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

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) $\binom{2022}{5}$

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