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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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]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
Hard geometry
Lukariman   1
N 3 minutes ago by Lukariman
Given triangle ABC, a line d intersects the sides AB, AC and the line BC at D, E, F respectively.

(a) Prove that the circles circumscribing triangles ADE, BDF and CEF pass through a point P and P belongs to the circumcircle of triangle ABC.

(b) Prove that the centers of the circles circumscribing triangles ADE, BDF, CEF and ABC are all on the circle.

(c) Let $O_a$,$ O_b$, $O_c$ be the centers of the circles circumscribing triangles ADE, BDF, CEF. Prove that the orthocenter of triangle $O_a$$O_b$$O_c$ belongs to d.

(d) Prove that the orthocenters of triangles ADE, ABC, BDF, CEF are collinear.
1 reply
Lukariman
Yesterday at 12:53 PM
Lukariman
3 minutes ago
Anything real in this system must be integer
Assassino9931   1
N 5 minutes ago by lksb
Source: Al-Khwarizmi International Junior Olympiad 2025 P1
Determine the largest integer $c$ for which the following statement holds: there exists at least one triple $(x,y,z)$ of integers such that
\begin{align*} x^2 + 4(y + z) = y^2 + 4(z + x) = z^2 + 4(x + y) = c \end{align*}and all triples $(x,y,z)$ of real numbers, satisfying the equations, are such that $x,y,z$ are integers.

Marek Maruin, Slovakia
1 reply
Assassino9931
Friday at 9:26 AM
lksb
5 minutes ago
geometry problem
kjhgyuio   0
20 minutes ago
........
0 replies
kjhgyuio
20 minutes ago
0 replies
Concurrency of two lines and a circumcircle
BR1F1SZ   1
N 26 minutes ago by MathLuis
Source: 2025 Francophone MO Juniors P3
Let $\triangle{ABC}$ be a triangle, $\omega$ its circumcircle and $O$ the center of $\omega$. Let $P$ be a point on the segment $BC$. We denote by $Q$ the second intersection point of the circumcircles of triangles $\triangle{AOB}$ and $\triangle{APC}$. Prove that the line $PQ$ and the tangent to $\omega$ at point $A$ intersect on the circumcircle of triangle $\triangle AOB$.
1 reply
BR1F1SZ
2 hours ago
MathLuis
26 minutes ago
IMO Shortlist 2009 - Problem A2
April   93
N 32 minutes ago by ezpotd
Let $a$, $b$, $c$ be positive real numbers such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = a+b+c$. Prove that:
\[\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}.\]
Proposed by Juhan Aru, Estonia
93 replies
April
Jul 5, 2010
ezpotd
32 minutes ago
Product of consecutive terms divisible by a prime number
BR1F1SZ   0
an hour ago
Source: 2025 Francophone MO Seniors P4
Determine all sequences of strictly positive integers $a_1, a_2, a_3, \ldots$ satisfying the following two conditions:
[list]
[*]There exists an integer $M > 0$ such that, for all indices $n \geqslant 1$, $0 < a_n \leqslant M$.
[*]For any prime number $p$ and for any index $n \geqslant 1$, the number
\[
a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p}
\]is a multiple of $p$.
[/list]


0 replies
BR1F1SZ
an hour ago
0 replies
Fixed and variable points
BR1F1SZ   0
an hour ago
Source: 2025 Francophone MO Seniors P3
Let $\omega$ be a circle with center $O$. Let $B$ and $C$ be two fixed points on the circle $\omega$ and let $A$ be a variable point on $\omega$. We denote by $X$ the intersection point of lines $OB$ and $AC$, assuming $X \neq O$. Let $\gamma$ be the circumcircle of triangle $\triangle AOX$. Let $Y$ be the second intersection point of $\gamma$ with $\omega$. The tangent to $\gamma$ at $Y$ intersects $\omega$ at $I$. The line $OI$ intersects $\omega$ at $J$. The perpendicular bisector of segment $OY$ intersects line $YI$ at $T$, and line $AJ$ intersects $\gamma$ at $P$. We denote by $Z$ the second intersection point of the circumcircle of triangle $\triangle PYT$ with $\omega$. Prove that, as point $A$ varies, points $Y$ and $Z$ remain fixed.
0 replies
BR1F1SZ
an hour ago
0 replies
Use 3d paper
YaoAOPS   7
N an hour ago by EGMO
Source: 2025 CTST p4
Recall that a plane divides $\mathbb{R}^3$ into two regions, two parallel planes divide it into three regions, and two intersecting planes divide space into four regions. Consider the six planes which the faces of the cube $ABCD-A_1B_1C_1D_1$ lie on, and the four planes that the tetrahedron $ACB_1D_1$ lie on. How many regions do these ten planes split the space into?
7 replies
YaoAOPS
Mar 6, 2025
EGMO
an hour ago
Cyclic ine
m4thbl3nd3r   2
N an hour ago by m4thbl3nd3r
Let $a,b,c>0$ such that $a^2+b^2+c^2=3$. Prove that $$\sum \frac{a^2}{b}+abc \ge 4$$
2 replies
m4thbl3nd3r
Yesterday at 3:34 PM
m4thbl3nd3r
an hour ago
Sequence inequality
BR1F1SZ   0
an hour ago
Source: 2025 Francophone MO Seniors P1
Let $a_1, a_2, a_3, \ldots$ be a sequence of positive integers satisfying the following property: for all positive integers $k < \ell$, for all distinct integers $m_1, m_2, \ldots, m_k$ and for all distinct integers $n_1, n_2, \ldots, n_\ell$,
\[
a_{m_1} + a_{m_2} + \cdots + a_{m_k} \leqslant a_{n_1} + a_{n_2} + \cdots + a_{n_\ell}.
\]Prove that there exist two integers $N$ and $b$ such that $a_n = b$ for all $n \geqslant N$.
0 replies
BR1F1SZ
an hour ago
0 replies
GCD and LCM operations
BR1F1SZ   0
2 hours ago
Source: 2025 Francophone MO Juniors P4
Charlotte writes the integers $1,2,3,\ldots,2025$ on the board. Charlotte has two operations available: the GCD operation and the LCM operation.
[list]
[*]The GCD operation consists of choosing two integers $a$ and $b$ written on the board, erasing them, and writing the integer $\operatorname{gcd}(a, b)$.
[*]The LCM operation consists of choosing two integers $a$ and $b$ written on the board, erasing them, and writing the integer $\operatorname{lcm}(a, b)$.
[/list]
An integer $N$ is called a winning number if there exists a sequence of operations such that, at the end, the only integer left on the board is $N$. Find all winning integers among $\{1,2,3,\ldots,2025\}$ and, for each of them, determine the minimum number of GCD operations Charlotte must use.

Note: The number $\operatorname{gcd}(a, b)$ denotes the greatest common divisor of $a$ and $b$, while the number $\operatorname{lcm}(a, b)$ denotes the least common multiple of $a$ and $b$.
0 replies
BR1F1SZ
2 hours ago
0 replies
Balanced grids
BR1F1SZ   0
2 hours ago
Source: 2025 Francophone MO Juniors/Seniors P2
Let $n \geqslant 2$ be an integer. We consider a square grid of size $2n \times 2n$ divided into $4n^2$ unit squares. The grid is called balanced if:
[list]
[*]Each cell contains a number equal to $-1$, $0$ or $1$.
[*]The absolute value of the sum of the numbers in the grid does not exceed $4n$.
[/list]
Determine, as a function of $n$, the smallest integer $k \geqslant 1$ such that any balanced grid always contains an $n \times n$ square whose absolute sum of the $n^2$ cells is less than or equal to $k$.
0 replies
BR1F1SZ
2 hours ago
0 replies
Radiant sets
BR1F1SZ   0
2 hours ago
Source: 2025 Francophone MO Juniors P1
A finite set $\mathcal S$ of distinct positive real numbers is called radiant if it satisfies the following property: if $a$ and $b$ are two distinct elements of $\mathcal S$, then $a^2 + b^2$ is also an element of $\mathcal S$.
[list=a]
[*]Does there exist a radiant set with a size greater than or equal to $4$?
[*]Determine all radiant sets of size $2$ or $3$.
[/list]
0 replies
1 viewing
BR1F1SZ
2 hours ago
0 replies
Classic Diophantine
Adywastaken   4
N 2 hours ago by mrtheory
Source: NMTC 2024/6
Find all natural number solutions to $3^x-5^y=z^2$.
4 replies
Adywastaken
Yesterday at 3:39 PM
mrtheory
2 hours ago
Collinearity with orthocenter
liberator   180
N Apr 26, 2025 by joshualiu315
Source: IMO 2013 Problem 4
Let $ABC$ be an acute triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ is the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$. Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM$, and let $Y$ be the point such that $WY$ is a diameter of $\omega_2$. Prove that $X,Y$ and $H$ are collinear.

Proposed by Warut Suksompong and Potcharapol Suteparuk, Thailand
180 replies
liberator
Jan 4, 2016
joshualiu315
Apr 26, 2025
Collinearity with orthocenter
G H J
G H BBookmark kLocked kLocked NReply
Source: IMO 2013 Problem 4
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liberator
95 posts
#1 • 26 Y
Y by Davi-8191, tenplusten, Arthur., Tawan, Wizard_32, Arshia.esl, OlympusHero, Instance, HWenslawski, donotoven, centslordm, jhu08, TheCollatzConjecture, mathlearner2357, tiendung2006, megarnie, RedFlame2112, rayfish, ImSh95, Lamboreghini, newinolympiadmath, Adventure10, Mango247, Rounak_iitr, farhad.fritl, ItsBesi
Let $ABC$ be an acute triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ is the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$. Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM$, and let $Y$ be the point such that $WY$ is a diameter of $\omega_2$. Prove that $X,Y$ and $H$ are collinear.

Proposed by Warut Suksompong and Potcharapol Suteparuk, Thailand
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High
876 posts
#2 • 9 Y
Y by Tawan, RedFlame2112, ImSh95, megarnie, Danielzh, lian_the_noob12, Adventure10, Mango247, Rounak_iitr
Just happened to stumble upon this problem recently; I guess I'll post a quick solution. Also, I think this problem needs to be added as the 2013 ISL G1 in the contest section.
Solution
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rkm0959
1721 posts
#3 • 6 Y
Y by Tawan, JUSTemom, RedFlame2112, ImSh95, Adventure10, Mango247
As $\angle BNC = \angle BMC = 90$, $B, N, M, C$ is cyclic. Let this circle be $\omega_3$. Let $Z(\not= W)  = \omega_1 \cap \omega_2$. Let $D = AH \cap BC$.
Now $BN$ is the radical axis of $\omega_1$ and $\omega_3$ and $CM$ is the radical axis of $\omega_2$ and $\omega_3$.
This implies that $A=BN \cap CM$ is the radical center of these three circles, so $A, Z, W$ are colinear.
Now by Power of a Point, we have $AN \cdot AB = AH \cdot AD= AZ \cdot AW = AM \cdot AC$, so we have $$\angle AEW = \angle AZH = \angle XZW = \angle WZY = \angle AZX = \angle AZY = 90$$by simple angle chasing - so $X, Y, Z, H$ are colinear.
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PROF65
2016 posts
#4 • 4 Y
Y by RedFlame2112, ImSh95, Adventure10, Mango247
Let $Z $ the second point of intersection of $\omega _1$ and $\omega _2$.
$(BCMN)$ are concyclic and the radical axes of its circumcircle , $\cal{C}$ $\ \text{and}\ \omega _1,$ $\cal{C}$ $  \text{and} \ \omega _2,\omega _1\ \text{and}\ \omega _2,$ are concurrent at $A$ . $\widehat{WZY}=\frac{\pi}{2},\widehat{WZX}=\frac{\pi}{2}$ thus $X,Y$ and $Z$ are colinear .if $H'$ is the intersection of $XY$ and $AH$ then $H'H_1ZW$ is cyclic where $H_1$ is the foot of $AH$.hence $AZ\cdot AW=AH_1\cdot AH'$ but $AZ\cdot AW=AN\cdot AB$ then $AH'\cdot AH_1=AB\cdot AN$ besides $BHH_1N$ is cyclic implies $AN\cdot AB=AH\cdot AH_1$ so $H=H'$ and the result follows.
R.HAS
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liberator
95 posts
#5 • 27 Y
Y by rkm0959, jam10307, vlohani, A_Math_Lover, Vfire, FadingMoonlight, Polynom_Efendi, zuss77, hara-huri, OlympusHero, sabrinamath, myh2910, mathlearner2357, prMoLeGend42, RedFlame2112, rayfish, ImSh95, megarnie, newinolympiadmath, Danielzh, Dimanas23, aqwxderf, Adventure10, Mango247, saltamonte, Math_.only., raffigm
The easiest way to do this is with Reim's theorem.
[asy]
/* IMO 2014 Question 4, free script by liberator, 12 August 2014 */
unitsize(2cm);
defaultpen(fontsize(10pt));
/* Initialize objects */
pair A = (-2.5, 2.5);
pair B = (-3.5, -0.5);
pair C = (0.5, -0.5);
pair H = orthocenter(A,B,C);
pair M = foot(B,C,A);
pair N = foot(C,A,B);
pair W = (-1.5, -0.5);
pair X = rotate(180, circumcenter(B,W,N))*W;
pair Y = rotate(180, circumcenter(C,W,M))*W;
pair P = intersectionpoint(H--Y, circumcircle(B,W,N)); 
/* Draw objects */
draw(A--B--C--cycle, rgb(0.4,0.6,0.8)+linewidth(1));
draw(X--Y, rgb(0.4,0.6,0.8)+dashed+linewidth(1));
draw(A--H, rgb(0.4,0.6,0.8));
draw(B--X, rgb(0.4,0.6,0.8));
draw(C--Y, rgb(0.4,0.6,0.8));
draw(circumcircle(B,W,N), red);
draw(circumcircle(C,W,M), red);
draw(circumcircle(A,M,N), red);
/* Place dots on and label each point, label circles */
dot(A); label("$A$", A, dir(90));
dot(B); label("$B$", B, dir(200));
dot(C); label("$C$", C, dir(340));
dot(H); label("$H$", H, dir(-90));
dot(M); label("$M$", M, dir(0));
dot(N); label("$N$", N, dir(180));
dot(P); label("$P$", P, dir(-70));
dot(W); label("$W$", W, 2.5*dir(-80));
dot(X); label("$X$", X, dir(180));
dot(Y); label("$Y$", Y, dir(60));
label("$\omega_1$", B--W, 15*dir(-90));
label("$\omega_2$", C--Y, 10*dir(0));
[/asy]
Let $P$ be the second intersection of $\omega_1, \omega_2$. Then $P$ is the Miquel point of $\triangle MNW$ w.r.t $\triangle ABC$, so $AMPHN$ is cyclic. $BX \parallel AH$, hence $X, H, P$ are collinear by Reim's theorem; similarly, $H,P,Y$ are collinear, and the result follows.
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K6160
276 posts
#6 • 4 Y
Y by Mr.Mister, RedFlame2112, Adventure10, Mango247
Let $Z$ be the second intersection of $\omega_1$ and $\omega_2$. Let $A'$ be the foot of the $A$-altitude. We have that $A$ lies on the radical axis of $\omega_1$ and $\omega_2$ because $BCMN$ is cyclic. Also by power of a point, $AZ\cdot AW=AN\cdot AB = AH\cdot AA'$, so $A'WZH$ is cyclic. Therefore, $\angle XZW=\angle YZW=\angle HZW=90^{\circ}$ and the conclusion follows. $\boxed{}$
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Kezer
986 posts
#7 • 7 Y
Y by MSTang, S117, RedFlame2112, Adventure10, Mango247, Rounak_iitr, Math_.only.
Different angle chasing approach without the radical axis/Reim

Was too noob after drawing in the Miquel point to think of proving that it lies on $XH$, so I needed Evan's hint from his book to do so. :blush:
This post has been edited 2 times. Last edited by Kezer, Mar 29, 2016, 8:39 PM
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MSTang
6012 posts
#8 • 5 Y
Y by donotoven, RedFlame2112, Adventure10, Mango247, Natrium
Spiral sims!
This post has been edited 2 times. Last edited by MSTang, Jun 18, 2016, 3:03 AM
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Reynan
634 posts
#9 • 3 Y
Y by RedFlame2112, Adventure10, Mango247
$AN\cdot AB=AM\cdot AC = AP\cdot AW\cdots (1)$

we have $\angle XPW=90=\angle WPX$ so $XPY$ collinear

let $D$ the perpendicular from $A$ to BC$
we have $NHDB$ and HDPW$ cylich from $(1)$
$\angle XHD=\angle DWP=180-\angle DHP$ so $XHP$ collinear
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the_executioner
533 posts
#10 • 3 Y
Y by Tawan, RedFlame2112, Adventure10
Let $Z$ be the second intersection point $\omega_1$ and $\omega_2$ . Now $\angle YZW$$+$$ \angle XZW $$=$$ 90^o+90^o=180^o$. Therefore $Y$, $Z$ and $X$ are collinear. So we are left to prove that $Z$, $H$ and $X$ are collinear. Now since $Z$ is the miquel point, we have $A$, $N$, $H$, $Z$, $M$ are concyclic. So$\angle AZM$$ +$$\angle MZW$ $= C+(A+B)=180^o$.
Therefore $A$, $Z$ and $W$ are collinear. Now, $\angle AZH$ $=90^o$ and thus $\angle HZW$$ =90^o$. But $\angle WZX$$ =90^o$, therefore $Z$, $H$ and $X$ are collinear and we are done.
This post has been edited 2 times. Last edited by the_executioner, Jul 1, 2016, 3:06 AM
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anantmudgal09
1980 posts
#11 • 4 Y
Y by Tawan, Aryan-23, RedFlame2112, Adventure10
A non standard way to do it...

Let $\ell_b,\ell_c$ be the lines perpendicular to line $BC$ at points $B,C$. Notice that as $W$ varies, $\angle WMY=\angle WNX=90^{\circ}$ and points $X,Y$ vary on lines $\ell_b,\ell_c$ respectively. Therefore, since this implies that $N$ and $NW$ are related projectively, i.e., the pencil and range have same cross ratio and essentially this is the same cross ratio for $X$ as it varies on the line $\ell_b$ we see that points $X$ and $W$ are projectively related to each other. Similarly, points $W$ and $Y$ are projectively related to each other. Thus, points $X$ and $Y$ are projectively related, i.e., there exist a projectivity mapping $X$ to $Y$. We prove that it is in fact a perspectivity. Thus, we only need to consider three positions of $W$ for which $X,Y,H$ are collinear. These are clearly true for $W=B,C,D$ where $D$ is the feet of altitude from $A$. Hence it holds for all $W$.
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Ghost_rider
35 posts
#12 • 2 Y
Y by RedFlame2112, Adventure10
Let $\omega_1 \cap \omega_2$ = $G$. $A$ has the same power with respect $\omega_1$ and $\omega_2$ so $A,G$ and $W$ are collinear.
Then $\angle NGA$ = $\angle ABW$ = $\angle NHA$, so $H,N,A$ and $G$ are concyclic ($\angle HGA$ = $90^\circ$).
Finally $X,H,G$ and $Y$ are collinear.
This post has been edited 1 time. Last edited by Ghost_rider, Jul 6, 2016, 2:43 AM
Reason: type
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jayme
9792 posts
#13 • 6 Y
Y by AlastorMoody, Kanep, Siddharth03, RedFlame2112, Adventure10, Mango247
Dear Mathlinkers,
just for history, this result comes from Mannhein in 1893... actually a rediscovery...

Sincerely
Jean-Louis
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Vexation
49 posts
#14 • 2 Y
Y by RedFlame2112, Adventure10
$K$ is the foot of the altitudes from $A$.

We need to check that $\frac{XB \cdot KC+YC \cdot KB}{BC} = {HK}$ ... (*)

From $NXB \sim NWC$ and $MYC \sim MWB$, (*) becomes $KH \cdot KA = KB \cdot KC$, which is true when $H$ is reflected in $BC$ by power of $K$ wrt the circumcircle.
This post has been edited 2 times. Last edited by Vexation, Aug 10, 2016, 6:44 AM
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CountofMC
838 posts
#15 • 4 Y
Y by RedFlame2112, Adventure10, Mango247, Rounak_iitr
Solution
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G
H
=
a