2017 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 2017 B3

f(x) = \sum_{i=0}^\infty c_ix^i

is a power series for which each coefficient

c_i

0

1

f(2/3) = 3/2

f(1/2)

must be irrational.

1

Putnam 2017 B6

Find the number of ordered

64

\{x_0,x_1,\dots,x_{63}\}

x_0,x_1,\dots,x_{63}

are distinct elements of

\{1,2,\dots,2017\}

x_0+x_1+2x_2+3x_3+\cdots+63x_{63}

is divisible by

2017.

1

Putnam 2017 B5

A line in the plane of a triangle

T

is called an equalizer if it divides

T

into two regions having equal area and equal perimeter. Find positive integers

a>b>c,

a

as small as possible, such that there exists a triangle with side lengths

a,b,c

that has exactly two distinct equalizers.

1

Putnam 2017 B4

Evaluate the sum

\sum_{k=0}^{\infty}\left(3\cdot\frac{\ln(4k+2)}{4k+2}-\frac{\ln(4k+3)}{4k+3}-\frac{\ln(4k+4)}{4k+4}-\frac{\ln(4k+5)}{4k+5}\right)

=3\cdot\frac{\ln 2}2-\frac{\ln 3}3-\frac{\ln 4}4-\frac{\ln 5}5+3\cdot\frac{\ln 6}6-\frac{\ln 7}7-\frac{\ln 8}8-\frac{\ln 9}9+3\cdot\frac{\ln 10}{10}-\cdots.

\ln x

denotes the natural logarithm of

x.

1

Putnam 2017 B2

Suppose that a positive integer

N

can be expressed as the sum of

k

consecutive positive integers

N=a+(a+1)+(a+2)+\cdots+(a+k-1)

k=2017

but for no other values of

k>1.

Considering all positive integers

N

with this property, what is the smallest positive integer

a

that occurs in any of these expressions?

1

Putnam 2017 B1

L_1

L_2

be distinct lines in the plane. Prove that

L_1

L_2

intersect if and only if, for every real number

\lambda\ne 0

and every point

P

L_1

L_2,

there exist points

A_1

L_1

A_2

L_2

\overrightarrow{PA_2}=\lambda\overrightarrow{PA_1}.

1

Putnam 2017 A6

30

edges of a regular icosahedron are distinguished by labeling them

1,2,\dots,30.

How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color?

1

Putnam 2017 A5

Each of the integers from

1

n

is written on a separate card, and then the cards are combined into a deck and shuffled. Three players,

A,B,

C,

take turns in the order

A,B,C,A,\dots

choosing one card at random from the deck. (Each card in the deck is equally likely to be chosen.) After a card is chosen, that card and all higher-numbered cards are removed from the deck, and the remaining cards are reshuffled before the next turn. Play continues until one of the three players wins the game by drawing the card numbered

1.

Show that for each of the three players, there are arbitrarily large values of

n

for which that player has the highest probability among the three players of winning the game.

1

Putnam 2017 A4

2N

students took a quiz, on which the possible scores were

0,1,\dots,10.

Each of these scores occurred at least once, and the average score was exactly

7.4.

Show that the class can be divided into two groups of

N

students in such a way that the average score for each group was exactly

7.4.

1

Putnam 2017 A3

a

b

be real numbers with

a<b,

f

g

be continuous functions from

[a,b]

(0,\infty)

\int_a^b f(x)\,dx=\int_a^b g(x)\,dx

f\ne g.

For every positive integer

n,

I_n=\int_a^b\frac{(f(x))^{n+1}}{(g(x))^n}\,dx.

I_1,I_2,I_3,\dots

is an increasing sequence with

\displaystyle\lim_{n\to\infty}I_n=\infty.

1

Putnam 2017 A2

Q_0(x)=1

Q_1(x)=x,

Q_n(x)=\frac{(Q_{n-1}(x))^2-1}{Q_{n-2}(x)}

n\ge 2.

Show that, whenever

n

is a positive integer,

Q_n(x)

is equal to a polynomial with integer coefficients.

1

Putnam 2017 A1

S

be the smallest set of positive integers such that
a)

2

S,

n

S

n^2

S,

(n+5)^2

S

n

S.

Which positive integers are not in

S?

S

is ``smallest" in the sense that

S

is contained in any other such set.)