2021 ITAMO

Part of ITAMO

Subcontests

(6)

1

Italian Mathematical Olympiad 2021 - Problem 3

A grid consists of

n\times n

n\in\mathbb{Z}^+

. In some of these points is a sentry. Every sentry chooses two directions, one perpendicular to the other (one vertical and the other horizontal) and watches over all the points that are found in the two chosen directions. Each sentry watches over her own point as well and the sentries on the edge of the grid can also watch the vacuum, depending on the directions they have chosen.
For instance, in the figure below, representing a disposition of

5

4\times 4

grid, the sentries in

A,\,B,\,C,\,D,\,E

1,\,3,\,4,\,5,\,7

points, respectively; points

D

E

are watched by one sentry, point

C

is watched by two sentries, points

A,\,B

F

are watched by three sentries.
(a) Prove that we can place

12

4\times 4

grid in such a way that every point of the grid is watched by at most two sentries. (b) Let

S(n)

be the maximum number of sentries we can place on an

n\times n

grid in such a way that every point of the grid is watched by at most two sentries. Prove that

3n\leq S(n)\leq 4n

n\geq 3

1

Periodic sequence

x_1, x_2, ..., x_n, ...

consists of an initial block of

p

positive distinct integers that then repeat periodically. This means that

\{x_1, x_2, \dots, x_p\}

p

distinct positive integers and

x_{n+p}=x_n

for every positive integer

n

. The terms of the sequence are not known and the goal is to find the period

p

. To do this, at each move it possible to reveal the value of a term of the sequence at your choice.
(a) Knowing that

1 \le p \le 10

, find the least

n

such that there is a strategy which allows to find

p

revealing at most

n

terms of the sequence. (b) Knowing that

p

is one of the first

k

prime numbers, find for which values of

k

there exist a strategy that allows to find

p

revealing at most

5

terms of the sequence.

1

Pirate sum

Given two fractions

a/b

c/d

we define their pirate sum as:

\frac{a}{b} \star \frac{c}{d} = \frac{a+c}{b+d}

where the two initial fractions are simplified the most possible, like the result. For example, the pirate sum of

2/7

4/5

1/2

. Given an integer

n \ge 3

, initially on a blackboard there are the fractions:

\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, ..., \frac{1}{n}

. At each step we choose two fractions written on the blackboard, we delete them and write at their place their pirate sum. Continue doing the same thing until on the blackboard there is only one fraction. Determine, in function of

n

, the maximum and the minimum possible value for the last fraction.

1

Perpendicularity and concurrence of two circles and a line

ABC

be an acute-angled triangle, let

M

be the midpoint of

BC

H

be the foot of the

B

Q

be the circumcenter of

ABM

X

be the intersection point between

BH

and the axis of

BC

.
Show that the circumcircles of the two triangles

ACM

AXH

CQ

pass through a same point if and only if

BQ

is perpendicular to

CQ

1

Two circumcircles are tangent

ABC

a triangle and let

I

be the center of its inscribed circle. Let

D

be the symmetric point of

I

with respect to

AB

E

be the symmetric point of

I

with respect to

AC

. Show that the circumcircles of the triangles

BID

CIE

are eachother tangent.

1

Zeros taker

A positive integer

m

is said to be \emph{zero taker} if there exists a positive integer

k

k

is a perfect square;

m

k

; the decimal expression of

k

contains at least

2021

'0' digits, but the last digit of

k

is not equal to

0

.
Find all positive integers that are zero takers.