1985 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 1985 B6

G

be a finite set of real

n \times n

\left\{M_{i}\right\}, 1 \leq i \leq r,

which form a group under matrix multiplication. Suppose that

\textstyle\sum_{i=1}^{r} \operatorname{tr}\left(M_{i}\right)=0,

\operatorname{tr}(A)

denotes the trace of the matrix

A .

\textstyle\sum_{i=1}^{r} M_{i}

n \times n

1

Putnam 1985 B5

\textstyle\int_{0}^{\infty} t^{-1 / 2} e^{-1985\left(t+t^{-1}\right)} d t.

You may assume that

\textstyle\int_{-\infty}^{\infty} e^{-x^{2}} d x=\sqrt{\pi}.

1

Putnam 1985 B4

C

be the unit circle

x^{2}+y^{2}=1 .

p

is chosen randomly on the circumference

C

and another point

q

is chosen randomly from the interior of

C

(these points are chosen independently and uniformly over their domains). Let

R

be the rectangle with sides parallel to the

x

y

-axes with diagonal

p q .

What is the probability that no point of

R

lies outside of

C ?

1

Putnam 1985 B3

\begin{array}{cccc}{a_{1,1}} & {a_{1,2}} & {a_{1,3}} & {\dots} \\ {a_{2,1}} & {a_{2,2}} & {a_{2,3}} & {\cdots} \\ {a_{3,1}} & {a_{3,2}} & {a_{3,3}} & {\cdots} \\ {\vdots} & {\vdots} & {\vdots} & {\ddots}\end{array}

be a doubly infinite array of positive integers, and suppose each positive integer appears exactly eight times in the array. Prove that

a_{m, n}>m n

for some pair of positive integers

(m, n) .

1

Putnam 1985 B2

Define polynomials

f_{n}(x)

n \geq 0

f_{0}(x)=1, f_{n}(0)=0

n \geq 1,

\frac{d}{d x} f_{n+1}(x)=(n+1) f_{n}(x+1)

n \geq 0 .

Find, with proof, the explicit factorization of

f_{100}(1)

into powers of distinct primes.

1

Putnam 1985 B1

k

be the smallest positive integer for which there exist distinct integers

m_{1}, m_{2}, m_{3}, m_{4}, m_{5}

such that the polynomial

p(x)=\left(x-m_{1}\right)\left(x-m_{2}\right)\left(x-m_{3}\right)\left(x-m_{4}\right)\left(x-m_{5}\right)

k

nonzero coefficients. Find, with proof, a set of integers

m_{1}, m_{2}, m_{3}, m_{4}, m_{5}

for which this minimum

k

1

Putnam 1985 A6

p(x)=a_{0}+a_{1} x+\cdots+a_{m} x^{m}

is a polynomial with real coefficients

a_{i},

\Gamma(p(x))=a_{0}^{2}+a_{1}^{2}+\cdots+a_{m}^{2}.

F(x)=3 x^{2}+7 x+2 .

Find, with proof, a polynomial

g(x)

with real coefficients such that
(i)

g(0)=1,

\Gamma\left(f(x)^{n}\right)=\Gamma\left(g(x)^{n}\right)

for every integer

n \geq 1.

1

Putnam 1985 A5

I_{m}=\textstyle\int_{0}^{2 \pi} \cos (x) \cos (2 x) \cdots \cos (m x) d x .

For which integers

m, 1 \leq m \leq 10

I_{m} \neq 0 ?

1

Putnam 1985 A4

Define a sequence

\left\{a_{i}\right\}

a_{1}=3

a_{i+1}=3^{a_{i}}

i \geq 1.

Which integers between

00

99

inclusive occur as the last two digits in the decimal expansion of infinitely many

a_{i} ?

1

Putnam 1985 A3

d

be a real number. For each integer

m \geq 0,

define a sequence

\left\{a_{m}(j)\right\}, j=0,1,2, \ldots

by the condition \begin{align*} a_{m}(0)&=d / 2^{m},\\ a_{m}(j+1)&=\left(a_{m}(j)\right)^{2}+2 a_{m}(j), j \geq 0. \end{align*} Evaluate

\lim _{n \rightarrow \infty} a_{n}(n).

1

Putnam 1985 A2

T

be an acute triangle. Inscribe a rectangle

R

T

with one side along a side of

T.

Then inscribe a rectangle

S

in the triangle formed by the side of

R

opposite the side on the boundary of

T,

and the other two sides of

T,

with one side along the side of

R.

For any polygon

X,

A(X)

denote the area of

X.

Find the maximum value, or show that no maximum exists, of

\tfrac{A(R)+A(S)}{A(T)},

T

ranges over all triangles and

R,S

over all rectangles as above.

1

Putnam 1985 A1

Determine, with proof, the number of ordered triples

\left(A_{1}, A_{2}, A_{3}\right)

of sets which have the property that
(i)

A_{1} \cup A_{2} \cup A_{3}=\{1,2,3,4,5,6,7,8,9,10\},

A_{1} \cap A_{2} \cap A_{3}=\emptyset.

Express your answer in the form

2^{a} 3^{b} 5^{c} 7^{d},

a, b, c, d

are nonnegative integers.