AoPS Academy
be a fixed acute triangle inscribed in a circle 
with center 
. A variable point 
is chosen on minor arc 
of , and segments 
and meet at 
. Denote by 
and 
the circumcenters of triangles 
and 
, respectively. Determine all points for which the area of triangle 
is minimized.
denote the sum of the 
th powers of the roots of the polynomial 
. In particular, 
, 
, and 
. Let 
, 
, and 
be real numbers such that 
for 
, 
, 
What is 
?
if the real numbers 
, 
, 
, and 
satisfy the equations ![\[ax + by = 3^{}_{},\]](http://latex.artofproblemsolving.com/c/d/5/cd552c7ada95d98bc5f7df21d63d11b3c4fbaa64.png)
![\[ax^2 + by^2 = 7^{}_{},\]](http://latex.artofproblemsolving.com/4/a/a/4aa8b398fa738219f22d7c40be8ac8888969079a.png)
![\[ax^3 + by^3 = 16^{}_{},\]](http://latex.artofproblemsolving.com/a/e/9/ae92af789a106a02e74702b5b7bb0fce7fbf3525.png)
![\[ax^4 + by^4 = 42^{}_{}.\]](http://latex.artofproblemsolving.com/a/c/6/ac6ea536de1ad0908972237419f7531073e2cbca.png)
if the real numbers , , , and satisfy the equations 
denote the sum of the th powers of the roots of the polynomial . In particular, , , and . Let , , and be real numbers such that for , , What is ?
, then we have 
. Summing over all 3 roots, we get an equation of the desired form, so the answer is 
where 
are the roots of 
in terms of the coefficients, iirc
Is that correct? / Is that what Newton's Sums are?
and get it. But this way is SO MUCH BETTER: 
, so our answer is 
. Fun fact: This is LITERALLY the proof for newton's sums. Bro i'm just using this now instead of the formula lmao.
It’s just a direct plug into the formula.
![\[s_k= 5s_{k-1}-8s_{k-2}+13s_{k-3}\]](http://latex.artofproblemsolving.com/3/3/2/3320cf956be5c04a3438127bd96b3e0f963da287.png)
so the answer is 10.
,
be 
,
and 
.
Adding the equations,
This motivates the solution to the problem: create a recurrence by considering each polynomial root separately. Suppose we want to find 
Adding the equations together again, we get 
.
,
for 
.
.
,
,
.
i am so dumb