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whoa....
by
bachkieu, Jan 31, 2025, 1:40 AM
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hello...
by
ethan2011, Jul 4, 2024, 5:13 PM
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2024 shout ftw
by
Shreyasharma, Feb 19, 2024, 10:28 PM
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time flies
by
Asynchrone, Dec 13, 2023, 9:29 PM
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first 2023 shout
by
gracemoon124, Aug 2, 2023, 4:58 AM
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offline.................
by
799786, Dec 27, 2021, 7:08 AM
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YOU SHALL NOT PASS! - liberator
by
OlympusHero, Aug 16, 2021, 4:10 AM
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Nice Blog!
by
geometry6, Jul 31, 2021, 1:39 PM
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First shout out in 2021
by
Aimingformygoal, May 31, 2021, 4:23 PM
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indeed a pr0 blog
by
Kanep, Dec 3, 2020, 10:46 PM
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pr0 blog !!
by
Hamroldt, Dec 2, 2020, 8:32 AM
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niice bloog!
by
Eliot, Oct 1, 2020, 3:27 PM
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nice blog
by
fukano_2, Aug 8, 2020, 7:49 AM
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Nice blog
by
Feridimo, Mar 31, 2020, 9:29 AM
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Very nice blog !
by
Kamran011, Oct 31, 2019, 5:48 PM
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1000 VISITS!!!
IN 10 DAYS!!! 
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liberator, Aug 23, 2014, 3:47 PM
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YOU ARE SO PRO!!!!
Have you been to the IMO? If you have, then on not bragging about it. If you haven't, HOW ARE YOU SO CLEVER??!!?
by
jlammy, Aug 23, 2014, 12:51 PM
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Cool blog, great geometry problems!
by
HYP135peppers, Aug 22, 2014, 6:25 PM
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Extremely good at Geometry
Could you consider joining an Online Math Open Team with me?
by
DanielL2000, Aug 22, 2014, 12:40 AM
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Nice blog! I like geo too.
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sjaelee, Aug 21, 2014, 10:00 PM
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thanks Cyclicduck!
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liberator, Aug 21, 2014, 9:56 PM
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Hi, you are very good at geometry!
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Cyclicduck, Aug 21, 2014, 8:29 PM
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Posts? What are those, exactly?
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kmw2001, Aug 19, 2014, 8:50 PM
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@kmw2001, have 5 posts first.
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bcp123, Aug 19, 2014, 8:19 PM
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How do i blog?
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kmw2001, Aug 19, 2014, 5:30 PM
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thx bcp123, I guess its because I only have 10 posts, not much to brag on about
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liberator, Aug 15, 2014, 10:26 PM
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why no one is writing here?
keep this going. nice collection of geoish things 
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bcp123, Aug 15, 2014, 10:14 PM
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thx!
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liberator, Aug 14, 2014, 2:56 PM
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nice blog
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bcp123, Aug 14, 2014, 2:39 PM
by liberator, Oct 5, 2018, 8:45 PM 
such that, for all integers 
that satisfy 
, the following equality holds:![\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]](//latex.artofproblemsolving.com/a/9/e/a9edf1a3207d29facc970b6ee288d4fabdd43a4b.png)
(Here 
denotes the set of integers.)
such that, for all integers 
that satisfy 
, the following equality holds:![\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]](http://latex.artofproblemsolving.com/a/9/e/a9edf1a3207d29facc970b6ee288d4fabdd43a4b.png)
(Here 
denotes the set of integers.)
,
and two pathological ones: 
for any 
.
implies that 
.
implies 
,![$$\big[f(a)+f(b)-f(a+b)\big]^2=4f(a)f(b).$$](http://latex.artofproblemsolving.com/9/6/2/9628037cf5e447b568f9c5eee86d2de4f04c1147.png)
Let 
denote the above equation. Then ![$P(a,a)\implies f(2a)\big[f(2a)-4f(a)\big]=0$](http://latex.artofproblemsolving.com/e/6/0/e600d7841028e324f93ed64811d153132dfb8ba2.png)
,
for each 
.
for some 
;
,
is periodic with period 
.
,
.
.
,
.
.![[asy]
unitsize(0.6cm);
real e = 1;
MP("k",origin,E);
D((e,0)--(2,0));
MP("4k",(2,0),E);
D((4,-1)--(2+e,0)--(4,1));
MP("k",(4,1),E);
MP("9k",(4,-1),E);
D((6,0)--(4+e,1)--(6,2));
D((6,0)--(4+e,-1)--(6,-2));
DPA((6,0.3)--(7,-0.3)^^(6,-0.3)--(7,0.3),red);
MP("0",(6,2),E);
MP("\color{red}4k",(6,0),E);
MP("16k",(6,-2),E);
MP("f(1)",(0,3),E);
MP("f(2)",(2,3),E);
MP("f(3)",(4,3),E);
MP("f(4)",(6,3),E);
[/asy]](http://latex.artofproblemsolving.com/4/c/b/4cbcba452f921e2ef6cad5f6377172915491d4fd.png)
and 
.
,
is the only solution in this case. Suppose that 
for 
.
and 
implies 
.
and 
implies 
.
and 
implies 
.
and 
implies 
.
October 2018
April 2018