# Morning Session ## Problem 1 (i) Show that a regular hexagon, six squares, and six equilateral triangles can be assembled without overlapping to form a regular dodecagon. (ii) Let $P_1, P_2, \ldots, P_{12}$ be the successive vertices of a regular dodecagon. Explain how the three diagonals $P_1 P_9, P_2 P_{11}$, and $P_4 P_{12}$ intersect. ## Problem 2 Let $\{f(n)\}$ be a strictly increasing sequence of positive integers such that $f(2)=2$ and $f(m n)=f(m) f(n)$ for every relatively prime pair of positive integers $m$ and $n$ (the greatest common divisor of $m$ and $n$ is equal to 1). Prove that $f(n)=n$ for every positive integer $n$. ## Problem 3 Find an integral formula for the solution of the differential equation $ \delta(\delta-1)(\delta-2) \cdots(\delta-n+1) y=f(x), \quad x \geq 1, $ for $y$ as a function of $x$ satisfying the initial conditions $y(1)=y^{\prime}(1)=\ldots=$ $y^{(n-1)}(1)=0$, where $f$ is continuous and $ \delta \equiv x \frac{d}{d x} $ ## Problem 4 Let $\left\{a_n\right\}$ be a sequence of positive real numbers. Show that $ \lim _{n \rightarrow \infty} \sup _n n\left(\frac{1+a_{n+1}}{a_n}-1\right) \geq 1 . $ Show that the number 1 on the right-hand side of this inequality cannot be replaced by any larger number. (The symbol lim sup is sometimes written (im.) ## Problem 5 (i) Prove that if a function $f$ is continuous on the closed interval $[0, \pi]$ and if $ \int_0^\pi f(\theta) \cos \theta d \theta=\int_0^\pi f(\theta) \sin \theta d \theta=0 $ then there exist points $\alpha$ and $\beta$ such that $ 0<\alpha<\beta<\pi \text {\quad and\quad} f(\alpha)=f(\beta)=0 . $ (ii) Let $R$ be any bounded convex open region in the Euclidean plane (that is, $R$ is a connected open set contained in some circular disk, and the line segment joining any two points of $R$ lies entirely in $R$ ). Prove with the help of part (i) that the centroid (center of gravity) of $R$ bisects at least three distinct chords of the boundary of $R$. ## Problem 6 Let $U$ and $V$ be any two distinct points on an ellipse, let $M$ be the midpoint of the chord $U V$, and let $A B$ and $C D$ be any two other chords through $M$. If the line $U V$ meets the line $A C$ in the point $P$ and the line $B D$ in the point $Q$, prove that $M$ is the midpoint of the segment $P Q$. # Afternoon Session ## Problem 1 For what integer $a$ does $x^2-x+a$ divide $x^{13}+x+90?$ ## Problem 2 Let $S$ be the set of all numbers of the form $2^m 3^n$, where $m$ and $n$ are integers, and let $P$ be the set of all positive real numbers. Is $S$ dense in $P$ ? ## Problem 3 Find every twice-differentiable real-valued function $f$ with domain the set of all real numbers and satisfying the functional equation$(f(x))^2-(f(y))^2=f(x+y) f(x-y)$for all real numbers $x$ and $y$. ## Problem 4 Let $C$ be a closed plane curve that has a continuously turning tangent and bounds a convex region. If $T$ is a triangle inscribed in $C$ with maximum perimeter, show that the normal to $C$ at each vertex of $T$ bisects the angle of $T$ at that vertex. If a triangle $T$ has the property just described, does it necessarily have maximum perimeter? What is the situation if $C$ is a circle? (A convex region is a connected open set such that the line segment joining any two points of the set lies entirely in the set.) ## Problem 5 Let $\left\{a_n\right\}$ be a sequence of real numbers satisfying the inequalities $0 \leq a_k \leq 100 a_n$ for $n \leq k \leq 2 n$ and $n=1,2, \ldots,$ and such that the series $\sum_{n=0}^{\infty} a_n$ converges. Prove that$\lim _{n \rightarrow \infty} n a_n=0 .$ ## Problem 6 Let $E$ be a Euclidean space of at most three dimensions. If $A$ is a nonempty subset of $E$, define $S(A)$ to be the set of all points that lie on closed segments joining pairs of points of $A$. For a given nonempty set $A_0$, define $A_n \equiv S\left(A_{n-1}\right)$ for $n=1,2, \ldots$. Prove that $A_2=A_3=\ldots$. (A one-point set should be considered to be a special case of a closed segment.)