2010 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 2010 B6

A

n\times n

matrix of real numbers for some

n\ge 1.

For each positive integer

k,

A^{[k]}

be the matrix obtained by raising each entry to the

k

th power. Show that if

A^k=A^{[k]}

k=1,2,\cdots,n+1,

A^k=A^{[k]}

k\ge 1.

1

Putnam 2010 B5

Is there a strictly increasing function

f:\mathbb{R}\to\mathbb{R}

f'(x)=f(f(x))

x?

1

Putnam 2010 B4

Find all pairs of polynomials

p(x)

q(x)

with real coefficients for which

p(x)q(x+1)-p(x+1)q(x)=1.

1

Putnam 2010 B3

There are 2010 boxes labeled

B_1,B_2,\dots,B_{2010},

2010n

balls have been distributed among them, for some positive integer

n.

You may redistribute the balls by a sequence of moves, each of which consists of choosing an

i

and moving exactly

i

B_i

into any one other box. For which values of

n

is it possible to reach the distribution with exactly

n

balls in each box, regardless of the initial distribution of balls?

1

Putnam 2010 B2

A,B,

C

are noncollinear points in the plane with integer coordinates such that the distances

AB,AC,

BC

are integers, what is the smallest possible value of

AB?

1

Putnam 2010 B1

Is there an infinite sequence of real numbers

a_1,a_2,a_3,\dots

a_1^m+a_2^m+a_3^m+\cdots=m

for every positive integer

m?

1

Putnam 2010 A6

f:[0,\infty)\to\mathbb{R}

be a strictly decreasing continuous function such that

\lim_{x\to\infty}f(x)=0.

\displaystyle\int_0^{\infty}\frac{f(x)-f(x+1)}{f(x)}\,dx

1

Putnam 2010 A5

G

be a group, with operation

*

. Suppose that
(i)

G

\mathbb{R}^3

*

need not be related to addition of vectors);
(ii) For each

\mathbf{a},\mathbf{b}\in G,

\mathbf{a}\times\mathbf{b}=\mathbf{a}*\mathbf{b}

\mathbf{a}\times\mathbf{b}=\mathbf{0}

(or both), where

\times

is the usual cross product in

\mathbb{R}^3.

\mathbf{a}\times\mathbf{b}=\mathbf{0}

\mathbf{a},\mathbf{b}\in G.

1

Putnam 2010 A4

Prove that for each positive integer

n,

10^{10^{10^n}}+10^{10^n}+10^n-1

1

Putnam 2010 A3

Suppose that the function

h:\mathbb{R}^2\to\mathbb{R}

has continuous partial derivatives and satisfies the equation

h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)

for some constants

a,b.

Prove that if there is a constant

M

|h(x,y)|\le M

(x,y)

\mathbb{R}^2,

h

is identically zero.

1

Putnam 2010 A2

Find all differentiable functions

f:\mathbb{R}\to\mathbb{R}

f'(x)=\frac{f(x+n)-f(x)}n

for all real numbers

x

and all positive integers

n.

1

Putnam 2010 A1

Given a positive integer

n,

what is the largest

k

such that the numbers

1,2,\dots,n

can be put into

k

boxes so that the sum of the numbers in each box is the same?
[When

n=8,

\{1,2,3,6\},\{4,8\},\{5,7\}

shows that the largest

k

is at least 3.]