2005 ITAMO

Part of ITAMO

Subcontests

(3)

2

1 or 2 is written in a 4X4 grid

In each cell of a

4 \times 4

1

2

is written. Suppose that the sum of the digits in each of the four

3 \times 3

sub-tables is divisible by

4

, but the sum of the digits in the entire table is not divisible by

4

. Find the greatest and the smallest possible value of the sum of the

16

CD bisects arc EF

\gamma_1, \gamma_2

in a plane, with centers

A

B

respectively, intersect at

C

D

. Suppose that the circumcircle of

ABC

\gamma_1

E

\gamma_2

F

, where the arc

EF

C

\gamma_1

\gamma_2

. Prove that this arc

EF

is bisected by the line

CD

2

At least 1 among 18 divisible by the sum of its digits

Prove that among any

18

consecutive positive integers not exceeding

2005

there is at least one divisible by the sum of its digits.

For which h, a_n=1

h

be a positive integer. The sequence

a_n

a_0 = 1

a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{ if } a_n \text{ is even }\\\\a_n+h \text{ otherwise }.\end{array}

h = 27

a_1=28, a_2 = 14, a_3 = 7, a_4 = 34

h

n > 0

a_n = 1

2

find all possible values of AH/CH

ABC

be a right angled triangle with hypotenuse

AC

H

be the foot of the altitude from

B

AC

. Knowing that there is a right-angled triangle with side-lengths

AB, BC, BH

, determine all the possible values of

\frac{AH}{CH}

find the greatest n

n \geq 3

for which there are

n

positive integers

a_1, \cdots , a_n

any two of which have a common divisor greater than

1

, but any three of which are coprime. Assuming that, moreover, the numbers

a_i

5000

, find the greatest possible

n