[/quote]
,
be a triangle with 
and 
.
and 
be the midpoints of 
and 
respectively. The point 
is inside of 
such that 
is equilateral. Let 
and 
.
.
obtuse. Let 
,
,
respectively. Let 
,
,
be arbitrary points on 
,
respectively. The circles with diameter 
,
,
are drawn. Show that the lengths of the tangents from the orthocentre of 
is inscribed in a circle. Given 
and 
.
and 
be the orthocentre and incentre of 
to 
.
is tangent to the sides 
of a triangle 
respectively. The lines 
and 
meet 
at points 
and 
.
and points 
.
,
.
,
lines intersect at point 
.
,
be a cyclic quadrilateral with circumcenter 
is not a diameter of its circumcircle. The lines 
and 
,
is acute and let 
and 
.
,
,
,
be the centers of the circles circumscribing triangles ADE, BDF, CEF. Prove that the orthocenter of triangle 
.
.
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)
Y by Danielzh, BS2012, EpicBird08, bachkieu
I got an email about new FedEx shipment earlier today. I never ordered anything and was pretty confused but it caught my interest because it shipped out of Elgin, IL, which is only ~15 miles from the place where MOP is taking place and was shipped directly to my name and the email I signed up to AMCs with (which I don't use for much other things).
This is a very stupid question and it might be a coincidence but did any other AoPSers waiting on MOP email receive this ;-;
This is a very stupid question and it might be a coincidence but did any other AoPSers waiting on MOP email receive this ;-;
This post has been edited 1 time. Last edited by Mathandski, Apr 18, 2025, 9:00 PM
