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>
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[[MIE 2016/Problem 1|Solution]]
  
 
===Problem 2===
 
===Problem 2===
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(e) <math>k\geq8</math>
 
(e) <math>k\geq8</math>
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[[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>
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[[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>
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[[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>
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[[MIE 2016/Day 1/Problem 5|Solution]]
  
 
===Problem 6===
 
===Problem 6===
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(e) <math>325</math>
 
(e) <math>325</math>
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[[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 <math>2x2</math> 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>.
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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>.
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 +
===Problem 2===
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Solve the inequation, where <math>x\in\mathbb{R}</math>.
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 +
<math>\frac{9x^2}{(1-\sqrt{3x+1})^2}>4</math>
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 +
===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>
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 +
===Problem 4===
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Classify the following system as determined, possible indetermined and impossible according to the real values of <math>m</math>.
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 +
<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>
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 +
===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>;
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 +
*The coordinates of the vertices <math>B</math> and <math>C</math>.
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 +
===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>.
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 +
===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.
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 +
[[File:MIE_2016_day_2_problem_10.png]]
  
  
  
 
{{stub}}
 
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Latest revision as of 09:46, 11 January 2018

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

Day 1

Problem 1

Choose the correct answer.

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

(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.

MIE 2016 problem 10.png


(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 $A=\{1,2,3,4\}$.
  • 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.

MIE 2016 day 2 problem 10.png


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