Difference between revisions of "MIE 2016"
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Revision as of 17:01, 7 January 2018
Contents
Problem 1
Choose the correct answer.
(a)
(b)
(c)
(d)
(e)
Problem 2
The following system has integer solutions. We can say that:
(a)
(b)
(c)
(d)
(e)
Problem 3
Let and be complex numbers such that is a pure imaginary number and . For any values of and that satisfies these conditions we have:
(a)
(b)
(c)
(d)
(e)
Problem 4
In the expansion of
the independent term (in other words, the term without ) is equal to . With being a real number such that and , the value of is:
(a)
(b)
(c)
(d)
(e)
Problem 5
Compute , knowing that .
(a)
(b)
(c)
(d)
(e)
Problem 6
Let be with . We know that . The sum of the values of that satisfies this condition is:
(a)
(b)
(c)
(d)
(e)
Note: is the determinant of the matrix .
Problem 7
The product of the real roots of the following equation is equal to:
(a)
(b)
(c)
(d)
(e)
Problem 8
Let . The minimum value of is in the interval:
(a)
(b)
(c)
(d)
(e)
Problem 9
Let , and be complex numbers that satisfies the following system:
Compute .
(a)
(b)
(c)
(d)
(e)
Problem 10
A hexagon is divided into 6 equilateral triangles. How many ways can we put the numbers from 1 to 6 in each triangle, without repetition, such that the sum of the numbers of three adjacent triangles is always a multiple of 3? Solutions obtained by rotation or reflection are differents, thus the following figures represent two distinct solutions.
(a)
(b)
(c)
(d)
(e)
Problem 11
Let be an arithmetic progression and , an geometric progression of integer terms, of ratio and , respectively, where and are positive integers, with and . We also know that and . The value of is:
(a)
(b)
(c)
(d)
(e)