2018 ITAMO

Part of ITAMO

Subcontests

(6)

1

ITAMO 2018 problem 3

x_1,x_2, ... , x_n

be positive integers,Asumme that in their decimal representations no

x_i

x_j

.For instance ,

123

12

459

4

124

does not prolog

123

\frac {1}{x_1}+\frac {1}{x_2}+...+\frac {1}{x_n} < 3

1

ITAMO 2018 problem 2

2.

ABC

be an acute-angeled triangle , non-isosceles and with barycentre

G

(which is , in fact , the intersection of the medians).Let

M

be the midpoint of

BC

, and let Ω be the circle with centre

G

GM

N

be the point of intersection between Ω and

BC

that is distinct from

M

S

be the symmetric point of

A

with respect to

N

, that is , the point on the line

AN

AN=NS

GS

is perpendicular to

BC

1

ITAMO 2018 problem 4

4.

N

be an integer greater than

1

x

the smallest positive integer with the following property:there exists a positive integer

y

strictly less than

x-1

x

N+y

.Prove that x is either

p^n

2p

p

is a prime number and

n

is a positive integer

1

ITAMO problem 5

5.

Let x be a real number with

0<x<1

0.c_1c_2c_3...

be the decimal expansion of x.Denote by

B(x)

the set of all subsequences of

c_1c_2c_3

that consist of 6 consecutive digits. For instance ,

B(\frac{1}{22})={045454,454545,545454}

Find the minimum number of elements of

B(x)

x

varies among all irrational numbers with

0<x<1

1

ITAMO problem 1

1.

A bottle in the shape of a cone lies on its base. Water is poured into the bottle until its level reaches a distance of 8 centimeters from the vertex of the cone (measured vertically). We now turn the bottle upside down without changing the amount of water it contains; This leaves an empty space in the upper part of the cone that is 2 centimeters high.
Find the height of the bottle.

1

Italian MO Problem 6

ABC

be a triangle with

AB=AC

I

be its incenter. Let

\Gamma

be the circumcircle of

ABC

BI

CI

\Gamma

in two new points,

M

N

respectively. Let

D

be another point on

\Gamma

BC

A

E,F

be the intersections of

AD

BI

CI

, respectively. Let

P,Q

be the intersections of

DM

CI

DN

BI

respectively.
(i) Prove that

D,I,P,Q

lie on the same circle

\Omega

(ii) Prove that lines

CE

BF

\Omega