[/quote]
,
such that
,
outside the circle, draw tangents 
,
is between 
)
intersects 
at 
;
is midpoint of 
.
,
.
intersects 
at 
.
lie inside the triangle 
such that: 
=
.
,
,
on circle 
is a isosceles triangle
,
-
of positive integers with 
satisfy 
mod 
be a point on the side 
and 
is the circumcircle of 
,
be the point on 
is a diameter of 
the circumcircle of triangle 
,
be the point such that 
is a diameter of 
and 
and 
Prove that
Let 
and 
Prove that
Let 
and 
Prove that![$$a + b \geq \frac{1+\sqrt[3]{28-3\sqrt {87}}+\sqrt[3]{28+3\sqrt {87}}}{3}$$](http://latex.artofproblemsolving.com/4/f/9/4f9529f5dec193ec87a99134d9628ae59078eed7.png)
with 
.![\[
\frac{a^5}{b^2 + 2c^3} + \frac{2b^5}{3c + a^6} + \frac{c^7}{a + b^4} \geq 2.
\]](http://latex.artofproblemsolving.com/f/f/5/ff5043705025592aeefc55a764a8b7170e13ff4b.png)
such that for all positive real numbers 
,![\[
f^2(x) \geq f(x+y)\left(f(x) + y\right).
\]](http://latex.artofproblemsolving.com/7/1/f/71fb0d51492ee4c3b2bf4049e8224882fec16020.png)
is called a good prime if there exists a positive integer 
such that![\[
n^2 + 1 \text{ is divisible by } p^{2025}.
\]](http://latex.artofproblemsolving.com/3/3/4/334cd7b5e4a4bfb6b0dd3a4607cdd793ecadd66c.png)
Prove that there are infinitely many good primes.
.
represented by tokens to players 
to 
so that they all have the same number of tokens in their urns.
beats 
when, when 
.
What is 
and 
Prove that
........
?
,
are complex numbers with absolute value 
define 
.
converges. Show that,
.
Y by
Q) let 
and 
then is this uniformly convergence on 
comment on uniformly convergence on ![$[0,1]$](//latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
where in general it is should be uniformly convergence.
My I am trying with some contradicton method like chose
and trying to solve
Next take a in (0,1) and chose a= 2^1/N but not solution
How to solve like this way help.
and 
then is this uniformly convergence on 
comment on uniformly convergence on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
where in general it is should be uniformly convergence.My I am trying with some contradicton method like chose
and trying to solve
Next take a in (0,1) and chose a= 2^1/N but not solution
How to solve like this way help.
.![$[0,a]$](http://latex.artofproblemsolving.com/6/8/b/68bacdc4b96052586744d16169ccd8c3233ed09f.png)
for any 
because 
.
