# Difference between revisions of "MIE 2016"

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(e) <math>(2\sqrt{2016})^{-1}<\sqrt{2017}-\sqrt{2016}<\sqrt{2016}-\sqrt{2015}</math> | (e) <math>(2\sqrt{2016})^{-1}<\sqrt{2017}-\sqrt{2016}<\sqrt{2016}-\sqrt{2015}</math> | ||

+ | |||

+ | [[MIE 2016/Problem 1|Solution]] | ||

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

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(e) <math>k\geq8</math> | (e) <math>k\geq8</math> | ||

+ | |||

+ | [[MIE 2016/Day 1/Problem 2|Solution]] | ||

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

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(e) <math>\text{Re}(Z_1)\geq\text{Im}(Z_2)</math> | (e) <math>\text{Re}(Z_1)\geq\text{Im}(Z_2)</math> | ||

+ | |||

+ | |||

+ | [[MIE 2016/Day 1/Problem 3|Solution]] | ||

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

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(e) <math>\frac{\pi}{24}</math> | (e) <math>\frac{\pi}{24}</math> | ||

+ | |||

+ | |||

+ | [[MIE 2016/Day 1/Problem 4|Solution]] | ||

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

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(e) <math>\frac{26}{25}</math> | (e) <math>\frac{26}{25}</math> | ||

+ | |||

+ | |||

+ | [[MIE 2016/Day 1/Problem 5|Solution]] | ||

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

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(e) <math>325</math> | (e) <math>325</math> | ||

+ | |||

+ | [[MIE 2016/Day 1/Problem 9|Solution]] | ||

===Problem 10=== | ===Problem 10=== | ||

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==Day 2== | ==Day 2== | ||

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

− | Let <math>M</math> be a | + | 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? | ||

+ | |||

+ | ===Problem 9=== | ||

+ | In a triangle <math>\Delta_{ABC}</math>, let <math>AD</math> be the intern angle bisector such that it's the geometric mean of the <math>BD</math> and <math>DC</math>, and the measure of the median <math>AM</math> is the geometric mean of the sides <math>AB</math> and <math>AC</math>. The points <math>D</math> and <math>M</math> are on the side <math>BC</math> of side <math>a</math>. Compute the sides <math>AB</math> and <math>AC</math> of <math>\Delta_{ABC}</math> em terms of <math>a</math>. | ||

+ | |||

+ | ===Problem 10=== | ||

+ | In a equilateral cone there are two inscribed spheres of radii <math>\frac{\sqrt3-1}{\sqrt3+1}R</math> and <math>R</math>, like in the image. A surface is secant to the cone such that it's tangent to both spheres. Determine in terms of <math>R</math> the largest segment line that there is between the two points of the curve formed by the intersection of the surface with the cone. | ||

+ | |||

+ | [[File:MIE_2016_day_2_problem_10.png]] | ||

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

## Latest revision as of 08:46, 11 January 2018

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

## 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?

### Problem 9

In a triangle , let be the intern angle bisector such that it's the geometric mean of the and , and the measure of the median is the geometric mean of the sides and . The points and are on the side of side . Compute the sides and of em terms of .

### Problem 10

In a equilateral cone there are two inscribed spheres of radii and , like in the image. A surface is secant to the cone such that it's tangent to both spheres. Determine in terms of the largest segment line that there is between the two points of the curve formed by the intersection of the surface with the cone.

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