1986 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 1986 B6

A,B,C,D

n \times n

matrices with entries in a field

F

, satisfying the conditions that

AB^T

CD^T

are symmetric and

AD^T - BC^T = I

I

n \times n

identity matrix, and if

M

n \times n

M^T

is its transpose. Prove that

A^T D - C^T B = I

1

Putnam 1986 B5

f(x,y,z) = x^2+y^2+z^2+xyz

p(x,y,z), q(x,y,z)

r(x,y,z)

be polynomials with real coefficients satisfying

f(p(x,y,z), q(x,y,z), r(x,y,z)) = f(x,y,z).

Prove or disprove the assertion that the sequence

p,q,r

consists of some permutation of

\pm x, \pm y, \pm z

, where the number of minus signs is

0

2.

1

Putnam 1986 B4

For a positive real number

r

G(r)

be the minimum value of

|r - \sqrt{m^2+2n^2}|

for all integers

m

n

. Prove or disprove the assertion that

\lim_{r\to \infty}G(r)

exists and equals

0.

1

Putnam 1986 B3

\Gamma

consist of all polynomials in

x

with integer coefficients. For

f

g

\Gamma

m

a positive integer, let

f \equiv g \pmod{m}

mean that every coefficient of

f-g

is an integral multiple of

m

n

p

be positive integers with

p

prime. Given that

f,g,h,r

s

\Gamma

rf+sg\equiv 1 \pmod{p}

fg \equiv h \pmod{p}

, prove that there exist

F

G

\Gamma

F \equiv f \pmod{p}

G \equiv g \pmod{p}

FG \equiv h \pmod{p^n}

1

Putnam 1986 B2

Prove that there are only a finite number of possibilities for the ordered triple

T=(x-y,y-z,z-x)

x,y,z

are complex numbers satisfying the simultaneous equations

x(x-1)+2yz = y(y-1)+2zx = z(z-1)+2xy,

and list all such triples

T

1

Putnam 1986 B1

Inscribe a rectangle of base

b

h

in a circle of radius one, and inscribe an isosceles triangle in the region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). For what value of

h

do the rectangle and triangle have the same area?

1

Putnam 1986 A6

a_1, a_2, \dots, a_n

be real numbers, and let

b_1, b_2, \dots, b_n

be distinct positive integers. Suppose that there is a polynomial

f(x)

satisfying the identity

(1-x)^n f(x) = 1 + \sum_{i=1}^n a_i x^{b_i}.

Find a simple expression (not involving any sums) for

f(1)

b_1, b_2, \dots, b_n

n

(but independent of

a_1, a_2, \dots, a_n

1

Putnam 1986 A5

f_1(x), f_2(x), \dots, f_n(x)

are functions of

n

x = (x_1, \dots, x_n)

with continuous second-order partial derivatives everywhere on

\mathbb{R}^n

. Suppose further that there are constants

c_{ij}

\frac{\partial f_i}{\partial x_j} - \frac{\partial f_j}{\partial x_i} = c_{ij}

i

j

1\leq i \leq n

1 \leq j \leq n

. Prove that there is a function

g(x)

\mathbb{R}^n

f_i + \partial g/\partial x_i

is linear for all

i

1 \leq i \leq n

. (A linear function is one of the form

a_0 + a_1 x_1 + a_2 x_2 + \cdots + a_n x_n.)

1

Putnam 1986 A4

A transversal of an

n\times n

A

n

A

, no two in the same row or column. Let

f(n)

be the number of

n \times n

A

satisfying the following two conditions:
(a) Each entry

\alpha_{i,j}

A

\{-1,0,1\}

. (b) The sum of the

n

entries of a transversal is the same for all transversals of

A

.
An example of such a matrix

A

A = \left( \begin{array}{ccc} -1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \end{array} \right).

Determine with proof a formula for

f(n)

f(n) = a_1 b_1^n + a_2 b_2^n + a_3 b_3^n + a_4,

a_i

b_i

's are rational numbers.

1

Putnam 1986 A3

\textstyle\sum_{n=0}^\infty \mathrm{Arccot}(n^2+n+1)

\mathrm{Arccot}\,t

t \geq 0

denotes the number

\theta

in the interval

0 < \theta \leq \pi/2

\cot \theta = t

1

Putnam 1986 A2

What is the units (i.e., rightmost) digit of

\left\lfloor \frac{10^{20000}}{10^{100}+3}\right\rfloor ?

1

Putnam 1986 A1

Find, with explanation, the maximum value of

f(x)=x^3-3x

on the set of all real numbers

x

x^4+36\leq 13x^2