MathDB

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Italian Mathematical Olympiad 2022 - Problem 6

ABC

be a non-equilateral triangle and let

R

be the radius of its circumcircle. The incircle of

ABC

has

I

as its centre and is tangent to side

CA

in point

D

and to side

CB

in point

E

. Let

A_1

be the point on line

EI

such that

A_1I=R

, with

I

being between

A_1

and

E

. Let

B_1

be the point on line

DI

such that

B_1I=R

, with

I

being between

B_1

and

D

. Let

P

be the intersection of lines

AA_1

and

BB_1

. (a) Prove that

P

belongs to the circumcircle of

ABC

. (b) Let us now also suppose that

AB=1

and

P

coincides with

C

. Determine the possible values of the perimeter of

ABC

.

5

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Italian Mathematical Olympiad 2022 - Problem 5

Robot "Mag-o-matic" manipulates

101

glasses, displaced in a row whose positions are numbered from

1

to

101

. In each glass you can find a ball or not. Mag-o-matic only accepts elementary instructions of the form

(a;b,c)

, which it interprets as "consider the glass in position

a

: if it contains a ball, then switch the glasses in positions

b

and

c

(together with their own content), otherwise move on to the following instruction" (it means that

a,\,b,\,c

are integers between

1

and

101

, with

b

and

c

different from each other but not necessarily different from

a

). A \emph{programme} is a finite sequence of elementary instructions, assigned at the beginning, that Mag-o-matic does one by one. A subset

S\subseteq \{0,\,1,\,2,\dots,\,101\}

is said to be \emph{identifiable} if there exists a programme which, starting from any initial configuration, produces a final configuration in which the glass in position

1

contains a ball if and only if the number of glasses containing a ball is an element of

S

. (a) Prove that the subset

S

of the odd numbers is identifiable. (b) Determine all subsets

S

that are identifiable.

4

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Italian Mathematical Olympiad 2022 - Problem 4

Alberto chooses

2022

integers

a_1,\,a_2,\dots,\,a_{2022}

(not necessarily positive and not necessarily distinct) and places them on a

2022\times 2022

table such that in the

(i,j)

cell is the number

a_k

, with

k=\max\{i,j\}

, as shown in figure (in which, for a better readability, we have denoted

a_{2022}

with

a_n

). Barbara does not know the numbers Alberto has chosen, but knows how they are displaced in the table. Given a positive integer

k

, with

1\leq k\leq 2022

, Barbara wants to determine the value of

a_k

(and she is not interested in determining the values of the other

a_i

's with

i\neq k

). To do so, Barbara is allowed to ask Alberto one or more questions, in each of which she demands the value of the sum of the numbers contained in the cells of a "path", where with the term "path" we indicate a sorted list of cells with the following characteristics: • the path starts from the top left cell and finishes with the bottom right cell, • the cells of the path are all distinct, • two consecutive cells of the path share a common side. Determine, as

k

varies, the minimum number of questions Barbara needs to find

a_k

.

3

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Italian Mathematical Olympiad 2022 - Problem 3

In a mathematical competition

n=10\,000

contestants participate. During the final party, in sequence, the first one takes

1/n

of the cake, the second one takes

2/n

of the remaining cake, the third one takes

3/n

of the cake that remains after the first and the second contestant, and so on until the last one, who takes all of the remaining cake. Determine which competitor takes the largest piece of cake.

2

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Italian Mathematical Olympiad 2022 - Problem 2

Let

ABC

be an acute triangle with

AB<AC

. Let then •

D

be the foot of the bisector of the angle in

A

, •

E

be the point on segment

BC

(different from

B

) such that

AB=AE

, •

F

be the point on segment

BC

(different from

B

) such that

BD=DF

, •

G

be the point on segment

AC

such that

AB=AG

. Prove that the circumcircle of triangle

EFG

is tangent to line

AC

.

1

Italian Mathematical Olympiad 2022 - Problem 1

Determine for which positive integers

n

there exists a positive integer

A

such that •

A

is divisible by

2022

, • the decimal expression of

A

contains only digits

0

and

7

, • the decimal expression of

A

contains exactly

n

times the digit

7

.

2022 ITAMO

Subcontests

Italian Mathematical Olympiad 2022 - Problem 6

Italian Mathematical Olympiad 2022 - Problem 5

Italian Mathematical Olympiad 2022 - Problem 4

Italian Mathematical Olympiad 2022 - Problem 3

Italian Mathematical Olympiad 2022 - Problem 2

Italian Mathematical Olympiad 2022 - Problem 1