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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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0 replies
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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shadow of a cylinder, shadow of a cone
vanstraelen   4
N 3 minutes ago by mathafou

a) Given is a right cylinder of height $2R$ and radius $R$.
The sun shines on this solid at an angle of $45^{\circ}$.
What is the area of the shadow that this solid casts on the plane of the botom base?

b) Given is a right cone of height $2R$ and radius $R$.
The sun shines on this solid at an angle of $45^{\circ}$.
What is the area of the shadow that this solid casts on the plane of the base?
4 replies
vanstraelen
May 9, 2025
mathafou
3 minutes ago
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Hard Problem help?
7168   5
N 40 minutes ago by Shan3t
Suppose that a,b < or = 2010 are positive integers with b * Lcm(a,b) < or = a * gcd(a,b). Let N be the smallest positive integer value of n such that either nb/a and na/b are integers. What, over all a,b, is the greatest possible value of N?

THx!
5 replies
7168
Feb 25, 2011
Shan3t
40 minutes ago
J
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Binomial stuff
Arne   2
N an hour ago by Speedysolver1
Source: Belgian IMO preparation
Let $p$ be prime, let $n$ be a positive integer, show that \[ \gcd\left({p - 1 \choose n - 1}, {p + 1 \choose n}, {p \choose n + 1}\right) = \gcd\left({p \choose n - 1}, {p - 1 \choose n}, {p + 1 \choose n + 1}\right). \]
2 replies
Arne
Apr 4, 2006
Speedysolver1
an hour ago
J
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Geometry hard problem
noneofyou34   1
N an hour ago by Lil_flip38
Let ABC be a triangle with incircle Γ. The tangency points of Γ with sides BC, CA, AB are A1, B1, C1 respectively. Line B1C1 intersects line BC at point A2. Similarly, points B2 and C2 are constructed. Prove that the perpendicular lines from A2, B2, C2 to lines AA1, BB1, CC1 respectively are concurret.
1 reply
noneofyou34
Today at 3:13 PM
Lil_flip38
an hour ago
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segment of projections is half as sidelength, right triangle inscribed in right
parmenides51   3
N 2 hours ago by NumberzAndStuff
Source: 2020 Austrian Federal Competition For Advanced Students, Part 1, p2
Let $ABC$ be a right triangle with a right angle in $C$ and a circumcenter $U$. On the sides $AC$ and $BC$, the points $D$ and $E$ lie in such a way that $\angle EUD = 90 ^o$. Let $F$ and $G$ be the projection of $D$ and $E$ on $AB$, respectively. Prove that $FG$ is half as long as $AB$.

(Walther Janous)
3 replies
parmenides51
Nov 22, 2020
NumberzAndStuff
2 hours ago
J
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Austrian Regional MO 2025 P2
BR1F1SZ   3
N 2 hours ago by NumberzAndStuff
Source: Austrian Regional MO
Let $\triangle{ABC}$ be an isosceles triangle with $AC = BC$ and circumcircle $\omega$. The line through $B$ perpendicular to $BC$ is denoted by $\ell$. Furthermore, let $M$ be any point on $\ell$. The circle $\gamma$ with center $M$ and radius $BM$ intersects $AB$ once more at point $P$ and the circumcircle $\omega$ once more at point $Q$. Prove that the points $P,Q$ and $C$ lie on a straight line.

(Karl Czakler)
3 replies
BR1F1SZ
Apr 18, 2025
NumberzAndStuff
2 hours ago
J
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Computing functions
BBNoDollar   5
N 2 hours ago by ICE_CNME_4
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 $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )
5 replies
BBNoDollar
Yesterday at 5:25 PM
ICE_CNME_4
2 hours ago
J
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Inequalities
sqing   7
N 2 hours ago by ytChen
Let $ a,b>0 $ . Prove that
$$ \frac{a}{a^2+a +2b+1}+ \frac{b}{b^2+2a +b+1} \leq \frac{2}{5} $$$$ \frac{a}{a^2+2a +b+1}+ \frac{b}{b^2+a +2b+1} \leq \frac{2}{5} $$
7 replies
sqing
May 13, 2025
ytChen
2 hours ago
J
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Reflections of lines through reflections of excenters
cjquines0   39
N 3 hours ago by awesomeming327.
Source: 2016 IMO Shortlist G7
Let $I$ be the incentre of a non-equilateral triangle $ABC$, $I_A$ be the $A$-excentre, $I'_A$ be the reflection of $I_A$ in $BC$, and $l_A$ be the reflection of line $AI'_A$ in $AI$. Define points $I_B$, $I'_B$ and line $l_B$ analogously. Let $P$ be the intersection point of $l_A$ and $l_B$.
[list=a]
[*] Prove that $P$ lies on line $OI$ where $O$ is the circumcentre of triangle $ABC$.
[*] Let one of the tangents from $P$ to the incircle of triangle $ABC$ meet the circumcircle at points $X$ and $Y$. Show that $\angle XIY = 120^{\circ}$.
[/list]
39 replies
cjquines0
Jul 19, 2017
awesomeming327.
3 hours ago
J
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2xy is perfect square and x^2 + y^2 is prime
parmenides51   4
N 3 hours ago by LeYohan
Source: Dutch NMO 2020 p4
Determine all pairs of integers $(x, y)$ such that $2xy$ is a perfect square and $x^2 + y^2$ is a prime number.
4 replies
parmenides51
Nov 23, 2020
LeYohan
3 hours ago
J
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Really classical inequatily from canada
shobber   79
N 3 hours ago by sharknavy75
Source: Canada 2002
Prove that for all positive real numbers $a$, $b$, and $c$,
\[ \frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq a+b+c \]
and determine when equality occurs.
79 replies
shobber
Mar 5, 2006
sharknavy75
3 hours ago
J
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Functional equation
Pmshw   18
N 3 hours ago by jasperE3
Source: Iran 2nd round 2022 P2
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any real value of $x,y$ we have:
$$f(xf(y)+f(x)+y)=xy+f(x)+f(y)$$
18 replies
Pmshw
May 8, 2022
jasperE3
3 hours ago
J
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f(x)f(yf(x)) = f(x+y)
ISHO95   5
N 3 hours ago by jasperE3
Find all functions $f:\mathbb R^+ \to \mathbb R^+$, for all $x,y \in \mathbb R^+$, \[ f(x)f(yf(x))=f(x+y). \]
5 replies
ISHO95
Jan 14, 2013
jasperE3
3 hours ago
J
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Vieta's Relation
P162008   1
N 4 hours ago by vanstraelen
If $\alpha, \beta$ and $\gamma$ are the roots of the cubic equation $x^3 - x^2 - 2x + 1 = 0$ then compute $\sum_{cyc} (\alpha + \beta)^{1/3}.$
1 reply
P162008
Today at 10:38 AM
vanstraelen
4 hours ago
J
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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
L
G H BBookmark kLocked kLocked NReply
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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?
Z K Y
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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
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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
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Raul_S_Baz
#16 h  May 2, 2025, 4:43 PM
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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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