# Morning Session ## Problem 1 Prove that there is no set of integers $m, n, p$ except $0,0,0$ for which $m$ $+n \sqrt{2}+p \sqrt{3}=0$. ## Problem 2 $A_1 A_2 \ldots A_n$ is a regular polygon inscribed in a circle of radius $r$ and center $O . P$ is a point on line $O A_1$ extended beyond $A_1$. Show that $\prod_{i=1}^n{\overline{P A_i}}=\overline{O P}^{\,n}-r^n.$ ## Problem 3 Suppose that $\sum_{i=1}^{\infty} x_i$ is a convergent series of positive terms which monotonically decrease (that is, $x_1 \geq x_2 \geq x_3 \geq \cdots$ ). Let $P$ denote the set of all numbers which are sums of some (finite or infinite) subseries of $\sum_{i=1}^{\infty} x_i$. Show that $P$ is an interval if and only if $x_n \leq \sum_{i=n+1}^{\infty} x_i \quad \text { for every integer } n.$ ## Problem 4 On a circle, $n$ points are selected and the chords joining them in pairs are drawn. Assuming that no three of these chords are concurrent (except at the endpoints), how many points of intersection are there? ## Problem 5 If a parabola is given in the plane, find a geometric construction (ruler and compass) for the focus. ## Problem 6 Find a necessary and sufficient condition on the positive integer $n$ that the equation $x^n+(2+x)^n+(2-x)^n=0$have a rational root. ## Problem 7 Consider the function $f$ defined by the differential equation $f^{\prime \prime}(x)=\left(x^3+a x\right) f(x)$ and the initial conditions $f(0)=1, f^{\prime}(0)=0$. Prove that the roots of $f$ are bounded above but unbounded below. # Afternoon Session ## Problem 1 A sphere rolls along two intersecting straight lines. Find the locus of its center. ## Problem 2 Suppose that $f$ is a function with two continuous derivatives and $f(0)=$ 0 . Prove that the function $g$, defined by $g(0)=f^{\prime}(0), g(x)=f(x)^{\prime} x$ for $x \neq$ 0 , has a continuous derivative. $\quad$ ## Problem 3 Prove that there exists no distance-preserving map of a spherical cap into the plane. (Distances on the sphere are to be measured along great circles on the surface.) ## Problem 4 Do there exist $1,000,000$ consecutive integers each of which contains a repeated prime factor? $\quad$ ## Problem 5 Given an infinite sequence of 0 's and 1 's and a fixed integer $k$, suppose that there are no more than $k$ distinct blocks of $k$ consecutive terms. Show that the sequence is eventually periodic. (For example, the sequence 11011010101 followed by alternating 0's and 1's indefinitely, which is periodic beginning with the fifth term.) ## Problem 6 Prove: If $f(x)>0$ for all $x$ and $f(x) \rightarrow 0$ as $x \rightarrow \infty$, then there exists at most a finite number of solutions of $ f(m)+f(n)+f(p)=1 $ in positive integers $m, n$, and $p$. ## Problem 7 Four forces acting on a body are in equilibrium. Prove that, if their lines of action are mutually skew, they are rulings of a hyperboloid.