# MIE 2016

Note: Anyone that solve any of the problems can post your solutions.

## Contents

## Day 1

### 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)

## Day 2

### Problem 1

Let be a 2x2 real matrix . Define a function that each element of the matrix moves to the next position in clockwise direction, in other words, if , we have . Find all 2x2 real symmetric matrixes such that .

### Problem 2

Solve the inequation, where .

### Problem 3

Solve the system, where .

### Problem 4

Classify the following system as determined, possible indetermined and impossible according to the real values of .

### Problem 5

Let the complex numbers and , such that . Find the value of , and , knowing that they are positive integers.

### Problem 6

A triangle has its vertex at the origin of the cartesian system, its centroid is the point and its circumcenter is the point . Determine:

- The equation of the circumcircle of ;

- The coordinates of the vertices and .

### Problem 7

If , then compute .

### Problem 8

Let .

- How many function from to have exactly 2 elements in its image set?

- Between the 256 functions from to , we draw the function and , knowing that can have repetition. What's the probability of be a constant function?

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