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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
Serbian selection contest for the IMO 2025 - P6
OgnjenTesic   5
N 12 minutes ago by JARP091
Source: Serbian selection contest for the IMO 2025
For an $n \times n$ table filled with natural numbers, we say it is a divisor table if:
- the numbers in the $i$-th row are exactly all the divisors of some natural number $r_i$,
- the numbers in the $j$-th column are exactly all the divisors of some natural number $c_j$,
- $r_i \ne r_j$ for every $i \ne j$.

A prime number $p$ is given. Determine the smallest natural number $n$, divisible by $p$, such that there exists an $n \times n$ divisor table, or prove that such $n$ does not exist.

Proposed by Pavle Martinović
5 replies
1 viewing
OgnjenTesic
May 22, 2025
JARP091
12 minutes ago
JBMO TST Bosnia and Herzegovina 2024 P3
FishkoBiH   2
N 34 minutes ago by Ianis
Source: JBMO TST Bosnia and Herzegovina 2024 P3
Let $ABC$ be a right-angled triangle where $ACB$=90°.Let $CD$ be an altitude of that triangle and points $M$ and $N$ be the midpoints of $CD$ and $BC$, respectively.If $S$ is the circumcenter of the triangle $AMN$, prove that $AS$ and $BC$ are paralel.
2 replies
FishkoBiH
4 hours ago
Ianis
34 minutes ago
A line parallel to the asymptote of a cubic
kosmonauten3114   0
an hour ago
Source: My own, but uninspiring...
Given a scalene triangle $\triangle{ABC}$, let $P$ be a point ($\neq \text{X(4)}$). Let $P'$ be the anticomplement of $P$, and let $Q$ be the $\text{X(1)}$-anticomplementary conjugate of $P$. Prove that the line $P'Q$ is parallel to the real asymptote of the circular pivotal isocubic with pivot $P$.
0 replies
kosmonauten3114
an hour ago
0 replies
My Unsolved Problem
ZeltaQN2008   1
N an hour ago by Tung-CHL
Source: IDK
Given a positive integer \( m \) and \( a > 1 \). Prove that there always exists a positive integer \( n \) such that \( m \mid (a^n + n) \).

P/s: I can prove the problem if $m$ is a power of a prime number, but for arbitrary $m$ then well.....
1 reply
ZeltaQN2008
Yesterday at 10:18 AM
Tung-CHL
an hour ago
Please be my guest!
Madunglecha   0
an hour ago
Find all the (a,b,c) are integers which suggests
a^4-6a^2b^2+b^4=-c^2
0 replies
Madunglecha
an hour ago
0 replies
sequence (.) eventually becomes constant.
N.T.TUAN   61
N an hour ago by BS2012
Source: USAMO 2007
Let $n$ be a positive integer. Define a sequence by setting $a_{1}= n$ and, for each $k > 1$, letting $a_{k}$ be the unique integer in the range $0\leq a_{k}\leq k-1$ for which $a_{1}+a_{2}+...+a_{k}$ is divisible by $k$. For instance, when $n = 9$ the obtained sequence is $9,1,2,0,3,3,3,...$. Prove that for any $n$ the sequence $a_{1},a_{2},...$ eventually becomes constant.
61 replies
N.T.TUAN
Apr 26, 2007
BS2012
an hour ago
2025 consecutive numbers are divisible by 2026
cuden   0
an hour ago
Source: Collect
Problem..
0 replies
cuden
an hour ago
0 replies
Lots of perpendiculars; compute HQ/HR
MellowMelon   55
N 2 hours ago by Stead
Source: USA TST 2011 P1
In an acute scalene triangle $ABC$, points $D,E,F$ lie on sides $BC, CA, AB$, respectively, such that $AD \perp BC, BE \perp CA, CF \perp AB$. Altitudes $AD, BE, CF$ meet at orthocenter $H$. Points $P$ and $Q$ lie on segment $EF$ such that $AP \perp EF$ and $HQ \perp EF$. Lines $DP$ and $QH$ intersect at point $R$. Compute $HQ/HR$.

Proposed by Zuming Feng
55 replies
MellowMelon
Jul 26, 2011
Stead
2 hours ago
Another right angled triangle
ariopro1387   0
2 hours ago
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 $BQ$.
0 replies
ariopro1387
2 hours ago
0 replies
diophantine equation
m4thbl3nd3r   1
N 2 hours ago by whwlqkd
Find all positive integers $n,k$ such that $$5^{2n+1}-5^n+1=k^2$$
1 reply
m4thbl3nd3r
Today at 10:34 AM
whwlqkd
2 hours ago
ISI UGB 2025 P2
SomeonecoolLovesMaths   11
N 2 hours ago by ProMaskedVictor
Source: ISI UGB 2025 P2
If the interior angles of a triangle $ABC$ satisfy the equality, $$\sin ^2 A + \sin ^2 B + \sin^2  C = 2 \left( \cos ^2 A + \cos ^2 B + \cos ^2 C \right),$$prove that the triangle must have a right angle.
11 replies
SomeonecoolLovesMaths
May 11, 2025
ProMaskedVictor
2 hours ago
Functional Inequaility
ariopro1387   2
N 2 hours ago by Triborg-V
Source: Own
Find all functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) such that for any real numbers \(x\) and \(y\), the following inequality holds:
\[
f\left(x^2+2y f(x)\right) + (f(y))^2 \leq (f(x+y))^2
\]
2 replies
ariopro1387
Apr 9, 2025
Triborg-V
2 hours ago
Problem 7
SlovEcience   4
N 2 hours ago by SlovEcience
Consider the sequence \((u_n)\) defined by \(u_0 = 5\) and
\[
u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}.
\]a) Prove that there exist infinitely many positive integers \(n\) such that \(u_n > 2020n\).

b) Compute
\[
\lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}.
\]
4 replies
SlovEcience
May 14, 2025
SlovEcience
2 hours ago
Orthocentres of triangles ABC and AB’C’
Stun   40
N 3 hours ago by mathwiz_1207
Source: IMO Shortlist 1995, G8
Suppose that $ ABCD$ is a cyclic quadrilateral. Let $ E = AC\cap BD$ and $ F = AB\cap CD$. Denote by $ H_{1}$ and $ H_{2}$ the orthocenters of triangles $ EAD$ and $ EBC$, respectively. Prove that the points $ F$, $ H_{1}$, $ H_{2}$ are collinear.

Original formulation:

Let $ ABC$ be a triangle. A circle passing through $ B$ and $ C$ intersects the sides $ AB$ and $ AC$ again at $ C'$ and $ B',$ respectively. Prove that $ BB'$, $CC'$ and $ HH'$ are concurrent, where $ H$ and $ H'$ are the orthocentres of triangles $ ABC$ and $ AB'C'$ respectively.
40 replies
Stun
Mar 13, 2005
mathwiz_1207
3 hours ago
Cyclic points and concurrency [1st Lemoine circle]
shobber   10
N Apr 24, 2025 by Ilikeminecraft
Source: China TST 2005
Let $\omega$ be the circumcircle of acute triangle $ABC$. Two tangents of $\omega$ from $B$ and $C$ intersect at $P$, $AP$ and $BC$ intersect at $D$. Point $E$, $F$ are on $AC$ and $AB$ such that $DE \parallel BA$ and $DF \parallel CA$.
(1) Prove that $F,B,C,E$ are concyclic.

(2) Denote $A_{1}$ the centre of the circle passing through $F,B,C,E$. $B_{1}$, $C_{1}$ are difined similarly. Prove that $AA_{1}$, $BB_{1}$, $CC_{1}$ are concurrent.
10 replies
shobber
Jun 27, 2006
Ilikeminecraft
Apr 24, 2025
Cyclic points and concurrency [1st Lemoine circle]
G H J
Source: China TST 2005
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shobber
3498 posts
#1 • 1 Y
Y by Adventure10
Let $\omega$ be the circumcircle of acute triangle $ABC$. Two tangents of $\omega$ from $B$ and $C$ intersect at $P$, $AP$ and $BC$ intersect at $D$. Point $E$, $F$ are on $AC$ and $AB$ such that $DE \parallel BA$ and $DF \parallel CA$.
(1) Prove that $F,B,C,E$ are concyclic.

(2) Denote $A_{1}$ the centre of the circle passing through $F,B,C,E$. $B_{1}$, $C_{1}$ are difined similarly. Prove that $AA_{1}$, $BB_{1}$, $CC_{1}$ are concurrent.
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yetti
2643 posts
#2 • 2 Y
Y by Adventure10, Mango247
AP is the A-symmedian of the triangle $\triangle ABC.$ Let O be the triangle circumcenter and K the symmedian point.

(1) AEDF is a parallelogram, hence its diagonals AD, EF cut each other in half. Since the midpoint of EF lies on the A-symmedian AD, EF is antiparallel to BC with respect to the angle $\angle A,$ wich means that the points B, C, E, F are concyclic.

(2) Let parallels to the B-, C-symmedians BK, CK through the foot $D \in BC$ of the A-symmedian $AK \equiv AD \equiv AP$ meet the rays (AB, (AC at B', C'. The triangles $\triangle AB'C' \sim \triangle ABC$ are centrally similar with the similarity center A and D is the symmedian point of the triangle $\triangle AB'C'.$ It immediately follows that the circumcircle $(A_{1})$ of the quadrilateral BCEF is the 1st Lemoine circle of the triangle $\triangle AB'C'$ centered at the midpoint X' of the segment DO', where O' is the circumcenter of this triangle. Therefore, $AA_{1}$ intersects the segment KO of the original triangle $\triangle ABC$ also at its midpoint X, the center of the 1st Lemoine circle of the original triangle. Simiarly, $BB_{1}, CC_{1}$ cut KO at its midpoint X, hence all three are concurrent at X.
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alpha-beta
20 posts
#3 • 2 Y
Y by Adventure10, Mango247
can someone define 1st Lemoine circle or give some links?
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mihai miculita
666 posts
#4 • 1 Y
Y by Adventure10
$ \mbox{The three parallels to the sides of a triangle ABC through the Lemoine point of the triangle ABC, }$
$ \mbox{ determine on the sides of triangle ABC, 6 concyclic points.}$
$ \mbox{The circle of the 6 points is the 1-st Lemoine circle of triangle ABC.}$
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Sardor
801 posts
#5 • 2 Y
Y by Adventure10, Mango247
What's Lamoine point?
Please help me .
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Dilshodbek
115 posts
#6 • 1 Y
Y by Adventure10
alpha-beta wrote:
can someone define 1st Lemoine circle or give some links?

can you explain me about Lemoin circle please
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ak12sr99
156 posts
#7 • 1 Y
Y by Adventure10
Here is my solution with some angle and length chasing

Disclaimer: This is definitely not as elegant as yetti's beautiful solution (:thumbup:), but it is much neater than I had originally expected it to be, which is the reason I decided to mention it anyway.

$(1):$

By Thales' Theorem, $\frac{BD}{BC} = \frac{BF}{BA}$ and $\frac{CD}{CB} = \frac{CE}{CA}$. As $ADP$ is the symmedian, $\frac{BD}{DP} = \frac{AB^2}{AC^2}$ (as the symmedian is the reflection of the median over the angle bisector).

This yields the following, where $a=BC$ etc. (we will use these in part $(2)$ as well):
$BF=\frac{c^3}{b^2+c^2} ...(1)\\ \\AF=\frac{cb^2}{b^2+c^2} ...(2)\\ \\AE=\frac{bc^2}{b^2+c^2} ...(3)\\ \\CE=\frac{b^3}{b^2+c^2}...(4)$

From here we get $AF.AB = AE.AC = \frac{b^2c^2}{b^2+c^2}$ and concyclicity follows.


$(2):$

Let the radius of circle $BFEC$ be $r$.

Let $\angle BCF=\alpha \implies \angle BA_1F=2\alpha \implies \angle A_1BF=\angle A_1FB=90-\alpha \implies \angle A_1BC = B+\alpha-90 = \angle A_1CB \implies \angle A_1CE= 90-\alpha-B+C = \angle A_1EC \implies \angle CA_1E = 2(\alpha+B-C)$.

Now, in $\Delta sA_1BF$ and $A_1CE$ we get, using equations $(1)$ and $(4)$ above,
$2r sin \alpha = \frac{c^3}{b^2+c^2}$ and $2r sin (\alpha+B-C) = \frac{b^3}{b^2+c^2}$
$\implies \frac{sin \alpha}{sin (\alpha+B-C)} = \frac{c^3}{b^3}    ...(5)$

Now we observe that,
$\frac {[ABA_1]}{ACA_1]} = \frac{\frac{1}{2}AB. AA_1 sin \angle BAA_1}{\frac{1}{2}AC.AA_1 sin \angle CAA_1} = \frac{c}{b}.\frac{sin \angle BAA_1}{sin \angle CAA_1} ...(6)$
and
$\frac {[ABA_1]}{ACA_1]} = \frac{\frac{1}{2} AB. BA_1 sin \angle ABA_1}{\frac{1}{2} AC.CA_1 sin \angle ACA_1} =  \frac{c}{b}.\frac{sin(90-\alpha)}{sin(90-\alpha-B+C)}  = \frac{c}{b}.\frac{cos \alpha}{cos (\alpha-B+C)}  ...(7)$

$(6)$ and $(7)$ together imply
$\frac{sin \angle BAA_1}{sin \angle CAA_1} = \frac{cos \alpha}{cos (\alpha-B+C)}  ...(8)$

Now after some elementary manipulations on relation $(5)$ we get,
$\frac{cos \alpha}{cos (\alpha-B+C)} = \frac{\frac{b^3}{c^3} - cos (B-C)}{\frac{b^3}{c^3}cos (B-C) - 1}    ...(9)$

Finally we use $cos \theta = cos^2 \frac{\theta}{2} - sin^2 \frac{\theta}{2} = \frac{1-tan^2 \frac{\theta}{2}}{1+tan^2 \frac{\theta}{2}}$ (on $\theta = B-C$ duh :P ) and $tan \frac{B-C}{2} = \frac{b-c}{b+c}.cot\frac{A}{2}$ in relations $(8)$ and $(9)$ to finish the proof by the trigonometric form of Ceva's theorem.
This post has been edited 5 times. Last edited by ak12sr99, Sep 16, 2017, 2:40 PM
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Sanjana42
21 posts
#8 • 1 Y
Y by kamatadu
(1) Define $E,F$ as follows. Let the line passing through the midpoint of $AD$ which is antiparallel to $BC$ w.r.t $AB,AC$ intersect $AB,AC$ at $F,E\implies FBCE$ cyclic. Since $AD$ is isogonal to the $A$-median in $\triangle ABC$, it must be the $A$-median in $\triangle AEF\implies$ the midpoint of $AD$ (which is on $FE$) is also the midpoint of $FE$, so $AFDE$ is a parallelogram, so $E,F$ are the same $E,F$ in the problem statement.

(2) Let $EF=a_A,AF=b_A,AE=c_A$. By similarity we get $a=BC=\frac{a_A(b_A^2+c_A^2)}{b_Ac_A}$ and $FB=\frac{c_A^2}{b_A}$.

Let $\angle FBE = \angle FCE = \theta_A$. Similarly define $\theta_B,\theta_C$. Sine rule in $\triangle FEB$ gives us $$\frac{\sin (C-\theta_A)}{\sin \theta_A}=\frac{c_A^2}{a_Ab_A}=\frac{c^2}{ab}=\frac{\sin (C-\theta_B)}{\theta_B}$$by symmetry. Therefore the corresponding $\theta$ is the same for all 3 vertices.

Let the feet from $A_1$ to $AB,AC$ be $M_a,N_a$. Note that $\angle FA_1M_a=\angle FEB=C-\theta$. $$\implies \frac{\sin \angle BAA_1}{\sin \angle CAA_1}=\frac{A_1M}{A_1N}=\frac{A_1M}{A_1F}\cdot\frac{A_1E}{A_1N}=\frac{\cos (C-\theta)}{\cos (B-\theta)}$$
Clearly the cyclic product of these is 1, so we're done by trig Ceva.
This post has been edited 1 time. Last edited by Sanjana42, Jan 5, 2025, 8:09 PM
Reason: typo
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cursed_tangent1434
642 posts
#9 • 1 Y
Y by stillwater_25
Solved with stilwater_25. Amazing problem! We realized what the concurrence point is but missed the slick Lemoine circle argument that can be done by shifting the reference triangle.

For part (1) note that since $AEDF$ is a parallelogram by definition, $\overline{AD}$ bisects $EF$. It is well known that the $A-$symmedian only bisects the antiparallels to $BC$, which implies that $BFEC$ is cyclic.

Now, we can move to the interesting part of the problem. We claim that these lines concur at $X_{182}$, the midpoint of $OK$ where $O$ and $K$ are the circumcenter and the symmedian point of $\triangle ABC$ respectively. We show that $\overline{AA_1}$ bisects segment $OK$ from which the result follows due to symmetry.

Let $M_a$ and $M$ denote the midpoints of segments $BC$ and $EF$ respectively. Let $X$ be the intersection of lines $\overline{EF}$ and $\overline{BC}$. Let $K_a$ denote the intersection of the $A-$symmedian with $(ABC)$. The key claim is the following.

Claim : Points $M$ , $A_1$ , $M_a$ and $K_a$ are concyclic.

Proof : It is clear that $XM_aA_1M$ is cyclic due to the right angles. Let $Y$ be the intersection of the $A-$tangent with $\overline{BC}$. Since any antiparallel to side $BC$ is parallel to the $A-$tangent, note that
\[-1=(EF;M\infty)\overset{A}{=}(BC;DY)\]Thus,
\[DY \cdot DM_a = DB \cdot DC \]Further, from the midpoint theorem it follows that $X$ is the midpoint of segment $YD$. Thus,
\[DM \cdot DK_a = \frac{DA\cdot DK_a}{2} = \frac{DB\cdot DC}{2} = \frac{DY \cdot DM_a}{2} = DX \cdot DM_a\]which implies that $MM_aK_aX$ is also cyclic. Putting these observations together proves the claim.

We now show the following.

Claim : Lines $\overline{OK}$ and $\overline{DA_1}$ are parallel.

Proof : This is a simple length chase. First remember that $(AK_a;DP)=-1$. Note that,
\[PA_1 \cdot PM_a = PK_a \cdot PM\]Also,
\[PM_a \cdot PO = PB^2\]This then implies,
\[\frac{PA_1}{PO} = \frac{PK_a \cdot PM}{PB^2} = \frac{PM}{PA}\]Now, let $K_c$ denote the intersection of the $C-$symmedian with $(ABC)$. Then,
\[-1=(AB;CK_a)\overset{C}{=}(AD;PK)\]Thus,
\[PD \cdot PA = PK \cdot PM\]Thus,
\[\frac{PA_1}{PO} = \frac{PM}{PA}=\frac{PD}{PK}\]which implies that $OK \parallel DA_1$ as claimed.

Now we are done since letting $X = \overline{AA_1} \cap \overline{OK}$ we have,
\[(OK;X\infty)\overset{A_1}{=}(PK;AD)=-1\]which implies that $X$ is indeed the midpoint of $OK$ and we are done.
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Batsuh
152 posts
#10
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(1) Let $E' = DE \cap PB$ and $F' = DF \cap PC$. By an easy angle chase we see that $BFCF'$ and $BECE'$ are cyclic. So by PoP we have
\[FD \cdot DF' = BD \cdot DC = ED \cdot DE'\]so the points $B, F, E, C, F', E'$ are cyclic.

(2) Let $Q$ be the Symmedian point of $ABC$ and let $O$ be the center of $\omega$. We'll show that $AA_1$ passes through the midpoint of $OQ$, after which we'll be done by symmetry.

[asy]
import geometry;
import olympiad;
size(9cm);
filldraw(unitcircle, purple+white+white, blue);
pair A = dir(110); pair B = dir(225); pair C = dir(315);
pair O = (0,0);
pair M = B / 2+ C / 2;
pair P = extension(B, B+rotate(90)*(B-O),O,M);
pair D = extension(A,P,B,C);
pair E = intersectionpoint(parallel(D,line(A,B)),line(A,C));
pair Ep = extension(E,D,B,P);
pair F = intersectionpoint(parallel(D,line(A,C)),line(A,B));
pair Fp = extension(F,D,C,P);
circle BFEC = circle(B,F,E);
pair A_1 = circumcenter(B,F,E);
pair N = B / 2 + Ep / 2;
pair Q = intersectionpoint(parallel(B,line(N,D)), line(A,P));



draw(A -- B -- C -- cycle);
draw(line(P, false, B));
draw(line(P, false, C));
draw(E -- Ep);
draw(F -- Fp);
draw(O -- P);
draw(Q -- O, darkblue+1);
draw(D -- A_1, darkblue+1);
draw(B -- Q, darkblue+1);
draw(N -- D, darkblue+1);
draw(A -- P);
draw(circumcircle(B,F,E), red);

dot("$A$", A, dir(A));
dot("$B$", B, dir(B));
dot("$C$", C, dir(C));
dot("$P$", P, dir(P));
dot("$D$", D, dir(D));
dot("$Q$", Q, NW);
dot("$E$", E, dir(E));
dot("$F$", F, dir(F));
dot("$E'$", Ep, dir(Ep));
dot("$F'$", Fp, dir(Fp));
dot("$O$", O, NW);
dot("$A_1$", A_1, SE);
dot("$N$", N, dir(N));


[/asy]

Let $N$ be the midpoint of $BE'$. Observe that triangles $\triangle BDE'$ and $\triangle ABC$ are inversely similar with parallel sides. This means that the $B$-symmedian in $\triangle ABC$ and the $D$-median in $\triangle BDE'$ are parallel. In other words, $BQ \parallel ND$. Therefore,
\[\frac{PA_1}{PO} = \frac{PM}{PB} = \frac{PD}{PQ}\]which implies that $QO \parallel DA_1$. Now,
\[-1 = (A,D;Q,P) \overset{A_1}{=} (AA_1 \cap QO, QO_{\infty}; Q, O)\]implies that $AA_1 \cap QO$ is the midpoint of $QO$ as needed.
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Ilikeminecraft
658 posts
#11
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For part one, we simply note that $EF$ and $AB$ are antiparallel since $AD$ is the $A$-median in $AEF.$

Let $O$ denote the center of $ABC.$ Let $L$ denote the Lemoine point(intersection of symmedians).
I claim that $AA_1$ passes through the midpoint of $LO.$

Let $E’, F’$ be the intersections of $BP, CP$ with $(BFEC).$
Observe that $\angle BE’E = \angle BCE = \angle AFE = \angle FED$ so $FB\parallel EE’,$ so $EDE’$ are collinear.
Similarly, $FDF’$ are collinear.
Let $N$ be the midpoint of $BE’.$
Next, note that $BDE’$ and $ABC$ are inversely similar, with $B$ corresponding to $D.$ Thus, the $B$ symmedian in $ABC$ must be parallel to the $D$-median in $BDE’.$ Hence, $BL\parallel ND.$
Furthermore, $BO\parallel NA_1.$
Thus, there is homothety centered at $P$ sending $BLO$ to $NDA_1.$
Thus, $LO\parallel DA_1.$
Finally, by Ceva-Menelaus, we have $-1 = (AD;LP).$ Projection through $A_1$ onto $LO$ finishes.
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