1981 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 1981 B6

C

be a fixed unit circle in the cartesian plane. For any convex polygon

P

, each of whose sides is tangent to

C

N( P, h, k)

be the number of points common to

P

and the unit circle with center at

(h, k).

H(P)

be the region of all points

(x, y)

N(P, x, y) \geq 1

F(P)

H(P).

Find the smallest number

u

\frac{1}{F(P)} \int \int N(P,x,y)\;dx \;dy <u

for all polygons

P

, where the double integral is taken over

H(P).

1

Putnam 1981 B5

B(n)

be the number of ones in the base two expression for the positive integer

n.

Determine whether

\exp \left( \sum_{n=1}^{\infty} \frac{ B(n)}{n(n+1)} \right)

is a rational number.

1

Putnam 1981 B4

V

5\times7

matrices, with real entries and closed under addition and scalar multiplication. Prove or disprove the following assertion: If

V

contains matrices of ranks

0, 1, 2, 4,

5

, then it also contains a matrix of rank

3

1

Putnam 1981 B2

Determine the minimum value of

(r-1)^2 + \left(\frac{s}{r}-1 \right)^2 + \left(\frac{t}{s}-1 \right)^{2} + \left( \frac{4}{t} -1 \right)^2

for all real numbers

1\leq r \leq s \leq t \leq 4.

1

Putnam 1981 B1

\lim_{n\to \infty} \frac{1}{n^5 } \sum_{h=1}^{n} \sum_{k=1}^{n} (5h^4 -18h^2 k^2 +5k^4).

1

Putnam 1981 A6

Suppose that each of the vertices of

ABC

is a lattice point in the

xy

-plane and that there is exactly one lattice point

P

in the interior of the triangle. The line

AP

is extended to meet

BC

E

. Determine the largest possible value for the ratio of lengths of segments

\frac{|AP|}{|PE|}.

1

Putnam 1981 A5

P(x)

be a polynomial with real coefficients and form the polynomial

Q(x) = ( x^2 +1) P(x)P'(x) + x(P(x)^2 + P'(x)^2 ).

Given that the equation

P(x) = 0

n

distinct real roots exceeding

1

, prove or disprove that the equation

Q(x)=0

2n - 1

distinct real roots.

1

Putnam 1981 A4

P

moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection. Let

N( t)

be the number of starting directions from a fixed interior point

P_0

P

t

units of time. Find the least constant

a

for which constants

b

c

exist such that

N(t) \leq at^2 +bt+c

t>0

and all initial points

P_0 .

1

Putnam 1981 A3

\lim_{t\to \infty} e^{-t} \int_{0}^{t} \int_{0}^{t} \frac{e^x -e^y }{x-y} \,dx\,dy,

or show that the limit does not exist.

1

Putnam 1981 A2

Two distinct squares of the

8\times8

C

are said to be adjacent if they have a vertex or side in common. Also,

g

C

-gap if for every numbering of the squares of

C

with all the integers

1, 2, \ldots, 64

there exist twoadjacent squares whose numbers differ by at least

g

. Determine the largest

C

g

1

Putnam 1981 A1

E(n)

denote the largest integer

k

5^k

1^{1}\cdot 2^{2} \cdot 3^{3} \cdot \ldots \cdot n^{n}.

\lim_{n\to \infty} \frac{E(n)}{n^2 }.

1

p|n^2 +3, p|k^2 +3

Prove that there are infinitely many positive

n

that for all prime divisors

p

n^2 + 3, \exists 0 \leq k \leq \sqrt{n}

p \mid k^2+3