MathDB

1

sum_{n=1}^{1.000.000}[ \sqrt{n} ]

\sum_{n=1}^{1.000.000}[ \sqrt{n} ]

. You may use the formula

\sum_{i=1}^{k} i^2=\frac{k(k +1)(2k +1)}{6}

without a proof.

6

1

increasing functions, multiplicative and completely multiplicative

We say that a function

f : N \to N

is increasing if

f(n) < f(m)

whenever

n < m

, multiplicative if

f(nm) = f(n)f(m)

whenever

n

and

m

are coprime, and completely multiplicative if

f(nm) = f(n)f(m)

for all

n,m

. (a) Prove that if

f

is increasing then

f(n) \ge n

for each

n

. (b) Prove that if

f

is increasing and completely multiplicative and

f(2) = 2

, then

f(n) = n

for all

n

. (c) Does (b) remain true if the word ”completely” is omitted?

5

1

P(a_1) = P(a_2) = P(a_3) = P(a_4) = 1, P(n) = 12 impossible, P(n) = 1998

Suppose

a_1,a_2,a_3,a_4

are distinct integers and

P(x)

is a polynomial with integer coefficients satisfying

P(a_1) = P(a_2) = P(a_3) = P(a_4) = 1

. (a) Prove that there is no integer

n

such that

P(n) = 12

. (b) Do there exist such a polynomial and

a_n

integer

n

such that

P(n) = 1998

?

4

1

BC^2 = CD \cdot CE wanted, trapezoid with perpendicular diagonals given,

Let

ABCD

be a trapezoid with the longer base

AB

such that its diagonals

AC

and

BD

are perpendicular. Let

O

be the circumcenter of the triangle

ABC

and

E

be the intersection of the lines

OB

and

CD

. Prove that

BC^2 = CD \cdot CE

.

3

1

italian poker game between 7 friends

Alberto wants to organize a poker game with his friends this evening. Bruno and Barbara together go to gym once in three evenings, whereas Carla, Corrado, Dario and Davide are busy once in two evenings (not necessarily the same day). Moreover, Dario is not willing to play with Davide, since they have a quarrel over a girl. A poker game requires at least four persons (including Alberto). What is the probability that the game will be played?

1998 ITAMO

Subcontests

sum_{n=1}^{1.000.000}[ \sqrt{n} ]

increasing functions, multiplicative and completely multiplicative

P(a_1) = P(a_2) = P(a_3) = P(a_4) = 1, P(n) = 12 impossible, P(n) = 1998

BC^2 = CD \cdot CE wanted, trapezoid with perpendicular diagonals given,

italian poker game between 7 friends