2023 Putnam

Part of Putnam

Subcontests

(12)

1

2023 Putnam B6

n

be a positive integer. For

i

j

\{1,2, \ldots, n\}

s(i, j)

be the number of pairs

(a, b)

of nonnegative integers satisfying

a i+b j=n

S

n

-by-n matrix whose

(i, j)

s(i, j)

. For example, when

n=5

S=\left[\begin{array}{lllll}6 & 3 & 2 & 2 & 2 \\ 3 & 0 & 1 & 0 & 1 \\ 2 & 1 & 0 & 0 & 1 \\ 2 & 0 & 0 & 0 & 1 \\ 2 & 1 & 1 & 1 & 2\end{array}\right]

. Compute the determinant of

S

1

2023 Putnam B5

Determine which positive integers

n

have the following property: For all integers

m

that are relatively prime to

n

, there exists a permutation

\pi:\{1,2, \ldots, n\} \rightarrow\{1,2, \ldots, n\}

\pi(\pi(k)) \equiv m k(\bmod n)

k \in\{1,2, \ldots, n\}

1

2023 Putnam B4

For a nonnegative integer

n

and a strictly increasing sequence of real numbers

t_0, t_1, \ldots, t_n

f(t)

be the corresponding real-valued function defined for

t \geq t_0

by the following properties: (a)

f(t)

is continuous for

t \geq t_0

, and is twice differentiable for all

t>t_0

t_1, \ldots, t_n

f\left(t_0\right)=1 / 2

\lim _{t \rightarrow t_k^{+}} f^{\prime}(t)=0

0 \leq k \leq n

0 \leq k \leq n-1

f^{\prime \prime}(t)=k+1

t_k<t<t_{k+1}

f^{\prime \prime}(t)=n+1

t>t_n

. Considering all choices of

n

t_0, t_1, \ldots, t_n

t_k \geq t_{k-1}+1

1 \leq k \leq n

, what is the least possible value of

T

f\left(t_0+T\right)=2023

1

2023 Putnam B3

y_1, y_2, \ldots, y_k

of real numbers is called

<span class='latex-italic'>zigzag</span>

k=1

y_2-y_1, y_3-y_2, \ldots, y_k-y_{k-1}

are nonzero and alternate in sign. Let

X_1, X_2, \ldots, X_n

be chosen independently from the uniform distribution on

[0,1]

a\left(X_1, X_2, \ldots, X_n\right)

be the largest value of

k

for which there exists an increasing sequence of integers

i_1, i_2, \ldots, i_k

X_{i_1}, X_{i_2}, \ldots X_{i_k}

is zigzag. Find the expected value of

a\left(X_1, X_2, \ldots, X_n\right)

n \geq 2

1

2023 Putnam B2

For each positive integer

n

k(n)

be the number of ones in the binary representation of

2023 \cdot n

. What is the minimum value of

k(n)

1

2023 Putnam B1

m

n

grid of unit squares, indexed by

(i, j)

1 \leq i \leq m

1 \leq j \leq n

(m-1)(n-1)

coins, which are initially placed in the squares

(i, j)

1 \leq i \leq m-1

1 \leq j \leq n-1

. If a coin occupies the square

(i, j)

i \leq m-1

j \leq n-1

and the squares

(i+1, j),(i, j+1)

(i+1, j+1)

are unoccupied, then a legal move is to slide the coin from

(i, j)

(i+1, j+1)

. How many distinct configurations of coins can be reached starting from the initial configuration by a (possibly empty) sequence of legal moves?

1

2023 Putnam A6

Alice and Bob play a game in which they take turns choosing integers from 1 to

n

. Before any integers are chosen, Bob selects a goal of "odd" or "even". On the first turn, Alice chooses one of the

n

integers. On the second turn, Bob chooses one of the remaining integers. They continue alternately choosing one of the integers that has not yet been chosen, until the

n

th turn, which is forced and ends the game. Bob wins if the parity of

\{k

k

was chosen on the

k

\}

matches his goal. For which values of

n

does Bob have a winning strategy?

1

2023 Putnam A5

For a nonnegative integer

k

f(k)

be the number of ones in the base 3 representation of

k

. Find all complex numbers

z

\sum_{k=0}^{3^{1010}-1}(-2)^{f(k)}(z+k)^{2023}=0

1

2023 Putnam A4

v_1, \ldots, v_{12}

be unit vectors in

\mathbb{R}^3

from the origin to the vertices of a regular icosahedron. Show that for every vector

v \in \mathbb{R}^3

\varepsilon>0

, there exist integers

a_1, \ldots, a_{12}

\left\|a_1 v_1+\cdots+a_{12} v_{12}-v\right\|<\varepsilon

1

2023 Putnam A3

Determine the smallest positive real number

r

such that there exist differentiable functions

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

g: \mathbb{R} \rightarrow \mathbb{R}

f(0)>0

g(0)=0

\left|f^{\prime}(x)\right| \leq|g(x)|

x

\left|g^{\prime}(x)\right| \leq|f(x)|

x

f(r)=0

1

2023 Putnam A2

n

be an even positive integer. Let

p

be a monic, real polynomial of degree

2 n

; that is to say,

p(x)=

x^{2 n}+a_{2 n-1} x^{2 n-1}+\cdots+a_1 x+a_0

for some real coefficients

a_0, \ldots, a_{2 n-1}

p(1 / k)=k^2

for all integers

k

1 \leq|k| \leq n

. Find all other real numbers

x

p(1 / x)=x^2

1

2023 Putnam A1

For a positive integer

n

f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)

. Find the smallest

n

\left|f_n^{\prime \prime}(0)\right|>2023