2020 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 2020 A6

For a positive integer

N

f_N

be the function defined by

f_N (x)=\sum_{n=0}^N \frac{N+1/2-n}{(N+1)(2n+1)} \sin\left((2n+1)x \right).

Determine the smallest constant

M

f_N (x)\le M

N

x

1

Putnam 2020 A5

a_n

be the number of sets

S

of positive integers for which

\sum_{k\in S}F_k=n,

where the Fibonacci sequence

(F_k)_{k\ge 1}

F_{k+2}=F_{k+1}+F_k

F_1=1

F_2=1

F_3=2

F_4=3

. Find the largest number

n

a_n=2020

1

Putnam 2020 A4

Consider a horizontal strip of

N+2

squares in which the first and the last square are black and the remaining

N

squares are all white. Choose a white square uniformly at random, choose one of its two neighbors with equal probability, and color tis neighboring square black if it is not already black. Repeat this process until all the remaining white squares have only black neighbors. Let

w(N)

be the expected number of white squares remaining. Find

\lim_{N\to\infty}\frac{w(N)}{N}.

1

Putnam 2020 B5

j \in \{ 1,2,3,4\}

z_j

be a complex number with

| z_j | = 1

z_j \neq 1

3 - z_1 - z_2 - z_3 - z_4 + z_1z_2z_3z_4 \neq 0.

1

Putnam 2020 B4

n

be a positive integer, and let

V_n

be the set of integer

(2n+1)

\mathbf{v}=(s_0,s_1,\cdots,s_{2n-1},s_{2n})

s_0=s_{2n}=0

|s_j-s_{j-1}|=1

j=1,2,\cdots,2n

q(\mathbf{v})=1+\sum_{j=1}^{2n-1}3^{s_j},

M(n)

be the average of

\frac{1}{q(\mathbf{v})}

\mathbf{v}\in V_n

M(2020)

1

Putnam 2020 B3

x_0=1

\delta

be some constant satisfying

0<\delta<1

. Iteratively, for

n=0,1,2,\dots

x_{n+1}

is chosen uniformly form the interval

[0,x_n]

Z

be the smallest value of

n

x_n<\delta

. Find the expected value of

Z

, as a function of

\delta

1

Putnam 2020 B6

n

be a positive integer. Prove that

\sum_{k=1}^n (-1)^{\lfloor k (\sqrt{2} - 1) \rfloor} \geq 0.

\lfloor x \rfloor

denotes the greatest integer less than or equal to

x

1

Putnam 2020 B2

k

n

be integers with

1\leq k<n

. Alice and Bob play a game with

k

pegs in a line of

n

holes. At the beginning of the game, the pegs occupy the

k

leftmost holes. A legal move consists of moving a single peg to any vacant hole that is further to the right. The players alternate moves, with Alice playing first. The game ends when the pegs are in the

k

rightmost holes, so whoever is next to play cannot move and therefore loses. For what values of

n

k

does Alice have a winning strategy?

1

Putnam 2020 A3

a_0=\pi /2

a_n=\sin (a_{n-1})

n\ge 1

. Determine whether

\sum_{n=1}^{\infty}a_n^2

1

Putnam 2020 B1

For a positive integer

n

d(n)

to be the sum of the digits of

n

when written in binary (for example,

d(13)=1+1+0+1=3

S=\sum_{k=1}^{2020}(-1)^{d(k)}k^3.

S

2020

1

Putnam 2020 A2

k

be a nonnegative integer. Evaluate

\sum_{j=0}^k 2^{k-j} \binom{k+j}{j}.

1

Putnam 2020 A1

How many positive integers

N

satisfy all of the following three conditions?\\ (i)

N

is divisible by

2020

N

2020

decimal digits.\\ (iii) The decimal digits of

N

are a string of consecutive ones followed by a string of consecutive zeros.