be a triangle with circumcircle 
.
on 
respectively. Circumcircle 
of triangle 
intersects 
.
intersects 
at 
.
cuts 
respectively. Construct parallelogram 
.
are collinear.
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)
by residue theorem? Should by something like![\[[z^n]\tan z\propto\oint _{|z|=N}\frac{\tan z}{z^{n+1}}\mathrm dz\]](http://latex.artofproblemsolving.com/8/5/c/85c34140e3cc61d93bd3188c22ae06649ca98c5a.png)
where 
tends to infty, but I'm not sure about details.
![\[\iint_{R} \sin(xy) \,dx\,dy, \quad R = \left[0, \frac{\pi}{2}\right] \times \left[0, \frac{\pi}{2}\right]\]](http://latex.artofproblemsolving.com/a/2/0/a205868dd99f4794de53ad1db50f34b17c70a923.png)
be a group, 
a subgroup of 
and 
an endomorphism with the following property:
such that for any 
there exist 
with 
and 
,
.
is abelian.
is a subgroup of 
such that there exists an integer polynomial 
with no integer roots and a constant 
such that
has at least 
distinct prime divisors for all positive integers 
is an integer, then there exist invertible matrices 
with non-zero entries such that:![\[A_1^{-1} + A_2^{-1} + \ldots + A_n^{-1} = (A_1 + A_2 + \ldots + A_n)^{-1}.\]](http://latex.artofproblemsolving.com/a/f/9/af9ff29771952abe37a00f3830c72e8c517e1cd6.png)
topic in College Math :coolspeak:
be a closed and countable set. Prove that if the analytic function 
is bounded, then 
is constant.
Y by MathsMadman, mathematicsy, cubres
Players A and B play a game. They are given a box with 
candies. A starts first. On a move, if in the box there are 
candies, the player chooses positive integer 
so that 
and 
, and eats candies from the box. The player who eats the last candy wins. Who has winning strategy, in terms of 
.
candies. A starts first. On a move, if in the box there are 
candies, the player chooses positive integer 
so that 
and 
, and eats candies from the box. The player who eats the last candy wins. Who has winning strategy, in terms of 
.This post has been edited 2 times. Last edited by VicKmath7, Mar 11, 2020, 11:19 AM
candies and eat them so he wins(except for 
where he obviously wins).for even 
.
is odd number.
candles remained.Second eats 1 and first eats the last one and wins.
is even.
candles and wins...
,
they simply take 
