2006 ITAMO

Part of ITAMO

Subcontests

(6)

1

Rightlines in spiral chessboard

The squares of an infinite chessboard are numbered

1,2,\ldots

along a spiral, as shown in the picture. A rightline is the sequence of the numbers in the squares obtained by starting at one square at going to the right. a) Prove that exists a rightline without multiples of

3

. b) Prove that there are infinitely many pairwise disjoint rightlines not containing multiples of

3

1

Coins in piles

Alberto and Barbara play the following game. Initially, there are some piles of coins on a table. Each player in turn, starting with Albert, performs one of the two following ways: 1) take a coin from an arbitrary pile; 2) select a pile and divide it into two non-empty piles. The winner is the player who removes the last coin on the table. Determine which player has a winning strategy with respect to the initial state.

1

Removing cards with sum 15

Two people play the following game: there are

40

cards numbered from

1

10

4

different signs. At the beginning they are given

20

cards each. Each turn one player either puts a card on the table or removes some cards from the table, whose sum is

15

. At the end of the game, one player has a

5

3

in his hand, on the table there's a

9

, the other player has a card in his hand. What is it's value?

1

Locus of orthocentre of triangle ABP

A

B

be two distinct points on the circle

\Gamma

, not diametrically opposite. The point

P

, distinct from

A

B

\Gamma

. Find the locus of the orthocentre of triangle

ABP

1

Itamo 2006/2

p^n+144=m^2

m,n\in \mathbb{N}

p

is a prime number.

1

Inequality true for positive reals

Consider the inequality

(a_1+a_2+\dots+a_n)^2\ge 4(a_1a_2+a_2a_3+\cdots+a_na_1).

n\ge 3

such that the inequality is true for positive reals. b) Find all

n\ge 3

such that the inequality is true for reals.