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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
2-var inequality
sqing   4
N 23 minutes ago by mathuz
Source: Own
Let $ a,b\geq 0    $. Prove that
$$ \frac{a }{a^2+2b^2+1}+ \frac{b }{b^2+2a^2+1}\leq \frac{1}{\sqrt{3}} $$$$   \frac{a }{2a^2+ b^2+2ab+1}+ \frac{b }{2b^2+ a^2+2ab+1}  \leq \frac{1}{\sqrt{5}} $$$$ \frac{a }{2a^2+ b^2+ ab+1}+ \frac{b }{2b^2+ a^2+ ab+1} \leq \frac{1}{2} $$$$\frac{a }{a^2+2b^2+2ab+1}+ \frac{b }{b^2+2a^2+2ab+1}\leq \frac{1}{2} $$
4 replies
+1 w
sqing
3 hours ago
mathuz
23 minutes ago
How many numbers
brokendiamond   0
38 minutes ago
How many 5-digit numbers can be formed using the digits 1, 3, 5, 7, 9 such that the smaller digits are not positioned between two larger digits?
0 replies
brokendiamond
38 minutes ago
0 replies
Sum of squares in 1865
Twoisaprime   2
N 44 minutes ago by EmptyMachine
Source: 2024 CWMO P1
For positive integer $n$, note $S_n=1^{2024}+2^{2024}+ \cdots +n^{2024}$.
Prove that there exists infinitely many positive integers $n$, such that $S_n$ isn’t divisible by $1865$ but $S_{n+1}$ is divisible by $1865$
2 replies
Twoisaprime
Aug 6, 2024
EmptyMachine
44 minutes ago
USA 97 [1/(b^3+c^3+abc) + ... >= 1/(abc)]
Maverick   47
N 44 minutes ago by justaguy_69
Source: USAMO 1997/5; also: ineq E2.37 in Book: Inegalitati; Authors:L.Panaitopol,V. Bandila,M.Lascu
Prove that, for all positive real numbers $ a$, $ b$, $ c$, the inequality
\[ \frac {1}{a^3 + b^3 + abc} + \frac {1}{b^3 + c^3 + abc} + \frac {1}{c^3 + a^3 + abc} \leq \frac {1}{abc}
\]
holds.
47 replies
Maverick
Sep 12, 2003
justaguy_69
44 minutes ago
Inspired by old results
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b>0 $ and $ \frac{a}{b^2}+\frac{2\sqrt{3}}{2a^2+b^2}\leq 3\sqrt 3. $ Prove that$$a^2+2b^2\geq 1$$
1 reply
sqing
an hour ago
sqing
an hour ago
Orthocenter
jayme   7
N an hour ago by Ianis
Dear Mathlinkers,

1. ABC an acuatangle triangle
2. H the orthcenter of ABC
3. DEF the orthic triangle of ABC
4. A* the midpoint of AH
5. X the point of intersection of AH and EF.

Prove : X is the orthocenter of A*BC.

Sincerely
Jean-Louis
7 replies
jayme
Mar 25, 2015
Ianis
an hour ago
something...
SunnyEvan   1
N an hour ago by SunnyEvan
Source: unknown
Try to prove : $$ \sum csc^{20} \frac{2^{i} \pi}{7} csc^{23} \frac{2^{j}\pi }{7} csc^{2023} \frac{2^{k} \pi}{7} $$is a rational number.
Where $ (i,j,k)=(1,2,3) $ and other permutations.
1 reply
SunnyEvan
3 hours ago
SunnyEvan
an hour ago
HK bisect QS
lssl   24
N 2 hours ago by LeYohan
Source: 1998 HK
In a concyclic quadrilateral $PQRS$,$\angle PSR=\frac{\pi}{2}$ , $H,K$ are perpendicular foot from $Q$ to sides $PR,RS$ , prove that $HK$ bisect segment$SQ$.
24 replies
lssl
Jan 5, 2012
LeYohan
2 hours ago
Points in general position
AshAuktober   3
N 3 hours ago by blackbluecar
Source: 2025 Nepal ptst p1 of 4
Shining tells Prajit a positive integer $n \ge 2025$. Prajit then tries to place n points such that no four points are concyclic and no $3$ points are collinear in Euclidean plane, such that Shining cannot find a group of three points such that their circumcircle contains none of the other remaining points. Is he always able to do so?

(Prajit Adhikari, Nepal and Shining Sun, USA)
3 replies
AshAuktober
Mar 15, 2025
blackbluecar
3 hours ago
Interesting inequalities
sqing   2
N 3 hours ago by sqing
Source: Own
Let $ a,b,c>0 $ and $ a+b\leq 16abc. $ Prove that
$$ a+b+kc\geq\sqrt{k}$$$$ a+b+kc^2\geq\frac{3\sqrt[3]{k}}{4}$$Where $ k>0. $
$$ a+b+c\geq1$$$$ a+b+4c\geq2$$$$ a+b+c^2\geq\frac{3}{4}$$$$ a+b+8c^2\geq\frac{3}{2}$$
2 replies
1 viewing
sqing
Yesterday at 12:23 PM
sqing
3 hours ago
IMO 2014 Problem 1
Amir Hossein   132
N 3 hours ago by maromex
Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that
\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]
Proposed by Gerhard Wöginger, Austria.
132 replies
Amir Hossein
Jul 8, 2014
maromex
3 hours ago
IMO Genre Predictions
ohiorizzler1434   31
N 3 hours ago by ohiorizzler1434
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
31 replies
ohiorizzler1434
Saturday at 6:51 AM
ohiorizzler1434
3 hours ago
weird FE
tobiSALT   10
N 3 hours ago by NicoN9
Source: Pan American Girls' Mathematical Olympiad 2024, P5
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$f(f(x+y) - f(x)) + f(x)f(y) = f(x^2) - f(x+y),$

for all real numbers $x, y$.
10 replies
tobiSALT
Nov 27, 2024
NicoN9
3 hours ago
2015 solutions for quotient function!
raxu   49
N 3 hours ago by blueprimes
Source: TSTST 2015 Problem 5
Let $\varphi(n)$ denote the number of positive integers less than $n$ that are relatively prime to $n$. Prove that there exists a positive integer $m$ for which the equation $\varphi(n)=m$ has at least $2015$ solutions in $n$.

Proposed by Iurie Boreico
49 replies
raxu
Jun 26, 2015
blueprimes
3 hours ago
angles in triangle
AndrewTom   33
N Apr 30, 2025 by zuat.e
Source: BrMO 2012/13 Round 2
The point $P$ lies inside triangle $ABC$ so that $\angle ABP = \angle PCA$. The point $Q$ is such that $PBQC$ is a parallelogram. Prove that $\angle QAB = \angle CAP$.
33 replies
AndrewTom
Feb 1, 2013
zuat.e
Apr 30, 2025
angles in triangle
G H J
Source: BrMO 2012/13 Round 2
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AndrewTom
12750 posts
#1 • 4 Y
Y by Adventure10, Mango247, Rounak_iitr, ItsBesi
The point $P$ lies inside triangle $ABC$ so that $\angle ABP = \angle PCA$. The point $Q$ is such that $PBQC$ is a parallelogram. Prove that $\angle QAB = \angle CAP$.
Z K Y
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Nilashis
132 posts
#2 • 3 Y
Y by AndrewTom, Adventure10, Mango247
Use sin rule on $\triangle ABQ$, $\triangle ACQ$, $\triangle BAP$, $\triangle CAP$ and comparing them we get that
$\frac{sin\angle BAQ}{sin\angle QAC}=\frac{sin\angle CAP}{sin\angle PAB}$. Now take $\angle BAQ=x$ and $\angle PAC=y$ then the equation reduces to $\frac{sinx}{sin(A-x)}=\frac{siny}{sin(A-y)}$
$2sin(A-x)siny=2sin(A-y)sinx$
$cos(A-x-y)-cos(A-x+y)=cos(A-y-x)-cos(A-y+x)$
$cos(A-x+y)=cos(A-y+x)$
$x-y=y-x$
$x=y$
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nsato
15654 posts
#3 • 2 Y
Y by Adventure10, ehuseyinyigit
This appears as an exercise in Geometry Revisited (Section 1.9, Exercise 3).
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sunken rock
4391 posts
#4 • 3 Y
Y by AndrewTom, Adventure10, Mango247
It has been posted here around few years ago, with a very nice synthetic solution!

Best regards,
sunken rock
Z K Y
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MMEEvN
252 posts
#5 • 12 Y
Y by sunken rock, AndrewTom, kprepaf, jlammy, 93051, v_Enhance, Med_Sqrt, hakN, Adventure10, Mango247, ohiorizzler1434, ehuseyinyigit
Let $R$ be the point such that $APBR$ is a parallelogram . Hence $AR || BP ||QC$ and $AR=BP=CQ$ Hence $ARQC$ is a parallelogram.$\angle ACQ = \angle ARQ$ . But $ \angle ACQ = \angle ABQ$ . Hence $ARBQ$ is cyclic.
.$\angle PAB=\angle ABR =\angle AQR= \angle QAC$. $ \Longrightarrow \angle QAB=\angle PAC$
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jlammy
1099 posts
#6 • 2 Y
Y by Adventure10, Mango247
sunken rock wrote:
It has been posted here around few years ago, with a very nice synthetic solution!

Best regards,
sunken rock

Can you specify the details of this "very nice synthetic solution"?
Z K Y
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sunken rock
4391 posts
#7 • 2 Y
Y by Adventure10, Mango247
@jlammy: Like MMEEvN did!

Best regards,
sunken rock
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IDMasterz
1412 posts
#8 • 1 Y
Y by Adventure10
no angle chasing: Let $P'$ be the $P$ isogonal conjugate and $P''$ be its reflection over $BC$. The angle bisectors of $BPC$ and $BAC$ are obviously parallel. $AP, PP''$ are antiparallel wrt $BPC$ so $AP' \parallel PP'' \parallel QP'$ since $PP'QP''$ form a parallelogram, so done.
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DottedCaculator
7346 posts
#9
Y by
Solution
This post has been edited 1 time. Last edited by DottedCaculator, Dec 10, 2021, 10:45 PM
Z K Y
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Project_Donkey_into_M4
148 posts
#11
Y by
AndrewTom wrote:
The point $P$ lies inside triangle $ABC$ so that $\angle ABP = \angle PCA$. The point $Q$ is such that $PBQC$ is a parallelogram. Prove that $\angle QAB = \angle CAP$.

For a non complicated solution unlike above here's a hint
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guptaamitu1
656 posts
#12 • 1 Y
Y by Rounak_iitr
Here's a different proof with similar triangles and homothety (plus reflection in angle bisector)
Let $E = \overline{BP} \cap \overline{AC}, F = \overline{CP} \cap \overline{AB}$. Then points $B,C,E,F$ are concyclic. Using $BPCQ$ is a parallelogram we get
$$ \angle QCB = \angle PBC = \angle EBC = \angle EFC = \angle EFP $$Similarly $\angle QBC = \angle FEP$. Hence,
$$ \triangle QBC \sim \triangle PEF $$[asy]
size(200);
pair B=dir(-160),C=dir(-20),E=dir(70),F=dir(135),A=extension(B,F,C,E),P=extension(B,E,C,F),Q=B+C-P;
draw(unitcircle,cyan);
fill(P--E--F--P--cycle,purple+grey+grey);
fill(B--Q--C--B--cycle,purple+grey+grey);
dot("$A$",A,dir(A));
dot("$B$",B,dir(B));
dot("$C$",C,dir(C));
dot("$E$",E,dir(E));
dot("$F$",F,dir(F));
dot("$P$",P,dir(-90));
dot("$Q$",Q,dir(Q));
draw(A--B--C--A,royalblue);
draw(B--E^^C--F,red);
draw(P--A--Q,green);
draw(B--Q--C^^E--F,brown);
[/asy]
Let $\mathbb H$ denote homothety at $A$ with scale $\frac{AF}{AC} = \frac{AE}{AB}$ followed by reflection in internal angle bisector of $\angle BAC$. Note $\mathbb H(F) = C$ and $\mathbb H(E) = B$. Thus $\mathbb H(P) = P'$ is a point such that $$\triangle PEF \sim \triangle P' BC$$Hence $P' \equiv Q$. As $\mathbb H$ also consists of reflection in internal angle bisector of $\angle BAC$, so $\angle BAP = \angle CAQ$ follows. $\blacksquare$
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TechnoLenzer
55 posts
#13 • 2 Y
Y by ike.chen, allin27x
Let $\infty_1 = BP \cap CQ$ and $\infty_2 = CP \cap BQ$. Since these are parallel pairs of lines, $\infty_1, \infty_2$ are the points at infinity for those pencils of parallel lines respectively. Note that $\measuredangle \infty_1AB = \measuredangle PBA = \measuredangle ACP = \measuredangle CA\infty_2$ by $A\infty_1 \; || \; CP$ and $A\infty_2 \; || \; BP$. Thus, $A\infty_1$ is isogonal to $A\infty_2$ wrt. $\triangle ABC$. Hence by DDIT on complete quadrilateral $P, B, Q, C, \infty_1, \infty_2$, there exists a projective involution swapping $(AB, AC)$, $(A\infty_1, A\infty_2)$, $(AP, AQ)$. This is taking the isogonal, and so $AP, AQ$ are isogonal. $\blacksquare$
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samrocksnature
8791 posts
#14
Y by
Any complex sols?
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AwesomeYRY
579 posts
#15
Y by
Consider the following two series of laws of sines:
\begin{align*}
    \frac{BQ}{\sin(\angle BAQ)} = \frac{AQ}{\sin(\angle ABP + \angle PBQ)} &= \frac{AQ}{\sin(\angle PCA + \angle ACP)} = \frac{CQ}{\sin (\angle QAC)},\\
    \frac{BP}{\sin(\angle BAP)} = \frac{AP}{\sin(\angle ABP)} &= \frac{AP}{\sin(\angle PCA)} = \frac{PC}{\sin(\angle PAC)}.
\end{align*}Putting them together, we get
\[\frac{\sin(\angle QAC)}{\sin(\angle BAQ)} = \frac{CQ}{BQ}= \frac{BP}{PC} = \frac{\sin(\angle BAP)}{\sin(\angle PAC)}\]and since $\angle WAC + \angle BAQ = \angle BAC = \angle BAP + \angle PAC$, we have that $\angle BAP = \angle QAC$ and we're done. $\blacksquare$.
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anurag27826
93 posts
#16
Y by
Amazing problem, although my solution is the same as of guptaamitu1's solution, but still im posting it for the sake of storage.

First of all we claim that $\triangle PEF \sim \triangle BQC$. Note that $\angle EPF = \angle BPC = \angle BQC$. Also note that $\angle PFE = \angle PBC = \angle QCB$. Both angle equalities are there since $BPCQ$ is a parallelogram. So, consider homothety $\psi$ under $A$ follow by reflection along the angle bisector of $\angle BAC$ with scale $\frac{AF}{AC}$. Note that $\psi$ sends $F$ to $C$ and $E$ to $B$. Then $\psi$ sends $P$ to $P'$ such that $\triangle CP'B \sim FPE \implies P' = Q$. So, it also implies that the line $AQ$ is a reflection of the line $AP$ along the angle bisector of $\angle BAC$, which implies $\angle BAP = \angle CAQ$. So, we're done.
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OronSH
1730 posts
#17
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Great problem.

We use directed angles. Let $P'$ be the isogonal conjugate of $P.$ Consider the lines through $P'$ parallel to $PB,PC.$ These lines intersect $AB$ at points $D,F$ with $D$ between $A$ and $F,$ and they intersect $AC$ at points $E,G$ with $E$ between $A$ and $G.$ Since $\angle DFE=\angle ABP=\angle PCA=\angle DGE,$ we have that $DEGF$ is cyclic. Also, since $\angle BFP'=\angle ABP=\angle PCA=\angle BCP',$ we have $BFCP'$ is cyclic, and similarly $BCGP'$ is cyclic, so $BFCG$ is cyclic, and by Reim's theorem we have $DE,BC$ parallel. Now, take a homothety at $A$ sending $DE$ to $BC.$ It is not hard to see that this takes $P'$ to $Q,$ so $A,P',Q$ are collinear, and we have $\angle QAB=\angle CAP.$
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huashiliao2020
1292 posts
#18
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It states that for a point P inside a triangle ABC with ABP=ACP, the point Q such that BPCQ is a parallelogram, AQ-AP are pairwise isogonal lines.

Here's the first one, quoted from my sl2012g2 (a good application for anyone who's reading!)
Sketch. If we let D be the point s.t. APBD is a parallelogram, we have that BDQ is congruent to APC by a translation of vector AD. (This is just done by length equalities and parallel lines.) Now, from DQB=ACP=ABP=DAB, ADBQ is cyclic. Then BAQ=BRQ=PAC, as desired. $\square$

Second solution I just came up with: Let BP,CP intersect AC,AB at E,F, respectively. It's obvious that $$F\in(BCE)\implies EFP=EBC=QCB,FEP=FCB=CBQ\implies EFP\sim CBQ,AEF\sim ABC\implies AEPF\sim ABQC\implies BAP=FAP=CAQ,$$as desired. $\blacksquare$
This post has been edited 2 times. Last edited by huashiliao2020, Aug 30, 2023, 11:01 PM
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Ianis
411 posts
#19
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Let $P'$ be the isogonal conjugate of $P$ in $ABC$, the angle condition implies that $P'$ is on the perpendicular bisector of $BC$. Use barycentric coordinates with respect to $ABC$ and let $P=(x,y,z)$. Then $P'=\left (\frac{a^2}{x}:\frac{b^2}{y}:\frac{c^2}{z}\right )$ and $Q=B+C-P=(-x,1-y,1-z)=(-x,z+x,x+y)$. We have\begin{align*}\begin{vmatrix}1&0&0\\\dfrac{a^2}{x}&\dfrac{b^2}{y}&\dfrac{c^2}{z}\\-x&z+x&x+y\end{vmatrix} & =\begin{vmatrix}\dfrac{b^2}{y}&\dfrac{c^2}{z}\\z+x&x+y\end{vmatrix} \\
& =b^2\frac{x+y}{y}-c^2\frac{z+x}{z} \\
& =\frac{b^2yz-c^2yz+b^2zx-c^2xy}{yz} \\
& =\frac{x}{a^2}\left ((b^2-c^2)\frac{a^2}{x}+a^2\left (\frac{b^2}{y}-\frac{c^2}{z}\right )\right ) \\
& =0,
\end{align*}where the last equality holds because $P'$ is on the perpendicular bisector of $BC$. Hence $A,P',Q$ are collinear, so $\angle QAB=\angle P'AB=\angle CAP$. Done.
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asdf334
7585 posts
#20
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hello;;; ddit with $A$ and $BPCQ$
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shendrew7
794 posts
#21
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Trivial by first isogonality lemma.

Let $X = BP \cap CA$ and $Y = CP \cap AB$. Then $BCXY$ is cyclic and $BPCQ$ is a parallelogram, so
\[\angle PXY = \angle BCP = \angle CBQ, \quad \angle PXY = \angle CBP = \angle BCQ.\]
Hence we have $\triangle PXY \sim \triangle QBC$ as well as $\triangle AXY \sim \triangle ABC$, so the quadrilaterals $AXPY$ and $ABQC$ are also similar with opposite orientation, which implies the desired. $\blacksquare$
This post has been edited 1 time. Last edited by shendrew7, Dec 31, 2023, 7:14 AM
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joshualiu315
2534 posts
#22
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Let $R$ be the reflection of $P$ across the midpoint of $\overline{AB}$. It is clear that $ARBP$ is a parallelogram. Then, notice that $\overline{AR} \parallel \overline{BP} \parallel \overline{QC}$, and $AR=BP=QC$. Hence, $ARQC$ is a parallelogram, meaning that $\angle ARQ = \angle ACQ$.

Also, we have

\[\angle ABQ = \angle PBQ+\angle ABP =\angle PCQ+\angle PCA = \angle ACQ.\]
This means that $\angle ABQ = \angle ARQ$, so $ARBQ$ is cyclic. Thus,

\[\angle BAP = \angle ABR = \angle AQR = \angle CAQ. \ \square\]
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EpicBird08
1751 posts
#23
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Let $H$ be the point such that $APCH$ is a parallelogram. First note that $AH = CP = BQ$ since $BPCQ$ is a parallelogram as well. Additionally, this gives $AH \parallel PC \parallel BQ,$ so $AHQB$ is also a parallelogram. Then $$\measuredangle AHQ = \measuredangle QBA = \measuredangle QBP + \measuredangle PBA = \measuredangle PCQ + \measuredangle ACP = \measuredangle ACQ,$$so $AHCQ$ is cyclic.

Next, note from our three parallelograms that $QC = BP, CH = AP,$ and $QH = AB,$ so $\triangle ABP \cong \triangle HQC.$ Finally, $\angle BAP = \angle QHC = \angle QAC,$ as desired.
This post has been edited 1 time. Last edited by EpicBird08, Jan 13, 2024, 5:09 PM
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dolphinday
1324 posts
#24
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Construct parallelograms $APCR$, $APBS$, and $BPCQ$.
And it follows that $ARQB$ is a parallelogram since $BQ \parallel PC \parallel AR$ and $AP \parallel RC$. Similarly, $ACQS$ is a parallelogram. Then we have $\angle SBP = 180^{\circ} = \angle SAP = \angle QAR = \angle BQA$ so $ASBQ$ is cyclic. Then $\angle CAQ = \angle{AQS} = \angle SBA = \angle{PAB}$ so we are done.
This post has been edited 2 times. Last edited by dolphinday, Feb 4, 2024, 3:53 PM
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Assassino9931
1321 posts
#25
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Denote $\angle BAC = \alpha$, $\angle ABP = \angle ACP = x$, $\angle BAP = y$, $\angle CAQ = z$. The Sine Law in $ABP$ and $ACP$ gives $\frac{BP}{\sin y} = \frac{AP}{\sin x} = \frac{CP}{\sin(\alpha - y)}$, i.e. $\frac{CP}{BP} = \frac{\sin(\alpha - y)}{\sin y}$. Analogously $\frac{BQ}{CQ} = \frac{\sin(\alpha - z)}{\sin z}$. However, $CP = BQ$ and $BP = CQ$ from the parallelogram $BPCQ$, thus $\frac{\sin(\alpha - y)}{\sin y} = \frac{\sin(\alpha - z)}{\sin z}$. Hence $\sin\alpha \cot y - \cos \alpha = \sin\alpha \cot z - \cos \alpha$, equivalent to $\cot y = \cot z$, i.e. $y=z$, as desired.
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dolphinday
1324 posts
#26
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Alternate solution involving DDIT;

Apply DDIT on quadrilateral $BC\infty_{BP}\infty_{BQ}$ from point $A$.
Then $B\infty_{BP} \cap C\infty_{BQ} = P$, and $B_\infty{BQ} \cap C\infty_{BP} = Q$, so we get that $(AB, AC)$, $(A\infty_{BP}, A\infty_{BQ})$, and $(AP, AQ)$ are involutions.
However $\angle \infty_{BQ}AC = \angle ACP = \angle ABP = \angle \infty_{BP}AB$, so $(AB, AC)$ and $(A\infty_{BP}, A\infty_{BQ})$ are both involutions around the angle bisector of $\angle BAC$. So it follows that $AP$ and $AQ$ are isogonal as desired.
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Martin2001
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#27
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Let the reflection of $P$ over the midpoint of $AC$ be $K.$ We see that $AKQB$ is a parralelogram. We show $AKCQ$ is cyclic because then $\angle QAC=\angle QKC=\angle BAQ.$ Let $\angle QAC=y, \angle ACP=x.$ Then $a-y=\angle BAQ=\angle KQA.$ Then note that $\angle CAK=x.$ Therefore $$180-a-x=\angle AKQ=\angle ACQ=b+c-x,$$as desired$.\blacksquare$
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ihatemath123
3446 posts
#28 • 1 Y
Y by OronSH
Vertically stretch about the angle bisector of $\angle BAC$ until $P$ is the orthocenter of $\triangle ABC$, then $Q$ is the antipode so it's obvious.
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bachkieu
136 posts
#29
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I think this works?
Let $CQ \cap AB = X, BQ \cap AC = Y$; it's easy to show that $\triangle ABC \sim \triangle AYX$ and that $P$ of $\triangle$ $ABC$ corresponds to $Q$ of $\triangle AYX$.
This post has been edited 1 time. Last edited by bachkieu, Sep 5, 2024, 12:14 AM
Reason: forgot period
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Lleeya
11 posts
#30 • 1 Y
Y by endless_abyss
This is Romanian Lemma, construct $EPFR$ to be parralelogram and trivial by similarity of $AEF$ and $ABC$.
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endless_abyss
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#31
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Here's my solution :)
Attachments:
This post has been edited 1 time. Last edited by endless_abyss, Nov 25, 2024, 5:31 PM
Reason: typo haha
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AndreiVila
210 posts
#32 • 2 Y
Y by trigadd123, Ciobi_
Lleeya wrote:
This is Romanian Lemma, construct $EPFR$ to be parralelogram and trivial by similarity of $AEF$ and $ABC$.

Now I can finally go to sleep knowing that we've achieved our goal at MOP. We won.
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AshAuktober
1003 posts
#33
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Yay!
The given is equivalent to $A\infty_{BP}$, $A\infty_{CP}$ being isogonal. From here Isogonal Line Lemma (or DDIT if you wish) gives the required.
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Tsikaloudakis
980 posts
#35
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see the figure:
Attachments:
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zuat.e
55 posts
#36
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Let $E,F=CP\cap AB, BP\cap AC$, then $EFBC$ is cyclic and $\triangle PEF\sim \triangle QCB$.
Now, we consider the transformation composed by the homothety of ratio $\frac{AC}{AF}$ and its reflection $w.r.t.$ the $\angle A$ angle bisector.
Clearly $E$ is sent to $B$ and $F$ to $C$ and as this transformation preserves similarity, it sends $P$ to $Q$, hence $AP$ and $AQ$ are isogonal.
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