Y by Adventure10
O -circumcenter, H - ortocenter and (OD) perpendicular (DE).
Prove that angle(DHE)=angle(ABC).
Prove that angle(DHE)=angle(ABC).
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Darboux cubic
srirampanchapakesan 1
N
by srirampanchapakesan
Source: Own
Let P be a point on the Darboux cubic (or the McCay Cubic ) of triangle ABC.
P1P2P3 is the circumcevian or pedal triangle of P wrt ABC.
Prove that P also lie on the Darboux cubic ( or the McCay Cubic) of P1P2P3 .
P1P2P3 is the circumcevian or pedal triangle of P wrt ABC.
Prove that P also lie on the Darboux cubic ( or the McCay Cubic) of P1P2P3 .
1 reply
9 ARML Location
deduck 42
N
by llddmmtt1
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!!! :D
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!!! :D
Current List:
42 replies
An FE. Who woulda thunk it?
nikenissan 117
N
by EpicBird08
Source: 2021 USAJMO Problem 1
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)
117 replies
9 Will I make JMO?
EaZ_Shadow 18
N
by mathkidAP
will I be able to make it... will the cutoffs will be pre-2024
18 replies
Past USAMO Medals
sdpandit 1
N
by CatCatHead
Does anyone know where to find lists of USAMO medalists from past years? I can find the 2025 list on their website, but they don't seem to keep lists from previous years and I can't find it anywhere else. Thanks!
1 reply
usamOOK geometry
KevinYang2.71 106
N
by jasperE3
Source: USAMO 2025/4, USAJMO 2025/5
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 
.
106 replies
Geo #3 EQuals FReak out
Th3Numb3rThr33 106
N
by BS2012
Source: 2018 USAJMO #3
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 
.
106 replies
Aime ll 2022 problem 5
Rook567 4
N
by Rook567
I don’t understand the solution. I got 220 as answer. Why does it insist, for example two primes must add to the third, when you can take 2,19,19 or 2,7,11 which for drawing purposes is equivalent to 1,1,2 and 2,7,9?
4 replies
USAJMO problem 2: Side lengths of an acute triangle
BOGTRO 59
N
by ostriches88
Source: Also USAMO problem 1
Find all integers 
such that among any 
positive real numbers 
with 
, there exist three that are the side lengths of an acute triangle.
such that among any 
positive real numbers 
with 
, there exist three that are the side lengths of an acute triangle.
59 replies
high tech FE as J1?!
imagien_bad 60
N
by SimplisticFormulas
Source: USAJMO 2025/1
Let 
be the set of integers, and let 
be a function. Prove that there are infinitely many integers 
such that the function 
defined by 
is not bijective.
Note: A function is bijective if for every integer 
, there exists exactly one integer such that 
.
be the set of integers, and let 
be a function. Prove that there are infinitely many integers 
such that the function 
defined by 
is not bijective.Note: A function
is bijective if for every integer 
, there exists exactly one integer such that 
.
60 replies
ANGLE EQUAL
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are two chords of a circle and 
,
,
,
if and only if 
.
intersect the circumcircle of 
again in 
.
,
intersects the circumcircle in 
,
,
)
.
is a parallelogramm, and consequently
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Y by PrimeSol, Adventure10
Dear Mathlinkers,
if you like butterflies, you can see
http://jl.ayme.pagesperso-orange.fr/Docs/Papillon.pdf
Sincerely
Jean-Louis
if you like butterflies, you can see
http://jl.ayme.pagesperso-orange.fr/Docs/Papillon.pdf
Sincerely
Jean-Louis
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K
Y
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Y by Adventure10, Mango247
Dear Mathlinkers,
just an interesting link with
http://www.artofproblemsolving.com/community/c6t48f6h1283140_two_perpendiculars
Sincerely
Jean-Louis
just an interesting link with
http://www.artofproblemsolving.com/community/c6t48f6h1283140_two_perpendiculars
Sincerely
Jean-Louis
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