# Difference between revisions of "MIE 2016"

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

## Day 1

### Problem 1

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

### 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) $k\geq8$

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

### 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) $\frac{\pi}{24}$

### 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) $\frac{26}{25}$

### 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) $325$

### 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 $M^2=f(M)$.