MIE 2015

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) $2$


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


$\sum_{k=1}^{15}$ 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.


Problem 7



(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 the X axis and the line between the centers of the circumferences is $\alpha$.

  • Find the coordinates of the point $P$ marked on $C'$ in terms of $\alpha$.
  • Find the equation of the cartesian coordinates of the geometric place of the point P when $\alpha$ varies in the interval $[0,2\pi)$.


Problem 9

A chord intersects the diameter of a circle of center $O$ at the point $C'$ according to an angle of 45°. Let $A$ and $B$ be the extreme points of this chord, and the distance $AC'$ is equal to $\sqrt3+1$ cm. The radius of the circle is 2 cm, and $C$ is the farther extremity of the diameter from $C'$. The extension of $AO$ intersects $BC$ at $A'$. Compute $\frac{BC}{A'}$.


Problem 10

A cone is inscribed into a cube $ABCDEFGH$ such that the base of the cone is the circle inscribed in the side $ABCD$ of the cube. The vertex of the cone is the center of the opposite face of the cube. The projection of the vertex H on the base $ABCD$ coincides with the vertex $D$. Compute the area of section of the cone by the surface $ABH$ in terms of $a$, the side of the cube.


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