[/quote]
,
prove that
such that for all naturals 
the function satisfies:![\[a + b \mid a^{f(a)} + b^{f(b)} \]](http://latex.artofproblemsolving.com/e/b/b/ebb44041cedd29de516192d983a334d46e7fe7d6.png)
Bob wants to find all possible functions Alice could have. Help Bob and find all functions that Alice could have.
-
be the medial triangle of an acute-angle triangle 
.
perpendicular to 
meet 
at 
.
analogously. Let 
.
and 
.
meet the 
,
and 
similarly. Let 
be the circumcenter of 
be the circle with diameter 
,
at 
.
and 
.
are collinear.
lie on a circle centered at 
are coaxial.
denote the set of all real numbers. Find all functions 
such that
for all real numbers 
,
.
![\[x^2-cx+1 = \dfrac{f(x)}{g(x)},\]](http://latex.artofproblemsolving.com/1/3/3/1339acdaf1999e06e1a4b7dec7d1217389e2afef.png)
where 
and 
are polynomials with nonnegative real coefficients. For each 
,
exist.
such that every collection of 
.
,
are in that order and 
and 
.
on line 
,
on line 
and 
are equidistant from the circumcenter of 
be six positive real numbers. Prove that![\[a^2+b^2+c^2+\frac{4(ax+by+cz)\sqrt{ab+bc+ca}}{x+y+z} \geqslant 2(ab+bc+ca).\]](http://latex.artofproblemsolving.com/f/8/0/f808f7857c2150960f2a430837286487666797d8.png)
be a cyclic quadrilateral with circumcentre 
and circumcircle 
.
be the intersection of tangents at 
and 
to 
be the circumcircle of triangle 
and let 
,
denote the second intersections of 
with 
and 
and 
be the circumcircle of triangle 
.
,
is also tangent to 
.
be the midpoint of 
be the intersection of the tangents from 
.
intersects 
.
and 
meet at 
,
and 
meet at 
.
Y by
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Hi everyone,
For an AIME score of 7-11, would you recommend MathPath or AMSP Level 2/3?
Thanks in advance!
Also people who have gone to them, please tell me more about the programs!
For an AIME score of 7-11, would you recommend MathPath or AMSP Level 2/3?
Thanks in advance!
Also people who have gone to them, please tell me more about the programs!
This post has been edited 1 time. Last edited by FuturePanda, Jan 30, 2025, 3:16 AM
