/BT
so you can see MN paralel to PQ

also we had another way here
BM // DC so CB/CQ = AD/CQ
also CD/CP = AB/CP
so here ABD=~CPQ
so hipo of ABD = BD
so hipo of CPQ = PQ
so BD//PQ

also I can check it with half angle here
if NM || PQ
m(QPB)=m(PBD)
m(PQD)=m(QDB)
so BD / QP= KP/KB = KQ/KD
also I was found KC is a half angle of 90degree
so KB/KP = BC/CP
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undefined-NaN
11 posts
undefined-NaN
#8 h  Apr 27, 2025, 11:52 AM
Y by
also I want to learn where did you find this question because I want more geometry problems like this
you can share with me any website or whatever

also AB // PD so
ABN=~DPN
so we can check 2 times
BN/NP=BM/MT

ALSO have another way to see BM/MT
let G a point of GADP you can complete the rectangle with G
so AN//GP so BN/NP = BA/AG also AG=PD so ABN=~DPN


waiting for your response
This post has been edited 2 times. Last edited by undefined-NaN, Apr 27, 2025, 12:09 PM
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undefined-NaN
11 posts
undefined-NaN
#9 h  Apr 27, 2025, 9:48 PM
Y by
Are there any problems?
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Raul_S_Baz
17 posts
Raul_S_Baz
#10 h  Apr 28, 2025, 7:39 PM
Y by
What kind of problems?
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undefined-NaN
11 posts
undefined-NaN
#11 h  Apr 28, 2025, 8:18 PM
Y by
is everything understandable?
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Raul_S_Baz
17 posts
Raul_S_Baz
#12 h  Yesterday at 7:57 AM
Y by
This problem originates from a problem that appeared in 'Gazeta Matematică.
Can you define the type of problems you are looking for?
This post has been edited 1 time. Last edited by Raul_S_Baz, Yesterday at 7:59 AM
Reason: Adding a question.
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george_54
1587 posts
george_54
#13 h  Yesterday at 10:29 AM
Y by
Prove that triangles $AMN, CPQ$ are similar.
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undefined-NaN
11 posts
undefined-NaN
#14 h  Yesterday at 4:38 PM
Y by
you must prove the paralel lines before say to AMN CPQ

but I was already prove the paralel lines with
ABN=~DPN
ABN=~CPB
also I found these from square in rectangle with equal lines
so if you didnt understand that I will continue with
1-Trigonometric rates
2-Area calculating
3-Water filling

I dont know who understand who didnt understand without any comment to my solving
so I just wanna see te comment like this "I didnt understand at here" , "You might not true at here"
...

I dont have any bad think in my mind I just want to see "who understand or not the this solving"
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george_54
1587 posts
george_54
#15 h  Today at 9:08 AM
Y by
With the symbols on the figure it is, $MB//DC \Leftrightarrow \frac{d}{a} = \frac{x}{{x + b}} \Leftrightarrow x = \frac{{bd}}{{a - d}}$

and in the same way I find $y = \frac{{ad}}{{b - d}}.$ So, $\frac{{CP}}{{CQ}} = \frac{{a + y}}{{b + x}} = \frac{{a - d}}{{b - d}} = \frac{{AM}}{{AN}}$

Hence, the right triangles $AMN, CPQ$ are similar. But, $AM//PC$ and $AN//QC,$ thus $\boxed{PQ//MN}$
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