2020 CIIM

Part of CIIM

Subcontests

(6)

1

Divergent reciprocals, with sum-set density zero

A

A + A = \{a + b: a, b \in A \}

. Determine whether there exists a set

A

of positive integers such that \sum_{a \in A} \frac{1}{a} = +\infty \text{and} \lim_{n \rightarrow +\infty} \frac{|(A+A) \cap \{1,2,\cdots,n \}|}{n}=0.
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1

Ratios of areas: Arch and Triangle

\alpha>1

and consider the function

f(x)=x^{\alpha}

x \ge 0

t>0

M(t)

as the largest area that a triangle with vertices

(0, 0), (s, f(s)), (t, f(t))

could reach, for

s \in (0,t)

A(t)

be the area of the region bounded by the segment with endpoints

(0, 0)

(t, f(t))

and the graph of

y =f(x)

. (a) Show that

A(t)/M(t)

does not depend on

t

. We denote this value by

c(\alpha)

c(\alpha)

. (b) Determine the range of values of

c(\alpha)

\alpha

varies in the interval

(1, +\infty)

.
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1

Breaking a polynomial in a sum of polynomials

For each polynomial

P(x)

with real coefficients, define

P_0=P(0)

P_j(x)=x^j\cdot P^{(j)}(x)

P^{(j)}

j

-th derivative of

P

j\geq 1

. Prove that there exists one unique sequence of real numbers

b_0, b_1, b_2, \dots

such that for each polynomial

P(x)

with real coefficients and for each

x

P(x)=b_0P_0+\sum_{k\geq 1}b_kP_k(x)=b_0P_0+b_1P_1(x)+b_2P_2(x)+\dots

1

2021 equations

Determine all the positive real numbers

x_1, x_2, x_3, \dots, x_{2021}

x_{i+1}=\frac{x_i^3+2}{3x_i^2}

i=1, 2, 3, \dots, 2020

x_{2021}=x_1

1

Sets and (maybe) graphs

(m,r,s,t)

be positive integers such that

m\geq s+1

r\geq t

m

A_1, A_2, \dots, A_m

r

elements each one. Suppose that, for each

1\leq i\leq m

, there exist at least

t

A_i

, such that each one(element) belongs to (at least)

s

A_j

j\neq i

. Determine the greatest quantity of elements in the following set

A_1 \cup A_2 \cup A_3 \dots \cup A_m

1

Cubic and Quadratic

Find all triples of positive integers

(a,b,c)

such that the following equations are both true: I-

a^2+b^2=c^2

a^3+b^3+1=(c-1)^3