# Difference between revisions of "MIE 2016"

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===Problem 1=== | ===Problem 1=== | ||

Let <math>M</math> be a 2x2 real matrix . Define a function <math>f(x)</math> that each element of the matrix moves to the next position in clockwise direction, in other words, if <math>M=\begin{pmatrix}a&b\\c&d\end{pmatrix}</math>, we have <math>f(M)=\begin{pmatrix}c&a\\d&b\end{pmatrix}</math>. Find all 2x2 real symmetric matrixes such that <math>M^2=f(M)</math>. | Let <math>M</math> be a 2x2 real matrix . Define a function <math>f(x)</math> that each element of the matrix moves to the next position in clockwise direction, in other words, if <math>M=\begin{pmatrix}a&b\\c&d\end{pmatrix}</math>, we have <math>f(M)=\begin{pmatrix}c&a\\d&b\end{pmatrix}</math>. Find all 2x2 real symmetric matrixes such that <math>M^2=f(M)</math>. | ||

+ | |||

+ | ===Problem 2=== | ||

+ | Solve the inequation, where <math>x\in\mathbb{R}</math>. | ||

+ | |||

+ | <math>\frac{9x^2}{(1-\sqrt{3x+1})^2}>4</math> | ||

+ | |||

+ | ===Problem 3=== | ||

+ | Solve the system, where <math>x,y\in\mathbb{R}</math>. | ||

+ | |||

+ | <math>\begin{cases}\log_{3}(\log_{\sqrt3}x)-\log_{\sqrt3}(\log_{3}y)=1\\(y\sqrt[3]{x})^2=3^{143} | ||

+ | \end{cases}</math> | ||

+ | |||

+ | ===Problem 4=== | ||

+ | Classify the following system as determined, possible indetermined and impossible according to the real values of <math>m</math>. | ||

+ | |||

+ | <math>\begin{cases}(m-2)x+2y-z=m+1\\2x+my+2z=m^2+2\\2mx+2(m+1)y+(m+1)z=m^3+3\end{cases}</math> | ||

+ | |||

+ | ===Problem 5=== | ||

+ | Let the complex numbers <math>z=a+bi</math> and <math>w=47+ci</math>, such that <math>z^3+w=0</math>. Find the value of <math>a</math>, <math>b</math> and <math>c</math>, knowing that they are positive integers. | ||

+ | |||

+ | ===Problem 6=== | ||

+ | A triangle <math>ABC</math> has its vertex at the origin of the cartesian system, its centroid is the point <math>D(3,2)</math> and its circumcenter is the point <math>E(55/18,5/6)</math>. Determine: | ||

+ | |||

+ | *The equation of the circumcircle of <math>\Delta_{ABC}</math>; | ||

+ | |||

+ | *The coordinates of the vertices <math>B</math> and <math>C</math>. | ||

+ | |||

+ | ===Problem 7=== | ||

+ | If <math>\frac{\cos x}{\cos y}+\frac{\sin x}{\sin y}=-1</math>, then compute <math>S</math>. | ||

+ | |||

+ | |||

+ | <math>S=\frac{3\cos{y}+\cos{3y}}{\cos{x}}+\frac{3\sin{y}-\sin{3y}}{\sin{x}}</math> | ||

+ | |||

+ | ===Problem 8=== | ||

+ | Let <math>A=\{1,2,3,4\}</math>. | ||

+ | |||

+ | *How many function from <math>A</math> to <math>A</math> have exactly 2 elements in its image set? | ||

+ | |||

+ | *Between the 256 functions from <math>A</math> to <math>A</math>, we draw the function <math>f</math> and <math>g</math>, knowing that can have repetition. What's the probability of <math>fog</math> be a constant function? | ||

{{stub}} | {{stub}} |

## Revision as of 18:15, 7 January 2018

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?

*This article is a stub. Help us out by expanding it.*