[/quote]
,
denote the set of positive integers
such that for all 
it holds that 
be a positive integer. There are 
distinct points and no 
circles pass through the same point. Turbo the snail slides along the circles in the following manner, leaving snail goo behind. Initially he moves on one of the circles in clockwise direction. He keeps sliding along until he reaches an intersection with another circle. Then, he continues his journey on this new circle and also changes the direction he is moving in. We define a snail orbit to be the covering of the whole surface of a circle with turbo's goo, and specifically only a single layer of it. Prove that for every odd 
such that for any two infinite sequences of positive integers 
and 
,
for all 
,
such that 
for all 
.
and 
be two isogonal conjugate with respect to triangle 
.
and 
be two points lying on the circle 
such that 
and 
are perpendicular and parallel to bisector of 
,
and 
bisect two arcs 
containing 
and not containing 
.
and 
be positive integers. Prove that there exists a positive integer 
such that for every odd integer 
,
are all greater than 
and 
.
.
be the orthocenter of acute triangle 
be the foot of the altitude from 
to 
,
intersects line 
and 
.
.
Y by
Every other thread on this forum is for USA(J)MO lol.
Anyways, to other USAPhO students, what are you doing to prepare? It seems too close to the test date (April 10) to learn new content, so I am just going through past USAPhO and BPhO exams to practice (untimed for now). How about you? Any predictions for what will be on the test this year? I'm completely cooked if there are any circuitry questions.
Anyways, to other USAPhO students, what are you doing to prepare? It seems too close to the test date (April 10) to learn new content, so I am just going through past USAPhO and BPhO exams to practice (untimed for now). How about you? Any predictions for what will be on the test this year? I'm completely cooked if there are any circuitry questions.
