1999 Putnam

Part of Putnam

Subcontests

(6)

2

Putnam 1999 A6

(a_n)_{n\geq 1}

a_1=1,a_2=2,a_3=24,

n\geq 4,

a_n=\dfrac{6a_{n-1}^2a_{n-3}-8a_{n-1}a_{n-2}^2}{a_{n-2}a_{n-3}}.

Show that, for all

n

a_n

is an integer multiple of

n

Putnam 1999 B6

S

be a finite set of integers, each greater than

1

. Suppose that for each integer

n

s\in S

\gcd(s,n)=1

\gcd(s,n)=s

. Show that there exist

s,t\in S

\gcd(s,t)

2

Putnam 1999 A5

Prove that there is a constant

C

p(x)

is a polynomial of degree

1999

|p(0)|\leq C\int_{-1}^1|p(x)|\,dx.

Putnam 1999 B5

n\geq 3

\theta=2\pi/n

. Evaluate the determinant of the

n\times n

I+A

I

n\times n

identity matrix and

A=(a_{jk})

a_{jk}=\cos(j\theta+k\theta)

j,k

2

Putnam 1999 A4

\sum_{m=1}^\infty\sum_{n=1}^\infty\dfrac{m^2n}{3^m(n3^m+m3^n)}.

Putnam 1999 B4

f

be a real function with a continuous third derivative such that

f(x)

f^\prime(x)

f^{\prime\prime}(x)

f^{\prime\prime\prime}(x)

are positive for all

x

f^{\prime\prime\prime}(x)\leq f(x)

x

f^\prime(x)<2f(x)

x

2

Putnam 1999 A2

p(x)

be a polynomial that is nonnegative for all real

x

. Prove that for some

k

, there are polynomials

f_1(x),f_2(x),\ldots,f_k(x)

p(x)=\sum_{j=1}^k(f_j(x))^2.

Putnam 1999 B2

P(x)

be a polynomial of degree

n

P(x)=Q(x)P^{\prime\prime}(x)

Q(x)

is a quadratic polynomial and

P^{\prime\prime}(x)

is the second derivative of

P(x)

P(x)

has at least two distinct roots then it must have

n

distinct roots.

2

Putnam 1999 A1

Find polynomials

f(x)

g(x)

h(x)

, if they exist, such that for all

x

|f(x)|-|g(x)|+h(x)=\begin{cases}-1 & \text{if }x<-1\\3x+2 &\text{if }-1\leq x\leq 0\\-2x+2 & \text{if }x>0.\end{cases}

Putnam 1999 B1

ABC

has right angle at

C

\angle BAC=\theta

D

AB

|AC|=|AD|=1

E

BC

\angle CDE=\theta

. The perpendicular to

BC

E

AB

F

\lim_{\theta\to 0}|EF|

2

Putnam 1999 A3

Consider the power series expansion

\dfrac{1}{1-2x-x^2}=\sum_{n=0}^\infty a_nx^n.

Prove that, for each integer

n\geq 0

, there is an integer

m

a_n^2+a_{n+1}^2=a_m.

Putnam 1999 B3

A=\{(x,y): 0\le x,y < 1\}.

(x,y)\in A,

S(x,y)=\sum_{\frac12\le\frac mn\le2}x^my^n,

where the sum ranges over all pairs

(m,n)

of positive integers satisfying the indicated inequalities. Evaluate

\lim_{(x,y)\to(1,1),(x,y)\in A}(1-xy^2)(1-x^2y)S(x,y).