# Morning Session ## Problem 1 Answer either `(i)` or `(ii)`: > [!i] Three straight lines pass through the three points $(0,-a, a),(a, 0$, $-a$ ), and $(-a, a, 0)$, parallel to the $x$-axis, $y$-axis, and $z$-axis, respectively; $a$ gt;0$. A variable straight line moves so that it has one point in common with each of the three given straight lines. Find the equation of the surface described by the variable line. > [!i] Which planes cut the surface $x y+x z+y z=0$ in (1) circles, (2) parabolas? ## Problem 2 We consider three vectors drawn from the same initial point $O$, of lengths $a, b$, and $c$, respectively. Let $E$ be the parallelepiped with vertex $O$ of which the given vectors are the edges and $H$ the parallelepiped with vertex $O$ of which these vectors are the altitudes. Show that the product of the volumes of $E$ and $H$ equals $(a b c)^2$, and generalize the theorem, with proof, to $n$ dimensions. ## Problem 3 Assume that the complex numbers $a_1, a_2, \ldots, a_n, \ldots$ are all different from zero, and that $\left|a_r-a_s\right|>1$ for $r \neq s$. Show that the series $\sum_{n=1}^{\infty} \frac{1}{a_n{ }^3}$ converges. ## Problem 4 Given that $P$ is a point inside a tetrahedron with vertices at $A, B, C$, and $D$, such that the sum of the distances $P A+P B+P C+P D$ is a minimum, show that the two angles $\angle A P B$ and $\angle C P D$ are equal and are bisected by the same straight line. What other pairs of angles must be equal? ## Problem 5 How many roots of the equation $z^6+6 z+10=0$ lie in each quadrant of the complex plane? ## Problem 6 Prove that for every real or complex $\boldsymbol{x}$ $ \prod_{k=1}^{\infty} \frac{1+2 \cos \frac{2 x}{3^k}}{3}=\frac{\sin x}{x} . $ # Afternoon Session ## Problem 1 Each rational number $p / q(p, q$ relatively prime positive integers) of the open interval $(0,1)$ is covered by a closed interval of length $1 / 2 q^2$, whose center is at $p / q$. Prove that $\sqrt{2} / 2$ is not covered by any of the above closed intervals. ## Problem 2 Answer either `(i)` or `(ii)`: > [!i] Prove that $\sum_{n=2}^{\infty} \frac{\cos (\log \log n)}{\log n}$ diverges. > [!ii] Assume that $p>0, a>0$, and $a c-b^2>0$, and show that $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{d x d y}{\left(p+a x^2+2 b x y+c y^2\right)^2}=\pi p^{-1}\left(a c-b^2\right)^{-\frac12}$ ## Problem 3 Let $K$ be a closed plane curve such that the distance between any two points of $K$ is always less than 1 . Show that $K$ lies inside a circle of radius $1 / \sqrt{3}$. ## Problem 4 Show that the coefficients $a_1, a_2, a_3, \ldots$ in the expansion $\frac{1}{4}[1+x-$ $\left.\left(1-6 x+x^2\right)^{12}\right]=a_1 x+a_2 x^2+a_3 x^3+\cdots$ are positive integers. ## Problem 5 Let $a_1, a_2, \ldots, a_n, \ldots$ be an arbitrary sequence of positive numbers. Show that $\lim _{n \rightarrow \infty} \sup \left(\frac{a_1+a_{n+1}}{a_n}\right)^n \geq e .$ ## Problem 6 Let $C$ be a closed convex curve with a continuously turning tangent and let $O$ be a point inside $C$. With each point $P$ on $C$ we associate the point $T(P)$ on $C$ which is defined as follows: Draw the tangent to $C$ at $P$ and from $O$ drop the perpendicular to that tangent. $T(P)$ is then the point at which this perpendicular intersects the curve $C$. Starting now with a point $P_0$ on $C$, we define points $P_n$ by the formula $P_n=$ $T\left(P_{n-1}\right), n \geq 1$. Prove that the points $P_n$ approach a limit, and characterize those points which can be limits of sequences $P_n$. (You may consider the facts that $T$ is a continuous transformation and that a convex curve lies on one side of each of its tangents as not requiring proofs.)