JMO<200?
by DreamineYT, May 10, 2025, 5:37 PM
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ARML Location
by deduck, May 6, 2025, 4:19 PM
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UNR -> Nevada
St Anselm -> New Hampshire
PSU -> Pennsylvania
WCU -> North Carolina
Put your USERNAME in the list ONLY IF YOU WANT TO!!!!
I'm going to UNR if anyone wants to meetup!!!
Current List:
St Anselm -> New Hampshire
PSU -> Pennsylvania
WCU -> North Carolina
Put your USERNAME in the list ONLY IF YOU WANT TO!!!!
I'm going to UNR if anyone wants to meetup!!!
Current List:
This post has been edited 11 times. Last edited by deduck, May 7, 2025, 4:55 PM
usamOOK geometry
by KevinYang2.71, Mar 21, 2025, 12:00 PM
Let 
be the orthocenter of acute triangle 
, let 
be the foot of the altitude from 
to 
, and let 
be the reflection of across 
. Suppose that the circumcircle of triangle 
intersects line at two distinct points 
and 
. Prove that is the midpoint of 
.
be the orthocenter of acute triangle 
, let 
be the foot of the altitude from 
to 
, and let 
be the reflection of across 
. Suppose that the circumcircle of triangle 
intersects line at two distinct points 
and 
. Prove that is the midpoint of 
.An FE. Who woulda thunk it?
by nikenissan, Apr 15, 2021, 5:13 PM
Let 
denote the set of positive integers. Find all functions 
such that for positive integers 
and 
![\[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]](//latex.artofproblemsolving.com/c/1/f/c1f8ffe04cfcc45b498f3931a4796d5c56dc04d0.png)
denote the set of positive integers. Find all functions 
such that for positive integers 
and 
![\[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]](http://latex.artofproblemsolving.com/c/1/f/c1f8ffe04cfcc45b498f3931a4796d5c56dc04d0.png)
IMO 2012 Problem 4
by liberator, Oct 5, 2018, 8:45 PM
This was a pretty tricky functional equation, especially for a P4...
Problem: Find all functions
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.)
Proposed by Liam Baker, South Africa
Problem: Find all functions
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.)Proposed by Liam Baker, South Africa
,
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 
.Geo #3 EQuals FReak out
by Th3Numb3rThr33, Apr 18, 2018, 11:00 PM
Let 
be a quadrilateral inscribed in circle 
with 
. Let 
and be the reflections of 
over lines 
and , respectively, and let be the intersection of lines 
and 
. Suppose that the circumcircle of 
meets at and 
, and the circumcircle of 
meets at and 
. Show that 
.
be a quadrilateral inscribed in circle 
with 
. Let 
and be the reflections of 
over lines 
and , respectively, and let be the intersection of lines 
and 
. Suppose that the circumcircle of 
meets at and 
, and the circumcircle of 
meets at and 
. Show that 
.This post has been edited 2 times. Last edited by Th3Numb3rThr33, Apr 18, 2018, 11:44 PM
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