. Prove that![$$ ab^2c (a^2 + b + c^2)\leq \frac{84589+12461\sqrt[3]{17}-8738\sqrt[3]{289}}{17496}$$](http://latex.artofproblemsolving.com/3/0/4/304d89aebd79434816624d25ecc61e22806177e6.png)
such that the following inequality holds:![\[
x^k y^k z^k (x^3 + y^3 + z^3) \leq 3
\]](http://latex.artofproblemsolving.com/2/3/0/230cc1b4378093eed4a28308584cc716337cd395.png)
for all positive real numbers 
satisfying the condition 
.
be a positive integer. We say a polynomial 
with integer coefficients is 
if there exists a polynomial 
of degree 
with integer coefficients such that 
is never divisible by 
.
be a scalene triangle with incenter 
.
at points 
,
to 
,
be the midpoint of 
.
and 
intersect the circumcircle of triangle 
and 
coincides with 
be the set of positive real numbers. Determine all functions 
and 
,
.
at 
,
.
and 
are tangent to each other.
be a cyclic quadrilateral inscribed in circle 
whose diagonals meet at 
and 
meet at 
.
intersects 
.
and 
and 
.
and 
and 
and 
meet at 
.
,
be the common point of the segment 
and the circle 
and 
.
at 
.
intersects 
,
and 
meet on 
such that 
divides 
for all positive integers 
and 
with 
.
is not a square.
.
be its circumcircle, 
its orthocenter, and 
such that 
and let 
.
,
,
and 
are tangent to each other.
be the set of real numbers. Determine all functions 
that satisfy the equation 
for all real numbers 
the set of all positive integers. Find all functions 
such that for all positive integers 
and 
is nonzero and divides 
.
be an odd prime number. Determine all pairs of polynomials 
and 
from ![$\mathbb{Z}[X]$](http://latex.artofproblemsolving.com/3/c/3/3c382e829ea7c95d18cfb978f22c8c5429d5c767.png)
such that
.
,
intersects 
and the line through 
,
is divisible by 
.
where 
satisfy 
.
equals 
,
.
Prove that
be positive real numbers such that
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
let 
the incenter of 
and 
Prove that 
are concyclic
such that 
,
and 
.
where 
is the intersection of 
.
Y by
For a finite set 
, let 
denote the number of elements in the set .
(a) Let
be the set of all functions
![\[ f : \{1, 2, \ldots, n\} \to \{1, 2, \ldots, k\} \quad \text{with } n \geq 3,\; k \geq 2 \]](//latex.artofproblemsolving.com/b/1/b/b1bf08934378882b9cd0b7d983f83b15453acca9.png)
satisfying the condition:
![\[ f(i) \ne f(i+1) \quad \text{for every } i,\; 1 \leq i \leq n-1. \]](//latex.artofproblemsolving.com/f/d/1/fd11b44d7fc482b5c6018e19161f6e30b7835c2d.png)
Show that
![\[ |F| = k(k-1)^{n-1}. \]](//latex.artofproblemsolving.com/a/8/8/a88a07984bad88a854de4dad8c24424aaad38365.png)
(b) Let
denote the number of functions in satisfying 
.
For
, show that
![\[ c(n, k) = k(k-1)^{n-1} - c(n-1, k). \]](//latex.artofproblemsolving.com/8/7/2/87285a27401b9853e72b45249196954082d9ec30.png)
(c) Using part (b), prove that for
,
![\[ c(n, k) = (k-1)^n - (-1)^n(k-1). \]](//latex.artofproblemsolving.com/2/f/c/2fcbcb385fcc1b3956232dadfd7ec4a31f4d71d6.png)
, let 
denote the number of elements in the set .(a) Let
be the set of all functions![\[ f : \{1, 2, \ldots, n\} \to \{1, 2, \ldots, k\} \quad \text{with } n \geq 3,\; k \geq 2 \]](http://latex.artofproblemsolving.com/b/1/b/b1bf08934378882b9cd0b7d983f83b15453acca9.png)
satisfying the condition:![\[ f(i) \ne f(i+1) \quad \text{for every } i,\; 1 \leq i \leq n-1. \]](http://latex.artofproblemsolving.com/f/d/1/fd11b44d7fc482b5c6018e19161f6e30b7835c2d.png)
Show that![\[ |F| = k(k-1)^{n-1}. \]](http://latex.artofproblemsolving.com/a/8/8/a88a07984bad88a854de4dad8c24424aaad38365.png)
(b) Let
denote the number of functions in satisfying 
.For
, show that![\[ c(n, k) = k(k-1)^{n-1} - c(n-1, k). \]](http://latex.artofproblemsolving.com/8/7/2/87285a27401b9853e72b45249196954082d9ec30.png)
(c) Using part (b), prove that for
,![\[ c(n, k) = (k-1)^n - (-1)^n(k-1). \]](http://latex.artofproblemsolving.com/2/f/c/2fcbcb385fcc1b3956232dadfd7ec4a31f4d71d6.png)
