1967 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 1967 B6

f

be a real-valued function having partial derivatives and which is defined for

x^2 +y^2 \leq1

and is such that

|f(x,y)|\leq 1.

Show that there exists a point

(x_0, y_0 )

in the interior of the unit circle such that

\left( \frac{ \partial f}{\partial x}(x_0 ,y_0 ) \right)^{2}+ \left( \frac{ \partial f}{\partial y}(x_0 ,y_0 ) \right)^{2} \leq 16.

1

Putnam 1967 B5

Show that the sum of the first

n

terms in the binomial expansion of

(2-1)^{-n}

\frac{1}{2},

n

is a positive integer.

1

Putnam 1967 B4

a) A certain locker room contains

n

lockers numbered

1,2,\ldots,n

and all are originally locked. An attendant performs a sequence of operations

T_1, T_2 ,\ldots, T_n

, whereby with the operation

T_k

the state of those lockers whose number is divisible by

k

is swapped. After all

n

operations have been performed, it is observed that all lockers whose number is a perfect square (and only those lockers) are open. Prove this. b) Investigate in a meaningful mathematical way a procedure or set of operations similar to those above which will produce the set of cubes, or the set of numbers of the form

2 m^2

, or the set of numbers of the form

m^2 +1

, or some nontrivial similar set of your own selection.

1

Putnam 1967 B3

f

g

are continuous and periodic functions with period

1

on the real line, then

\lim_{n\to \infty} \int_{0}^{1} f(x)g (nx)\; dx =\left( \int_{0}^{1} f(x)\; dx\right)\left( \int_{0}^{1} g(x)\; dx\right).

1

Putnam 1967 B2

0\leq p,r\leq 1

and consider the identities

a)\; (px+(1-p)y)^{2}=a x^2 +bxy +c y^2, \;\;\;\, b)\; (px+(1-p)y)(rx+(1-r)y) =\alpha x^2 + \beta xy + \gamma y^2.

a)\; \max(a,b,c) \geq \frac{4}{9}, \;\;\;\; b)\; \max( \alpha, \beta , \gamma) \geq \frac{4}{9}.

1

Putnam 1967 B1

ABCDEF

be a hexagon inscribed in a circle of radius

r.

AB=CD=EF=r,

then the midpoints of

BC, DE

FA

are the vertices of an equilateral triangle.

1

Putnam 1967 A6

Given real numbers

(a_i)

(b_i)

i=1,2,3,4

a_1 b _2 \ne a_2 b_1 .

Consider the set of all solutions

(x_1 ,x_2 ,x_3 , x_4)

of the simultaneous equations

a_1 x_1 +a _2 x_2 +a_3 x_3 +a_4 x_4 =0 \;\; \text{and}\;\; b_1 x_1 +b_2 x_2 +b_3 x_3 +b_4 x_4 =0

x_i

is zero. Each such solution generates a

4

-tuple of plus and minus signs (by considering the sign of

x_i

).
[*] Determine, with proof, the maximum number of distinct

4

-tuples possible. [*] Investigate necessary and sufficient conditions on

(a_i)

(b_i)

such that the above maximum of distinct

4

-tuples is attained.

1

Putnam 1967 A5

Show that in a convex region in the plane whose boundary contains at most a finite number of straight line segments and whose area is greater than

\frac{\pi}{4}

there is at least one pair of points a unit distance apart.

1

Putnam 1967 A4

\lambda > \frac{1}{2}

there does not exist a real-valued function

u(x)

such that for all

x

in the closed interval

[0,1]

the following holds:

u(x)= 1+ \lambda \int_{x}^{1} u(y) u(y-x) \; dy.

1

Putnam 1967 A3

Consider polynomial functions

ax^2 -bx +c

with integer coefficients which have two distinct zeros in the open interval

(0,1).

Exhibit with proof the least positive integer value of

a

for which such a polynomial exists.

1

Putnam 1967 A2

S_0

1.

n \geq 1

S_n

be the number of

n\times n

matrices whose elements are nonnegative integers with the property that

a_{ij}=a_{ji}

i,j=1,2,\ldots, n

\sum_{i=1}^{n} a_{ij}=1

j=1,2,\ldots, n

). Prove that a)

S_{n+1}=S_{n} +nS_{n-1}.

\sum_{n=0}^{\infty} S_{n} \frac{x^{n}}{n!} =\exp \left(x+\frac{x^{2}}{2}\right).

1

Putnam 1967 A1

f(x)= a_1 \sin x + a_2 \sin 2x+\cdots +a_{n} \sin nx

a_1 ,a_2 ,\ldots,a_n

are real numbers and where

n

is a positive integer. Given that

|f(x)| \leq | \sin x |

x,

|a_1 +2a_2 +\cdots +na_{n}|\leq 1.