2021 CIIM

Part of CIIM

Subcontests

(6)

1

Calculate the overlapping area

0 < \alpha < 1

R(\alpha)

be the region in

\mathbb{R}^2

whose boundary is the convex pentagon of vertices

(0,1-\alpha), (\alpha, 0), (1, 0), (1,1)

(0, 1)

R

be the set of points that belong simultaneously to each of the regions

R(\alpha)

0 < \alpha < 1

R =\bigcap_{0<\alpha<1} R(\alpha)

.
Determine the area of

R

1

The sign of integration f(x)x^k cannot be alternating

0 \le a < b

be real numbers. Prove that there is no continuous function

f : [a, b] \to \mathbb{R}

such that \int_a^b f(x)x^{2n} \mathrm dx>0 \text{and} \int_a^b f(x)x^{2n+1} \mathrm dx <0 for every integer

n \ge 0

1

Liouville function on two certain primes

For every positive integer

n

s(n)

be the sum of the exponents of

71

97

in the prime factorization of

n

s(2021) = s(43 \cdot 47) = 0

s(488977) = s(71^2 \cdot 97) = 3

f(n)=(-1)^{s(n)}

, prove that the limit

\lim_{n \to +\infty} \frac{f(1) + f(2) + \cdots+ f(n)}{n}

exists and determine its value.

1

function, inequality, function, ine...

\mathbb{Z}^{+}

be the set of positive integers. a) Prove that there is only one function

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

, strictly increasing, such that

f(f(n))=2n+1

n\in \mathbb{Z}^{+}

. b) For the function in a. Prove that for every

n\in \mathbb{Z}^{+}

\frac{4n+1}{3}\leq f(n)\leq \frac{3n+1}{2}

c) Prove that in each inequality side of b the equality can reach by infinite positive integers

n

1

Table modulo N

m,n

N

be positive integers and

\mathbb{Z}_{N}=\{0,1,\dots,N-1\}

a set of residues modulo

N

. Consider a table

m\times n

such that each one of the

mn

cells has an element of

\mathbb{Z}_{N}

. A move is choose an element

g\in \mathbb{Z}_{N}

, a cell in the table and add

+g

to the elements in the same row/column of the chosen cell(the sum is modulo

N

). Prove that if

N

is coprime with

m-1,n-1,m+n-1

then any initial arrangement of your elements in the table cells can become any other arrangement using an finite quantity of moves.

1

Real equality in Poly...

r>s

be positive integers. Let

P(x)

Q(x)

be distinct polynomials with real coefficients, non-constant(s), such that

P(x)^r-P(x)^s=Q(x)^r-Q(x)^s

x\in \mathbb{R}

(r,s)=(2,1)