2015 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 2015 B6

For each positive integer

k,

A(k)

be the number of odd divisors of

k

in the interval

\left[1,\sqrt{2k}\right).

\sum_{k=1}^{\infty}(-1)^{k-1}\frac{A(k)}k.

1

Putnam 2015 B5

P_n

be the number of permutations

\pi

\{1,2,\dots,n\}

|i-j|=1\text{ implies }|\pi(i)-\pi(j)|\le 2

i,j

\{1,2,\dots,n\}.

n\ge 2,

P_{n+5}-P_{n+4}-P_{n+3}+P_n

does not depend on

n,

and find its value.

1

Putnam 2015 B4

T

be the set of all triples

(a,b,c)

of positive integers for which there exist triangles with side lengths

a,b,c.

\sum_{(a,b,c)\in T}\frac{2^a}{3^b5^c}

as a rational number in lowest terms.

1

Putnam 2015 B3

S

be the set of all

2\times 2

M=\begin{pmatrix}a&b\\c&d\end{pmatrix}

a,b,c,d

(in that order) form an arithmetic progression. Find all matrices

M

S

for which there is some integer

k>1

M^k

S.

1

Putnam 2015 B2

Given a list of the positive integers

1,2,3,4,\dots,

take the first three numbers

1,2,3

6

and cross all four numbers off the list. Repeat with the three smallest remaining numbers

4,5,7

16.

Continue in this way, crossing off the three smallest remaining numbers and their sum and consider the sequence of sums produced:

6,16,27, 36, \dots.

Prove or disprove that there is some number in this sequence whose base 10 representation ends with

2015.

1

Putnam 2015 B1

f

be a three times differentiable function (defined on

\mathbb{R}

and real-valued) such that

f

has at least five distinct real zeros. Prove that

f+6f'+12f''+8f'''

has at least two distinct real zeros.

1

Putnam 2015 A6

n

be a positive integer. Suppose that

A,B,

M

n\times n

matrices with real entries such that

AM=MB,

A

B

have the same characteristic polynomial. Prove that

\det(A-MX)=\det(B-XM)

n\times n

X

with real entries.

1

Putnam 2015 A5

q

be an odd positive integer, and let

N_q

denote the number of integers

a

0<a<q/4

\gcd(a,q)=1.

N_q

is odd if and only if

q

p^k

k

a positive integer and

p

a prime congruent to

5

7

8.

1

Putnam 2015 A4

For each real number

x,

f(x)=\sum_{n\in S_x}\frac1{2^n}

S_x

is the set of positive integers

n

\lfloor nx\rfloor

is even.
What is the largest real number

L

f(x)\ge L

x\in [0,1)

\lfloor z\rfloor

denotes the greatest integer less than or equal to

z.

1

Putnam 2015 A3

\log_2\left(\prod_{a=1}^{2015}\prod_{b=1}^{2015}\left(1+e^{2\pi iab/2015}\right)\right)

i

is the imaginary unit (that is,

i^2=-1

1

Putnam 2015 A2

a_0=1,a_1=2,

a_n=4a_{n-1}-a_{n-2}

n\ge 2.

Find an odd prime factor of

a_{2015}.

1

Putnam 2015 A1

A

B

be points on the same branch of the hyperbola

xy=1.

P

is a point lying between

A

B

on this hyperbola, such that the area of the triangle

APB

is as large as possible. Show that the region bounded by the hyperbola and the chord

AP

has the same area as the region bounded by the hyperbola and the chord

PB.