2016 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 2016 B6

\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\sum_{n=0}^{\infty}\frac{1}{k2^n+1}.

1

Putnam 2016 B5

Find all functions

f

from the interval

(1,\infty)

(1,\infty)

with the following property: if

x,y\in(1,\infty)

x^2\le y\le x^3,

(f(x))^2\le f(y) \le (f(x))^3.

1

Putnam 2016 B4

A

2n\times 2n

matrix, with entries chosen independently at random. Every entry is chosen to be

0

1,

each with probability

1/2.

Find the expected value of

\det(A-A^t)

(as a function of

n

A^t

is the transpose of

A.

1

Putnam 2016 B3

S

is a finite set of points in the plane such that the area of triangle

\triangle ABC

1

A,B,

C

S.

Show that there exists a triangle of area

4

that (together with its interior) covers the set

S.

1

Putnam 2016 B2

Define a positive integer

n

to be squarish if either

n

is itself a perfect square or the distance from

n

to the nearest perfect square is a perfect square. For example,

2016

is squarish, because the nearest perfect square to

2016

45^2=2025

2025-2016=9

is a perfect square. (Of the positive integers between

1

10,

6

7

are not squarish.)
For a positive integer

N,

S(N)

be the number of squarish integers between

1

N,

inclusive. Find positive constants

\alpha

\beta

\lim_{N\to\infty}\frac{S(N)}{N^{\alpha}}=\beta,

or show that no such constants exist.

1

Putnam 2016 B1

x_0,x_1,x_2,\dots

be the sequence such that

x_0=1

n\ge 0,

x_{n+1}=\ln(e^{x_n}-x_n)

(as usual, the function

\ln

is the natural logarithm). Show that the infinite series

x_0+x_1+x_2+\cdots

converges and find its sum.

1

Putnam 2016 A6

Find the smallest constant

C

such that for every real polynomial

P(x)

3

that has a root in the interval

[0,1],

\int_0^1|P(x)|\,dx\le C\max_{x\in[0,1]}|P(x)|.

1

Putnam 2016 A5

G

is a finite group generated by the two elements

g

h,

where the order of

g

is odd. Show that every element of

G

can be written in the form

g^{m_1}h^{n_1}g^{m_2}h^{n_2}\cdots g^{m_r}h^{n_r}

1\le r\le |G|

m_n,n_1,m_2,n_2,\dots,m_r,n_r\in\{1,-1\}.

|G|

is the number of elements of

G.

1

Putnam 2016 A4

(2m-1)\times(2n-1)

rectangular region, where

m

n

are integers such that

m,n\ge 4.

The region is to be tiled using tiles of the two types shown: \begin{picture}(140,40)
\put(0,0){\line(0,1){40}} \put(0,0){\line(1,0){20}} \put(0,40){\line(1,0){40}} \put(20,0){\line(0,1){20}} \put(20,20){\line(1,0){20}} \put(40,20){\line(0,1){20}} \multiput(0,20)(5,0){4}{\line(1,0){3}} \multiput(20,20)(0,5){4}{\line(0,1){3}}
\put(80,0){\line(1,0){40}} \put(120,0){\line(0,1){20}} \put(120,20){\line(1,0){20}} \put(140,20){\line(0,1){20}} \put(80,0){\line(0,1){20}} \put(80,20){\line(1,0){20}} \put(100,20){\line(0,1){20}} \put(100,40){\line(1,0){40}} \multiput(100,0)(0,5){4}{\line(0,1){3}} \multiput(100,20)(5,0){4}{\line(1,0){3}} \multiput(120,20)(0,5){4}{\line(0,1){3}}
\end{picture} (The dotted lines divide the tiles into

1\times 1

squares.) The tiles may be rotated and reflected, as long as their sides are parallel to the sides of the rectangular region. They must all fit within the region, and they must cover it completely without overlapping.
What is the minimum number of tiles required to tile the region?

1

Putnam 2016 A3

f

is a function from

\mathbb{R}

\mathbb{R}

f(x)+f\left(1-\frac1x\right)=\arctan x

x\ne 0.

y=\arctan x

-\pi/2<y<\pi/2

\tan y=x.

\int_0^1f(x)\,dx.

1

Putnam 2016 A2

Given a positive integer

n,

M(n)

be the largest integer

m

\binom{m}{n-1}>\binom{m-1}{n}.

\lim_{n\to\infty}\frac{M(n)}{n}.

1

Putnam 2016 A1

Find the smallest positive integer

j

such that for every polynomial

p(x)

with integer coefficients and for every integer

k,

p^{(j)}(k)=\left. \frac{d^j}{dx^j}p(x) \right|_{x=k}

j

-th derivative of

p(x)

k

) is divisible by

2016.