MathDB

1

(m,n) =a f(m,n-1)+(1- a)f(m -1,n-1), | f(m,n)| < 1989

\alpha

, a function

f

is defined on pairs of nonnegative integers by

f(0,0) = 1, f(m,0) = f(0,m) = 0

for

m > 0

,

f(m,n) = \alpha f(m,n-1)+(1- \alpha)f(m -1,n-1)

for

m,n > 0

. Find the values of

\alpha

such that

| f(m,n)| < 1989

holds for any integers

m,n \ge 0

.

5

1

1 marker for every ”head” and 2markers for every ”tail”, with a fair coin

A fair coin is repeatedly tossed. We receive one marker for every ”head” and two markers for every ”tail”. We win the game if, at some moment, we possess exactly

100

markers. Is the probability of winning the game greater than, equal to, or less than

2/3

?

4

1

cyclic wanted, 5 points concyclic given

Points

A,M,B,C,D

are given on a circle in this order such that

A

and

B

are equidistant from

M

. Lines

MD

and

AC

intersect at

E

and lines

MC

and

BD

intersect at

F

. Prove that the quadrilateral

CDEF

is inscridable in a circle.

3

1

tetrahedra PABC,PABD,PACD,PBCD have equal volumes, P interior of tetr.ABCD

Prove that, for every tetrahedron

ABCD

, there exists a unique point

P

in the interior of the tetrahedron such that the tetrahedra

PABC,PABD,PACD,PBCD

have equal volumes.

2

1

30 men with their 30 wives sitting at a round table

There are

30

men with their

30

wives sitting at a round table. Show that there always exist two men who are on the same distance from their wives. (The seats are arranged at vertices of a regular polygon.)

1

x^2 +xy+y^2 = 2 in Q

Determine whether the equation

x^2 +xy+y^2 = 2

has a solution

(x,y)

in rational numbers.

1989 ITAMO

Subcontests

(m,n) =a f(m,n-1)+(1- a)f(m -1,n-1), | f(m,n)| < 1989

1 marker for every ”head” and 2markers for every ”tail”, with a fair coin

cyclic wanted, 5 points concyclic given

tetrahedra PABC,PABD,PACD,PBCD have equal volumes, P interior of tetr.ABCD

30 men with their 30 wives sitting at a round table

x^2 +xy+y^2 = 2 in Q

1989 ITAMO

Subcontests

(m,n) =a f(m,n-1)+(1- a)f(m -1,n-1), | f(m,n)| &lt; 1989

1 marker for every &rdquo;head&rdquo; and 2markers for every &rdquo;tail&rdquo;, with a fair coin

cyclic wanted, 5 points concyclic given

tetrahedra PABC,PABD,PACD,PBCD have equal volumes, P interior of tetr.ABCD

30 men with their 30 wives sitting at a round table

x^2 +xy+y^2 = 2 in Q

(m,n) =a f(m,n-1)+(1- a)f(m -1,n-1), | f(m,n)| < 1989

1 marker for every ”head” and 2markers for every ”tail”, with a fair coin