[/quote]
,
a point in the space and 
the centroid of a tetrahedron 
.
be a 
on it, line 
makes a fixed angle with the tangent to 
be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.
on the spiral, the polar of 
be a monic polynomial so that there exists another real coefficients 
that satisfy![\[P(x^2-2)=P(x)Q(x)\]](http://latex.artofproblemsolving.com/6/9/7/697faa929e4fda7a6e9b1cd97849bd42ffc14306.png)
Determine all complex roots that are possible from 
with exactly 9 elements that is subset of 
.
and 
that satisfy the following:
is an empty subset.
is 
and 
.
to 
is 
and 
.
and 
is 
is 
is tangent to the 
and 
.
and 
intersect at 
.
be the midpoints of sides 
,
be points on segment 
and 
,
is the angle bisector of 
and 
is the angle bisector of 
.
is parallel to 
if and only if 
divides 
(i dont remember exactly if it was 8032038 or some other number, so correct me on this)
(i posted this at 12:29) lol
?
for which, for all 
,![\[
f(x) f(yf(x)) + y f(xy) = \frac{f\left(\frac{x}{y}\right)}{y} + \frac{f\left(\frac{y}{x}\right)}{x}
\]](http://latex.artofproblemsolving.com/9/0/c/90c180110402e1a32b70edb2b0a03a28727457d1.png)
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
,![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
Y by
Hii I am looking for a USAPhO tutor for next year's season. I think I have tried literally everything possible to improve but I feel like I just hit a massive roadblock right now.
It would be ideal if I can find someone who have a lot of experience with physics olympiads. My goal is medal/gold in usapho next year, and I am very determined & willing to put in a lot of hours, especially more so in the summer. Please recommend anyone or dm in aops, thank you.
Have qualified usapho before (last year), took both physics c and sufficient higher math.
It would be ideal if I can find someone who have a lot of experience with physics olympiads. My goal is medal/gold in usapho next year, and I am very determined & willing to put in a lot of hours, especially more so in the summer. Please recommend anyone or dm in aops, thank you.
Have qualified usapho before (last year), took both physics c and sufficient higher math.
This post has been edited 1 time. Last edited by physicsplease, Apr 11, 2025, 7:47 AM
