1970 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 1970 B6

Show that if a circumscribable quadrilateral of sides

a,b,c,d

A= \sqrt{abcd},

then it is also inscribable.

1

Putnam 1970 B5

u_n

denote the ramp function

u_n (x) =\begin{cases} -n \;\; \text{for} \;\; x \leq -n, \\ \; x \;\;\; \text{for} \;\; -n \leq x \leq n,\\ \;n \;\; \; \text{for} \;\; n \leq x, \end{cases}

f

be a real function of a real variable. Show that

f

is continuous if and only if

u_n \circ f

is continuous for all

n.

1

Putnam 1970 B4

An automobile starts from rest and ends at rest, traversing a distance of one mile in one minute, along a straight road. If a governor prevents the speed of the car from exceeding

90

miles per hour, show that at some time of the traverse the acceleration or deceleration of the car was at least

6.6

1

Putnam 1970 B3

A closed subset

S

\mathbb{R}^{2}

a<x<b

. Show that its projection on the

y

-axis is closed.

1

Putnam 1970 B2

The time-varying temperature of a certain body is given by a polynomial in the time of degree at most three. Show that the average temperature of the body between

9

3

pm can always be found by taking the average of the temperatures at two fixed times, which are independent of the polynomial. Also, show that these two times are

10\colon \! 16

1\colon \!44

pm to the nearest minute.

1

Putnam 1970 B1

\lim_{n\to \infty} \frac{1}{n^4 } \prod_{i=1}^{2n} (n^2 +i^2 )^{\frac{1}{n}}.

1

Putnam 1970 A6

Three numbers are chosen independently at random, one from each of the three intervals

[0, L_i ]

i=1,2,3

). If the distribution of each random number is uniform with respect to the length of the interval it is chosen from, determine the expected value of the smallest number chosen.

1

Putnam 1970 A5

Determine the radius of the largest circle which can lie on the ellipsoid

\frac{x^2 }{a^2 } +\frac{ y^2 }{b^2 } +\frac{z^2 }{c^2 }=1 \;\;\;\; (a>b>c).

1

Putnam 1970 A4

Given a sequence

(x_n )

\lim_{n\to \infty} x_n - x_{n-2}=0,

\lim_{n\to \infty} \frac{x_n -x_{n-1}}{n}=0.

1

Putnam 1970 A2

Consider the locus given by the real polynomial equation

Ax^2 +Bxy+Cy^2 +Dx^3 +E x^2 y +F xy^2 +G y^3=0,

B^2 -4AC <0.

Prove that there is a positive number

\delta

such that there are no points of the locus in the punctured disk

0 <x^2 +y^2 < \delta^2.

1

Putnam 1970 A1

Show that the power series for the function

e^{ax} \cos bx,

a,b >0

, has either no zero coefficients or infinitely many zero coefficients.

1

Longest Tail on Perfect Squares

12^2=144

4

38^2=1444

4

s. Determine the length of the longest string of equal nonzero digits in which the square of an integer can end.