MIE 2016

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

Day 1

Problem 1

Choose the correct answer.

(a) $\sqrt{2016}-\sqrt{2015}<\sqrt{2017}-\sqrt{2016}<(2\sqrt{2016})^{-1}$

(b) $\sqrt{2017}-\sqrt{2016}<\sqrt{2016}-\sqrt{2015}<(2\sqrt{2016})^{-1}$

(c) $\sqrt{2017}-\sqrt{2016}<(2\sqrt{2016})^{-1}<\sqrt{2016}-\sqrt{2015}$

(d) $\sqrt{2016}-\sqrt{2015}<(2\sqrt{2016})^{-1}<\sqrt{2017}-\sqrt{2016}$

(e) $(2\sqrt{2016})^{-1}<\sqrt{2017}-\sqrt{2016}<\sqrt{2016}-\sqrt{2015}$

Solution

Problem 2

The following system has $k$ integer solutions. We can say that:

$\begin{cases}\frac{x^2-2x-14}{x}>3\\\\x\leq12\end{cases}$

(a) $0\leq k\leq 2$

(b) $2\leq k\leq 4$

(c) $4\leq k\leq6$

(d) $6\leq k\leq8$

(e) $k\geq8$

Solution

Problem 3

Let $Z_1$ and $Z_2$ be complex numbers such that $Z_2$ is a pure imaginary number and $|Z_1-Z_2|=|Z_2|$ . For any values of $Z_1$ and $Z_2$ that satisfies these conditions we have:

(a) $\mbox{Im}(Z_2)>0$

(b) $\text{Im}(Z_2)\leq0$

(c) $|Z_1|\leq2|Z_2|$

(d) $\text{Re}(Z_1)\geq0$

(e) $\text{Re}(Z_1)\geq\text{Im}(Z_2)$

Solution

Problem 4

In the expansion of

$\left(x\sin2\beta+\frac{1}{x}\cos2\beta\right)^{10}$

the independent term (in other words, the term without $x$ ) is equal to $63/256$ . With $\beta$ being a real number such that $0< \beta<\pi/8$ and $x\neq0$ , the value of $\beta$ is:

(a) $\frac{\pi}{9}$

(b) $\frac{\pi}{12}$

(c) $\frac{\pi}{16}$

(d) $\frac{\pi}{18}$

(e) $\frac{\pi}{24}$

Solution

Problem 5

Compute $\frac{\sin^4\alpha+\cos^4\alpha}{\sin^6\alpha+\cos^6\alpha}$ , knowing that $\sin\alpha\cos\alpha=\frac{1}{5}$ .

(a) $\frac{22}{21}$

(b) $\frac{23}{22}$

(c) $\frac{25}{23}$

(d) $\frac{13}{12}$

(e) $\frac{26}{25}$

Solution

Problem 6

Let $A$ be $A=\begin{vmatrix}1&a&-2\\a-2&1&1\\2&-3&1\end{vmatrix}$ with $a\in\mathbb{R}$ . We know that $\text{det}(A^2-2A+I)=16$ . The sum of the values of $a$ that satisfies this condition is:

(a) $0$

(b) $1$

(c) $2$

(d) $3$

(e) $4$

Note: $\text{det}(X)$ is the determinant of the matrix $X$ .

Problem 7

The product of the real roots of the following equation is equal to:

$y^{\log_3\sqrt{3y}}=y^{\log_33y}-6,~~y>0$

(a) $\frac{1}{3}$

(b) $\frac{1}{2}$

(c) $\frac{3}{4}$

(d) $2$

(e) $3$

Problem 8

Let $f(x)=\sqrt{|x-1|+|x-2|+|x-3|+...+|x-2017|}$ . The minimum value of $f(x)$ is in the interval:

(a) $(-\infty,1008]$

(b) $(1008,1009]$

(c) $(1009,1010]$

(d) $(1010,1011]$

(e) $(1011,+\infty)$

Problem 9

Let $x$ , $y$ and $z$ be complex numbers that satisfies the following system:

$\begin{cases}x+y+z=7\\x^2+y^2+z^2=25\\\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{4}\end{cases}$

Compute $x^3+y^3+z^3$ .

(a) $210$

(b) $235$

(c) $250$

(d) $320$

(e) $325$

Solution

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) $12$

(b) $24$

(c) $36$

(d) $48$

(e) $96$

Problem 11

Let $(a_1,a_2,a_3,a_4,...)$ be an arithmetic progression and $(b_1,b_2,b_3,b_4,...)$ , an geometric progression of integer terms, of ratio $r$ and $q$ , respectively, where $r$ and $q$ are positive integers, with $q>2$ and $b_1>0$ . We also know that $a_1+b_2=3$ and $a_4+b_3=26$ . The value of $b_1$ is:

(a) $1$

(b) $2$

(c) $3$

(d) $4$

(e) $5$

Day 2

Problem 1

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

Problem 2

Solve the inequation, where $x\in\mathbb{R}$ .

$\frac{9x^2}{(1-\sqrt{3x+1})^2}>4$

Problem 3

Solve the system, where $x,y\in\mathbb{R}$ .

$\begin{cases}\log_{3}(\log_{\sqrt3}x)-\log_{\sqrt3}(\log_{3}y)=1\\(y\sqrt[3]{x})^2=3^{143} \end{cases}$

Problem 4

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

$\begin{cases}(m-2)x+2y-z=m+1\\2x+my+2z=m^2+2\\2mx+2(m+1)y+(m+1)z=m^3+3\end{cases}$

Problem 5

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

Problem 6

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

The equation of the circumcircle of $\Delta_{ABC}$ ;

The coordinates of the vertices $B$ and $C$ .

Problem 7

If $\frac{\cos x}{\cos y}+\frac{\sin x}{\sin y}=-1$ , then compute $S$ .

$S=\frac{3\cos{y}+\cos{3y}}{\cos{x}}+\frac{3\sin{y}-\sin{3y}}{\sin{x}}$

Problem 8

Let .

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

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

Problem 9

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

Problem 10

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

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MIE 2016

Contents

Day 1

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Day 2

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10