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this guy is an absolute legend. much love wherever you are Michael
by
LeonidasTheConquerer, Aug 5, 2024, 9:37 PM
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amazing blog
by
anurag27826, Jun 17, 2023, 7:20 AM
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hi i randomly found this
by
purplepenguin2, Mar 1, 2023, 8:43 AM
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can i be a contributor please?
by
cinnamon_e, Mar 10, 2022, 6:58 PM
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orzorzorzorzorzorozo
by
samrocksnature, Jul 16, 2021, 8:25 PM
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2021 post
by
the_mathmagician, May 5, 2021, 3:28 PM
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Let
be an equilateral triangle of side length 
. Let 
be the point such that 
is the midpoint of 
, and let 
be the incenter of triangle 
. Let 
be the point on line 
such that 
and 
are perpendicular. $ \
by
ARay10, Aug 25, 2020, 5:55 PM
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Nice blog!
by
User526797, Jan 12, 2020, 4:48 PM
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oh my gosh it's been so longggggg.... contrib? what does that mean?
by
adiarasel, Dec 1, 2019, 8:31 PM
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2019 post
by
piphi, Aug 10, 2019, 6:32 AM
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hi contrib please
by
Emathmaster, Dec 27, 2018, 5:38 PM
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hihihihihi contrib plzzzzz
by
haha0201, Aug 20, 2018, 3:58 PM
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contrib please
by
Max0815, Aug 1, 2018, 12:35 AM
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contrib /charmander
by
mathmaster2000, Apr 16, 2017, 4:59 PM
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for contrib
by
SomethingNeutral, Mar 30, 2017, 7:57 PM
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all contrib requests added I usually get to posting every night after running on the treadmill and showering, posting my favorite problem(s) I did during the day.
by
shiningsunnyday, Jun 16, 2016, 4:37 AM
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Since problems is plural, he might post multiple problems through different posts in one day...
by
zephyrcrush78, Jun 15, 2016, 8:37 PM
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hi can i have contrib?
by
wu2481632, Jun 15, 2016, 8:36 PM
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Hello it is called problems of the DAY
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Temp456, Jun 15, 2016, 8:17 PM
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10th shout lol
How often will this blog be updated?
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zephyrcrush78, Jun 15, 2016, 5:24 PM
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why are we counting shouts
contrib pls
by
MathAwesome123, Jun 15, 2016, 2:55 PM
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eighth shout
contrib pls
by
skipiano, Jun 15, 2016, 2:21 PM
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seventh shout
Can I be contrib?
Though probably not because it's not like I'll post anything anyways :/ :/
by
zephyrcrush78, Jun 15, 2016, 1:20 AM
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sixth shout
new goal: get the shouts that are congruent to 1 mod 5
by
DeathLlama9, Jun 15, 2016, 12:33 AM
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fifth shout >
by
budu, Jun 14, 2016, 10:04 PM
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4th yay
by
EpicSkills32, Jun 14, 2016, 9:10 PM
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third shout
by
skipiano, Jun 14, 2016, 2:35 PM
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Second shout
by
zephyrcrush78, Jun 14, 2016, 12:06 AM
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omg yesssss
by
DeathLlama9, Jun 13, 2016, 2:19 PM
by shiningsunnyday, Jul 5, 2016, 1:18 PM 
be the circumcenter and 
the orthocenter of an acute triangle 
. Prove that the area of one of the triangles 
, 
and 
is equal to the sum of the areas of the other two.
be the circumcenter and 
the orthocenter of an acute triangle 
. Prove that the area of one of the triangles 
, 
and 
is equal to the sum of the areas of the other two.
as the base, and thus it suffices to prove 
where 
denotes the distance between 
and Euler's line. 
is impossible to compute each on their own, but note how we can drop an altitude from the midpoint 
of 
onto Euler's line and obtain a segment with length 
![[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(13.735451454550365cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -7.144678668733572, xmax = 10.590772785816794, ymin = -3.12446898938282, ymax = 6.281329930169833; /* image dimensions */
/* draw figures */
draw(circle((1.72,1.56), 4.430031288579984));
draw((-2.215626569177502,-0.4737209065606236)--(5.670709900056741,-0.44426258344341996));
draw((5.670709900056741,-0.44426258344341996)--(-0.6453206064565732,5.305722313048519));
draw((-0.6453206064565732,5.305722313048519)--(-2.215626569177502,-0.4737209065606236));
draw((-0.6453206064565732,5.305722313048519)--(-0.6237545040170589,-0.4677746878265274));
draw((-2.215626569177502,-0.4737209065606236)--(1.373051677820604,3.468237497526557));
draw((5.670709900056741,-0.44426258344341996)--(-1.6659986766365313,1.5491607943722665));
draw((1.4801807028527427,1.5301775345114093)--(1.7275416654396194,-0.4589917450020218), dotted);
draw((-0.6453206064565732,5.305722313048519)--(1.7275416654396194,-0.4589917450020218));
draw((-0.6302372755773338,1.2677388230444755)--(1.72,1.56), linetype("4 4"));
draw((-0.6453206064565732,5.305722313048519)--(-0.15059868128281953,1.3273837540216564), linetype("4 4"));
/* dots and labels */
dot((1.72,1.56),dotstyle);
label("$O$", (1.7160590950055392,1.7810814129395882), NE * labelscalefactor);
dot((-0.6453206064565732,5.305722313048519),dotstyle);
label("$A$", (-0.6318966531146037,5.596509503634795), NE * labelscalefactor);
dot((-2.215626569177502,-0.4737209065606236),dotstyle);
label("$B$", (-2.5326227349261483,-0.6786817517576883), NE * labelscalefactor);
dot((5.670709900056741,-0.44426258344341996),dotstyle);
label("$C$", (5.908837216648652,-0.580850262252683), NE * labelscalefactor);
dot((-0.6302372755773338,1.2677388230444755),linewidth(3.pt) + dotstyle);
label("$H$", (-0.911415194557478,1.5155387985688595), NE * labelscalefactor);
dot((-1.6659986766365313,1.5491607943722665),linewidth(3.pt) + dotstyle);
dot((1.373051677820604,3.468237497526557),linewidth(3.pt) + dotstyle);
dot((-0.6237545040170589,-0.4677746878265274),linewidth(3.pt) + dotstyle);
dot((1.7275416654396194,-0.4589917450020218),linewidth(3.pt) + dotstyle);
label("$M$", (1.6601553867169645,-0.7485613871184064), NE * labelscalefactor);
dot((1.4801807028527427,1.5301775345114093),linewidth(3.pt) + dotstyle);
label("$X$", (1.338709064057659,1.683249923434583), NE * labelscalefactor);
dot((-0.15059868128281953,1.3273837540216564),linewidth(3.pt) + dotstyle);
label("$Y$", (-0.08683549730099917,1.5434906527131467), NE * labelscalefactor);
dot((0.936587574807555,1.4625796076814916),linewidth(3.pt) + dotstyle);
label("$G$", (0.6259367833783299,1.6133702880738647), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/6/9/c/69c8af108110cf2fcc32818c60450ca2fa69fb54.png)
where we let 
be the foot of the latitude from 
suspiciously reminds us of 
which lies on the intersection of 
and 
Now we're home. 
is the image of the homothety of 
with scale factor 
and the result follows.
May 2017
April 2017