1991 Nordic

Part of Nordic

Subcontests

(4)

1

integer f, f(n),f(n+1),... , f(n+k-1) are not divisible by k

f(x)

be a polynomial with integer coefficients. We assume that there exists a positive integer

k

k

consecutive integers

n, n+1, ... , n+k -1

so that none of the numbers

f(n), f(n+ 1),... , f(n + k - 1)

is divisible by

k

. Show that the zeroes of

f(x)

are not integers.

1

prove that 1/2^2 +1/3^2 +...+1/n^2 <2/3

\frac{1}{2^2} +\frac{1}{3^2} +...+\frac{1}{n^2} <\frac{2}{3}

n \ge 2

1

find point so that area inside a trapezium is largest

In the trapezium

ABCD

AB

CD

are parallel, and

E

is a fixed point on the side

AB

. Determine the point

F

CD

so that the area of the intersection of the triangles

ABF

CDE

is as large as possible.

1

last 2 digits 2^5 + 2^{5^2}} + 2^{5^{3}} +... + 2^{5^{1991}}

Determine the last two digits of the number

2^5 + 2^{5^{2}} + 2^{5^{3}} +... + 2^{5^{1991}}

, written in decimal notation.