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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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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
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
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
Inequality em981
oldbeginner   17
N 6 minutes ago by sqing
Source: Own
Let $a, b, c>0, a+b+c=3$. Prove that
\[\sqrt{a+\frac{9}{b+2c}}+\sqrt{b+\frac{9}{c+2a}}+\sqrt{c+\frac{9}{a+2b}}+\frac{2(ab+bc+ca)}{9}\ge\frac{20}{3}\]
17 replies
oldbeginner
Sep 22, 2016
sqing
6 minutes ago
primes,exponentials,factorials
skellyrah   6
N 12 minutes ago by skellyrah
find all primes p,q such that $$ \frac{p^q+q^p-p-q}{p!-q!} $$is a prime number
6 replies
skellyrah
Apr 30, 2025
skellyrah
12 minutes ago
Serbian selection contest for the IMO 2025 - P5
OgnjenTesic   2
N 19 minutes ago by GreenTea2593
Source: Serbian selection contest for the IMO 2025
Determine the smallest positive real number $\alpha$ such that there exists a sequence of positive real numbers $(a_n)$, $n \in \mathbb{N}$, with the property that for every $n \in \mathbb{N}$ it holds that:
\[
        a_1 + \cdots + a_{n+1} < \alpha \cdot a_n.
    \]Proposed by Pavle Martinović
2 replies
OgnjenTesic
May 22, 2025
GreenTea2593
19 minutes ago
centroid wanted, point that minimizes sum of squares of distances from sides
parmenides51   1
N 23 minutes ago by SuperBarsh
Source: Oliforum Contest V 2017 p9 https://artofproblemsolving.com/community/c2487525_oliforum_contes
Given a triangle $ABC$, let $ P$ be the point which minimizes the sum of squares of distances from the sides of the triangle. Let $D, E, F$ the projections of $ P$ on the sides of the triangle $ABC$. Show that $P$ is the barycenter of $DEF$.

(Jack D’Aurizio)
1 reply
parmenides51
Sep 29, 2021
SuperBarsh
23 minutes ago
Strictly monotone polynomial with an extra condition
Popescu   11
N 33 minutes ago by Iveela
Source: IMSC 2024 Day 2 Problem 2
Let $\mathbb{R}_{>0}$ be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions $f:\mathbb{R}_{>0} \to \mathbb{R}$ such that there exists a two-variable polynomial $P(x, y)$ with real coefficients satisfying
$$
f(xy)=P(f(x), f(y))
$$for all $x, y\in\mathbb{R}_{>0}$.

Proposed by Navid Safaei, Iran
11 replies
Popescu
Jun 29, 2024
Iveela
33 minutes ago
Hard geometry
Lukariman   1
N 38 minutes ago by ricarlos
Given triangle ABC, any line d intersects AB at D, intersects AC at E, intersects BC at F. Let O1,O2,O3 be the centers of the circles circumscribing triangles ADE, BDF, CFE. Prove that the orthocenter of triangle O1O2O3 lies on line d.
1 reply
Lukariman
May 12, 2025
ricarlos
38 minutes ago
Russian Diophantine Equation
LeYohan   1
N 44 minutes ago by Natrium
Source: Moscow, 1963
Find all integer solutions to

$\frac{xy}{z} + \frac{xz}{y} + \frac{yz}{x} = 3$.
1 reply
LeYohan
Yesterday at 2:59 PM
Natrium
44 minutes ago
Simple Geometry
AbdulWaheed   6
N an hour ago by Gggvds1
Source: EGMO
Try to avoid Directed angles
Let ABC be an acute triangle inscribed in circle $\Omega$. Let $X$ be the midpoint of the arc $\overarc{BC}$ not containing $A$ and define $Y, Z$ similarly. Show that the orthocenter of $XYZ$ is the incenter $I$ of $ABC$.
6 replies
AbdulWaheed
May 23, 2025
Gggvds1
an hour ago
Bosnia and Herzegovina JBMO TST 2013 Problem 4
gobathegreat   4
N an hour ago by FishkoBiH
Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2013
It is given polygon with $2013$ sides $A_{1}A_{2}...A_{2013}$. His vertices are marked with numbers such that sum of numbers marked by any $9$ consecutive vertices is constant and its value is $300$. If we know that $A_{13}$ is marked with $13$ and $A_{20}$ is marked with $20$, determine with which number is marked $A_{2013}$
4 replies
gobathegreat
Sep 16, 2018
FishkoBiH
an hour ago
A geometry problem
Lttgeometry   2
N an hour ago by Acrylic3491
Triangle $ABC$ has two isogonal conjugate points $P$ and $Q$. The circle $(BPC)$ intersects circle $(AP)$ at $R \neq P$, and the circle $(BQC)$ intersects circle $(AQ)$ at $S\neq Q$. Prove that $R$ and $S$ are isogonal conjugates in triangle $ABC$.
Note: Circle $(AP)$ is the circle with diameter $AP$, Circle $(AQ)$ is the circle with diameter $AQ$.
2 replies
Lttgeometry
Today at 4:03 AM
Acrylic3491
an hour ago
anglechasing , circumcenter wanted
parmenides51   1
N an hour ago by Captainscrubz
Source: Sharygin 2011 Final 9.2
In triangle $ABC, \angle B = 2\angle C$. Points $P$ and $Q$ on the medial perpendicular to $CB$ are such that $\angle CAP = \angle PAQ = \angle QAB = \frac{\angle A}{3}$ . Prove that $Q$ is the circumcenter of triangle $CPB$.
1 reply
parmenides51
Dec 16, 2018
Captainscrubz
an hour ago
Nice FE over R+
doanquangdang   4
N an hour ago by jasperE3
Source: collect
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that
\[x+f(yf(x)+1)=xf(x+y)+yf(yf(x))\]for all $x,y>0.$
4 replies
doanquangdang
Jul 19, 2022
jasperE3
an hour ago
right triangle, midpoints, two circles, find angle
star-1ord   0
an hour ago
Source: Estonia Final Round 2025 8-3
In the right triangle $ABC$, $M$ is the midpoint of the hypotenuse $AB$. Point $D$ is chosen on the leg $BC$ so that the line segment $DM$ meets $(ACD)$ again at $K$ ($K\neq D$). Let $L$ be the reflection of $K$ in $M$. The circles $(ACD)$ and $(BCL)$ meet again at $N$ ($N\neq C$). Find the measure of $\angle KNL$.
0 replies
star-1ord
an hour ago
0 replies
interesting functional equation
tabel   3
N an hour ago by waterbottle432
Source: random romanian contest
Determine all functions \( f : (0, \infty) \to (0, \infty) \) that satisfy the functional equation:
\[
f(f(x)(1 + y)) = f(x) + f(xy), \quad \forall x, y > 0.
\]
3 replies
tabel
3 hours ago
waterbottle432
an hour ago
CWMO 2010, Day 1, Problem 2
chaotic_iak   13
N Feb 13, 2021 by Keith50
$AB$ is a diameter of a circle with center $O$. Let $C$ and $D$ be two different points on the circle on the same side of $AB$, and the lines tangent to the circle at points $C$ and $D$ meet at $E$. Segments $AD$ and $BC$ meet at $F$. Lines $EF$ and $AB$ meet at $M$. Prove that $E,C,M$ and $D$ are concyclic.
13 replies
chaotic_iak
Oct 30, 2010
Keith50
Feb 13, 2021
CWMO 2010, Day 1, Problem 2
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chaotic_iak
2932 posts
#1 • 2 Y
Y by Adventure10, Mango247
$AB$ is a diameter of a circle with center $O$. Let $C$ and $D$ be two different points on the circle on the same side of $AB$, and the lines tangent to the circle at points $C$ and $D$ meet at $E$. Segments $AD$ and $BC$ meet at $F$. Lines $EF$ and $AB$ meet at $M$. Prove that $E,C,M$ and $D$ are concyclic.
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lssl
240 posts
#2 • 2 Y
Y by Adventure10, Mango247
This problem is famous , it is from Russia , in that problem , you have to prove $EF$ is perpendicular to $AB$ and this is just the key step to solve this CWMO problem.
Let $AC$ and $BD$ meet at point $L$ , easy to get :$\angle ALB=\frac{\pi}{2}-\angle CAD$ , $\angle CED=\pi-2\angle EDC =\pi-2\angle CAD=2\angle CLD$ , Notice that $EC=DE$ , hence $E$ is the circumcentre of triangle $LCD$ .because $\angle FCL=\angle FDL$ ,hence $L,E,F$ is collinear , Obviously $F$ is the orthocentre of triangle $LAB$ ,hence $EF$ is perpendicular to $AB$ .
Join $CM,DM$ , $C,A,M,F$ ; $M,F,D,B$ are both concyclic , $\Longrightarrow$ $\angle EDC=\angle CAF=\angle CMF$ , Hence $E,C,M,D$ are concyclic .
QED.
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dgreenb801
1896 posts
#3 • 2 Y
Y by Adventure10, Mango247
Solution 1
Note that $\angle OCA= \angle OAC = \angle BCE = \alpha$ and
$\angle ODB= \angle OBD= \angle ADE = \beta$
Note that since $CE=DE$, $E$ lies on the perpendicular bisector of $CD$.
Also,
$\angle CFD= \angle FCA + \angle CAF= 90+ \frac{1}{2} \angle COD= 90+ \frac{1}{2} (180- \angle CED)= \newline 180- \frac{1}{2} \angle CED$
Thus $E$ is the circumcenter of $\triangle CFD$, so $\angle EFC= \angle ECF= \alpha$ and thus $\angle CEF= \angle COA$. Thus $CEOM$ is cyclic, similarly $DEMO$ is cyclic, so $CEDOM$ is cyclic.

In fact, here is a more interesting and quicker solution:
Solution 2
Rotate $\triangle DOB$ about $O$ until $B$ coincides with $A$, let $D$ rotate to $D'$. Note that we have $CO=OD'$ and $CE=ED$. Also,
$\angle COD'=180- \angle COD =\angle CED$. Next,
$\angle ACO=\angle CAO= \angle ECF$ and $\angle AD'O= \angle ODB= \angle OBD= \angle EDF$. Thus, $ECFD \sim OCAD'$. It follows that $\angle CEF= \angle COA$, so $CEOM$ is cyclic, similarly $DEMO$ is cyclic, so $CEDOM$ is cyclic.
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mahanmath
1354 posts
#4 • 4 Y
Y by Hieuamaster, Adventure10, Mango247, and 1 other user
Just apply Pascal theorem to $ACCBDD$ :wink:
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Andy Loo
35 posts
#5 • 2 Y
Y by Adventure10, Mango247
mahanmath wrote:
Just apply Pascal theorem to $ACCBDD$ :wink:
Brilliant idea!

I actually went to this year's CWMO, and to be honest, almost the entire Hong Kong team used coordinate geometry to solve both geometry problems - i guess that's our style. :)
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jgnr
1343 posts
#6 • 2 Y
Y by Adventure10, Mango247
Andy Loo wrote:
mahanmath wrote:
Just apply Pascal theorem to $ACCBDD$ :wink:
Brilliant idea!

I actually went to this year's CWMO, and to be honest, almost the entire Hong Kong team used coordinate geometry to solve both geometry problems - i guess that's our style. :)
Wow, that's quite amazing. :D I also used Pascal to solve this. This is my friend's solution:

$\angle CFD=\frac12(\widehat{CD}+\widehat{AB})=\frac12\widehat{CD}+90^{\circ}$ and $\angle CED_{maj}=360^{\circ}-\frac12(\widehat{CD}_{maj}-\widehat{CD})=360^{\circ}-\frac12(360^{\circ}-2\widehat{CD})=180^{\circ}+\widehat{CD}$. Thus $\angle CED_{maj}=2\angle CFD$, therefore $C,F,D$ lie on a circle centered at $E$, which gives $EC=EF=ED$. So $\angle BFM=\angle CFE=\angle FCE=\frac12\widehat{BC}=90^{\circ}-\frac12\widehat{AC}=90^{\circ}-\angle FBM$, thus $\angle FMB=90^{\circ}$. So $\angle EMO=\angle EDO=\angle ECO=90^{\circ}$, which implies that $E,D,M,O,C$ are cyclic.
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jayme
9801 posts
#7 • 2 Y
Y by Adventure10, Mango247
Dear mathlinkers,
yes, the idea of using Pascal is good in order to have a synthetic proof.
After, we can can show thet this circle goes through the midpoint of AB...
Sincerely
Jean-Louis
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vslmat
154 posts
#8 • 1 Y
Y by Adventure10
Let $CD$ cut $AB$ at $K$, $AC$ cut $BD$ at $G$.
$GF$ is $k$, the polar of $K$
$CD$ is $e$, the polar of $E$. As $K$ is on $e$, $E$ must be on $k$, the polar of $K$. Thus $G, E, F$ are all on $k$, they are collinear and $GM \perp KO$.
It follows that $E, C, M, O, D$ are concyclic.
Attachments:
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sunken rock
4401 posts
#9 • 2 Y
Y by Adventure10, Mango247
From figure above, $\angle AFC=\angle AGB\implies \angle AGB=\frac{\angle COA+\angle BOD}{2}$ $=90^\circ-\frac{\angle COD}{2}=\angle CDO=\angle DCO$, so $CO, DO$ are tangent to the circle $(CDG)$, hence $E$ is its circumcenter, $\iff GE\bot AB$.

Best regards,
sunken rock
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jred
290 posts
#10 • 2 Y
Y by Adventure10, Mango247
There's another simple solution. Suppose $AC$ and $BD$ meet at $G$ (still using the figure above), we easliy get $F$ is the orthocenter of $\triangle ABG$. Let $E'$ be the midpoint of $GF$, then we have $\angle GCE'=\angle CGE'=\angle ABF=\angle BCO$, hence $\angle E'CO=\angle GCF=90^\circ$, which means $E'C$ is tangent to circle $O$ at $C$. Similarly, we have $E'D$ is tangent to circle $O$ at $D$, so $E'$ is $E$, thus $EM\bot AB$. Finally, we have $E, C, M, D$ are on the nine-point-circle of $\triangle  ABG$.
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sayantanchakraborty
505 posts
#11 • 1 Y
Y by Adventure10
Solution:

Extend $CD$ to meet $AB$ at $X$.Note that $CD$ is the polar of $E$ and it passes through $X$.So the polar of $X$ must pass through $E$.Also it is well known that the polar of $X$ passes through $F$.So $EF$ is the polar of $X$ which means that $EF \perp OX \implies EF \perp AB$.Now the rest is easy:Note that $MFDB$ concyclic(why?) which means $\angle{ECD}=\angle{CBD}=\angle{FBD}=\angle{FMD}=\angle{EMD}$ and everything is oK.
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Devastator
348 posts
#12 • 2 Y
Y by Adventure10, Mango247
Just a question, do u still need to deal with the config issue where F could be inside or outside the semicircle, or it's ok to just deal with the inside case?
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L-.Lawliet
19 posts
#13 • 3 Y
Y by RudraRockstar, RAMUGAUSS, Adventure10
Let $AB \cap CD=P$ and $AC \cap DB=Q$. Then by Brocard's theorem $QF$ is the polar of $P$ , and hence $QF \perp AB$. Now $CD$ is the polar of $E \implies P$ lies on the polar of $E$. By La hire's theorem the polar of $P$ which is $QF$ passes through $E$. So $Q,E,F,M$ collinear. Hence $\angle EMA =90$. So $\angle CME=\angle CAD=\angle ECD \implies C,E,D,M$ is concyclic.
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Keith50
464 posts
#14
Y by
https://i.imgur.com/VrXKcrx.png
We claim that $E$ is the circumcenter of triangle $CDF$, indeed \[\angle DEC=2 \angle DFC \iff 180^{\circ}-\angle EDC-\angle ECD=360^{\circ}-2\angle A-2\angle B\iff \angle A+\angle B-\angle CBD=\angle A+\angle DBA=90^{\circ}\]which is true as $AB$ is the diameter of the circle with center $O$. So, we now prove that quadrilateral $OCEM$ is cyclic, \[\angle COM=2\angle B=2\angle CDF=\angle CEF \implies \textrm{OCEM is cyclic. $\square$}\]Then, we show that quadrilateral $OCED$ is cyclic, \[\angle CED=180^{\circ}-\angle EDC-\angle ECD=180^{\circ}-2\angle CBD=180^{\circ}-\angle COD \implies \textrm{OCED is cyclic. $\square$} \]Therefore, since quadrilaterals $OCEM$ and $OCED$ are cyclic, we have quadrilateral $ECMD$ is cyclic. $\blacksquare$
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