2018 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 2018 B6

S

be the set of sequences of length 2018 whose terms are in the set

\{1, 2, 3, 4, 5, 6, 10\}

and sum to 3860. Prove that the cardinality of

S

2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}.

1

Putnam 2018 B5

f = (f_1, f_2)

be a function from

\mathbb{R}^2

\mathbb{R}^2

with continuous partial derivatives

\tfrac{\partial f_i}{\partial x_j}

that are positive everywhere. Suppose that

\frac{\partial f_1}{\partial x_1} \frac{\partial f_2}{\partial x_2} - \frac{1}{4} \left(\frac{\partial f_1}{\partial x_2} + \frac{\partial f_2}{\partial x_1} \right)^2 > 0

everywhere. Prove that

f

1

Putnam 2018 B4

Given a real number

a

, we define a sequence by

x_0 = 1

x_1 = x_2 = a

x_{n+1} = 2x_nx_{n-1} - x_{n-2}

n \ge 2

. Prove that if

x_n = 0

n

, then the sequence is periodic.

1

Putnam 2018 B3

Find all positive integers

n < 10^{100}

for which simultaneously

n

2^n

n-1

2^n - 1

n-2

2^n - 2

1

Putnam 2018 B2

n

be a positive integer, and let

f_n(z) = n + (n-1)z + (n-2)z^2 + \dots + z^{n-1}

f_n

has no roots in the closed unit disk

\{z \in \mathbb{C}: |z| \le 1\}

1

Putnam 2018 B1

\mathcal{P}

be the set of vectors defined by

\mathcal{P} = \left\{\begin{pmatrix} a \\ b \end{pmatrix} \, \middle\vert \, 0 \le a \le 2, 0 \le b \le 100, \, \text{and} \, a, b \in \mathbb{Z}\right\}.

\mathbf{v} \in \mathcal{P}

such that the set

\mathcal{P}\setminus\{\mathbf{v}\}

obtained by omitting vector

\mathbf{v}

\mathcal{P}

can be partitioned into two sets of equal size and equal sum.

1

Putnam 2018 A6

A

B

C

D

are distinct points, no three of which lie on a line, in the Euclidean plane. Show that if the squares of the lengths of the line segments

AB

AC

AD

BC

BD

CD

are rational numbers, then the quotient

\frac{\mathrm{area}(\triangle ABC)}{\mathrm{area}(\triangle ABD)}

is a rational number.

1

Putnam 2018 A5

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

be an infinitely differentiable function satisfying

f(0) = 0

f(1) = 1

f(x) \ge 0

x \in \mathbb{R}

. Show that there exist a positive integer

n

and a real number

x

f^{(n)}(x) < 0

1

Putnam 2018 A4

m

n

be positive integers with

\gcd(m, n) = 1

a_k = \left\lfloor \frac{mk}{n} \right\rfloor - \left\lfloor \frac{m(k-1)}{n} \right\rfloor

k = 1, 2, \dots, n

g

h

are elements in a group

G

gh^{a_1} gh^{a_2} \cdots gh^{a_n} = e,

e

is the identity element. Show that

gh = hg

\lfloor x \rfloor

denotes the greatest integer less than or equal to

x

1

Putnam 2018 A3

Determine the greatest possible value of

\sum_{i = 1}^{10} \cos(3x_i)

for real numbers

x_1, x_2, \dots, x_{10}

\sum_{i = 1}^{10} \cos(x_i) = 0

1

Putnam 2018 A2

S_1, S_2, \dots, S_{2^n - 1}

be the nonempty subsets of

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

in some order, and let

M

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

(i, j)

m_{ij} = \left\{ \begin{array}{cl} 0 & \text{if $S_i \cap S_j = \emptyset$}, \\ 1 & \text{otherwise}. \end{array} \right.

Calculate the determinant of

M

1

Putnam 2018 A1

Find all ordered pairs

(a, b)

of positive integers for which

\frac{1}{a} + \frac{1}{b} = \frac{3}{2018}.