AT-OEMO-FED1 2011

Determine all integer triplets $(x,y,z)$ such that $$x^4+x^2=7^zy^2\text{.}$$

For a positive integer $k$ and real numbers $x$ and $y$ , let $$f_k(x,y)=(x+y)-\left(x^{2k+1}+y^{2k+1}\right)\text{.}$$ If $x^2+y^2=1$ , then determine the maximal possible value $c_k$ of $f_k(x,y)$ .

A set of three elements is called arithmetic if one of its elements is the arithmetic mean of the other two. Likewise, a set of three elements is called harmonic if one of its elements is the harmonic mean of the other two.

How many three-element subsets of the set of integers $\left\{z\in\mathbb{Z}\mid -2011<z<2011\right\}$ are arithmetic and harmonic?

(Remark: The arithmetic mean $A(a,b)$ and the harmonic mean $H(a,b)$ are defined as $$A(a,b)=\frac{a+b}{2} \text{and} H(a,b)=\frac{2ab}{a+b}=\frac{2}{\frac{1}{a}+\frac{1}{b}}\text{,}$$ respectively, where $H(a,b)$ is not defined for some $a$ , $b$ .)

Inside or on the faces of a tetrahedron with five edges of length $2$ and one edge of lenght $1$ , there is a point $P$ having distances $a, b, c, d$ to the four faces of the tetrahedron. Determine the locus of all points $P$ such that $a+b+c+d$ is minimal and the locus of all points $P$ such that $a+b+c+d$ is maximal.