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Difference between revisions of "MIE 2015"

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(e) <math>2</math>
 
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[[MIE 2015/Day 1/Problem 2|Solution]]
  
  
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(e) <math>\binom{2022}{5}</math>
 
(e) <math>\binom{2022}{5}</math>
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==Day 2==
 +
===Problem 1===
 +
The ingeters <math>a_1,a_2,a_3,...,a_{25}</math> are in arithmetic progression of ratio not equal to zero. The sequences <math>(a_1,a_2,a_{10})</math> and <math>(a_6,a_j,a_{25})</math> are both in geometric progression. Find <math>j</math>.
 +
 +
 +
===Problem 2===
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Let the functions <math>f_n</math>, for <math>n\in\{0,1,2,3,...\}</math>, such that <math>f_0(x)=\frac{1}{1-x}</math> and <math>f_n(x)=f_0(f_{n-1}(x))</math>, for every <math>n\geq1</math>.
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Compute <math>f_{2016}(2016)</math>.
 +
 +
 +
===Problem 3===
 +
Let <math>Z</math> be a complex number such that <math>\frac{2Z}{\overline{Z}i}</math> has an argument <math>(\theta)</math> equal to <math>\frac{3\pi}{4}</math> and <math>\log_3(2Z+2\overline{Z}+1)=2</math>. Find the complex number <math>Z</math>.
 +
 +
 +
===Problem 4===
 +
Define <math>A</math> as the matrix 2016 x 2016, such that its elements satisfy the equality:
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 +
<math>a_{i,j}=\begin{pmatrix}i+j-2\\j-1\end{pmatrix}</math>, for <math>i,j\in\{1,2,...,2016\}</math>.
 +
 +
Compute det(A).
 +
 +
 +
===Problem 5===
 +
Find the solution of the equation:
 +
 +
<math>\sin x\left(1+\tan x\tan\frac{x}{2}\right)=4-\cot x</math>
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 +
 +
===Problem 6===
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Let the equation <math>n^2-7m^2=(5m-2n)^2+49</math>. Find all the pairs of integers <math>(m,n)</math> that satisfy this equation.
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 +
 +
===Problem 7===
 +
Three player sits around a table and play, alternatively, a dice of six faces. The first player throw the dice, then the player at its left plays, and then it goes, until the finish of the game. The one that throw the dice and get the side with number 6, wins the game and it ends. If a player gets the number 1, he loses its chance and the player at its right will throw the dice. The game will continue until someone gets the number 6. What's the probability of the first player to throw the dice to win?
 +
 +
 +
===Problem 8===
 +
The equation of the circumference <math>C</math> is <math>x^2+y^2=16</math>. Let <math>C'</math> be a circumference of radius 1 wich moves internally tangentiating the circumference <math>C</math>, without slipping between the points of contact, in other words, <math>C'</math> internally rolls on <math>C</math>.
 +
 +
 +
[[Image:MIE_2015_day_1_problem_8.png]]
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 +
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Let the point <math>P</math> be on <math>C'</math> such that in the beginning of the movement of <math>C'</math> the point <math>P</math> is at the tangent point <math>(4,0)</math>, like in the figure a. After some movement, the angle between
  
  
 
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Revision as of 13:32, 11 January 2018

Day 1

Problem 1

Given any three sets $F$, $G$ and $H$. The set $G-H$ is equal to the set:

(a) $(G\cup F)-(F-H)$

(b) $(G\cup H)-(H-F)$

(c) $(G\cup(H-F))\cap\overline{H}$

(d) $\overline{G}\cup(H\cap F)$

(e) $(\overline{H}\cap G)\cap(G-F)$


Problem 2

The polynomial $x^3+ax^2+bx+c$ has real roots $\alpha$, $-\alpha$ and $\frac{1}{\alpha}$. Thus the value of the sum of $b+c^2+ac+\frac{b}{c^2}$ is:

(a) $-2$

(b) $-1$

(c) $0$

(d) $1$

(e) $2$


Solution


Problem 3

Let $m$ and $n$ be positive integers such that $3^m+14400=n^2$. Find the remainder of the division of $m+n$ by $5$.

(a) $0$

(b) $1$

(c) $2$

(d) $3$

(e) $4$


Problem 4

Compute

$\sum_{k=1}^{15}$ Img$\left(\right.$cis$\left.^{2k-1}\frac{\pi}{36}\right)$

(a) $\frac{2+\sqrt3}{4\sin\frac{\pi}{36}}$

(b) $\frac{2-\sqrt3}{4\sin\frac{\pi}{36}}$

(c) $\frac{1}{4\sin\frac{\pi}{36}}$

(d) $\sin\frac{\pi}{36}$

(e) $\frac{1}{4}$

Note: Img(w) is the imaginary part of w.


Problem 5

Let $P(x)=x^2+ax+b$. It's known that $P(x)$ and $P(P(P(x)))$ have a root in common. Therefore, we can say that for every value of $a$ and $b$:

(a) $P(-1)P(1)<0$

(b) $P(-1)P(1)=0$

(c) $P(-1)+P(1)=2$

(d) $P(0)P(1)=0$

(e) $P(0)+P(1)=0$


Problem 6

Let $(a,b,c)$ be a geometric progression and $\log\left(\frac{5c}{a}\right)$, $\log\left(\frac{3b}{5c}\right)$ and $\log\left(\frac{a}{3b}\right)$ be a arithmetic progression, both in these order, so we can say that $a$, $b$ and $c$:

(a) are the sides of a obtusangle triangle.

(b) are the sides of a acutangle triangle that's not equilateral.

(c) are the sides of a equilateral triangle.

(d) are the sides of a right triangle.

(e) can't be the sides of a triangle.

Solution

Problem 7

Compute


$\binom{2016}{5}+\binom{2017}{5}+\binom{2018}{5}+\binom{2019}{5}+\binom{2020}{5}+\binom{2016}{6}$


(a) $\binom{2020}{6}$


(b) $\binom{2020}{7}$


(c) $\binom{2021}{5}$


(d) $\binom{2021}{6}$


(e) $\binom{2022}{5}$


Day 2

Problem 1

The ingeters $a_1,a_2,a_3,...,a_{25}$ are in arithmetic progression of ratio not equal to zero. The sequences $(a_1,a_2,a_{10})$ and $(a_6,a_j,a_{25})$ are both in geometric progression. Find $j$.


Problem 2

Let the functions $f_n$, for $n\in\{0,1,2,3,...\}$, such that $f_0(x)=\frac{1}{1-x}$ and $f_n(x)=f_0(f_{n-1}(x))$, for every $n\geq1$.

Compute $f_{2016}(2016)$.


Problem 3

Let $Z$ be a complex number such that $\frac{2Z}{\overline{Z}i}$ has an argument $(\theta)$ equal to $\frac{3\pi}{4}$ and $\log_3(2Z+2\overline{Z}+1)=2$. Find the complex number $Z$.


Problem 4

Define $A$ as the matrix 2016 x 2016, such that its elements satisfy the equality:

$a_{i,j}=\begin{pmatrix}i+j-2\\j-1\end{pmatrix}$, for $i,j\in\{1,2,...,2016\}$.

Compute det(A).


Problem 5

Find the solution of the equation:

$\sin x\left(1+\tan x\tan\frac{x}{2}\right)=4-\cot x$


Problem 6

Let the equation $n^2-7m^2=(5m-2n)^2+49$. Find all the pairs of integers $(m,n)$ that satisfy this equation.


Problem 7

Three player sits around a table and play, alternatively, a dice of six faces. The first player throw the dice, then the player at its left plays, and then it goes, until the finish of the game. The one that throw the dice and get the side with number 6, wins the game and it ends. If a player gets the number 1, he loses its chance and the player at its right will throw the dice. The game will continue until someone gets the number 6. What's the probability of the first player to throw the dice to win?


Problem 8

The equation of the circumference $C$ is $x^2+y^2=16$. Let $C'$ be a circumference of radius 1 wich moves internally tangentiating the circumference $C$, without slipping between the points of contact, in other words, $C'$ internally rolls on $C$.


MIE 2015 day 1 problem 8.png


Let the point $P$ be on $C'$ such that in the beginning of the movement of $C'$ the point $P$ is at the tangent point $(4,0)$, like in the figure a. After some movement, the angle between


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