PUTNAM-84-A1

(Geometry) Let A be a solid $a\times b\times c$ rectangular brick in three dimensions, where a,b,c>0. Let B be the set of all points which are a distance at most one from some point of A (in particular, B contains A). Express the volume of B as a polynomial in a, b,and c.

PUTNAM-83-B1

(Geometry) Let v be a vertex (corner) of a cube C with edges of length 4. Let S be the largest sphere that can be inscribed in C. Let R be the region consisting of all points p between S and C such that pis closer to v than to any other vertex of the cube. Find the volume of R.

PUTNAM-91-A1

(Geometry) A $2\times 3$ rectangle has vertices at (0,0), (2,0), (0,3), and (2,3). It rotates $90^{\circ}$clockwise about the point (2,0). It then rotates $90^{\circ}$ clockwise about the point (5,0) then $90^{\circ}$ about the point (7,0), and finally, $90^{\circ}$ clockwise about the point (10,0). (the side originally on the x-axis is not back on the x-axis.) Find the area of the region above the x-axis and below the curve traced out by the point whose initial position is (1,1).

PUTNAM-88-A1

(Geometry) Let R be the region consisting of the points (x,y)of the cartesian plane satisfying both $\vert x\vert-\vert y\vert\leq 1$and $\vert y\vert\leq 1$. Sketch the region R and find its area.

PUTNAM-79-B1

(Geometry) Prove or disprove: there is at least one straight line normal to the graph of $y=\cosh x$ at a point $(a, \cosh a)$ and also normal to the graph of $y=\sinh x$ at a point $(c,\sinh c)$.

[At a point on a graph, the normal line is the perpendicular to the tangent at that point. Also, $\cosh x=(e^x + e^{-x})/2$ and $\sinh x=(e^x-e^{-x})/ 2$.]

PUTNAM-91-B1

(Number Theory) For each integer $n\geq 0$, let S(n) = n-m², where m is the greatest integer with $m^2\leq n$. Define a sequence $(a_k)_{k=0}^{\infty}$ by a₀=A and a_k+1=a_k+S(a_k) for $k\geq 0$. For what positive integers A is this sequence eventually constant?

PUTNAM-74-A1

(Number Theory) Call a set of positive integers ``conspiratorial'' if no three of them are pairwise relatively prime. (A set of integers is ``pairwise relatively prime'' if no pair of them has a common divisor greater than 1.) What is the largest number of elements in any ``conspiratorial'' subset of the integers 1 through 16?

PUTNAM-83-A1

(Number Theory) How many positive integers n are there such that n is an exact divisor of at least one of the numbers

$$10^{40},\ 20^{30}?$$

PUTNAM-77-A1

(Polynomials) Consider all lines which meet the graph of

y-2x⁴+7x³+3x-5

in four distinct points, say (x_i, y_i), i=1,2,3,4. Show that $\dfrac{x_1+x_2+x_3+x_4}{4}$is independent of the line and find its value.

PUTNAM-87-A1

(Polynomials) Curves A, B, C, and D are defined in the plane as follows:
$$\begin{align*}A&=\left\{ (x,y): x^2-y^2= \frac{x}{x^2 +y^2} \right\},\\ B&=\lef... ...-3xy^2+3y=1 \right\},\\ D&=\left\{ (x,y): 3x^2y-3x-y^3=0 \right\}. \end{align*}$$
Prove that $A \cap B = C \cap D.$

PUTNAM-84-B1

(Recurrence) Let n be a positive integer, and define

$$f(n)=1!+2!+\dots+n!.$$

Find polynomials P(x) and Q(x) such that

f(n+2)=P(n)f(n+1)+Q(n)f(n),

for all $n\geq 1$.