2014 Putnam

Part of Putnam

Subcontests

(6)

2

Putnam 2014 A6

n

be a positive integer. What is the largest

k

for which there exist

n\times n

M_1,\dots,M_k

N_1,\dots,N_k

with real entries such that for all

i

j,

the matrix product

M_iN_j

has a zero entry somewhere on its diagonal if and only if

i\ne j?

Putnam 2014 B6

f:[0,1]\to\mathbb{R}

be a function for which there exists a constant

K>0

|f(x)-f(y)|\le K|x-y|

x,y\in [0,1].

Suppose also that for each rational number

r\in [0,1],

there exist integers

a

b

f(r)=a+br.

Prove that there exist finitely many intervals

I_1,\dots,I_n

f

is a linear function on each

I_i

[0,1]=\bigcup_{i=1}^nI_i.

2

Putnam 2014 A5

P_n(x)=1+2x+3x^2+\cdots+nx^{n-1}.

Prove that the polynomials

P_j(x)

P_k(x)

are relatively prime for all positive integers

j

k

j\ne k.

Putnam 2014 B5

In the 75th Annual Putnam Games, participants compete at mathematical games. Patniss and Keeta play a game in which they take turns choosing an element from the group of invertible

n\times n

matrices with entries in the field

\mathbb{Z}/p\mathbb{Z}

of integers modulo

p,

n

is a fixed positive integer and

p

is a fixed prime number. The rules of the game are:
(1) A player cannot choose an element that has been chosen by either player on any previous turn.
(2) A player can only choose an element that commutes with all previously chosen elements.
(3) A player who cannot choose an element on his/her turn loses the game.
Patniss takes the first turn. Which player has a winning strategy?

2

Putnam 2014 A4

X

is a random variable that takes on only nonnegative integer values, with

E[X]=1,

E[X^2]=2,

E[X^3]=5.

E[Y]

denotes the expectation of the random variable

Y.

) Determine the smallest possible value of the probability of the event

X=0.

Putnam 2014 B4

Show that for each positive integer

n,

all the roots of the polynomial

\sum_{k=0}^n 2^{k(n-k)}x^k

are real numbers.

2

Putnam 2014 A3

a_0=5/2

a_k=a_{k-1}^2-2

k\ge 1.

\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)

in closed form.

Putnam 2014 B3

A

m\times n

matrix with rational entries. Suppose that there are at least

m+n

distinct prime numbers among the absolute values of the entries of

A.

Show that the rank of

A

2.

2

Putnam 2014 A2

A

n\times n

matrix whose entry in the

i

j

\frac1{\min(i,j)}

1\le i,j\le n.

\det(A).

Putnam 2014 B2

f

is a function on the interval

[1,3]

-1\le f(x)\le 1

x

\displaystyle \int_1^3f(x)\,dx=0.

\displaystyle\int_1^3\frac{f(x)}x\,dx

2

Putnam 2014 A1

Prove that every nonzero coefficient of the Taylor series of

(1-x+x^2)e^x

x=0

is a rational number whose numerator (in lowest terms) is either

1

or a prime number.

Putnam 2014 B1

A base 10 over-expansion of a positive integer

N

is an expression of the form

N=d_k10^k+d_{k-1}10^{k-1}+\cdots+d_0 10^0

d_k\ne 0

d_i\in\{0,1,2,\dots,10\}

i.

For instance, the integer

N=10

has two base 10 over-expansions:

10=10\cdot 10^0

and the usual base 10 expansion

10=1\cdot 10^1+0\cdot 10^0.

Which positive integers have a unique base 10 over-expansion?