[/quote]
,
be a positive integer. Find the number of reduced fractions 
for which 
,
denotes the fractional part of 
.
,
,
be the centers of the circles circumscribing triangles ADE, BDF, CEF. Prove that the orthocenter of triangle 
is tangent to sides 
and 
at 
and 
respectively, and 
is the circumcenter of 
.
.
denote the maximum number of 
-
board such that each 
such that![\[
f(N) \leq cN^2
\]](http://latex.artofproblemsolving.com/7/4/8/748109c3d63203f9a67a167434e10a909f38e83f.png)
for all positive integers 
.
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
such that for all positive real numbers 
and 
,![\[
\frac{x + f(y)}{x f(y)} = f\left( \frac{1}{y} + f\left( \frac{1}{x} \right) \right)
\]](http://latex.artofproblemsolving.com/7/d/f/7df25755bc33bef970fffb392fb5b6b789390881.png)
such that for all 
and 
,
such that 
or 
,![\[
1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p
\]](http://latex.artofproblemsolving.com/6/8/e/68e3d73edec43b2cc480832adf23bbbb5f4c7bee.png)
such that![\[
\prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}.
\]](http://latex.artofproblemsolving.com/2/9/e/29edf6ab8cee2a526e7753a140c5a291e987856a.png)
be distinct real numbers such that![\[
a + b + c + \frac{1}{abc} = \frac{19}{2}
\]](http://latex.artofproblemsolving.com/5/1/e/51eef77358db2b6c98f0fb0fb4e30db837a03642.png)
Find the maximum possible value of 
.
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)
be the orthocenter of acute triangle 
be the foot of the altitude from 
to 
be the reflection of 
.
intersects line 
and 
.
.
Y by
I am hoping to participate in the USAMTS this fall, and I wanted to know of any resources to help me study for it, as I can't really find anything online about preparing for USAMTS. I have no experience in proofs (except two column lol), so I want some resources for that. I also looked at some of the problems and they looked really hard and heard the problems are AIME/USA(J)MO level, and that is nowhere near my level. For reference, when I mocked the 2024 amc 10b, I got an 81 (but I sillied number one). I do plan to finish the intro series this year (assuming I don't procrastinate), so maybe that will help. Also, are the puzzles for question one difficult? I know you don't have to provide a proof for those questions.
