MathDB

1

The n-th term of a sequence is coprime with n

(a_{n})_{n \ge 1}

is defined by

a_{1}= 2

,

a_{n+1}= 2a_{n}^{2}-1

. Prove that for each positive integer n,

n

and

a_{n}

are coprime.

6

1

Minimizing a sum of inverses when variables have product 1

a) For each

n \ge 2

, find the maximum constant

c_{n}

such that

\frac 1{a_{1}+1}+\frac 1{a_{2}+1}+\ldots+\frac 1{a_{n}+1}\ge c_{n}

for all positive reals

a_{1},a_{2},\ldots,a_{n}

such that

a_{1}a_{2}\cdots a_{n}= 1

. b) For each

n \ge 2

, find the maximum constant

d_{n}

such that

\frac 1{2a_{1}+1}+\frac 1{2a_{2}+1}+\ldots+\frac 1{2a_{n}+1}\ge d_{n}

for all positive reals

a_{1},a_{2},\ldots,a_{n}

such that

a_{1}a_{2}\cdots a_{n}= 1

.

3

1

BC, a midpoint and a point on the median are concyclic

Let ABC be a triangle, G its centroid, M the midpoint of AB, D the point on the line

AG

such that

AG = GD, A \neq D

, E the point on the line

BG

such that

BG = GE, B \neq E

. Show that the quadrilateral BDCM is cyclic if and only if

AD = BE

.

1

Locus of points minimizing sums of distances with hexagon

It is given a regular hexagon in the plane. Let P be a point of the plane. Define s(P) as the sum of the distances from P to each side of the hexagon, and v(P) as the sum of the distances from P to each vertex. a) Find the locus of points P that minimize s(P) b) Find the locus of points P that minimize v(P)

4

1

Alberto plays a game for Barbara's birthday

Today is Barbara's birthday, and Alberto wants to give her a gift playing the following game. The numbers 0,1,2,...,1024 are written on a blackboard. First Barbara erases

2^{9}

numbers, then Alberto erases

2^{8}

numbers, then Barbara

2^{7}

and so on, until there are only two numbers a,b left. Now Barbara earns

|a-b|

euro. Find the maximum number of euro that Barbara can always win, independently of Alberto's strategy.

2

1

Common divisors of symilar polynomials

We define two polynomials with integer coefficients P,Q to be similar if the coefficients of P are a permutation of the coefficients of Q. a) if P,Q are similar, then

P(2007)-Q(2007)

is even b) does there exist an integer

k > 2

such that

k \mid P(2007)-Q(2007)

for all similar polynomials P,Q?

2007 ITAMO

Subcontests

The n-th term of a sequence is coprime with n

Minimizing a sum of inverses when variables have product 1

BC, a midpoint and a point on the median are concyclic

Locus of points minimizing sums of distances with hexagon

Alberto plays a game for Barbara's birthday

Common divisors of symilar polynomials