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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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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
J
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integral of product with monomial
segment   3
N 33 minutes ago by MathIQ.
Source: Own
$f$ is a polynomial with degree $k$. For $i=0,\cdots,k$, $$\int^b_a x^if(x)dx=0$$where $a,b$ is fixed real numbers such that $a<b$. Find all $f$.
3 replies
segment
Jul 18, 2024
MathIQ.
33 minutes ago
J
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Determining when an integral function is eventually constant
freshestcheese   2
N 35 minutes ago by MathIQ.
Source: My creation
Let
$$f\left(a\right)=\int_{0}^{1}\frac{\sin\left(2023\pi ax\right)\sin\left(2023\pi x\right)\cos\left(2024\pi x\right)\cos\left(2024\pi ax\right)}{\sin\left(\pi ax\right)\sin\left(\pi x\right)}dx$$Determine the smallest positive integer $N$ such that for all positive integers $m, n > N, f(m) = f(n).$
2 replies
freshestcheese
Oct 3, 2024
MathIQ.
35 minutes ago
J
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Romania TST 2022 Day 4 P2
oVlad   4
N 38 minutes ago by Andyexists
Source: Romania TST 2022
Fix a nonnegative integer $a_0$ to define a sequence of integers $a_0,a_1,\ldots$ by letting $a_k,k\geq 1$ be the smallest integer (strictly) greater than $a_{k-1}$ making $a_{k-1}+a_k{}$ into a perfect square. Let $S{}$ be the set of positive integers not expressible as the difference of two terms of the sequence $(a_k)_{k\geq 0}.$ Prove that $S$ is finite and determine its size in terms of $a_0.$
4 replies
oVlad
Jun 3, 2022
Andyexists
38 minutes ago
J
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A diophantine equation
crazyfehmy   14
N 43 minutes ago by MathIQ.
Source: Turkey Junior National Olympiad 2012 P1
Let $x, y$ be integers and $p$ be a prime for which

\[ x^2-3xy+p^2y^2=12p \]
Find all triples $(x,y,p)$.
14 replies
crazyfehmy
Dec 12, 2012
MathIQ.
43 minutes ago
J
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f(2) = 7, find all integer functions [Taiwan 2014 Quizzes]
v_Enhance   59
N an hour ago by MathIQ.
Find all increasing functions $f$ from the nonnegative integers to the integers satisfying $f(2)=7$ and \[ f(mn) = f(m) + f(n) + f(m)f(n) \] for all nonnegative integers $m$ and $n$.
59 replies
v_Enhance
Jul 18, 2014
MathIQ.
an hour ago
J
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f(x^3 + y^3 + z^3) = f(x)^3 + f(y)^3 + f(z)^3
pigfly   15
N an hour ago by MathIQ.
Source: VietNam TST 2005, problem 3
Find all functions $f: \mathbb{Z} \mapsto \mathbb{Z}$ satisfying the condition: $f(x^3 +y^3 +z^3 )=f(x)^3+f(y)^3+f(z)^3.$
15 replies
pigfly
Aug 4, 2004
MathIQ.
an hour ago
J
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geometry problem
invt   1
N an hour ago by Diamond-jumper76
In a triangle $ABC$ with $\angle B<\angle C$, denote its incenter and midpoint of $BC$ by $I$, $M$, respectively. Let $C'$ be the reflected point of $C$ wrt $AI$. Let the lines $MC'$ and $CI$ meet at $X$. Suppose that $\angle XAI=\angle XBI=90^{\circ}$. Prove that $\angle C=2\angle B$.
1 reply
invt
Yesterday at 11:59 AM
Diamond-jumper76
an hour ago
J
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Gergonne point Harmonic quadrilateral
niwobin   3
N an hour ago by niwobin
Triangle ABC has incircle touching the sides at D, E, F as shown.
AD, BE, CF concurrent at Gergonne point G.
BG and CG cuts the incircle at X and Y, respectively.
AG cuts the incircle at K.
Prove: K, X, D, Y form a harmonic quadrilateral. (KX/KY = DX/DY)
3 replies
niwobin
Yesterday at 8:17 PM
niwobin
an hour ago
J
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Computing functions
BBNoDollar   2
N an hour ago by alinazarboland
Let $f : [0, \infty) \to [0, \infty)$, $f(x) = \dfrac{ax + b}{cx + d}$, with $a, d \in (0, \infty)$, $b, c \in [0, \infty)$. Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
\[ f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0. \](For $n \in \mathbb{N}^*$ and $x \geq 0$, the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )
2 replies
BBNoDollar
4 hours ago
alinazarboland
an hour ago
J
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A "side chase" for juniors
Lukaluce   3
N 2 hours ago by lksb
Source: 2025 Junior Macedonian Mathematical Olympiad P5
Let $M$ be the midpoint of side $BC$ in $\triangle ABC$, and $P \neq B$ is such that the quadrilateral $ABMP$ is cyclic and the circumcircle of $\triangle BPC$ is tangent to the line $AB$. If $E$ is the second common point of the line $BP$ and the circumcircle of $\triangle ABC$, determine the ratio $BE: BP$.
3 replies
Lukaluce
6 hours ago
lksb
2 hours ago
J
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Diophantine Equation (cousin of Mordell)
urfinalopp   2
N 2 hours ago by urfinalopp
Find pairs of integers $(x;y)$ such that:

$x^2=y^5+32$
2 replies
+1 w
urfinalopp
3 hours ago
urfinalopp
2 hours ago
J
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p+2^p-3=n^2
tom-nowy   1
N 2 hours ago by urfinalopp
Let $n$ be a natural number and $p$ be a prime number. How many different pairs $(n, p)$ satisfy the equation:
$$p + 2^p - 3 = n^2 .$$
Inspired by https://artofproblemsolving.com/community/c4h3560823
1 reply
tom-nowy
Today at 11:16 AM
urfinalopp
2 hours ago
J
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Perfect cubes
Entrepreneur   6
N 3 hours ago by NamelyOrange
Find all ordered pairs of positive integers $(a,b,c)$ such that $\overline{abc}$ and $\overline{cab}$ are both perfect cubes.
6 replies
Entrepreneur
3 hours ago
NamelyOrange
3 hours ago
J
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Inequalities
sqing   13
N 4 hours ago by ytChen
Let $a,b,c >2 $ and $ ab+bc+ca \leq 75.$ Show that
$$\frac{1}{a-2}+\frac{1}{b-2}+\frac{1}{c-2}\geq 1$$Let $a,b,c >2 $ and $ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{6}{7}.$ Show that
$$\frac{1}{a-2}+\frac{1}{b-2}+\frac{1}{c-2}\geq 2$$
13 replies
sqing
May 13, 2025
ytChen
4 hours ago
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J
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A problem with a rectangle
Raul_S_Baz   16
N May 10, 2025 by Raul_S_Baz
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
16 replies
Raul_S_Baz
Apr 26, 2025
Raul_S_Baz
May 10, 2025
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A problem with a rectangle
G H J
Reveal topic tags
geometry
rectangle
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Raul_S_Baz
22 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
5 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
22 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
3001 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
22 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
22 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
22 posts
Raul_S_Baz
#12 h  May 1, 2025, 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, May 1, 2025, 7:59 AM
Reason: Adding a question.
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george_54
1587 posts
george_54
#13 h  May 1, 2025, 10:29 AM
Y by
Prove that triangles $AMN, CPQ$ are similar.
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undefined-NaN
11 posts
undefined-NaN
#14 h  May 1, 2025, 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  May 2, 2025, 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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Raul_S_Baz
22 posts
Raul_S_Baz
#16 h  May 2, 2025, 4:43 PM
Y by
Great! Thank you!
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Raul_S_Baz
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Raul_S_Baz
#17 h  May 10, 2025, 7:09 AM
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If T is the intersection of BP and DQ, can you prove that CT is the angle bisector of BCD?"
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