MIE 96/97
Contents
Problem 1
Solve the following system
Problem 2
Find the maximum term of the expansion of
Problem 3
Given the points 
and 
of the plane, find the equation of the geometric place of the points 
of the plane such that the ratio of the distances between from to and from to are given by a constant 
. Justify your answer discussing every possibility for .
Problem 4
On each of the six faces of a cube it has been drawn a circumference, where has been marked 
points. Considering that four points can't be on the same face and can't be coplanars, find how many lines and triangles, that aren't on the faces of this cube, are determined by the points.
Problem 5
Consider the function 
. Answer to the following questions:
(a) If 
, what's the relation between the curves of 
and 
?
(b) Can we say that the function defined by 
is a primitive form for the function 
?
Problem 6
If 
and 
are the roots of 
, then compute, in terms of 
and 
, the value of:
Consider 
and 
.
Problem 7
Consider the successively written odd numbers, like the following image, where the -line has numbers. Find in terms of , in this line, the sum of all written numbers.
Problem 8
Find the remainder of the division of the polynomial 
by 
, where is a natural number.
