The complex plane is given an orthonormal coordinate system ( $\mathrm { O } , \vec { u } , \vec { v }$ ). To every point $M$ with affixe $z$ in the plane, we associate the point $M ^ { \prime }$ with affixe $z ^ { \prime }$ defined by:
$$z ^ { \prime } = z ^ { 2 } + 4 z + 3 .$$

1. A point $M$ is called invariant when it coincides with the associated point $M ^ { \prime }$.
   Prove that there exist two invariant points. Give the affixe of each of these points in algebraic form, then in exponential form.
2. Let A be the point with affixe $\frac { - 3 - \mathrm { i } \sqrt { 3 } } { 2 }$ and B the point with affixe $\frac { - 3 + \mathrm { i } \sqrt { 3 } } { 2 }$.
   Show that OAB is an equilateral triangle.
3. Determine the set $\mathcal { E }$ of points $M$ with affixe $z = x + \mathrm { i } y$ where $x$ and $y$ are real, such that the associated point $M ^ { \prime }$ lies on the real axis.
4. In the complex plane, represent the points A and B as well as the set $\mathcal { E }$.

Question 161
A equação $x^2 - 5x + 6 = 0$ tem como raízes
(A) $x = 1$ e $x = 6$ (B) $x = 2$ e $x = 3$ (C) $x = -2$ e $x = -3$ (D) $x = 1$ e $x = -6$ (E) $x = -1$ e $x = 6$

Question 173
Um número inteiro positivo de dois algarismos é tal que a soma de seus algarismos é 9 e o produto de seus algarismos é 18. Esse número é
(A) 27 (B) 36 (C) 45 (D) 63 (E) 72

O número de diagonais de um polígono convexo de $n$ lados é dado pela fórmula $D = \dfrac{n(n-3)}{2}$. Um polígono convexo tem 20 diagonais. O número de lados desse polígono é
(A) 6 (B) 7 (C) 8 (D) 9 (E) 10

Um retângulo tem perímetro de 36 cm e área de 80 cm$^2$. As dimensões do retângulo são
(A) 8 cm e 10 cm (B) 9 cm e 9 cm (C) 6 cm e 12 cm (D) 5 cm e 16 cm (E) 4 cm e 20 cm

The temperature $T$ of an oven (in degrees Celsius) is reduced by a system from the moment it is turned off ($t = 0$) and varies according to the expression $T(t) = -\frac{t^{2}}{4} + 400$, with $t$ in minutes. For safety reasons, the oven lock is only released for opening when the oven reaches a temperature of $39^{\circ}C$.
What is the minimum waiting time, in minutes, after turning off the oven, for the door to be opened?
(A) 19.0 (B) 19.8 (C) 20.0 (D) 38.0 (E) 39.0

QUESTION 146
The function $f(x) = 2x^2 - 3x + 1$ has roots
(A) $x = 1$ and $x = \frac{1}{2}$
(B) $x = -1$ and $x = \frac{1}{2}$
(C) $x = 1$ and $x = -\frac{1}{2}$
(D) $x = -1$ and $x = -\frac{1}{2}$
(E) $x = 2$ and $x = \frac{1}{2}$

QUESTION 168
The equation $x^2 - 5x + 6 = 0$ has roots
(A) $x = 1$ and $x = 6$
(B) $x = 2$ and $x = 3$
(C) $x = -2$ and $x = -3$
(D) $x = 1$ and $x = -6$
(E) $x = -1$ and $x = 6$

A club has a soccer field with a total area of $8000 \mathrm{~m}^{2}$, corresponding to the grass. Usually, the grass mowing of this field is done by two machines owned by the club for this service. Working at the same pace, the two machines mow together $200 \mathrm{~m}^{2}$ per hour. Due to the urgency of holding a soccer match, the field administrator will need to request machines from the neighboring club equal to his own to do the mowing work in a maximum time of 5 h.
Using the two machines that the club already has, what is the minimum number of machines that the field administrator should request from the neighboring club?
(A) 4
(B) 6
(C) 8
(D) 14
(E) 16

A store is offering a 20\% discount on all products. After the discount, a product costs R\$\,80.00. What was the original price of the product, in reais?
(A) R\$\,96.00
(B) R\$\,100.00
(C) R\$\,104.00
(D) R\$\,108.00
(E) R\$\,112.00

List every solution of the following equation. You need not simplify your answer(s). $$\sqrt[3]{x+4} - \sqrt[3]{x} = 1$$

Answer the following questions
(a) Find all real solutions of the equation $$\left(x^{2}-2x\right)^{x^{2}+x-6} = 1$$ Explain why your solutions are the only solutions.
(b) The following expression is a rational number. Find its value. $$\sqrt[3]{6\sqrt{3}+10} - \sqrt[3]{6\sqrt{3}-10}$$

Find all solutions of the following equation where it is required that $x, k, y, n$ are positive integers with the exponents $k$ and $n$ both $> 1$. $$20x^k + 24y^n = 2024$$

For the irrational equation $$\sqrt { 4 x ^ { 2 } - 5 x + 7 } - 4 x ^ { 2 } + 5 x = 1$$ What is the product of all real roots? [3 points]
(1) $- \frac { 1 } { 2 }$
(2) $- \frac { 3 } { 2 }$
(3) $- \frac { 5 } { 2 }$
(4) $- \frac { 7 } { 2 }$
(5) $- \frac { 9 } { 2 }$

The product of all real roots of the irrational equation $x ^ { 2 } - 2 x + 2 \sqrt { x ^ { 2 } - 2 x } = 8$ is? [3 points]
(1) - 5
(2) - 4
(3) - 3
(4) - 2
(5) - 1

Find the product of all real roots of the irrational equation $\sqrt { 2 x ^ { 2 } - 6 x } = x ^ { 2 } - 3 x - 4$, and call it $k$. Find the value of $k ^ { 2 }$. [3 points]

For the irrational equation $x ^ { 2 } - 6 x - \sqrt { x ^ { 2 } - 6 x - 1 } = 3$, let $k$ be the product of all real roots. Find the value of $k ^ { 2 }$. [3 points]

20. (15 points) Let the function $f ( x ) = x ^ { 2 } + a x + b , ( a , b \in R )$ .
(1) When $b = \frac { a ^ { 2 } } { 4 } + 1$ , find the expression for the minimum value $g ( a )$ of the function $f ( x )$ on $[ - 1,1 ]$ ;
(2) Given that the function $f ( x )$ has a zero on $[ - 1,1 ]$ , $0 \leq b - 2 a \leq 1$ , find the range of values for $b$ .

Given set $A = \left\{ x \mid x ^ { 2 } - 3 x - 4 < 0 \right\} , B = \{ - 4,1,3,5 \}$ , then $A \cap B =$
A. $\{ - 4,1 \}$
B. $\{ 1,5 \}$
C. $\{ 3,5 \}$
D. $\{ 1,3 \}$

For each $k \in \mathbb { R }$, a polynomial function defined on $\mathbb { R }$ is given by
$$u ( x ) = x ^ { 3 } - 3 k \cdot x + k ^ { 2 } - 1$$
(1) Give the value of $k$ for which the corresponding function $u$ coincides with the function $f$.
(2) Determine all values of $k$ for which $u ( 2 ) = 2$ holds.

We set $\theta = \operatorname { arcosh } \left( \frac { \lambda _ { N } + \lambda _ { 1 } } { \lambda _ { N } - \lambda _ { 1 } } \right) > 0$ and $\alpha = e ^ { - \theta }$. Show that $\alpha$ is a root of the polynomial $$X ^ { 2 } - 2 \frac { \lambda _ { N } + \lambda _ { 1 } } { \lambda _ { N } - \lambda _ { 1 } } X + 1$$ and deduce the expression of $\alpha$ in terms of the quantity $\beta = \frac { \lambda _ { N } + \lambda _ { 1 } } { \lambda _ { N } - \lambda _ { 1 } }$.

We denote by $\kappa = \lambda _ { N } / \lambda _ { 1 }$. Show that the real number $\alpha$ from question 22 equals $\alpha = \frac { \sqrt { \kappa } - 1 } { \sqrt { \kappa } + 1 }$ and deduce that $$\left\| e _ { k } \right\| _ { A } \leq 2 \left\| e _ { 0 } \right\| _ { A } \left( \frac { \sqrt { \kappa } - 1 } { \sqrt { \kappa } + 1 } \right) ^ { k }$$

In this question, we examine the special case of a polynomial function of degree two $f$ defined by the formula $f ( x ) = ( x - \alpha ) ( x - \beta )$ where $\alpha$ and $\beta$ are real and $\alpha > \beta$. We take $I = ] ( \alpha + \beta ) / 2 , + \infty [$.
For $x \in \mathbb { R }$ we define $h ( x ) = \frac { x - \alpha } { x - \beta }$, with the convention $h ( \beta ) = \infty$.
(a) For $x \in \mathbb { R }$ show that we have $| h ( x ) | < 1$ if and only if $x \in I$.
(b) Explicitly state the recurrence relation satisfied by the sequence $u _ { n } : = h \left( x _ { n } \right)$ and deduce that the sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ is well defined for any $x _ { 0 }$ and $x _ { 1 }$ in $I$.
(c) Show that the sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ tends to 0 and deduce that $\left( x _ { n } \right) _ { n \geqslant 0 }$ tends to $\alpha$.
(d) Let $\phi = \frac { 1 + \sqrt { 5 } } { 2 }$. Show that there exists a strictly negative real number $s$ such that $$x _ { n } - \alpha = O \left( e ^ { s \phi ^ { n } } \right) .$$

141- The side lengths of a right triangle are $x+1$, $2x+1$, and $2x+3$. The area of the triangle is:
\[ (1)\quad 60 \qquad (2)\quad 56 \qquad (3)\quad 45 \qquad (4)\quad 39 \]

152 -- The four-digit number $\overline{aabb}$, whose square root is the two-digit number $\overline{cc}$, and $\overline{cc} = a - b$. What is $a - b$?

• [(1)] $2$
• [(2)] $3$
• [(3)] $4$
• [(4)] $5$