Putnam 1963

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 vertices of a regular dodecagon. Prove that the three diagonals $P_{1}P_{9}, P_{2}P_{11}$ and $P_{4}P_{12}$ intersect.

Let $f:\mathbb{N}\rightarrow \mathbb{N}$ be a strictly increasing function such that $f(2)=2$ and $f(mn)=f(m)f(n)$ for every pair of relatively prime positive integers $m$ and $n$ . Prove that $f(n)=n$ for every positive integer $n$ .

Find an integral formula for the solution of the differential equation $$\delta (\delta-1)(\delta-2) \cdots(\delta -n +1) y= f(x), \;\; x\geq 1,$$ for $y$ as a function of $f$ satisfying the initial conditions $y(1)=y'(1)=\ldots= y^{(n-1)}(1)=0$ , where $f$ is continuous and $\delta$ is the differential operator $x \frac{d}{dx}.$

Let $(a_n)$ be a sequence of positive real numbers. Show that $$\limsup_{n \to \infty} n \left(\frac{1 +a_{n+1}}{a_n } -1 \right) \geq 1$$ and prove that $1$ cannot be replaced by any larger number.

i) Prove that if a function $f$ is continuous on the closed interval $[0, \pi]$ and $$\int_{0}^{\pi} f(t) \cos t dt= \int_{0}^{\pi} f(t) \sin t dt=0,$$ then there exist points $0 < \alpha < \beta < \pi$ such that $f(\alpha) =f(\beta) =0.$ ii) Let $R$ be a bounded, convex, and open region in the Euclidean plane. Prove with the help of i) that the centroid of $R$ bisects at least three different chords of the boundary of $R.$

Let $U$ and $V$ be any two distinct points on an ellipse, let $M$ be the midpoint of the chord $UV$ , and let $AB$ and $CD$ be any two other chords through $M$ . If the line $UV$ meets the line $AC$ in the point $P$ and the line $BD$ in the point $Q$ , prove that $M$ is the midpoint of the segment $PQ.$

For what integers $a$ does $x^2 -x+a$ divide $x^{13}+ x +90$ ?

Let $S$ be the set of all numbers of the form $2^m 3^n$ , where $m$ and $n$ are integers. Is $S$ dense in the set of positive real numbers?

Find every twice-differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfies the functional equation $$f(x)^2 -f(y)^2 =f(x+y)f(x-y)$$ for all $x,y \in \mathbb{R}.$

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?

Let $(a_n )$ be a sequence of real numbers satisfying the inequalities $$0 \leq a_k \leq 100a_n \; \text{for} \; n \leq k \leq 2n \; \text{and} \; n=1,2,\ldots,$$ and such that the series $$\sum_{n=0}^{\infty} a_n$$ converges. Prove that $$\lim_{n\to \infty} n a_n = 0.$$

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 points that lie on closed segments joining pairs of points of $A$ (a one-point set should be considered to be a special case of a closed segment). For a given nonempty set $A_0$ , define $A_n =S(A_{n-1})$ for $n=1,2,\ldots$ Prove that $A_2 =A_3 =\ldots.$