AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
such that for all irrational numbers 
and 
,
compute the smallest possible value of ![\[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\]](http://latex.artofproblemsolving.com/5/6/2/562c27c7779285b6439c4087f52933a41f29ee2e.png)
over all permutations 
of 
with incenter 
, the incircle 
touches sides 
and 
at points 
and 
, respectively. A circle passing through 
and 
touches at point 
. The circumcircle of 
meets 
at 
. Prove that 
is parallel to 
.
with incircle . Let 
be the point on such that the circumcircle of 
is tangent to and let the 
-excircle be tangent to at 
. Prove that the tangent from to and the tangent from to (distinct from ) meet on the line parallel to and passing through .
and 
Prove that
Let and 
Prove that
Let and 
Prove that
of positive integers such that ![\[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]](http://latex.artofproblemsolving.com/3/5/3/353bbb15bac205053c26208af8cac7c32a296f55.png)
Proposed by Gabriel Chicas Reyes, El Salvador
satisfying ![\[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]](http://latex.artofproblemsolving.com/2/6/e/26edbd87f49eb68ec9b662b602db712e5151dc9c.png)
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.
where 
and 
.
satisfy that 
.![$$\sum_{cyc} \sqrt[4]{\frac{a^3}{b+c}}$$](http://latex.artofproblemsolving.com/e/f/0/ef089c4f2915bbf75b0f60b044bbe9273b301a81.png)
,
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