2012 ITAMO

Part of ITAMO

Subcontests

(6)

1

color every positive integer with A and B

Determine all pairs

\{a, b\}

of positive integers with the property that, in whatever manner you color the positive integers with two colors

A

B

, there always exist two positive integers of color

A

having their difference equal to

a

B

having their difference equal to

b

1

locus of P: APB+CPD=180

ABCD

is a square. Describe the locus of points

P

, different from

A, B, C, D

, on that plane for which

\widehat{APB}+\widehat{CPD}=180^\circ

1

Knights and scoundrels 2

n

be an integer greater than or equal to

2

n

people in one line, each of which is either a scoundrel (who always lie) or a knight (who always tells the truth). Every person, except the first, indicates a person in front of him/her and says "This person is a scoundrel" or "This person is a knight." Knowing that there are strictly more scoundrel than knights, seeing the statements show that it is possible to determine each person whether he/she is a scoundrel or a knight.

1

x_ax_b is a perfect square

x_1,x_2,x_3, \cdots

be a sequence defined by the following recurrence relation:

\begin{cases}x_{1}&= 4\\ x_{n+1}&= x_{1}x_{2}x_{3}\cdots x_{n}+5\text{ for }n\ge 1\end{cases}

The first few terms of the sequence are

x_1=4,x_2=9,x_3=41 \cdots

Find all pairs of positive integers

\{a,b\}

x_a x_b

is a perfect square.

1

300 times the sum of digits

Determine all positive integers that are equal to

300

times the sum of their digits.

1

1/x = 1/AB +1/AC

On the sides of a triangle

ABC

right angled at

A

D, E

F

BC, AC

AB

) are chosen so that the quadrilateral

AFDE

is a square. If

x

is the length of the side of the square, show that

\frac{1}{x}=\frac{1}{AB}+\frac{1}{AC}