Question 5: An unknown code consists of 8 characters. Each character can be a letter or a digit. There are therefore 36 usable characters for each position.
A code-breaking software tests approximately one hundred million codes per second. In how much time at most can the software discover the code?

\begin{tabular}{l} a. approximately 0.3

seconds

& b. approximately 8 hours & c. approximately 3 hours &

d. approximately 470

hours

\hline \end{tabular}

O número de anagramas da palavra AMOR é
(A) 12 (B) 16 (C) 20 (D) 24 (E) 32

Portuguese designer Miguel Neiva created a system of symbols that allows colorblind people to identify colors. The system consists of using symbols that identify primary colors (blue, yellow, and red). Furthermore, the juxtaposition of two of these symbols allows the identification of secondary colors (such as green, which is yellow combined with blue). Black and white are identified by small squares: the one that symbolizes black is filled, while the one that symbolizes white is empty. The symbols that represent black and white can also be associated with the symbols that identify colors, meaning whether these are light or dark.
According to the text, how many colors can be represented by the proposed system?
(A) 14
(B) 18
(C) 20
(D) 21
(E) 23

A bank asked its customers to create a personal six-digit password, formed only by digits from 0 to 9, for access to their checking account via the internet.
However, an expert in electronic security systems recommended to the bank's management to re-register its users, requesting, for each one of them, the creation of a new six-digit password, now allowing the use of the 26 letters of the alphabet, in addition to digits from 0 to 9. In this new system, each uppercase letter was considered distinct from its lowercase version. Furthermore, the use of other types of characters was prohibited.
One way to evaluate a change in the password system is to verify the improvement coefficient, which is the ratio of the new number of password possibilities to the old one.
The improvement coefficient of the recommended change is
(A) $\frac{62^{6}}{10^{6}}$ (B) $\frac{62!}{10!}$ (C) $\frac{62! \cdot 4!}{10! \cdot 56!}$ (D) $62! - 10!$ (E) $62^{6} - 10^{6}$

A jewellery artisan has at his disposal Brazilian stones of three colours: red, blue and green.
He intends to produce jewellery made of a metal alloy, based on a mould in the shape of a non-square rhombus with stones at its vertices, so that two consecutive vertices always have stones of different colours.
The figure illustrates a piece of jewellery, produced by this artisan, whose vertices $A$, $B$, $C$ and $D$ correspond to the positions occupied by the stones.
Based on the information provided, how many different pieces of jewellery, in this format, can the artisan obtain?
(A) 6 (B) 12 (C) 18 (D) 24 (E) 36

QUESTION 157
The number of ways to arrange 4 people in a row is
(A) 12
(B) 16
(C) 20
(D) 24
(E) 28

QUESTION 178
The value of $5! - 3!$ is
(A) 108
(B) 112
(C) 114
(D) 116
(E) 120

To stimulate his daughter's reasoning, a father made the following drawing and gave it to the child along with three colored pencils. He wants the girl to paint only the circles, so that those connected by a segment have different colors.
In how many different ways can the child do what the father asked?
(A) 6
(B) 12
(C) 18
(D) 24
(E) 72

A company will build its page on the internet and expects to attract an audience of approximately one million customers. To access this page, a password with a format to be defined by the company will be required. There are five format options offered by the programmer, described in the table, where ``L'' and ``D'' represent, respectively, uppercase letter and digit.

Option	Format
I	LDDDDD
II	DDDDDD
III	LLDDDD
IV	DDDDD
V	LLLDD

The letters of the alphabet, among the 26 possible ones, as well as the digits, among the 10 possible ones, can be repeated in any of the options.
The company wants to choose a format option whose number of possible distinct passwords is greater than the expected number of customers, but such that this number is not greater than twice the expected number of customers.
The option that best suits the company's conditions is
(A) I.
(B) II.
(C) III.
(D) IV.
(E) V.

How many ways can 4 people be arranged in a line?
(A) 8
(B) 12
(C) 16
(D) 20
(E) 24

Ten couples founded a dance group and decided to establish a board of directors with three positions: president, secretary, and treasurer. For greater representation, it was decided that at most one person per couple could hold a position on this board.
How many different boards can be formed by these 10 couples?
(A) $10 \times 9 \times 8$
(B) $20 \times 18 \times 16$
(C) $20 \times 19 \times 18$
(D) $10 \times 9 \times 8 \times 2$
(E) $20 \times 18 \times 16 \times 2$

A father bought eight different gifts (among which, a bicycle and a cell phone) to give to his three children. He intends to distribute the gifts so that the oldest and youngest children receive three gifts each, and the middle one receives the two remaining gifts. The oldest will receive, among his gifts, either a bicycle or a cell phone, but not both.
In how many distinct ways can the distribution of gifts be made?
(A) 36
(B) 53
(C) 300
(D) 360
(E) 560

The word MATHEMATICS consists of 11 letters. The number of distinct ways to rearrange these letters is
(A) $11 !$
(B) $\frac { 11 ! } { 3 }$
(C) $\frac { 11 ! } { 6 }$
(D) $\frac { 11 ! } { 8 }$

In a business meeting, each person shakes hands with each other person, with the exception of Mr. L. Since Mr. L arrives after some people have left, he shakes hands only with those present. If the total number of handshakes is exactly 100 , how many people left the meeting before Mr. L arrived? (Nobody shakes hands with the same person more than once.)

Let S be the set of all 5-digit numbers that contain the digits $1,3,5,7$ and 9 exactly once (in usual base 10 representation). Show that the sum of all elements of S is divisible by 11111. Find this sum.

The format for car license plates in a small country is two digits followed by three vowels, e.g. 04 IOU. A license plate is called ``confusing'' if the digit 0 (zero) and the vowel O are both present on it. For example $04\,IOU$ is confusing but $20\,AEI$ is not. (i) How many distinct number plates are possible in all? (ii) How many of these are not confusing?

A step starting at a point $P$ in the $XY$-plane consists of moving by one unit from $P$ in one of three directions: directly to the right or in the direction of one of the two rays that make the angle of $\pm 120^{\circ}$ with positive $X$-axis. (An opposite move, i.e. to the left/southeast/northeast, is not allowed.) A path consists of a number of such steps, each new step starting where the previous step ended. Points and steps in a path may repeat.
Find the number of paths starting at $(1,0)$ and ending at $(2,0)$ that consist of
(i) exactly 6 steps
(ii) exactly 7 steps.

10 mangoes are to be placed in 5 distinct boxes labeled $\mathrm{U}, \mathrm{V}, \mathrm{W}, \mathrm{X}, \mathrm{Y}$. A box may contain any number of mangoes including no mangoes or all the mangoes. What is the number of ways to distribute the mangoes so that exactly two of the boxes contain exactly two mangoes each?

An alien script has $n$ letters $b_{1}, \ldots, b_{n}$. For some $k < n/2$ assume that all words formed by any of the $k$ letters (written left to right) are meaningful. These words are called $k$-words. Such a $k$-word is considered sacred if:
i) no letter appears twice and,
ii) if a letter $b_{i}$ appears in the word then the letters $b_{i-1}$ and $b_{i+1}$ do not appear. (Here $b_{n+1} = b_{1}$ and $b_{0} = b_{n}$.)
For example, if $n = 7$ and $k = 3$ then $b_{1}b_{3}b_{6}, b_{3}b_{1}b_{6}, b_{2}b_{4}b_{6}$ are sacred 3-words. On the other hand $b_{1}b_{7}b_{4}, b_{2}b_{2}b_{6}$ are not sacred. What is the total number of sacred $k$-words? Use your formula to find the answer for $n = 10$ and $k = 4$.

Let $\pi = \pi_{1}\pi_{2}\ldots\ldots\pi_{n}$ be a permutation of the numbers $1,2,3,\ldots,n$. We say $\pi$ has its first ascent at position $k < n$ if $\pi_{1} > \pi_{2} \ldots > \pi_{k}$ and $\pi_{k} < \pi_{k+1}$. If $\pi_{1} > \pi_{2} > \ldots > \pi_{n-1} > \pi_{n}$ we say $\pi$ has its first ascent in position $n$. For example when $n = 4$ the permutation 2134 has its first ascent at position 2.
The number of permutations which have their first ascent at position $k$ is ......

For a natural number $n$ denote by $\operatorname{Map}(n)$ the set of all functions $f : \{1,2,\ldots,n\} \rightarrow \{1,2,\ldots,n\}$. For $f, g \in \operatorname{Map}(n)$, $f \circ g$ denotes the function in $\operatorname{Map}(n)$ that sends $x$ to $f(g(x))$.
(a) Let $f \in \operatorname{Map}(n)$. If for all $x \in \{1,\ldots,n\}$ $f(x) \neq x$, show that $f \circ f \neq f$.
(b) Count the number of functions $f \in \operatorname{Map}(n)$ such that $f \circ f = f$.

You are supposed to create a 7-character long password for your mobile device.
(i) How many 7-character passwords can be formed from the 10 digits and 26 letters? (Only lowercase letters are taken throughout the problem.) Repeats are allowed, e.g., 0001a1a is a valid password.
(ii) How many of the passwords contain at least one of the 26 letters and at least one of the 10 digits? Write your answer in the form: (Answer to part i) $-$ (something).
(iii) How many of the passwords contain at least one of the 5 vowels, at least one of the 21 consonants and at least one of the 10 digits? Extend your method for part ii to write a formula and explain your reasoning.
(iv) Now suppose that in addition to the lowercase letters and digits, you can also use 12 special characters. How many 7-character passwords are there that contain at least one of the 5 vowels, at least one of the 21 consonants, at least one of the 10 digits and at least one of the 12 special characters? Write only the final formula analogous to your answer to part iii. Do NOT explain.

Let $N = \{1,2,3,4,5,6,7,8,9\}$ and $L = \{a,b,c\}$.
Statements
(29) Suppose we arrange the 12 elements of $L \cup N$ in a line such that all three letters appear consecutively (in any order). The number of such arrangements is less than $10! \times 5$. (30) More than half of the functions from $N$ to $L$ have the element $b$ in their range. (31) The number of one-to-one functions from $L$ to $N$ is less than 512. (32) The number of functions from $N$ to $L$ that do not map consecutive numbers to consecutive letters is greater than 512. (e.g., $f(1) = b$ and $f(2) = a$ or $c$ is not allowed. $f(1) = a$ and $f(2) = c$ is allowed. So is $f(1) = f(2)$.)

Suppose that cards numbered $1, 2, \ldots, n$ are placed on a line in some sequence (with each integer $i \in [1,n]$ appearing exactly once). A move consists of interchanging the card labeled 1 with any other card. If it is possible to rearrange the cards in increasing order by doing a series of moves, we say that the given sequence can be rectified.
Statements
(37) The sequence 912345678 can be rectified in 8 moves and no fewer moves. (38) The sequence 134567892 can be rectified in 8 moves and no fewer moves. (39) The sequence 134295678 cannot be rectified. (40) There exists a sequence of 99 cards that cannot be rectified.

There are $n$ students in a class and no two of them have the same height. The students stand in a line, one behind another, in no particular order of their heights.
(a) How many different orders are there in which the shortest student is not in the first position and the tallest student is not in the last position?
(b) The badness of an ordering is the largest number $k$ with the following property. There is at least one student $X$ such that there are $k$ students taller than $X$ standing ahead of $X$. Find a formula for $g _ { k } ( n ) =$ number of orderings of $n$ students with badness $k$.
Example: The ordering $64\,61\,67\,63\,62\,66\,65$ (the numbers denote heights) has badness 3 as the student with height 62 has three taller students (with heights 64, 67 and 63) standing ahead in the line and nobody has more than 3 taller students standing ahead.
Possible hints for (b): It may be useful to first count orderings of badness 1 and/or to find $f _ { k } ( n ) =$ the number of orderings of $n$ students with badness less than or equal to $k$.