Let $k$ be a real number. Define on the set of reals the operation $x*y$= $\frac{xy}{x+y+k}$ whenever $x+y$ does not equal $-k$. Let $x_1<x_2<x_3<x_4$ be the roots of $t^4=27(t^2+t+1)$.suppose that $[(x_1*x_2)*x_3]*x_4=1$. Find all possible values of $k$

Let $D$ be a point in the interior of $ABC$. Let $BD$ and $AC$ intersect at $E$ while $CD$ and $AB$ intersect at $F$. Let $EF$ intersect $BC$ at $G$. Let $H$ be an arbitrary point on $AD$. Let $HF$ and $BD$ intersect at $I$. Let $HE$ and $CD$ intersect at $J$ . prove that $G$,$I$ and $J$ are collinear

$N$ straight lines are drawn on a plane. The $N$ lines can be partitioned into set of lines such that if a line $l$ belongs to a partition set then all lines parallel to $l$ make up the rest of that set. For each $n>=1$,let $a_n$ denote the number of partition sets of size $n$. Now that $N$ lines intersect at certain points on the plane. For each $n>=2$ let $b_n$ denote the number of points that are intersection of exactly $n$ lines. Show that $\sum_{n>= 2}(a_n+b_n)$$\binom{n}{2}$ $=$ $\binom{N}{2}$

Let $N>= 2$ be an integer. Show that $4n(N-n)+1$ is never a perfect square for each natural number $n$ less than $N$ if and only if $N^2+1$ is prime.

Given that $f(x)=(3+2x)^3(4-x)^4$ on the interval $\frac{-3}{2}<x<4$. Find the a. Maximum value of $f(x)$ b. The value of $x$ that gives the maximum in (a).