2004 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 2004 B6

A

be a nonempty set of positive integers, and let

N(x)

denote the number of elements of

A

x

B

denote the set of positive integers

b

that can be written in the form

b=a-a^{\prime}

a\in A

a^{\prime}\in A

b_1<b_2<\cdots

be the members of

B

, listed in increasing order. Show that if the sequence

b_{i+1}-b_i

is unbounded, then

\lim_{x\to \infty}\frac{N(x)}{x}=0

1

Putnam 2004 B5

\lim_{x\to 1^-}\prod_{n=0}^{\infty}\left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}

1

Putnam 2004 B4

n

be a positive integer,

n \ge 2

\theta=\frac{2\pi}{n}

. Define points

P_k=(k,0)

in the xy-plane, for

k=1,2,\dots,n

R_k

be the map that rotates the plane counterclockwise by the angle

\theta

about the point

P_k

R

denote the map obtained by applying in order,

R_1

R_2

R_n

. For an arbitrary point

(x,y)

, find and simplify the coordinates of

R(x,y)

1

Putnam 2004 B3

Determine all real numbers

a>0

for which there exists a nonnegative continuous function

f(x)

[0,a]

with the property that the region

R=\{(x,y): 0\le x\le a, 0\le y\le f(x)\}

k

k

square units for some real number

k

1

Putnam 2004 B2

m

n

be positive integers. Show that

\frac{(m+n)!}{(m+n)^{m+n}} < \frac{m!}{m^m}\cdot\frac{n!}{n^n}

1

Putnam 2004 B1

P(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_0

be a polynomial with integer coefficients. Suppose that

r

is a rational number such that

P(r)=0

. Show that the

n

c_nr, c_nr^2+c_{n-1}r, c_nr^3+c_{n-1}r^2+c_{n-1}r, \dots, c_nr^n+c_{n-1}r^{n-1}+\cdots+c_1r

are all integers.

1

Putnam 2004 A6

f(x,y)

is a continuous real-valued function on the unit square

0\le x\le1,0\le y\le1.

\int_0^1\left(\int_0^1f(x,y)dx\right)^2dy + \int_0^1\left(\int_0^1f(x,y)dy\right)^2dx

\le\left(\int_0^1\int_0^1f(x,y)dxdy\right)^2 + \int_0^1\int_0^1\left[f(x,y)\right]^2dxdy.

1

Putnam 2004 A5

m\times n

checkerboard is colored randomly: each square is independently assigned red or black with probability

\frac12.

we say that two squares,

p

q

, are in the same connected monochromatic region if there is a sequence of squares, all of the same color, starting at

p

q,

in which successive squares in the sequence share a common side. Show that the expected number of connected monochromatic regions is greater than

\frac{mn}8.

1

Putnam 2004 A4

Show that for any positive integer

n

there is an integer

N

such that the product

x_1x_2\cdots x_n

can be expressed identically in the form

x_1x_2\cdots x_n=\sum_{i=1}^Nc_i(a_{i1}x_1+a_{i2}x_2+\cdots +a_{in}x_n)^n

c_i

are rational numbers and each

a_{ij}

is one of the numbers,

-1,0,1.

1

Putnam 2004 A3

Define a sequence

\{u_n\}_{n=0}^{\infty}

u_0=u_1=u_2=1,

and thereafter by the condition that

\det\begin{vmatrix} u_n & u_{n+1} \\ u_{n+2} & u_{n+3} \end{vmatrix}=n!

n\ge 0.

u_n

is an integer for all

n.

(By convention,

0!=1

1

Putnam 2004 A2

i=1,2,

T_i

be a triangle with side length

a_i,b_i,c_i,

A_i.

a_1\le a_2, b_1\le b_2, c_1\le c_2,

T_2

is an acute triangle. Does it follow that

A_1\le A_2

1

Putnam 2004 A1

Basketball star Shanille O'Keal's team statistician keeps track of the number,

S(N),

of successful free throws she has made in her first

N

attempts of the season. Early in the season,

S(N)

was less than 80% of

N,

but by the end of the season,

S(N)

was more than 80% of

N.

Was there necessarily a moment in between when

S(N)

was exactly 80% of

N