PL-OM 1985

Find the largest $k$ such that for every positive integer $n$ we can find at least $k$ numbers in the set $\{n+1, n+2, ... , n+16\}$ which are coprime with $n(n+17)$ .

Given a square side $1$ and $2n$ positive reals $a_1, b_1, ... , a_n, b_n$ each $\le 1$ and satisfying $\sum a_ib_i \ge 100$ . Show that the square can be covered with rectangles $R_i$ with sides length $(a_i, b_i)$ parallel to the square sides.

The function $f : R \to R$ satisfies $f(3x) = 3f(x) - 4f(x)^3$ for all real $x$ and is continuous at $x = 0$ . Show that $|f(x)| \le 1$ for all $x$ .

$P$ is a point inside the triangle $ABC$ is a triangle. The distance of $P$ from the lines $BC, CA, AB$ is $d_a, d_b, d_c$ respectively. If $r$ is the inradius, show that $$\frac{2}{ \frac{1}{d_a} + \frac{1}{d_b} + \frac{1}{d_c}} < r < \frac{d_a + d_b + d_c}{2}$$

$p(x,y)$ is a polynomial such that $p(cos t, sin t) = 0$ for all real $t$ . Show that there is a polynomial $q(x,y)$ such that $p(x,y) = (x^2 + y^2 - 1) q(x,y)$ .

There is a convex polyhedron with $k$ faces. Show that if more than $k/2$ of the faces are such that no two have a common edge, then the polyhedron cannot have an inscribed sphere.