2022 Putnam

Part of Putnam

Subcontests

(12)

1

2022 Putnam B6

Find all continuous functions

f:\mathbb{R}^+\rightarrow \mathbb{R}^+

f(xf(y))+f(yf(x))=1+f(x+y)

x, y>0.

1

2022 Putnam B5

0 \leq p \leq 1/2,

X_1, X_2, \ldots

be independent random variables such that

X_i=\begin{cases} 1 & \text{with probability } p, \\ -1 & \text{with probability } p, \\ 0 & \text{with probability } 1-2p, \end{cases}

i \geq 1.

Given a positive integer

n

b,a_1, \ldots, a_n,

P(b, a_1, \ldots, a_n)

denote the probability that

a_1X_1+ \ldots + a_nX_n=b.

For which values of

p

is it the case that

P(0, a_1, \ldots, a_n) \geq P(b, a_1, \ldots, a_n)

for all positive integers

n

and all integers

b, a_1,\ldots, a_n?

1

2022 Putnam B4

Find all integers

n

n \geq 4

for which there exists a sequence of distinct real numbers

x_1, \ldots, x_n

such that each of the sets

\{x_1, x_2, x_3\}, \{x_2, x_3, x_4\},\ldots,\{x_{n-2}, x_{n-1}, x_n\}, \{x_{n-1}, x_n, x_1\},\text{ and } \{x_n, x_1, x_2\}

forms a 3-term arithmetic progression when arranged in increasing order.

1

2022 Putnam B3

Assign to each positive real number a color, either red or blue. Let

D

be the set of all distances

d>0

such that there are two points of the same color at distance

d

apart. Recolor the positive reals so that the numbers in

D

are red and the numbers not in

D

are blue. If we iterate the recoloring process, will we always end up with all the numbers red after a finite number of steps?

1

2022 Putnam B2

\times

represent the cross product in

\mathbb{R}^3.

For what positive integers

n

does there exist a set

S \subset \mathbb{R}^3

n

elements such that

S=\{v \times w: v, w \in S\}?

1

2022 Putnam B1

P(x)=a_1x+a_2x^2+\ldots+a_nx^n

is a polynomial with integer coefficients, with

a_1

odd. Suppose that

e^{P(x)}=b_0+b_1x+b_2x^2+\ldots

x.

b_k

is nonzero for all

k \geq 0.

1

2022 Putnam A6

n

be a positive integer. Determine, in terms of

n,

the largest integer

m

with the following property: There exist real numbers

x_1,\ldots, x_{2n}

-1<x_1<x_2<\ldots<x_{2n}<1

such that the sum of the lengths of the

n

[x_1^{2k-1},x_2^{2k-1}], [x_3^{2k-1},x_4^{2k-1}], \ldots, [x_{2n-1}^{2k-1},x_{2n}^{2k-1}]

is equal to 1 for all integers

k

1\leq k \leq m.

1

2022 Putnam A5

Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. By rule, a tile must not cover a square that is already covered by another tile. The game ends when no tile can be placed according to this rule. Alice's goal is to maximize the number of uncovered squares when the game ends; Bob's goal is to minimize it. What is the greatest number of uncovered squares that Alice can ensure at the end of the game, no matter how Bob plays?

1

2022 Putnam A4

X_1, X_2, \ldots

are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let

S=\sum_{i=1}^kX_i/2^i,

k

is the least positive integer such that

X_k<X_{k+1},

k=\infty

if there is no such integer. Find the expected value of

S.

1

2022 Putnam A3

p

be a prime number greater than 5. Let

f(p)

denote the number of infinite sequences

a_1, a_2, a_3,\ldots

a_n \in \{1, 2,\ldots, p-1\}

a_na_{n+2}\equiv1+a_{n+1}

p

n\geq 1.

f(p)

is congruent to 0 or 2 (mod 5).

1

2022 Putnam A2

n

be an integer with

n\geq 2.

Over all real polynomials

p(x)

n,

what is the largest possible number of negative coefficients of

p(x)^2?

1

2022 Putnam A1

Determine all ordered pairs of real numbers

(a,b)

such that the line

y=ax+b

intersects the curve

y=\ln(1+x^2)

in exactly one point.