MathDB

1

Find the limit of a sequence of vectors

Define in the plane the sequence of vectors

v_1, v_2, \ldots

with initial values

v_1 = (1, 0)

,

v_2 = (-1/\sqrt{2}, 1/\sqrt{2})

and satisfying the relationship

v_n=\frac{v_{n-1}+v_{n-2}}{\lVert v_{n-1}+v_{n-2}\rVert},

for

n \geq 3

. Show that the sequence is convergent and determine its limit.
Note: The expression

\lVert v \rVert

denotes the length of the vector

v

.

4

1

Find the number of permutations!

Given a positive integer

n

, determine how many permutations

\sigma

of the set

\{1, 2, \ldots , 2022n\}

have the following property: for each

i \in \{1, 2, \ldots , 2021n + 1\}

, the number

\sigma(i) + \sigma(i + 1) + \cdots + \sigma(i + n - 1)

is a multiple of

n

.

3

1

Bounds of the last point of the operation

Danielle draws a point

O

on the plane and a set of points

\mathcal P = \{P_0, P_1, \ldots , P_{2022}\}

such that

\angle{P_0OP_1} = \angle{P_1OP_2} = \cdots = \angle{P_{2021}OP_{2022}} = \alpha, \hspace{5pt} 0 < \alpha < \pi,

where the angles are measured counterclockwise and for

0 \leq n \leq 2022

,

OP_n = r^n

, where

r > 1

is a given real number. Then, obtain new sets of points in the plane by iterating the following process: given a set of points

\{A_0, A_1, \ldots , A_n\}

in the plane, it is built a new set of points

\{B_0, B_1, \ldots , B_{n-1}\}

such that

A_kA_{k+1}B_k

is an equilateral triangle oriented clockwise for

0 \leq k \leq n - 1

. After carrying out the process

2022

times from the set

P

, Danielle obtains a single point

X

. If the distance from

X

to point

O

is

d

, show that

(r-1)^{2022} \leq d \leq (r+1)^{2022}.

1

Area of quadratic function

Given the function

f(x) = x^2

, the sector of

f

from

a

to

b

is defined as the limited region between the graph of

y = f(x)

and the straight line segment that joins the points

(a, f(a))

and

(b, f(b))

. Define the increasing sequence

x_0

,

x_1, \cdots

with

x_0 = 0

and

x_1 = 1

, such that the area of the sector of

f

from

x_n

to

x_{n+1}

is constant for

n \geq 0

. Determine the value of

x_n

in function of

n

.

2

1

Linear Algebra CIIM 2022

Let

v \in \mathbb{R}^2

a vector of length 1 and

A

a

2 \times 2

matrix with real entries such that: (i) The vectors

A v, A^2 v

y

A^3 v

are also of length 1. (ii) The vector

A^2 v

isn't equal to

\pm v

nor to

\pm A v

. Prove that

A^t A=I_2

.

6

1

Interesting tau inequality involving factorials

Prove that

\tau ((n+1)!) \leq 2 \tau (n!)

for all positive integers

n

.

2022 CIIM

Subcontests

Find the limit of a sequence of vectors

Find the number of permutations!

Bounds of the last point of the operation

Area of quadratic function

Linear Algebra CIIM 2022

Interesting tau inequality involving factorials