2020 ITAMO

Part of ITAMO

Subcontests

(6)

1

Classical "knight/liar" problem

In each cell of a table

8\times 8

lives a knight or a liar. By the tradition, the knights always say the truth and the liars always lie. All the inhabitants of the table say the following statement "The number of liars in my column is (strictly) greater than the number of liars in my row". Determine how many possible configurations are compatible with the statement.

1

Italian functions.

S

be the set of positive integers greater than or equal to

2

f: S\rightarrow S

f

satifies all the following three conditions: 1)

f

is surjective 2)

f

is increasing in the prime numbers(that is, if

p_1<p_2

are prime numbers, then

f(p_1)<f(p_2)

n\in S

f(n)

is the product of

f(p)

p

varies among all the primes which divide

n

f(360)=f(2^3\cdot 3^2\cdot 5)=f(2)\cdot f(3)\cdot f(5)

). Determine the maximum and the minimum possible value of

f(2020)

f

varies among all italian functions.

1

Isosceles and Concurrency

ABC

be an acute-angled triangle with

AB=AC

D

be the foot of perpendicular, of the point

C

AB

M

is the midpoint of

AC

. Finally, the point

E

is the second intersection of the line

BC

and the circumcircle of

\triangle CDM

. Prove that the lines

AE, BM

CD

are concurrents if and only if

CE=CM

1

Finitely (many) numbers

a_1, a_2, \dots, a_{2020}

b_1, b_2, \dots, b_{2020}

be real numbers(not necessarily distinct). Suppose that the set of positive integers

n

for which the following equation:

|a_1|x-b_1|+a_2|x-b_2|+\dots+a_{2020}|x-b_{2020}||=n

(1) has exactly two real solutions, is a finite set. Prove that the set of positive integers

n

for which the equation (1) has at least one real solution, is also a finite set.

1

Simple Italian NT

Determine all the pairs

(a,b)

of positive integers, such that all the following three conditions are satisfied: 1-

b>a

b-a

is a prime number 2- The last digit of the number

a+b

3

ab

is a square of an integer.

1

5 points and tangent(s)

\omega

be a circle and let

A,B,C,D,E

be five points on

\omega

in this order. Define

F=BC\cap DE

, such that the points

F

A

are on opposite sides, with regard to the line

BE

AE

is tangent to the circumcircle of the triangle

BFE

. a) Prove that the lines

AC

DE

are parallel b) Prove that

AE=CD