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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
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
Inspired by 2025 Beijing
sqing   10
N an hour ago by ytChen
Source: Own
Let $ a,b,c,d >0  $ and $ (a^2+b^2+c^2)(b^2+c^2+d^2)=36. $ Prove that
$$ab^2c^2d \leq 8$$$$a^2bcd^2 \leq 16$$$$ ab^3c^3d \leq \frac{2187}{128}$$$$ a^3bcd^3 \leq \frac{2187}{32}$$
10 replies
sqing
Yesterday at 4:56 PM
ytChen
an hour ago
Three operations make any number
awesomeming327.   0
3 hours ago
Source: own
The number $3$ is written on the board. Anna, Boris, and Charlie can do the following actions: Anna can replace the number with its floor. Boris can replace any integer number with its factorial. Charlie can replace any nonnegative number with its square root. Prove that the three can work together to make any positive integer in finitely many steps.
0 replies
1 viewing
awesomeming327.
3 hours ago
0 replies
IMO 2017 Problem 4
Amir Hossein   116
N 3 hours ago by cj13609517288
Source: IMO 2017, Day 2, P4
Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$. Point $T$ is such that $S$ is the midpoint of the line segment $RT$. Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$. Line $AJ$ meets $\Omega$ again at $K$. Prove that the line $KT$ is tangent to $\Gamma$.

Proposed by Charles Leytem, Luxembourg
116 replies
Amir Hossein
Jul 19, 2017
cj13609517288
3 hours ago
A sharp one with 3 var
mihaig   10
N 3 hours ago by mihaig
Source: Own
Let $a,b,c\geq0$ satisfying
$$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$$Prove
$$ab+bc+ca+abc\geq4.$$
10 replies
mihaig
May 13, 2025
mihaig
3 hours ago
Another right angled triangle
ariopro1387   1
N 3 hours ago by lolsamo
Source: Iran Team selection test 2025 - P7
Let $ABC$ be a right angled triangle with $\angle A=90$. Point $M$ is the midpoint of side $BC$ And $P$ be an arbitrary point on $AM$. The reflection of $BP$ over $AB$ intersects lines $AC$ and $AM$ at $T$ and $Q$, respectively. The circumcircles of $BPQ$ and $ABC$ intersect again at $F$. Prove that the center of the circumcircle of $CFT$ lies on $BQ$.
1 reply
ariopro1387
Today at 4:13 PM
lolsamo
3 hours ago
four points lie on a circle
pohoatza   78
N 4 hours ago by ezpotd
Source: IMO Shortlist 2006, Geometry 2, AIMO 2007, TST 1, P2
Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} = \angle{BCD}\qquad\text{and}\qquad \angle{CQD} = \angle{ABC}.\]Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic.

Proposed by Vyacheslev Yasinskiy, Ukraine
78 replies
pohoatza
Jun 28, 2007
ezpotd
4 hours ago
JBMO TST Bosnia and Herzegovina 2023 P4
FishkoBiH   2
N 4 hours ago by Stear14
Source: JBMO TST Bosnia and Herzegovina 2023 P4
Let $n$ be a positive integer. A board with a format $n*n$ is divided in $n*n$ equal squares.Determine all integers $n$3 such that the board can be covered in $2*1$ (or $1*2$) pieces so that there is exactly one empty square in each row and each column.
2 replies
FishkoBiH
Today at 1:38 PM
Stear14
4 hours ago
Does there exist 2011 numbers?
cyshine   8
N 4 hours ago by TheBaiano
Source: Brazil MO, Problem 4
Do there exist $2011$ positive integers $a_1 < a_2 < \ldots < a_{2011}$ such that $\gcd(a_i,a_j) = a_j - a_i$ for any $i$, $j$ such that $1 \le i < j \le 2011$?
8 replies
cyshine
Oct 20, 2011
TheBaiano
4 hours ago
D1036 : Composition of polynomials
Dattier   1
N 4 hours ago by Dattier
Source: les dattes à Dattier
Find all $A \in \mathbb Q[x]$ with $\exists Q \in \mathbb Q[x], Q(A(x))= x^{2025!+2}+x^2+x+1$ and $\deg(A)>1$.
1 reply
Dattier
Yesterday at 1:52 PM
Dattier
4 hours ago
number sequence contains every large number
mathematics2003   3
N 4 hours ago by sttsmet
Source: 2021ChinaTST test3 day1 P2
Given distinct positive integer $ a_1,a_2,…,a_{2020} $. For $ n \ge 2021 $, $a_n$ is the smallest number different from $a_1,a_2,…,a_{n-1}$ which doesn't divide $a_{n-2020}...a_{n-2}a_{n-1}$. Proof that every number large enough appears in the sequence.
3 replies
mathematics2003
Apr 13, 2021
sttsmet
4 hours ago
IMO ShortList 2002, geometry problem 2
orl   28
N 4 hours ago by ezpotd
Source: IMO ShortList 2002, geometry problem 2
Let $ABC$ be a triangle for which there exists an interior point $F$ such that $\angle AFB=\angle BFC=\angle CFA$. Let the lines $BF$ and $CF$ meet the sides $AC$ and $AB$ at $D$ and $E$ respectively. Prove that \[ AB+AC\geq4DE. \]
28 replies
orl
Sep 28, 2004
ezpotd
4 hours ago
Arc Midpoints Form Cyclic Quadrilateral
ike.chen   56
N 4 hours ago by ezpotd
Source: ISL 2022/G2
In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$.
Prove that $B, C, X,$ and $Y$ are concyclic.
56 replies
ike.chen
Jul 9, 2023
ezpotd
4 hours ago
Non-linear Recursive Sequence
amogususususus   4
N 4 hours ago by GreekIdiot
Given $a_1=1$ and the recursive relation
$$a_{i+1}=a_i+\frac{1}{a_i}$$for all natural number $i$. Find the general form of $a_n$.

Is there any way to solve this problem and similar ones?
4 replies
amogususususus
Jan 24, 2025
GreekIdiot
4 hours ago
Russian Diophantine Equation
LeYohan   2
N 4 hours ago by RagvaloD
Source: Moscow, 1963
Find all integer solutions to

$\frac{xy}{z} + \frac{xz}{y} + \frac{yz}{x} = 3$.
2 replies
LeYohan
Yesterday at 2:59 PM
RagvaloD
4 hours ago
IMO Problem 4
iandrei   106
N Yesterday at 8:04 AM by alexanderchew
Source: IMO ShortList 2003, geometry problem 1
Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.
106 replies
iandrei
Jul 14, 2003
alexanderchew
Yesterday at 8:04 AM
Source: IMO ShortList 2003, geometry problem 1
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iandrei
138 posts
#1 • 13 Y
Y by Davi-8191, nguyendangkhoa17112003, TurtleKing123, HWenslawski, Adventure10, centslordm, megarnie, proxima1681, Mahmood.sy, Mango247, Rounak_iitr, and 2 other users
Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.
Attachments:
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sebadollahi
1 post
#2 • 4 Y
Y by Adventure10, centslordm, Mango247, and 1 other user
in two cyclic quadrilatrals APRD & DRCQ ,AD & CD are diameters respectively and we have:
RQ/sin(<RCQ)=CD/2 or RQ/sin(<BCA)=CD/2
PR/sin(<PAR)=AD/2 or PR/sin(<BAC)=AD/2
By dividing:
CD/AD= sin(<BAC)/sin(<BCA)=BC/AB
Since CD/AD=BC/AB thus bisectors of <CBA and <ADC intersect AC in the same poin.
S.Ebadollahi
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galois
400 posts
#3 • 2 Y
Y by Adventure10, Mango247
i must admit that this problem was quite easy by imo standards.my solution is based on a standard trick using pedal triangles and projections which is damn easy and pretty similar to the proof posted here.
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Lagrangia
1326 posts
#4 • 5 Y
Y by Inconsistent, Adventure10, Mango247, and 2 other users
here is an interesting thing about this problem and other 2 IMO problems in the past! it's about the so called Pedal Triangle Trick!

this was posted on a forum by: fritue2000 and I thought it would be interesting to post it here also!

"The previous two IMO problems solved by the Pedal Triangle Trick.
(1996 IMO) Let P be a point inside the triangle ABC such that angle
APB - angle ACB = angle APC - angle ABC. Let D, E be the incenters of
triangles APB, APC respectively. Show that AP, BD, CE meet at a
point.

As I know, at the jury meeting at IMO 1996, there is a discussion
for this problem becasuse its solution is very similar to problem
2 of the 34th IMO. The techniques include 'inversion' and 'PTT', etc.
For example, http://home1.pacific.net.sg/~slwee/imo96/imo96op.htm )

(1993 IMO) Let D be a point inside the acute-angled triangle ABC such
that angle ADB = angle ACB + 90 degrees, and AC*BD = AD*BC.

(a) Calculate the ratio AB*CD/(AC*BD).

The well-known 'Pedal Triangle Trick' is "For any point D, let
X, Y, Z be feet of the altitudes from D to AB, BC and CA. Then,
YZ = (DA*BC)/2r, etc, where r is the circumradius of ABC."
The proof is very easy, since D, A, Y, Z lies on a circle with
diameter DA, by the law of sines, YZ = DA sin A = DA*(BC/2r).
(2003 IMO) Given is a cyclic quadrilateral ABCD and let P, Q, R
be feet of the altitudes from D to AB, BC and CA respectively.

Prove that if PR = RQ then the interior angle bisectors of the
angles <ABC and <ADC are concurrent on AC.
Solution) By PTT, PR=RQ implies (DA*BC)/2r = (DC/AB)/2r, so,
CD/DA=BC/AB implies the results. q.e.d.

The most noticeable thing is in the above solution, we did not
used the condtion 'ABCD is cyclic'. And as an IMO problem, it
is not so intersting because all three problems 1993, 1996,
2003 solved by exactly the same TWO ways (PTT and inversion).
Well, the PTT is really well-known, for example, it appears
at "Geometry Revisited" by Coexter. It also used for a proof
of Ptolemy's Theorem.

In a cyclic qudrilateral ABCD, the three points are on the
simson line, we have PR+RQ=RP, with the same notation of 2003
imo. Then, (DA*BC)/2r + (DC*AB)/2r = (DB*AC)/2r or
DA*BC + DC*AB = DB*AC. "
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Anonymous
334 posts
#5 • 2 Y
Y by Adventure10, Mango247
A bit of standard angle chasing shows that triangles DPR and DBC are similar and so are DQR and DBA. Thus PR=QR gives DC/CB = DR/RP = DR/RQ = DA/AB, and the result follows from the angle bisector theorem.
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Peter
3615 posts
#6 • 4 Y
Y by Adventure10, Mango247, Dream6068., ehuseyinyigit
Yeah, I had something like that too, was quite easy :D solved in about 30 minutes (which is extremely few for me)
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Philip_Leszczynski
327 posts
#7 • 3 Y
Y by Adventure10, Adventure10, Mango247
Lemma: Let $ABCD$ be a cyclic quadrilateral. Let $DR$ and $DP$ be the altitudes from $D$ to $AB$ and $DC$, respectively. Let $RP$ intersect $AC$ at $Q$. Then $\angle AQD$ is a right angle.

Proof of Lemma:

$\angle DRB = \angle DPB = \pi / 2$, so $DRPB$ is cyclic. Then $\angle DRP = \angle DBP$. From cyclic quadrilateral $ABCD$, $\angle DBP = \angle DAP$. $\angle DRP = \angle DAP$, so $RAQD$ is cyclic. Thus $\angle AQD = \pi - \angle ARD = \pi / 2$.

Proof:

Let the bisector of $\angle ABC$ meet $AC$ at S. Let the bisector of $\angle ADC$ meet $AC$ at T.
By the Lemma, $R,Q,P$ are collinear.
Let $\angle DRQ = \alpha$. $\angle APD = \pi - \alpha - \angle RDP = \pi - \alpha - (\pi - \angle ABC) = \angle ABC - \alpha$.
$\frac{RQ}{\sin \angle RDQ} = \frac{DQ}{\sin \alpha}$, $\frac{PQ}{\sin \angle PDQ} = \frac{DQ}{\sin (\angle ABC - \alpha)}$.
$\frac{RQ}{PQ} \cdot \frac{\sin PDQ}{\sin RDQ} = \frac{sin(\angle ABC - \alpha)}{\sin \alpha}$.
$\angle RDQ = \pi - \angle RAQ = \angle BAC$. Also $\angle PDQ = \angle BCA$.
$\frac{RQ}{PQ} \cdot \frac{\sin \angle BCA}{\sin \angle BAC} = \frac{\sin (\angle ABC - \alpha)}{\sin \alpha}$.
$\frac{RQ}{PQ} \cdot \frac{AB}{BC} = \frac{\sin \angle QCD}{\sin \angle QAD} = \frac{AD}{DC}$.
So $RQ=PQ$ if and only if $\frac{AB}{BC} = \frac{AD}{DC}$.

By the Angle Bisector Theorem,
$\frac{AS}{SC} = \frac{AB}{BC}$ and $\frac{AT}{TC} = \frac{AD}{DC}$.
So $\frac{AB}{BC} = \frac{AD}{DC}$ if and only if $\frac{AS}{SC} = \frac{AT}{TC}$.
This can happen if and only if $S=T$.
So $PQ=QR$ if and only if the angle bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$. QED.
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beta
3001 posts
#8 • 2 Y
Y by Adventure10, Mango247
Yay I exploded this problem with like 20 cyclic quadrilaterals.

Okay first well-know fact: Simson's Line, so we know P, Q, R are collinear.

i'll go with the only if direction
Let X be the intersection of the angle bisectors. Extend BX, DX to Y, Z, where Y and Z lies on the circumcircle of ABCD. Y, Z lies on the perpendicular bisector of AC, hence YZ is a diameter, and it goes through midpoint M of AC. angle AMY = 90, angle ZDY = 90 because YZ is a diameter, so MYDX is cyclic, so angle ZYB = MDX = ADB. DX bisects ADC hence ADM = BDC = BAC = PDQ.
angle QAD=QPD. Hence triangle PQD is similar to triangle AMD, and triangle ADC is simlar to PDR. DM is a median hence DQ is a median => Q is the midpoint of PR as desired.

If direction:
Define X to be on AC such that DX bisects ADC, DX intersect the circumcircle at Z, M is the midpoint of AC, Y is the intersection of MZ and the circumcircle. triangle ADC is simlar to PDR. DM is a median and DQ is a median, so triangle PQD is similar to triangle AMD, angle ADM = PDQ=BAC = BDC. By definition XD bisects ADC hence XD bisects MDB so MDX = XDQ. QMY=90, YZ is still diameter, MYDX is cyclic, ZYX = MDX = ZDB. Now let BX intersect the circumcircle at Y'. XY'Z = XYZ, now Y' is unique(consider the locus of points Y' such that XY'Z = XYZ, which is a circle, and Y' lies on the cirucmcircle of ABCD. Two circle intersects in two points, one of which is Z, hence there's only one such point left and thus it's unique), hence Y = Y', thus B, X, Y are collinear. Y lies on the perpendicular bisector of AC so YA=YC, thus angle ABY = YBC.

QED
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FelixD
588 posts
#9 • 2 Y
Y by Adventure10, Mango247
The following will be useful: Let $ \triangle ABC$ be a triangle, $ D$ an arbitrary point and $ X$, $ Y$, $ Z$ the pedal points wrt the sides $ BC$, $ CA$, $ AB$. Then $ YZ= \frac{AD \cdot BC}{2R}$, where $ R$ denotes the circrumradius of $ \triangle ABC$. That's very easy to prove, I won't write it down now :) .
Using the lemma shown above, we have $ PQ= \frac{CD \cdot AB}{2R}$ and $ QR= \frac{AD \cdot BC}{2R}$. Hence, $ PQ =QR \Leftrightarrow \frac{AB}{BC} = \frac{AD}{DC}$. Let $ S$ denote the intersection of the angle bisectors of $ \angle ABC$ and $ \angle CDA$. If $ S \in AC$, then $ \frac{AB}{BC} = \frac{AS}{SC} = \frac{AD}{DC}$. If $ PQ=QR$, then $ \frac{AS_1}{S_1C} = \frac{AB}{BC} = \frac{AD}{DC} = \frac{AS_2}{S_2C}$, hence $ S_2=S_1=S$, where $ S_1$ and $ S_2$ denote the intersections of the angle bisectors of angles $ \angle ABC$ and $ \angle CDA$ with the side $ AC$. Thus, the problem is proved.
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serialk11r
1449 posts
#10 • 1 Y
Y by Adventure10
Hmmm I came up with this, which uses a bit more of Simson line properties.

Let the angle bisector of $ \angle{ABC}$ intersect $ AC$ at $ N$, let the midpoint of $ AC$ be $ M$, let $ E$ and $ F$ be points diametrically opposite on the circumcircle of $ ABCD$ such that $ EF$ is the perpendicular bisector of $ AC$, with $ E$ on the same side of $ AC$ as $ B$. Let $ B'$ be the point diametrically opposite to $ B$.

$ FDQN$ is cyclic, $ FDMN$ is a rectangle (cyclic), so $ \angle{FND} = \angle{FQD} = \angle{MDQ}$.
$ \angle{FND} = \angle{FBD} + \angle{BFE}$
$ \angle{B'BF} = \angle{BFE}$, since both are diameters.
Thus $ \angle{B'BD} = \angle{BNE} = \angle{MDQ}$.

$ P,Q,R$ are collinear and are the Simson line of $ D$. $ AC$ is the Simson line of $ B'$. By a well known property of Simson lines, $ \angle{PQA} = \frac12\overarc{B'D}$, but $ \frac12\overarc{B'D} = \angle{B'BD} = \angle{BNE} = \angle{MDQ}$ and $ \angle{PQA} = \angle{PDA} = \angle{CDR}$, since $ PDR$ is directly similar to $ ADC$. Thus we have a spiral similarity with center $ D$ mapping $ A,M,C$ to $ P,Q,R$ respectively, and since $ M$ is the midpoint of $ AC$, $ Q$ is the midpoint of $ PR$ so we are done.

The argument should work in reverse as well, with some tweaking here and there.
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hatchguy
555 posts
#11 • 4 Y
Y by AllanTian, Adventure10, Mango247, LeYohan
The concurrent with $AC$ bisectors condition is equivalent to $\frac{AB}{BC}=\frac{AD}{DC}$ $=> \frac{AD*BC}{AB*DC} =1$

Since $\angle DAB+ \angle DCB= 180$ and $\angle DCP+ \angle DCB= 180$ we have $\angle DAB= \angle DCP$.

Therefore we have that $ARD$ is similar to $CPD$ and therefore $\frac{AR}{PC}=\frac{AD}{DC}$. (1)

By Menelaus theorem for transversal $CQA$ in triangle $BRP$ we obtain:

$\frac{AB*RQ*PC}{AR*PQ*BC}=1$. Therefore $\frac{RQ}{QP}=\frac{AR*BC}{AB*PC}=\frac{AD*BC}{AB*DC}$ (because of (1)). The conclusion is now obvious.
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MariusBocanu
429 posts
#12 • 1 Y
Y by Adventure10
Use that a quadrilateral is harmonic if and only if the diagonals are symmedians(easy to prove with polarity and crossed ratio).
Now, the problem is equivalent to this:
$\triangle{ABC}$,$D$ is on the circumcircle of $\triangle{ABC}$(on the arc$BC$). Denote $X,Y,Z$ the projections of $D$ onto $AB,BC,CA$. $XY=YZ$if and only if $D$ is on the symmedian with respect to $A$.
Proof: See that $BXDY$ and $DYZC$ are inscribed in circles with diameter$BD$,respectively$CD$. We have from Simson's line that $X-Y-Z$are collinear. In $\triangle{XBY}$(aplying law of sines) $XY=BDsinB$, and in $\triangle{YZC}$ we have$YZ=DCsinC$. So, $XY=YZ$ if and only if $\frac{BD}{CD}=\frac{sinC}{sinB}$, but it happens only for symmedian(note that $\frac{sinBAM}{sinCAM}=\frac{sinB}{sinC}$, where $M$ is the midpoint of $BC$.
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arshakus
769 posts
#13 • 1 Y
Y by Adventure10
hey guys)
I solved this problem in about 10 minutes, but I think there is some thing wrong because I didn't use the fact that $ABCD$ is cyclic.
Can it be so?
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nickthegreek
35 posts
#14 • 2 Y
Y by Adventure10, Mango247
A different approach:


To begin with, it is clear that triangles $\bigtriangleup  DRP$ and $\bigtriangleup  DAC$ similar. (This follows by a very simple angle chasing, as in the above posts). This also gives us the fact that triangles $\bigtriangleup  DAR$ and $\bigtriangleup  DPC$ are similar, therefore $\frac{DA}{DC}=\frac{RA}{PC} (1)$

Also, points $P,Q,R$ are collinear (Simson's line)

It remains to prove the equivalence $PQ=QR \Leftrightarrow \frac{BA}{BC}=\frac{DA}{DC}$

Let's focus on triangle $\bigtriangleup RBP$ . Points $A,Q,C $ are collinear and lie on the lines $BR,RP,PB$ respectively. Applying Menelaus' Theorem, we have:
$\frac{AR}{AB}\cdot \frac{CB}{CP}\cdot \frac{QP}{QR}=1$

Therefore, $QP=QR\Leftrightarrow \frac{QP}{QR}=1\Leftrightarrow \frac{AB}{BC}=\frac{RA}{PC}\Leftrightarrow \frac{AB}{BC}= \frac{DA}{DC}$,

since $\frac{DA}{DC}=\frac{RA}{PC}$ by relation (1). The conclusion follows immediately.

Nick
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StefanS
149 posts
#15 • 1 Y
Y by Adventure10
nickthegreek wrote:
A different approach:
Your solution is actually identical to hatchguy's. Take a look:
hatchguy wrote:
The concurrent with $AC$ bisectors condition is equivalent to $\frac{AB}{BC}=\frac{AD}{DC}$ $=> \frac{AD*BC}{AB*DC} =1$

Since $\angle DAB+ \angle DCB= 180$ and $\angle DCP+ \angle DCB= 180$ we have $\angle DAB= \angle DCP$.

Therefore we have that $ARD$ is similar to $CPD$ and therefore $\frac{AR}{PC}=\frac{AD}{DC}$. (1)

By Menelaus theorem for transversal $CQA$ in triangle $BRP$ we obtain:

$\frac{AB*RQ*PC}{AR*PQ*BC}=1$. Therefore $\frac{RQ}{QP}=\frac{AR*BC}{AB*PC}=\frac{AD*BC}{AB*DC}$ (because of (1)). The conclusion is now obvious.
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