2022 Olympic Revenge

Part of Olympic Revenge

Subcontests

(5)

1

Du&scaron;an Đukić (?!)

Prove that there exists a positive integer

x<5^{2022}

\{\varphi\sqrt[3]{x}\}<\varphi^{-2022}.

1

How to solve this?

\{a_n\}_{n=1}^{\infty}

be a sequence of positive integers such that

a_1=1

n \geq 1

a_{n+1}

is the smallest positive integer, distinct from

a_1,a_2,...,a_n

\gcd(a_{n+1}a_n+1,a_i)=1

i=1,2,...,n

. Prove that every positive integer appears in

\{a_n\}_{n=1}^{\infty}

1

Vengeful numbers!

C

n-vengeful

if it is possible to color the cells of an

n \times n

chessboard such that: i) There is an equal number of cells of each color. ii) In each row or column, at least

Cn

cells have the same color. a) Show that

\frac{3}{4}

n-vengeful

for infinitely many values of

n

.
b) Show that it does not exist

n

\frac{4}{5}

n-vengeful

1

Bramerican Geo?

ABC

be a triangle and

\Omega

its circumcircle. Let the internal angle bisectors of

\angle BAC, \angle ABC, \angle BCA

BC,CA,AB

D,E,F

, respectively. The perpedincular line to

EF

D

EF

X

AD

EF

Z

. The circle internally tangent to

\Omega

AB,AC

\Omega

Y

(XYZ)

\Omega

1

Good pairs

(a,b)

of positive integers is good if

\gcd(a,b)=1

and for each pair of sets

A,B

of positive integers such that

A,B

are, respectively, complete residues system modulo

a,b

x \in A, y \in B

\gcd(x+y,ab)=1

. For each pair of positive integers

a,k

f(N)

b \leq N

b

k

distinct prime factors and

(a,b)

is good. Prove that

\liminf_{n \to \infty} f(n)/\frac{n}{(\log n)^k}\ge e^{k}