SE-SKOLORNAS 1978

Let $a,b,c,d$ be real numbers such that $a>b>c>d\geq 0$ and $a + d = b + c$ . Show that $$x^a + x^d \geq x^b + x^c$$ for $x>0$ .

Let $s_m$ be the number $66\cdots 6$ with $m$ digits $6$ . Find $$s_1 + s_2 + \cdots + s_n$$

Two satellites are orbiting the earth in the equatorial plane at an altitude $h$ above the surface. The distance between the satellites is always $d$ , the diameter of the earth. For which $h$ is there always a point on the equator at which the two satellites subtend an angle of $90^\circ$ ?

$b_0, b_1, b_2, \dots$ is a sequence of positive reals such that the sequence $b_0,c b_1, c^2b_2,c^3b_3,\dots$ is convex for all $c > 0$ . (A sequence is convex if each term is at most the arithmetic mean of its two neighbors.) Show that $\ln b_0, \ln b_1, \ln b_2, \dots$ is convex.

$k > 1$ is fixed. Show that for $n$ sufficiently large for every partition of $\{1,2,\dots,n\}$ into $k$ disjoint subsets we can find $a \neq b$ such that $a$ and $b$ are in the same subset and $a+1$ and $b+1$ are in the same subset. What is the smallest $n$ for which this is true?

$p(x)$ is a polynomial of degree $n$ with leading coefficient $c$ , and $q(x)$ is a polynomial of degree $m$ with leading coefficient $c$ , such that $$p(x)^2 = \left(x^2 - 1\right)q(x)^2 + 1$$ Show that $p'(x) = nq(x)$ .