1997 Abels Math Contest (Norwegian MO)

Part of Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round

Subcontests

(6)

1

integer polynomial, f(a) = b, f(b) = a, f(x) = x has at most 1 integer root

p(x)

be a polynomial with integer coefficients. Suppose that there exist different integers

a

b

f(a) = b

f(b) = a

. Show that the equation

f(x) = x

has at most one integer solution.

1

91 students in a school are distributed in three classes

Ninety-one students in a school are distributed in three classes. Each student took part in a competition. It is known that among any six students of the same sex some two got the same number of points. Show that here are four students of the same sex who are in the same class and who got the same number of points.

1

Each subset of 97 out of 1997 given real numbers has positive sum

97

1997

given real numbers has positive sum. Show that the sum of all the

1997

numbers is positive.

1

product AE x AD is independent of the choice of E

A,B,C

be different points on a circle such that

AB = AC

E

lies on the segment

BC

D \ne A

is the intersection point of the circle and line

AE

. Show that the product

AE \cdot AD

is independent of the choice of

E

1

sum PQ+PR+PS is independent of the choice of P interior of equilateral

P

be an interior point of an equilateral triangle

ABC

Q,R,S

be the feet of perpendiculars from

P

AB,BC,CA

, respectively. Show that the sum

PQ+PR+PS

is independent of the choice of

P

1

happy naturals , a^2+b^2 = n, sum of perfect squares

We call a positive integer

n

happy if there exist integers

a,b

a^2+b^2 = n

t

is happy, show that (a)

2t

3t