There are 2 houses, numbered 1 to 2 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`
 - Each person has a unique hobby: `gardening`, `photography`
 - People have unique favorite book genres: `science fiction`, `mystery`
 - People have unique favorite music genres: `rock`, `pop`
 - Each person has a unique birthday month: `april`, `sept`

## Clues:
1. The person who loves mystery books is the person who loves rock music.
2. Arnold is not in the first house.
3. The person who loves mystery books is the person who enjoys gardening.
4. The person whose birthday is in April is Arnold.
5. The person who loves mystery books is in the first house.
 What is the value of attribute Birthday for the person whose attribute BookGenre is mystery? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


Vertex \(E\) of equilateral \(\triangle{ABE}\) is in the interior of unit square \(ABCD\). Let \(R\) be the region consisting of all points inside \(ABCD\) and outside \(\triangle{ABE}\) whose distance from \(\overline{AD}\) is between \(\frac{1}{3}\) and \(\frac{2}{3}\). Find the area of \(R\). The original answer is in the form \(\frac{k-m\sqrt{3}}{n}\), please provide the value of k + m + n.

Answer: 53


In the adjoining figure, \( \triangle CDE \) is an equilateral triangle and \( \square ABCD \) and \( \square DEFG \) are squares. If the measure of \( \angle GDA \) is \( x^\circ \), what is the value of \( x \)?

Answer: 120


If the angles $a, b, c, d,$ and $e$ are known, what is the expression for the angle $u$ in terms of $a$, $b$, $c$, $d$, and $e$? The original answer is in the form of m*a + n*c + p*d. Please provide the value of m + n + p.

Answer: 3


The symbol $R_k$ stands for an integer whose base-ten representation is a sequence of $k$ ones. For example, $R_3=111$, $R_5=11111$, etc. When $R_{24}$ is divided by $R_4$, the quotient $Q=\frac{R_{24}}{R_4}$ is an integer whose base-ten representation is a sequence containing only ones and zeros. Find the number of zeros in $Q$.

Answer: 15


Kayla rolls four fair 6-sided dice. Find the probability that at least one of the numbers Kayla rolls is greater than 4 and at least two of the numbers she rolls are greater than 2. Express your answer as a fraction in simplest form, \(\frac{k}{m}\). Please provide the value of k + m.

Answer: 142


Three cubes with volumes $1$, $8$, and $27$ are glued together at their faces. Find the smallest possible surface area of the resulting configuration.

Answer: 72


Let \(ABCD\) be an isosceles trapezoid with parallel bases \(\overline{AB}\) and \(\overline{CD}\), where \(AB > CD\). Line segments from a point inside \(ABCD\) to the vertices divide the trapezoid into four triangles with areas \(2\), \(3\), \(4\), and \(5\), starting with the triangle with base \(\overline{CD}\) and moving clockwise. The original answer for the ratio \(\frac{AB}{CD}\) is in the form \(k + \sqrt{m}\). Please calculate the value of \(k + m\).

Answer: 4


Let $K$ be the point of intersection of $AB$ and the line touching the circumcircle of $\triangle ABC$ at $C$ where $m(\widehat {A}) > m(\widehat {B})$. Let $L$ be a point on $[BC]$ such that $m(\widehat{ALB})=m(\widehat{CAK})$, $5|LC|=4|BL|$, and $|KC|=12$. Find the length of $|AK|$ in centimeters, denoted as x. Please provide the value of x.

Answer: 8


A very large number of small particles forms a spherical cloud. Initially, they are at rest, have uniform mass density per unit volume $\rho_0$, and occupy a region of radius $r_0$. The cloud collapses due to gravitation; the particles do not interact with each other in any other way.

Find the time that passes until the cloud collapses fully, expressed in the form $\frac{k}{\sqrt{G \rho_0}}$. Please give the value of $10000k$.

Answer: 5427


Alex is a cybersecurity expert who spends part of his week analyzing misinformation threats online. Each day, he reviews 50 online articles to determine if they contain misinformation. On Monday, he identified 12 articles with misinformation. On Tuesday, he found 16 such articles, and on Wednesday, he discovered 9 misinformation articles. Alex's goal is to reduce the number of misinformation articles by 20% each day for the rest of the week. How many misinformation articles should Alex aim to find on Thursday if he wants to achieve his goal?

Answer: 7


A PhD candidate in sociology is working on improving their academic writing and publication skills. To do this, they decide to attend a series of workshops. Each workshop lasts 3 hours and they plan to attend a total of 8 workshops. For every 2 workshops they attend, they write one article draft. Additionally, they spend 5 hours revising and refining each draft into a final paper. How many total hours will the PhD candidate spend on writing and refining their papers after attending all the workshops?

Answer: 20


Jamie has been a Jaguars season ticket holder for 10 years. Every season, there are 8 home games. Jamie's parents have been attending games with Jamie since childhood. If Jamie attended his first game at the age of 5, and he is now 25 years old, how many total home games have Jamie and his parents attended together?

Answer: 160


Jamie is a parent of two teenagers, Alex and Taylor, living in San Francisco. Each weekend, the family likes to explore different parts of the city. Last Saturday, they visited the Golden Gate Park, where they rented bicycles. The rental shop charges $15 per hour for each bicycle. Alex and Taylor each rented a bicycle for 3 hours, while Jamie decided to rent one for 2 hours. How much did Jamie spend in total for the bicycle rentals?

Answer: 120


A disappointed Seinäjoki Crocodiles fan attended a football game where the Crocodiles were playing against their rivals. During the first half of the game, the Crocodiles scored 3 touchdowns, and each touchdown is worth 6 points. In the second half, they only managed to score 2 field goals, with each field goal worth 3 points. Meanwhile, their rivals scored a total of 28 points by the end of the game. How many more points did the Crocodiles need to score to win the game?

Answer: 5


Mr. Thompson, a wise retiree who has successfully managed his mortgage and medical debts, loves helping others with their financial planning. Inspired by his experience, he decides to host a workshop. During the workshop, he shares a strategy where he advises paying off a small debt first before moving on to larger ones.

Mr. Thompson has a total of $1,000 to allocate towards three debts. The first debt is $200, the second is $350, and the third is $450. He plans to pay off the smallest debt entirely first, then use the remaining money to pay as much as possible on the second debt.

How much money will Mr. Thompson have left to put towards the second debt after fully paying off the first debt? How much total debt will he have left after making these payments?

Answer: 0


Alex is a member of an online discussion forum dedicated to civil, evidence-based debates. Every day, Alex reads 12 evidence-based articles to prepare for discussions and spends 15 minutes on each article. After reading, Alex participates in 4 debates, spending 20 minutes on each one. If Alex dedicates a total of 3 hours a day to this forum, how much time does Alex have left after reading the articles and participating in the debates?

Answer: -80


Sarah is a hardcore fan of The Telescopes and has a grand collection of their records and merchandise. In her collection, she has 25 vinyl albums, 12 EPs, 18 singles, and 5 exclusive merchandise items. She recently attended a fan convention where she bought 3 more rare vinyl albums, 2 exclusive EPs, and traded away 1 single for a limited edition poster. After the convention, Sarah decided to organize a display shelf where she can showcase her collection, with each type of item on a separate shelf. If each shelf can hold exactly 5 items, how many shelves does Sarah need to properly display her entire collection?

Answer: 14


Given that $A, B, C, D$ are points along a circle in that order, and $AC$ intersects $BD$ at point $X$. If $BC = 6$, $BX = 4$, $XD = 5$, and $AC = 11$, find the length of $AB$.

Answer: 6


The equation $x^3 - 3x^2 - 7x - 1 = 0$ has three distinct real roots $a$, $b$, and $c$. If \[\left( \frac{1}{\sqrt[3]{a}-\sqrt[3]{b}} + \frac{1}{\sqrt[3]{b}-\sqrt[3]{c}} + \frac{1}{\sqrt[3]{c}-\sqrt[3]{a}} \right)^2 = \frac{p\sqrt[3]{q}}{r}\] where $p$, $q$, $r$ are positive integers such that $\gcd(p, r) = 1$ and $q$ is not divisible by the cube of a prime, find $100p + 10q + r$.

Answer: 1913


Frankie the Frog starts his morning at the origin in $\mathbb{R}^2$. He decides to go on a leisurely stroll, consisting of $3^1 + 3^{10} + 3^{11} + 3^{100} + 3^{111} + 3^{1000}$ moves, starting with the first move. On the $n$th move, he hops a distance of $$\max\{k\in\mathbb{Z}:3^k|n\}+1,$$ then turns $90^{\circ}$ counterclockwise. What is the square of the distance from his final position to the origin?

Answer: 0


Frank mistakenly believes that the number $1011$ is prime and for some integer $x$ writes down:
\[
(x+1)^{1011} \equiv x^{1011} + 1 \pmod{1011}.
\]
However, it turns out that for Frank's choice of $x$, this statement is actually true. If $x$ is positive and less than $1011$, what is the sum of the possible values of $x$?

Answer: 3030


A shipping company charges $0.30l + 0.40w + 0.50h$ dollars to process a right rectangular prism-shaped box with dimensions $l, w, h$ in inches. The customers themselves are allowed to label the three dimensions of their box with $l, w, h$ for the purpose of calculating the processing fee. A customer finds that there are two different ways to label the dimensions of their box $B$ to get a fee of $8.10, and two different ways to label $B$ to get a fee of $8.70. None of the faces of $B$ are squares. Find the surface area of $B$, in square inches.

Answer: 276


Let $W = \ldots x_{-1}x_0x_1x_2 \ldots$ be an infinite periodic word consisting of only the letters $a$ and $b$. The minimal period of $W$ is $2^{2016}$. Say that a word $U$ appears in $W$ if there are indices $k \le \ell$ such that $U = x_kx_{k+1} \ldots x_{\ell}$. A word $U$ is called special if $Ua, Ub, aU, bU$ all appear in $W$. (The empty word is considered special.) You are given that there are no special words of length greater than 2015.

Let $N$ be the minimum possible number of special words. Find the remainder when $N$ is divided by 1000.

Answer: 535


A polynomial $P$ of degree $2015$ satisfies the equation $P(n) = \frac{1}{n^2}$ for $n = 1, 2, \dots, 2016$. Find $\lfloor 2017P(2017) \rfloor$.

Answer: -9


Answer: 665280


A magician has a hat that contains $a$ white rabbits and $b$ black rabbits. The magician repeatedly draws pairs of rabbits chosen at random from the hat, without replacement. Call a pair of rabbits *checkered* if it consists of one white rabbit and one black rabbit. Given that the magician eventually draws out all the rabbits without ever drawing out an unpaired rabbit and that the expected value of the number of checkered pairs that the magician draws is $2020$, compute the number of possible pairs $(a,b)$.

Answer: 16


Plusses and minuses are inserted in the expression:
\[ \pm 1 \pm 2 \pm 3 \dots \pm 2016 \]
such that when evaluated, the result is divisible by 2017. Let there be $N$ ways for this to occur. Compute the remainder when $N$ is divided by 503.

Answer: 52


Call a polynomial $f$ with positive integer coefficients "triangle-compatible" if any three coefficients of $f$ satisfy the triangle inequality. For instance, $3x^3 + 4x^2 + 6x + 5$ is triangle-compatible, but $3x^3 + 3x^2 + 6x + 5$ is not. Given that $f$ is a degree $20$ triangle-compatible polynomial with $-20$ as a root, what is the least possible value of $f(1)$?

Answer: 420


For a regular hexahedron and a regular octahedron, all their faces are regular triangles with each side of length $a$. Let their inradii be $r_1$ and $r_2$, respectively. If $\frac{r_1}{r_2} = \frac{m}{n}$, where $\gcd(m, n) = 1$, then find the value of $mn$.

Answer: 2


Let $\mathcal{S}$ be a regular 18-gon. For two vertices in $\mathcal{S}$, define the \textit{distance} between them as the length of the shortest path along the edges of $\mathcal{S}$ between them (e.g., adjacent vertices have distance 1). Find the number of ways to choose three distinct vertices from $\mathcal{S}$ such that no two of them have distances 1, 8, or 9.

Answer: 240


A triangle $\triangle A_0A_1A_2$ in the plane has side lengths $A_0A_1 = 7$, $A_1A_2 = 8$, $A_2A_0 = 9$. For $i \ge 0$, given $\triangle A_iA_{i+1}A_{i+2}$, let $A_{i+3}$ be the midpoint of $A_iA_{i+1}$ and let $G_i$ be the centroid of $\triangle A_iA_{i+1}A_{i+2}$. Let point $G$ be the limit of the sequence of points $\{G_i\}^{\infty}_{i=0}$. If the distance between $G$ and $G_0$ can be written as $\frac{a\sqrt{b}}{c}$, where $a$, $b$, $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime, find $a^2 + b^2 + c^2$.

Answer: 422


Two different numbers are randomly selected from the set $T = \{2, 3, 5, 7, 9, 11, 13, 15, 17, 19\}$. What is the probability that their sum equals 18, and which number, if removed from $T$, would increase this probability the most?

Answer: 9


The surface area of a sphere is \(400\pi \text{ cm}^2\). Calculate the volume of the sphere in cubic centimeters and express your answer in terms of \(\pi\).

Answer: \frac{4000}{3}\pi \text{ cm}^3


A dodecahedron consists of two identical pentagonal-based pyramids glued together along their pentagonal bases to form a polyhedron with twelve faces. An ant starts at the top vertex of the dodecahedron and walks to one of the five adjacent vertices, which it randomly selects and calls vertex A. From vertex A, the ant then walks to one of the five adjacent vertices it randomly selects and calls vertex B. What is the probability that vertex B will be the bottom vertex of the dodecahedron? Express your answer as a common fraction.

Answer: \frac{1}{5}


The surface area of a sphere with radius $r$ is $4\pi r^2$. What is the total surface area of a hemisphere with radius 10 cm, including the area of its circular base and an additional circular cutout at the top with radius 3 cm?

Answer: 291\pi


In the diagram below, $PQRS$ is a trapezoid with $\overline{PQ} \parallel \overline{RS}$ and $\overline{PR} \perp \overline{RS}$. If $RS = 15$, $\tan S = 2$, and $\tan Q = 3$, what is the length of $PQ$?

Answer: 10


A pizza with 24 slices is made with three toppings: ham, pineapple, and jalapeno. Every slice has at least one topping. Exactly 15 slices are topped with ham, 10 slices are topped with pineapple, and 14 slices are topped with jalapeno. If we know that the number of slices having all three toppings is equal to the number of slices with only jalapeno, determine how many slices have all three toppings.

Answer: 5


In triangles $\triangle ABC\sim\triangle FGH$, the length of $BC$ is 24 cm, and side $FG$ measures 16 cm. If the length of $GH$ is 15 cm, determine how long side $AC$ is. Provide your answer as a decimal rounded to the nearest tenth.

Answer: 22.5 \text{ cm}


The expression $25x^2 - 130x - 120$ can be written as $(5x + a)(5x + b),$ where $a$ and $b$ are integers. What is $a + 3b$?

Answer: -86


Let $g(n)$ be the sum of all the divisors of a positive integer $n$. If $g(g(n)) = n+3$, define $n$ as hyperdeficient. Determine how many hyperdeficient positive integers there are.

Answer: 0


The integers 342 and 78 are expressed in base 5 and added. What is the resulting sum expressed in base 5?

Answer: 3140_5


Let \( x \) and \( y \) be two-digit integers such that:
1. \( x \) is an integer between 10 and 99, inclusive;
2. \( y \) is the number formed by reversing the digits of \( x \);
3. Let \( z = |x - y| \).

How many distinct values of \( z \) are possible?

Answer: 10


Triangle $PQR$ has $\angle Q = 45^{\circ}$ and $QR = 6$. Point $S$ is the midpoint of $QR$. If the altitude from point $P$ to $QR$ is 3 units, what is the largest possible value of $\tan{\angle PSQ}$?

Answer: 1


Let \( S \) be the set of positive real numbers. Define a function \( f: S \to \mathbb{R} \) such that
\[ f(x)f(y) = f(xy) + 1001 \left( \frac{1}{x} + \frac{1}{y} + 1000 \right) \]
for all \( x, y > 0 \).
Determine \( n \), the number of possible values of \( f(2) \), and \( s \), the sum of all possible values of \( f(2) \). Compute \( n \times s \).

Answer: 1001.5


The surface area of a sphere is $256\pi\text{ cm}^2$. Find the volume of the sphere in terms of $\pi$.

Answer: \frac{2048}{3}\pi \text{ cm}^3


A trapezoid is situated on a sloping hill, making the vertical elevations from the base to a midpoint in height unequal. The lower base of the trapezoid is four times the overall height $h$, while the upper base is five times as large. Moreover, the height to the midpoint from the lower base is $h/2$. Find the total area of this trapezoid considering a uniform slope.

Answer: \frac{9h^2}{2}


The total amount of money in John’s wallet is $6.80$, consisting of only dimes and quarters. There are four more dimes than quarters. How many dimes does John have?

Answer: 22


Consider the geometric sequence starting with $4$, followed by $\dfrac{16}{3}$, $\dfrac{64}{9}$, and so forth. Find the tenth and fifth terms of this sequence, expressing your answers as common fractions.

Answer: \frac{1024}{81}


Consider a rectangle and four circles, each with a radius of 3 cm. The circles are arranged at the corners of the rectangle in such a way that each circle touches two adjacent sides of the rectangle. What is the area, in square centimeters, of the rectangle?

Answer: 144 \text{ cm}^2


Let an ordered pair of positive integers $(m, n)$ be called regimented if for all nonnegative integers $k$, the numbers $m^k$ and $n^k$ have the same number of positive integer divisors. Let $N$ be the smallest positive integer such that $(2016^{2016}, N)$ is regimented. Compute the largest positive integer $v$ such that $2^v$ divides the difference $2016^{2016} - N$.

Answer: 10087


Let $S=\left\{ \frac{p}{q} \mid q \leq 2009, \frac{p}{q} < \frac{1257}{2009}, p,q \in \mathbb{N} \right\}$. If the maximum element of $S$ is $\frac{p_0}{q_0}$ in reduced form, find $p_0+q_0$.

Answer: 595


You are standing on one of the faces of a cube. Each turn, you randomly choose another face that shares an edge with the face you are standing on with equal probability, and move to that face. Let $F(n)$ be the probability that you land on the starting face after $n$ turns. Suppose that $F(8) = \frac{43}{256}$, and $F(10)$ can be expressed as a simplified fraction $\frac{a}{b}$. Find $a+b$.

Answer: 1195


In a certain country, there are $2023$ islands and $2022$ bridges. Each bridge connects two different islands, and any two islands share at most one bridge. It is possible to travel from any island to any other island using these bridges. If, among any three islands, there exists an island connected by bridges to each of the other two islands, these three islands form an "island group." It is known that any two "island groups" share at least one common island. What is the minimum number of islands with only one bridge connected to it?

Answer: 1011


There are two rows of seats with three side-by-side seats in each row. Two little boys, two little girls, and two adults sit in the six seats such that neither little boy sits next to either little girl. Determine the number of different ways these six people can be seated.

Answer: 176


The New York Public Library requires patrons to choose a 4-digit Personal Identification Number (PIN) to access its online system. (Leading zeros are allowed.) The PIN is not allowed to contain either of the following two forbidden patterns:

1. A digit that is repeated 3 or more times in a row. For example, $0001$ and $5555$ are not PINs, but $0010$ is a PIN.
2. A pair of digits that is duplicated. For example, $1212$ and $6363$ are not PINs, but $1221$ and $6633$ are PINs.

How many distinct possible PINs are there?

Answer: 9720


Andy wishes to open an electronic lock with a keypad containing all digits from $0$ to $9$. He knows that the password registered in the system is $2469$. Unfortunately, he is also aware that exactly two different buttons (but he does not know which ones) $\underline{a}$ and $\underline{b}$ on the keypad are broken $-$ when $\underline{a}$ is pressed, the digit $b$ is registered in the system, and when $\underline{b}$ is pressed, the digit $a$ is registered in the system. Find the least number of attempts Andy needs to surely be able to open the lock.

Answer: 31


The two numbers $0$ and $1$ are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the number $a+b$ between all pairs of consecutive numbers $a$, $b$ on the board. How many odd numbers will be on the board after $10$ such operations?

Answer: 683


Alice is planning a trip from the Bay Area to one of $5$ possible destinations, each serviced by only $1$ airport. She wants to book two flights: one to her destination and one returning. There are $3$ airports within the Bay Area from which she may leave and to which she may return. In how many ways can she plan her flight itinerary?

Answer: 45


Cyclic quadrilateral $ABCD$ satisfies $\angle ADC = 2 \cdot \angle BAD = 80^\circ$ and $\overline{BC} = \overline{CD}$. Let the angle bisector of $\angle BCD$ meet $AD$ at $P$. What is the measure, in degrees, of $\angle BPD$?

Answer: 60


Let $P_1, P_2, \dots, P_n$ be points on the plane. There is an edge between distinct points $P_x, P_y$ if and only if $x \mid y$. Find the largest $n$, such that the graph can be drawn with no crossing edges.

Answer: 15


Farmer John owns 2013 cows. Some cows are enemies of each other, and Farmer John wishes to divide them into as few groups as possible such that each cow has at most 3 enemies in her group. Each cow has at most 61 enemies. Compute the smallest integer $G$ such that, no matter which enemies they have, the cows can always be divided into at most $G$ such groups.

Answer: 16


Suppose $A_1, A_2, A_3, \ldots, A_{20}$ is a 20-sided regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon, but the sides are not the sides of the polygon?

Answer: 640


We are given the following function $f$, which takes a list of integers and outputs another list of integers. (Note that here the list is zero-indexed.)

\[
\begin{array}{l}
1: \textbf{FUNCTION} \ f(A) \\
2: \quad \textbf{FOR} \ i=1,\ldots, \operatorname{length}(A)-1: \\
3: \quad\quad A[i]\leftarrow A[A[i]] \\
4: \quad\quad A[0]\leftarrow A[0]-1 \\
5: \quad \textbf{RETURN} \ A
\end{array}
\]

Suppose the list $B$ is equal to $[0,1,2,8,2,0,1,7,0]$. In how many entries do $B$ and $f(B)$ differ?

Answer: 4


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Eric`, `Peter`, `Alice`, `Arnold`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`, `stew`, `stir fry`
 - The people keep unique animals: `bird`, `horse`, `cat`, `fish`, `dog`
 - Each person has a unique level of education: `master`, `doctorate`, `high school`, `associate`, `bachelor`

## Clues:
1. Alice is not in the fifth house.
2. The person who is a pizza lover is somewhere to the right of Peter.
3. Peter is in the first house.
4. The person with a bachelor's degree is the person who is a pizza lover.
5. The cat lover is the person with a master's degree.
6. The person who loves stir fry is the person with a doctorate.
7. The person with a high school diploma is directly left of the person with a master's degree.
8. The dog owner is the person who loves stir fry.
9. There is one house between Arnold and the person with an associate's degree.
10. The person who loves eating grilled cheese is the fish enthusiast.
11. Arnold is in the second house.
12. The person who keeps horses is Arnold.
13. Eric is directly left of the person with an associate's degree.
14. The person who loves the spaghetti eater is the cat lover.
 What is the value of attribute Animal for the person whose attribute Education is associate? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Bob`, `Peter`
 - Each person prefers a unique type of vacation: `cruise`, `city`, `mountain`, `camping`, `beach`
 - Each mother is accompanied by their child: `Fred`, `Bella`, `Meredith`, `Samantha`, `Timothy`
 - Each person has a unique type of pet: `bird`, `fish`, `dog`, `hamster`, `cat`
 - People have unique favorite sports: `basketball`, `swimming`, `tennis`, `baseball`, `soccer`
 - Each person has a unique favorite drink: `tea`, `coffee`, `water`, `milk`, `root beer`

## Clues:
1. The person who keeps a pet bird is the person who loves baseball.
2. The person with a pet hamster is not in the fourth house.
3. The person's child is named Bella is in the third house.
4. The person who prefers city breaks is directly left of the person's child is named Samantha.
5. The person's child is named Meredith is the person who enjoys camping trips.
6. There is one house between the person who prefers city breaks and Peter.
7. The person who loves tennis is the person's child is named Fred.
8. The person who owns a dog is Bob.
9. Bob is the person who loves beach vacations.
10. Bob is directly left of the person who has a cat.
11. The person's child is named Bella is the root beer lover.
12. The person who likes going on cruises is the person who likes milk.
13. The coffee drinker is not in the fourth house.
14. The one who only drinks water is the person who loves swimming.
15. The person who loves basketball is the person who has a cat.
16. The person who loves swimming is not in the fourth house.
17. The person who loves baseball is Eric.
18. There are two houses between Bob and the person's child is named Meredith.
19. The person who likes milk is the person who is the mother of Timothy.
20. The person's child is named Fred is directly left of Arnold.
21. The person who keeps a pet bird is directly left of Bob.
 What is the value of attribute Children for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: Samantha


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`, `Carol`
 - People have unique favorite sports: `volleyball`, `soccer`, `basketball`, `baseball`, `swimming`, `tennis`
 - Each person has a unique hobby: `gardening`, `cooking`, `painting`, `woodworking`, `knitting`, `photography`
 - Each mother is accompanied by their child: `Bella`, `Meredith`, `Alice`, `Samantha`, `Fred`, `Timothy`
 - Each person has a unique birthday month: `feb`, `jan`, `mar`, `may`, `sept`, `april`
 - Everyone has something unique for lunch: `stir fry`, `grilled cheese`, `pizza`, `spaghetti`, `stew`, `soup`

## Clues:
1. The person who loves soccer is somewhere to the right of Eric.
2. The person who paints as a hobby is somewhere to the left of the woodworking hobbyist.
3. The person whose birthday is in February is not in the third house.
4. The person who is the mother of Timothy is directly left of Arnold.
5. The person who loves cooking is somewhere to the left of the person's child is named Fred.
6. Alice is the person who paints as a hobby.
7. The person's child is named Alice is the person who loves cooking.
8. The person whose birthday is in March is the person's child is named Samantha.
9. The person who loves the soup is the person whose birthday is in May.
10. Carol is somewhere to the right of the person whose birthday is in May.
11. The person whose birthday is in March and the person who is a pizza lover are next to each other.
12. The person whose birthday is in February is not in the fourth house.
13. There are two houses between the person who loves baseball and the person who loves the stew.
14. The person whose birthday is in May is somewhere to the left of the person who loves volleyball.
15. The person whose birthday is in January is somewhere to the right of Arnold.
16. The person who loves cooking is not in the third house.
17. The person who enjoys knitting is the person whose birthday is in September.
18. The person who enjoys gardening is the person's child is named Meredith.
19. The person who is a pizza lover is the person who loves swimming.
20. There is one house between the person who loves basketball and the person whose birthday is in May.
21. The person who loves the soup is directly left of the person who loves the spaghetti eater.
22. The person who loves the stew is the person's child is named Bella.
23. The person who enjoys knitting is somewhere to the left of the person who loves the soup.
24. Bob is somewhere to the left of the person who loves the stew.
25. Arnold is the person who loves stir fry.
26. The woodworking hobbyist is directly left of the person who loves the soup.
 What is the value of attribute Children for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: Meredith


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Alice`, `Arnold`
 - Each mother is accompanied by their child: `Bella`, `Fred`, `Meredith`, `Samantha`
 - People have unique hair colors: `red`, `brown`, `black`, `blonde`
 - They all have a unique favorite flower: `carnations`, `roses`, `daffodils`, `lilies`
 - The people keep unique animals: `cat`, `bird`, `fish`, `horse`

## Clues:
1. Alice is somewhere to the right of the person who has red hair.
2. The person who loves a bouquet of daffodils is directly left of Arnold.
3. The fish enthusiast is the person's child is named Samantha.
4. Alice is directly left of the bird keeper.
5. The cat lover is the person who loves the rose bouquet.
6. The person who loves a carnations arrangement is somewhere to the right of the person who has brown hair.
7. The person who loves a bouquet of daffodils is directly left of the bird keeper.
8. Eric is the person's child is named Meredith.
9. There are two houses between the person who has red hair and the person who has black hair.
10. Eric is in the fourth house.
11. The person's child is named Bella is the person who loves a bouquet of daffodils.
 What is the value of attribute House for the person whose attribute Flower is roses? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Carol`, `Arnold`, `Peter`, `Alice`
 - Everyone has a unique favorite cigar: `blends`, `prince`, `dunhill`, `yellow monster`, `blue master`, `pall mall`
 - People have unique hair colors: `auburn`, `gray`, `blonde`, `black`, `red`, `brown`
 - Each person has a unique favorite drink: `boba tea`, `root beer`, `water`, `tea`, `coffee`, `milk`
 - Each person has an occupation: `artist`, `nurse`, `teacher`, `doctor`, `engineer`, `lawyer`

## Clues:
1. Bob is the tea drinker.
2. The Dunhill smoker is not in the second house.
3. There is one house between Carol and Arnold.
4. The Prince smoker is the person who has black hair.
5. The person who has auburn hair is not in the third house.
6. Arnold is in the fourth house.
7. The Prince smoker is not in the fourth house.
8. The person who is an artist is the person partial to Pall Mall.
9. The person who is an engineer is directly left of the root beer lover.
10. The boba tea drinker is the person who has brown hair.
11. The person who is a lawyer is in the fifth house.
12. The person who is an engineer is the person who smokes Blue Master.
13. The person who likes milk is the person who has auburn hair.
14. Alice is somewhere to the left of the person who is a teacher.
15. The person who has blonde hair is the person who smokes Yellow Monster.
16. The person who has black hair is the person who is a doctor.
17. Arnold is somewhere to the right of the person who has gray hair.
18. The person who has auburn hair is not in the first house.
19. The one who only drinks water is the person who smokes many unique blends.
20. The person who smokes many unique blends is somewhere to the right of the person who has brown hair.
21. Arnold is the root beer lover.
22. The person who has brown hair is not in the third house.
23. The person who smokes many unique blends is somewhere to the left of Peter.
24. The person who likes milk is somewhere to the left of the person who has blonde hair.
 What is the value of attribute House for the person whose attribute HairColor is brown? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Peter`, `Carol`, `Eric`
 - Each person prefers a unique type of vacation: `mountain`, `city`, `cultural`, `camping`, `beach`, `cruise`
 - Everyone has a favorite smoothie: `lime`, `desert`, `cherry`, `blueberry`, `dragonfruit`, `watermelon`
 - People have unique favorite sports: `tennis`, `volleyball`, `basketball`, `baseball`, `swimming`, `soccer`
 - Everyone has a unique favorite cigar: `blends`, `yellow monster`, `dunhill`, `pall mall`, `blue master`, `prince`
 - People have unique heights: `tall`, `very tall`, `average`, `super tall`, `very short`, `short`

## Clues:
1. Peter is the Prince smoker.
2. The person who smokes many unique blends is the person who enjoys mountain retreats.
3. The person who loves baseball is the person who is tall.
4. The person who goes on cultural tours is the person who drinks Blueberry smoothies.
5. The person who likes Cherry smoothies is not in the fourth house.
6. The person who loves soccer is the person who enjoys camping trips.
7. The person who is super tall is somewhere to the right of the person who likes going on cruises.
8. The person who drinks Blueberry smoothies is the person who is tall.
9. The person who is very tall is the Dunhill smoker.
10. The person who enjoys mountain retreats is Eric.
11. The person who is super tall is Alice.
12. Bob is the person who smokes Blue Master.
13. The person who loves volleyball is in the first house.
14. Peter and the person who drinks Lime smoothies are next to each other.
15. Carol is the person who likes Cherry smoothies.
16. Peter is the person who is short.
17. The person who smokes many unique blends is somewhere to the right of the Dragonfruit smoothie lover.
18. The person who loves swimming is Carol.
19. The person who is very short is the person who smokes Blue Master.
20. The person who loves tennis is somewhere to the left of the person who is very short.
21. The person who loves swimming is directly left of the person who enjoys camping trips.
22. The person who drinks Blueberry smoothies is not in the third house.
23. Alice is somewhere to the right of the person who drinks Lime smoothies.
24. The person who loves tennis is the person who is super tall.
25. The person who smokes Yellow Monster and the person who prefers city breaks are next to each other.
26. The Watermelon smoothie lover is the person who is very short.
27. The person who loves beach vacations is somewhere to the left of the person who is very tall.
 What is the value of attribute House for the person whose attribute Vacation is mountain? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - Each mother is accompanied by their child: `Bella`, `Meredith`, `Fred`
 - People use unique phone models: `samsung galaxy s21`, `iphone 13`, `google pixel 6`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`
 - Everyone has a unique favorite cigar: `blue master`, `prince`, `pall mall`
 - People have unique heights: `short`, `very short`, `average`

## Clues:
1. Eric is directly left of the Prince smoker.
2. The person who has an average height is somewhere to the right of Peter.
3. The person who uses a Samsung Galaxy S21 is in the second house.
4. The person who smokes Blue Master is in the third house.
5. The person who uses a Google Pixel 6 is Eric.
6. Arnold is somewhere to the right of the person who loves science fiction books.
7. The person who uses an iPhone 13 is the person's child is named Bella.
8. The person who is short is the Prince smoker.
9. The person who loves mystery books is the person who uses a Google Pixel 6.
10. The person's child is named Meredith and the person who is short are next to each other.
 What is the value of attribute Height for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: short


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Bob`, `Alice`, `Arnold`, `Carol`, `Eric`
 - The people are of nationalities: `swede`, `chinese`, `norwegian`, `german`, `brit`, `dane`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`, `chevrolet silverado`, `honda civic`, `bmw 3 series`
 - Each person has a unique favorite drink: `root beer`, `milk`, `tea`, `water`, `coffee`, `boba tea`

## Clues:
1. The boba tea drinker is the person who owns a Ford F-150.
2. The British person is directly left of Eric.
3. Alice is the person who owns a Honda Civic.
4. The tea drinker is not in the first house.
5. Eric is the Chinese.
6. The tea drinker is the person who owns a BMW 3 Series.
7. The German is in the second house.
8. The person who owns a Chevrolet Silverado is somewhere to the left of the person who likes milk.
9. Bob is in the fifth house.
10. The person who owns a Toyota Camry is directly left of the one who only drinks water.
11. The root beer lover is directly left of the Norwegian.
12. The Dane is somewhere to the right of the person who owns a BMW 3 Series.
13. Arnold is in the second house.
14. The person who owns a Honda Civic is directly left of the person who owns a Ford F-150.
15. Peter is not in the third house.
16. The person who owns a Toyota Camry is in the third house.
 What is the value of attribute CarModel for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: toyota camry


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Bob`, `Peter`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `dragonfruit`, `lime`, `desert`
 - People have unique favorite music genres: `hip hop`, `classical`, `rock`, `pop`, `jazz`
 - Everyone has something unique for lunch: `grilled cheese`, `pizza`, `spaghetti`, `stir fry`, `stew`
 - Each person has a unique favorite drink: `tea`, `root beer`, `milk`, `water`, `coffee`

## Clues:
1. The coffee drinker is not in the first house.
2. The person who loves jazz music is the Desert smoothie lover.
3. The Watermelon smoothie lover is in the fourth house.
4. The coffee drinker is directly left of Peter.
5. The one who only drinks water is the person who loves hip-hop music.
6. The Dragonfruit smoothie lover is directly left of the one who only drinks water.
7. The Desert smoothie lover is in the fifth house.
8. The person who loves stir fry is somewhere to the right of Bob.
9. The person who is a pizza lover is the person who likes Cherry smoothies.
10. The person who loves classical music is the person who loves the stew.
11. The one who only drinks water is Bob.
12. The root beer lover is the Dragonfruit smoothie lover.
13. There is one house between Arnold and the person who likes Cherry smoothies.
14. The person who loves the spaghetti eater is in the fifth house.
15. The person who likes milk is in the fifth house.
16. The person who likes Cherry smoothies is Bob.
17. The Dragonfruit smoothie lover is the person who loves the stew.
18. Eric and Peter are next to each other.
19. The person who loves rock music is not in the first house.
 What is the value of attribute Drink for the person whose attribute Food is grilled cheese? Please reason step by step, and put your final answer within \boxed{}

Answer: tea


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Bob`, `Peter`, `Alice`, `Carol`
 - People have unique heights: `very tall`, `tall`, `average`, `short`, `super tall`, `very short`
 - Each person has a unique birthday month: `sept`, `may`, `jan`, `april`, `feb`, `mar`
 - People have unique hair colors: `blonde`, `red`, `black`, `auburn`, `gray`, `brown`
 - People have unique favorite book genres: `science fiction`, `romance`, `biography`, `mystery`, `fantasy`, `historical fiction`
 - Each person has a unique type of pet: `bird`, `hamster`, `fish`, `dog`, `cat`, `rabbit`

## Clues:
1. The person with an aquarium of fish is the person who loves mystery books.
2. The person who has black hair is the person who owns a dog.
3. The person who has gray hair is the person who loves science fiction books.
4. The person who loves historical fiction books is somewhere to the left of the person who has black hair.
5. The person who loves historical fiction books is directly left of the person who owns a rabbit.
6. The person whose birthday is in February is the person who owns a dog.
7. Bob is the person who keeps a pet bird.
8. The person who keeps a pet bird is the person who loves romance books.
9. Alice is directly left of the person who loves fantasy books.
10. The person who owns a rabbit is directly left of the person who is very short.
11. The person who is short is somewhere to the left of the person who is super tall.
12. The person who has an average height is in the sixth house.
13. The person who has red hair is not in the first house.
14. The person who is very tall is somewhere to the right of the person who is super tall.
15. There are two houses between the person who loves romance books and Eric.
16. The person with an aquarium of fish is in the sixth house.
17. The person who has blonde hair is the person who loves historical fiction books.
18. The person who is tall is somewhere to the right of the person whose birthday is in February.
19. The person who has auburn hair is Carol.
20. The person with an aquarium of fish is the person whose birthday is in March.
21. The person who loves science fiction books is directly left of the person with an aquarium of fish.
22. The person whose birthday is in September is not in the first house.
23. The person who loves biography books is directly left of the person who has a cat.
24. The person whose birthday is in January is the person who loves science fiction books.
25. Arnold is the person whose birthday is in May.
 What is the value of attribute Birthday for the person whose attribute HairColor is red? Please reason step by step, and put your final answer within \boxed{}

Answer: may


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`, `Alice`, `Bob`
 - Each person has a unique level of education: `associate`, `doctorate`, `high school`, `bachelor`, `master`
 - Each mother is accompanied by their child: `Fred`, `Meredith`, `Bella`, `Timothy`, `Samantha`
 - Everyone has a unique favorite cigar: `prince`, `pall mall`, `blue master`, `dunhill`, `blends`
 - The people are of nationalities: `swede`, `brit`, `dane`, `german`, `norwegian`
 - People have unique favorite book genres: `romance`, `mystery`, `biography`, `science fiction`, `fantasy`

## Clues:
1. The person with a master's degree is the person partial to Pall Mall.
2. The person with a high school diploma is in the fourth house.
3. The person's child is named Samantha is in the first house.
4. The Norwegian and the person with a master's degree are next to each other.
5. The German is the Dunhill smoker.
6. The person who is the mother of Timothy is directly left of the person with a master's degree.
7. Arnold and the Dane are next to each other.
8. The Swedish person is the person who loves mystery books.
9. The Swedish person is Alice.
10. Arnold is the person with an associate's degree.
11. The person with a doctorate is in the second house.
12. Peter is somewhere to the right of the person who loves biography books.
13. The person who smokes many unique blends is Alice.
14. The person who loves fantasy books is in the first house.
15. Peter is the Norwegian.
16. The person with a doctorate is the person's child is named Meredith.
17. The Prince smoker is somewhere to the left of the person who smokes many unique blends.
18. The person partial to Pall Mall is the person's child is named Bella.
19. The person who loves romance books is Bob.
 What is the value of attribute Education for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: master


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Bob`, `Eric`
 - Each person has a favorite color: `white`, `blue`, `yellow`, `red`, `green`
 - Everyone has a unique favorite cigar: `blue master`, `dunhill`, `prince`, `pall mall`, `blends`
 - People have unique favorite book genres: `biography`, `mystery`, `romance`, `fantasy`, `science fiction`

## Clues:
1. There are two houses between the Prince smoker and the person who loves yellow.
2. The person who loves white is Bob.
3. The person who smokes Blue Master is directly left of Bob.
4. Alice is somewhere to the left of the person who loves science fiction books.
5. The person who loves biography books is not in the first house.
6. Arnold is somewhere to the left of the person who smokes Blue Master.
7. The person who loves mystery books and the person who loves biography books are next to each other.
8. The person who loves fantasy books is Bob.
9. The person whose favorite color is green is the person who smokes Blue Master.
10. The person partial to Pall Mall is in the fourth house.
11. The person who loves science fiction books is the Dunhill smoker.
12. The person whose favorite color is red is not in the fifth house.
13. Peter is directly left of the Dunhill smoker.
 What is the value of attribute BookGenre for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: romance


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Eric`, `Arnold`
 - Each person has an occupation: `doctor`, `engineer`, `artist`, `teacher`
 - Each person has a unique birthday month: `sept`, `april`, `jan`, `feb`
 - Each person has a favorite color: `red`, `white`, `green`, `yellow`

## Clues:
1. Arnold is in the first house.
2. The person whose birthday is in February and the person who is a teacher are next to each other.
3. The person who is a teacher is the person whose favorite color is red.
4. The person whose favorite color is red is not in the third house.
5. The person who loves yellow is directly left of Peter.
6. The person whose favorite color is green and the person whose favorite color is red are next to each other.
7. The person who is an engineer is the person whose favorite color is green.
8. Eric is the person who is a teacher.
9. The person who loves yellow is not in the second house.
10. The person whose birthday is in September is somewhere to the left of the person whose favorite color is green.
11. The person who is a doctor is somewhere to the right of the person whose birthday is in April.
 What is the value of attribute Color for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: red


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Peter`, `Arnold`, `Bob`, `Carol`
 - Each mother is accompanied by their child: `Meredith`, `Fred`, `Bella`, `Samantha`, `Alice`, `Timothy`
 - Each person has a unique birthday month: `jan`, `feb`, `mar`, `may`, `sept`, `april`
 - Everyone has a unique favorite cigar: `dunhill`, `blends`, `blue master`, `prince`, `yellow monster`, `pall mall`

## Clues:
1. The person's child is named Meredith and the person who smokes Yellow Monster are next to each other.
2. Carol is directly left of Arnold.
3. The person's child is named Meredith is the person who smokes many unique blends.
4. The person's child is named Samantha is the person whose birthday is in March.
5. The person partial to Pall Mall is somewhere to the right of the Dunhill smoker.
6. Eric is the Dunhill smoker.
7. There is one house between the person whose birthday is in May and the person who is the mother of Timothy.
8. The person who smokes Blue Master is the person's child is named Bella.
9. The person's child is named Samantha is somewhere to the right of the person whose birthday is in January.
10. The person's child is named Samantha is in the fifth house.
11. The person whose birthday is in May is the person's child is named Meredith.
12. The person's child is named Alice is the person whose birthday is in February.
13. There is one house between Alice and Arnold.
14. Bob is the person's child is named Meredith.
15. Carol is the person whose birthday is in April.
16. The person partial to Pall Mall is the person whose birthday is in April.
17. The person whose birthday is in May is somewhere to the left of the person whose birthday is in January.
 What is the value of attribute House for the person whose attribute Children is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`
 - People have unique favorite music genres: `rock`, `jazz`, `pop`, `classical`
 - Everyone has a favorite smoothie: `dragonfruit`, `watermelon`, `desert`, `cherry`
 - People have unique hair colors: `red`, `black`, `brown`, `blonde`
 - People have unique favorite book genres: `science fiction`, `fantasy`, `romance`, `mystery`

## Clues:
1. The person who has red hair is not in the third house.
2. There is one house between Eric and the person who likes Cherry smoothies.
3. The person who loves science fiction books is Peter.
4. The person who loves fantasy books is Alice.
5. The Dragonfruit smoothie lover is the person who loves pop music.
6. The Dragonfruit smoothie lover is the person who has red hair.
7. The Watermelon smoothie lover is somewhere to the right of the person who loves classical music.
8. The person who has black hair is the person who likes Cherry smoothies.
9. There is one house between the person who loves pop music and the person who loves fantasy books.
10. The person who loves romance books is somewhere to the right of the person who loves jazz music.
11. The person who has red hair is directly left of the person who has black hair.
12. The person who has brown hair and the Dragonfruit smoothie lover are next to each other.
13. The person who loves science fiction books is the Dragonfruit smoothie lover.
 What is the value of attribute MusicGenre for the person whose attribute BookGenre is fantasy? Please reason step by step, and put your final answer within \boxed{}

Answer: rock


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`
 - Each person prefers a unique type of vacation: `mountain`, `cruise`, `beach`, `city`
 - Each person has a unique birthday month: `jan`, `feb`, `sept`, `april`
 - The people are of nationalities: `swede`, `dane`, `brit`, `norwegian`

## Clues:
1. The person who enjoys mountain retreats is Alice.
2. The Norwegian is somewhere to the left of the person whose birthday is in September.
3. The person who prefers city breaks is the British person.
4. Alice and Eric are next to each other.
5. The person who likes going on cruises is not in the fourth house.
6. Alice is somewhere to the left of the person whose birthday is in January.
7. The person who enjoys mountain retreats is the Norwegian.
8. Alice is directly left of the Swedish person.
9. Eric is in the third house.
10. The person whose birthday is in April is Peter.
11. The Dane is the person whose birthday is in September.
 What is the value of attribute House for the person whose attribute Vacation is city? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Arnold`, `Peter`, `Alice`
 - Each person has a unique hobby: `painting`, `knitting`, `cooking`, `photography`, `gardening`

## Clues:
1. Alice is in the fifth house.
2. Bob is directly left of the person who enjoys gardening.
3. The person who loves cooking and the person who paints as a hobby are next to each other.
4. The person who paints as a hobby is not in the third house.
5. The photography enthusiast is not in the third house.
6. Arnold is in the fourth house.
7. Bob is in the first house.
8. The person who enjoys knitting is directly left of Eric.
 What is the value of attribute House for the person whose attribute Hobby is painting? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`, `Alice`
 - Each person prefers a unique type of vacation: `mountain`, `cruise`, `beach`, `city`
 - Each person lives in a unique style of house: `craftsman`, `ranch`, `victorian`, `colonial`
 - The mothers' names in different houses are unique: `Kailyn`, `Holly`, `Aniya`, `Janelle`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`
 - Each person has a favorite color: `green`, `white`, `yellow`, `red`

## Clues:
1. Alice is directly left of the person whose favorite color is red.
2. The person who loves a bouquet of daffodils is The person whose mother's name is Janelle.
3. Eric is not in the first house.
4. The person who loves the boquet of lilies is directly left of the person living in a colonial-style house.
5. The person residing in a Victorian house is in the second house.
6. The person who loves the rose bouquet is the person who loves yellow.
7. The person whose mother's name is Aniya is the person who loves yellow.
8. The person who enjoys mountain retreats is in the first house.
9. The person who likes going on cruises is the person whose favorite color is red.
10. The person who loves yellow is in the second house.
11. The person who loves white is the person who prefers city breaks.
12. The person whose mother's name is Holly is not in the fourth house.
13. The person whose favorite color is red is directly left of Peter.
14. The person who loves a bouquet of daffodils is the person in a ranch-style home.
 What is the value of attribute Color for the person whose attribute Flower is lilies? Please reason step by step, and put your final answer within \boxed{}

Answer: red


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`
 - Each person has a unique birthday month: `sept`, `april`, `jan`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`
 - People have unique hair colors: `brown`, `black`, `blonde`
 - The mothers' names in different houses are unique: `Aniya`, `Janelle`, `Holly`

## Clues:
1. Arnold is the person whose birthday is in January.
2. The person whose mother's name is Janelle is the person who has brown hair.
3. The person who has black hair is not in the second house.
4. The person who has black hair is the person whose birthday is in April.
5. The person whose mother's name is Aniya is Eric.
6. The person whose mother's name is Janelle is the person who owns a Toyota Camry.
7. The person who has black hair is The person whose mother's name is Aniya.
8. The person whose birthday is in September and the person who owns a Tesla Model 3 are next to each other.
9. Arnold is the person who has brown hair.
10. The person who owns a Toyota Camry is somewhere to the left of the person who owns a Ford F-150.
 What is the value of attribute Birthday for the person whose attribute CarModel is ford f150? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Arnold`, `Eric`
 - The people keep unique animals: `horse`, `bird`, `fish`, `cat`
 - Each person has a unique level of education: `high school`, `bachelor`, `associate`, `master`
 - Each person has a unique favorite drink: `tea`, `water`, `coffee`, `milk`
 - People have unique favorite book genres: `fantasy`, `science fiction`, `romance`, `mystery`
 - Each person has a unique birthday month: `april`, `feb`, `sept`, `jan`

## Clues:
1. Peter is the person whose birthday is in February.
2. The person with a master's degree is the bird keeper.
3. The person who keeps horses is not in the second house.
4. The person whose birthday is in January is the person who loves science fiction books.
5. The person who loves mystery books is Alice.
6. There is one house between the person who likes milk and Alice.
7. The person whose birthday is in April is the person who loves fantasy books.
8. The person with a high school diploma is not in the second house.
9. Eric is directly left of the person who likes milk.
10. The fish enthusiast is the person whose birthday is in February.
11. The tea drinker is somewhere to the left of Peter.
12. The one who only drinks water is the person with a master's degree.
13. The person with a bachelor's degree is somewhere to the right of the person whose birthday is in January.
14. Peter is in the third house.
15. The person who loves science fiction books is Arnold.
 What is the value of attribute BookGenre for the person whose attribute Birthday is jan? Please reason step by step, and put your final answer within \boxed{}

Answer: science fiction


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Carol`, `Bob`, `Eric`, `Peter`
 - People use unique phone models: `iphone 13`, `samsung galaxy s21`, `xiaomi mi 11`, `oneplus 9`, `google pixel 6`, `huawei p50`
 - Each person has a unique birthday month: `april`, `mar`, `feb`, `may`, `jan`, `sept`
 - Each person has an occupation: `nurse`, `doctor`, `engineer`, `teacher`, `artist`, `lawyer`

## Clues:
1. The person who uses a Google Pixel 6 is directly left of the person whose birthday is in March.
2. The person who is a nurse is Eric.
3. Arnold is directly left of the person whose birthday is in April.
4. Alice and the person whose birthday is in January are next to each other.
5. The person who is an artist is the person whose birthday is in May.
6. The person whose birthday is in September is not in the sixth house.
7. There is one house between the person whose birthday is in March and the person who is a nurse.
8. The person whose birthday is in March is the person who is a lawyer.
9. The person who uses a Xiaomi Mi 11 is directly left of the person who is a nurse.
10. The person who is a teacher is not in the sixth house.
11. Eric is the person who uses an iPhone 13.
12. The person who is an engineer and Peter are next to each other.
13. The person whose birthday is in January is in the third house.
14. The person who uses a Samsung Galaxy S21 is Bob.
15. The person who is a lawyer and the person whose birthday is in April are next to each other.
16. The person who uses a OnePlus 9 is somewhere to the right of the person who is a doctor.
 What is the value of attribute Occupation for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}

Answer: teacher


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Bob`, `Eric`, `Carol`, `Alice`
 - The people keep unique animals: `horse`, `rabbit`, `fish`, `cat`, `bird`, `dog`
 - Each person has an occupation: `engineer`, `nurse`, `lawyer`, `teacher`, `artist`, `doctor`
 - People have unique favorite sports: `basketball`, `volleyball`, `soccer`, `tennis`, `baseball`, `swimming`
 - People have unique heights: `average`, `tall`, `short`, `very short`, `very tall`, `super tall`

## Clues:
1. The person who is an engineer is the dog owner.
2. The person who has an average height is somewhere to the left of the person who is short.
3. The person who has an average height is directly left of the rabbit owner.
4. The person who is tall is somewhere to the left of the person who is very short.
5. Arnold is the cat lover.
6. The person who keeps horses is the person who is a teacher.
7. Carol is the person who loves soccer.
8. The person who is tall is the person who loves volleyball.
9. The person who is a lawyer is in the fifth house.
10. The person who loves tennis is the person who is a teacher.
11. The person who has an average height is the person who loves swimming.
12. The person who loves baseball is directly left of the person who is an engineer.
13. Peter is the person who is a nurse.
14. Bob is somewhere to the right of the person who is an artist.
15. The person who is a teacher is directly left of the person who loves soccer.
16. The rabbit owner is Alice.
17. The fish enthusiast is Carol.
18. The person who loves baseball is in the first house.
19. The cat lover is somewhere to the right of the person who is very short.
20. The person who is super tall is in the fifth house.
 What is the value of attribute FavoriteSport for the person whose attribute Animal is rabbit? Please reason step by step, and put your final answer within \boxed{}

Answer: volleyball


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Eric`
 - The mothers' names in different houses are unique: `Holly`, `Kailyn`, `Janelle`, `Aniya`
 - They all have a unique favorite flower: `carnations`, `roses`, `lilies`, `daffodils`

## Clues:
1. Alice is The person whose mother's name is Kailyn.
2. The person whose mother's name is Janelle is somewhere to the right of Arnold.
3. Peter is somewhere to the right of the person who loves a carnations arrangement.
4. Eric is the person who loves a bouquet of daffodils.
5. Arnold is The person whose mother's name is Holly.
6. The person who loves a carnations arrangement is somewhere to the right of The person whose mother's name is Holly.
7. The person who loves the boquet of lilies is directly left of Alice.
8. Alice is in the third house.
 What is the value of attribute Flower for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: lilies


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Peter`, `Carol`, `Alice`, `Eric`
 - Everyone has something unique for lunch: `stir fry`, `stew`, `soup`, `pizza`, `grilled cheese`, `spaghetti`
 - Each person has an occupation: `doctor`, `artist`, `engineer`, `teacher`, `lawyer`, `nurse`
 - Each person has a unique type of pet: `dog`, `bird`, `hamster`, `cat`, `fish`, `rabbit`
 - People have unique favorite music genres: `pop`, `country`, `jazz`, `classical`, `rock`, `hip hop`
 - Everyone has a favorite smoothie: `blueberry`, `desert`, `watermelon`, `cherry`, `lime`, `dragonfruit`

## Clues:
1. The person who loves the spaghetti eater is the Dragonfruit smoothie lover.
2. The person who drinks Lime smoothies is in the second house.
3. Carol is somewhere to the left of the Desert smoothie lover.
4. The person who is an engineer is not in the second house.
5. The person who loves the soup is in the second house.
6. The person who owns a dog is not in the second house.
7. The person who is a pizza lover is Alice.
8. Arnold is somewhere to the left of Eric.
9. The person who loves classical music is the Dragonfruit smoothie lover.
10. The person who owns a rabbit is somewhere to the left of the Desert smoothie lover.
11. The person who is a nurse is not in the sixth house.
12. There are two houses between the person who is a nurse and the person who loves the spaghetti eater.
13. The person who likes Cherry smoothies is the person with an aquarium of fish.
14. The person who has a cat is Alice.
15. The person who likes Cherry smoothies is somewhere to the right of the Desert smoothie lover.
16. There is one house between the person who drinks Blueberry smoothies and the person who loves eating grilled cheese.
17. The person who loves classical music is somewhere to the left of the person who loves the stew.
18. The person who is a lawyer is the person who loves classical music.
19. The person who loves jazz music is Bob.
20. Bob is the person who drinks Lime smoothies.
21. The person who loves pop music is the person who is a teacher.
22. The person who is an engineer is directly left of the person who loves rock music.
23. The person who loves hip-hop music is not in the first house.
24. The person with an aquarium of fish is Peter.
25. The person who drinks Blueberry smoothies is the person who keeps a pet bird.
26. The person who has a cat is in the first house.
27. The person who is a doctor is not in the sixth house.
28. The person who owns a rabbit is not in the second house.
29. Carol is not in the fourth house.
 What is the value of attribute Pet for the person whose attribute Food is spaghetti? Please reason step by step, and put your final answer within \boxed{}

Answer: rabbit


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Bob`, `Alice`, `Carol`
 - The people keep unique animals: `rabbit`, `dog`, `fish`, `bird`, `cat`, `horse`
 - Each person has a unique hobby: `cooking`, `knitting`, `painting`, `woodworking`, `photography`, `gardening`
 - People have unique favorite sports: `basketball`, `soccer`, `swimming`, `tennis`, `baseball`, `volleyball`
 - Each person prefers a unique type of vacation: `cruise`, `cultural`, `city`, `camping`, `mountain`, `beach`
 - Each person has a unique favorite drink: `water`, `root beer`, `boba tea`, `tea`, `milk`, `coffee`

## Clues:
1. The person who enjoys camping trips is the person who loves cooking.
2. The person who loves baseball is the photography enthusiast.
3. The person who loves volleyball is somewhere to the left of the person who prefers city breaks.
4. Carol is the person who loves tennis.
5. The dog owner is the person who loves swimming.
6. The photography enthusiast is the tea drinker.
7. The person who loves baseball is directly left of the person who goes on cultural tours.
8. Peter is the person who enjoys camping trips.
9. The person who enjoys knitting is the person who keeps horses.
10. The coffee drinker is directly left of the person who loves soccer.
11. The person who enjoys mountain retreats is the person who loves soccer.
12. The boba tea drinker is not in the fourth house.
13. The cat lover is somewhere to the left of the dog owner.
14. The person who loves beach vacations is Bob.
15. The coffee drinker is somewhere to the right of Alice.
16. The tea drinker is the fish enthusiast.
17. The person who likes milk is Carol.
18. There is one house between the rabbit owner and the woodworking hobbyist.
19. The person who loves cooking is not in the third house.
20. The person who loves tennis is the rabbit owner.
21. The dog owner is directly left of the person who likes going on cruises.
22. The woodworking hobbyist is the dog owner.
23. There are two houses between the person who paints as a hobby and the person who keeps horses.
24. Arnold is the root beer lover.
 What is the value of attribute Hobby for the person whose attribute Vacation is mountain? Please reason step by step, and put your final answer within \boxed{}

Answer: gardening


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Eric`, `Bob`
 - Each person has a unique favorite drink: `root beer`, `tea`, `water`, `coffee`, `milk`
 - People have unique favorite music genres: `pop`, `hip hop`, `classical`, `jazz`, `rock`
 - The people are of nationalities: `dane`, `norwegian`, `swede`, `german`, `brit`
 - Everyone has a unique favorite cigar: `dunhill`, `prince`, `blue master`, `pall mall`, `blends`

## Clues:
1. The Dunhill smoker is not in the second house.
2. The person who loves rock music is directly left of the person who loves classical music.
3. Peter is the person who loves jazz music.
4. Arnold is directly left of the Dane.
5. The root beer lover is the German.
6. The tea drinker is the Swedish person.
7. The Swedish person is the person who loves classical music.
8. The Dunhill smoker is not in the fourth house.
9. The person who loves jazz music is the Prince smoker.
10. Eric and the person who loves hip-hop music are next to each other.
11. There are two houses between the person partial to Pall Mall and the Swedish person.
12. Arnold is somewhere to the right of Eric.
13. The coffee drinker is somewhere to the right of the Dunhill smoker.
14. The Norwegian is somewhere to the right of the person who smokes Blue Master.
15. There is one house between the one who only drinks water and the person who smokes many unique blends.
16. Alice is the tea drinker.
 What is the value of attribute Drink for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: milk


What is the largest area of a regular hexagon that can be drawn inside an equilateral triangle with a side length of \(3\)? Provide your answer in the form of \(\frac{a\sqrt{b}}{c}\), where \(a\), \(b\), and \(c\) are integers. Please find the value of a + b + c.

Answer: 8


Let $P(x)=x^3+ax^2+bx+c$ where $a,b,c$ are positive real numbers. If $P(1)\geq 2$ and $P(3)\leq 31$, find how many integers $P(4)$ can take.

Answer: 4


A circle is inscribed in an equilateral triangle. Three nested sequences of circles are then constructed as follows: each circle touches the previous circle and has two edges of the triangle as tangents.

Find the ratio of the area of the largest circle to the combined area of all the other circles. The original answer is in the format \(\frac{k}{m}\), where \(k\) and \(m\) are integers. Please calculate the value of \(k + m\).

Answer: 11


If an integer of two digits is \(k\) times the sum of its digits, find the expression that represents the number formed by interchanging the digits as the sum of the digits multiplied by \(x\). The original answer is in the form \(m - k\). Please find the value of \(m + n\).

Answer: 11


Exercise 7. Let $n \geqslant 2$ be an integer. Clara has a $3 n \times 3 n$ board, similar to a chessboard. She has just invented a new piece, the leopard, which she can move as follows: one square up, one square to the right, or one square diagonally, down and to the left. Clara placed her leopard on one of the squares of the board, then moved it so that it never passes through the same square twice, until it returns to its starting square.

What is the maximum number of moves Clara could have made in this way?

Answer: 9n^2-3


9. Use three rectangles to form a larger rectangle (without gaps or overlaps), where two of the rectangles are $3 \mathrm{~cm} \times 8 \mathrm{~cm}$ and $2 \mathrm{~cm} \times 5 \mathrm{~cm}$, respectively. The dimensions of the third rectangle can be $\qquad$ kinds.

Answer: 4


Example 5 In a regular pentagon $A B C D E$, there is a point $P$ inside. It is known that $\angle A B P=6^{\circ}, \angle A E P=12^{\circ}$. Find the degree measure of $\angle P A C$.

Answer: 12^{\circ}


$[$ Mathematical logic (miscellaneous).]

Authors: Folklore, Folklore

In a school of witchcraft, there are 13 students. Before the clairvoyance exam, the teacher seated them at a round table and asked them to guess who would receive the clairvoyant diploma. They modestly kept quiet about themselves and their two neighbors, but wrote about everyone else: "None of these ten will get it!" Of course, all those who passed the exam guessed correctly, while all the others were wrong. How many wizards received the diploma?

#

Answer: 2


Let $ABCD$ be a unit square in the plane. Points $X$ and $Y$ are chosen independently and uniformly at random on the perimeter of $ABCD$. If the expected value of the area of triangle $\triangle AXY$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.

Answer: 113


$n$ coins are arranged in a circle. If two neighboring coins both show heads or both show tails, you can flip both coins. How many distinct arrangements of the coins are there such that no arrangement can be transformed into another by performing these operations?

Answer: 2


There are $10^{2015}$ planets in an Intergalactic empire. Every two planets are connected by a two-way space line served by one of $2015$ travel companies. The Emperor would like to close $k$ of these companies such that it is still possible to reach any planet from any other planet. Find the maximum value of $k$ for which this is always possible.

Answer: 1007


81. Select several numbers from $1,2,3,4,5,6$ so that the sum of the selected numbers is a multiple of 3 but not a multiple of 5. There are $\qquad$ different ways to do this.

Answer: 19


Two positive integers $a$ and $b$ are prime-related if $a = pb$ or $b = pa$ for some prime $p$. Find all positive integers $n$, such that $n$ has at least three divisors, and all the divisors can be arranged without repetition in a circle so that any two adjacent divisors are prime-related.

Note that $1$ and $n$ are included as divisors.

Answer: n


Given a permutation $\left(a_{0}, a_{1}, \ldots, a_{n}\right)$ of the sequence $0,1, \ldots, n$. A transposition of $a_{i}$ with $a_{j}$ is called legal if $a_{i}=0$ for $i>0$, and $a_{i-1}+1=a_{j}$. The permutation $\left(a_{0}, a_{1}, \ldots, a_{n}\right)$ is called regular if after a number of legal transpositions it becomes $(1,2, \ldots, n, 0)$. For which numbers $n$ is the permutation $(1, n, n-1, \ldots, 3,2,0)$ regular?

Answer: $n=2$ and $n=2^{k}-1, k$ positive integer.

#

Answer: 2


2. 7511 sets $M_{1}, M_{2}, \cdots, M_{11}$. Each set has 5 elements, and the intersection of any two sets is non-empty. Find the minimum possible value of the maximum number of sets that have a common element.

Answer: 4


5. Islandland consists of ten islands, some of which are connected by two-way air routes. If you choose any 9 islands, you can fly around them one by one and return to the starting island at the end. Find the minimum number of air routes that can exist in this country.

Answer: 15


Samantha, an aspiring AFLW coach, is studying the winning patterns of successful coaches. She has compiled data on the last 5 seasons of the league, focusing on two renowned coaches: Coach A and Coach B.

In her research, she found that Coach A's team has won 60% of their 25 games each season, while Coach B's team has won 70% of their 20 games each season.

Samantha wants to know how many more games Coach B's team won than Coach A's team over the past 5 seasons combined. Can you help her figure this out?

Answer: -5


Mr. Thompson, the local bookstore owner in West Kootenay, received a new shipment of books. He received 250 books in total, divided into three categories: fiction, non-fiction, and children's books. The fiction books make up 40% of the shipment, and the non-fiction books account for 35% of the shipment. How many children's books did Mr. Thompson receive in this shipment?

Answer: 63


A researcher named Alex is working on a documentary about historical battles and needs to find shooting locations that are both historically accurate and visually captivating. Alex has identified three potential sites: Site A, Site B, and Site C.

- Site A is 15 miles from the production studio, Site B is 25 miles away, and Site C is 35 miles away.
- The team plans to visit each site twice to evaluate its suitability for filming.
- For each visit, the team will incur a travel cost of $2 per mile.

Calculate the total travel cost for visiting all three sites twice each.

Answer: 600


Jin, a quiet musician from South Korea, finds peace and comfort in playing the guitar. Each day, he spends 3 hours practicing new songs and 2 hours playing his favorite tunes for relaxation. Over the weekend, Jin dedicates 5 hours each day solely to learning new techniques on the guitar.

If Jin practices for six days a week, including both weekdays and weekends, how many total hours does he spend playing the guitar in one week?

Answer: 30


A famous Red Sox player, known for giving exclusive interviews, talked about his game strategies and revealed some interesting numbers. He mentioned that during a recent game, he hit a total of 3 home runs. In the first interview, he shared that each home run brought in 2 additional runs, including himself. In the second interview, he revealed that he also made 4 successful base steals during the game. If each home run and each base steal contributes 1 point to the team's score, how many total points did the Red Sox player contribute to his team's score in that game?

Answer: 10


Alex is a broadcasting history enthusiast who loves sending in questions and suggestions for podcast episodes. This month, Alex has decided to focus on his favorite decade, the 1960s. He plans to send in 3 questions and 2 suggestions for each year of that decade. However, he realized he can only manage to send half of that amount for the year 1965 due to a busy schedule. How many total questions and suggestions will Alex send in for the entire 1960s decade?

Answer: 47


Maria is a single mother who has been able to go back to school thanks to a scholarship program for underprivileged students funded by a generous businessman. As part of her studies, she receives a monthly stipend of $500 to help cover her expenses.

Every month, Maria allocates her stipend as follows:
- 40% for her child's education expenses.
- 25% for groceries.
- The remainder is saved for emergencies.

In the first three months, Maria saved a total of $240.

How much money did Maria spend on her child's education expenses and groceries combined during these three months?

Answer: 1260


An esteemed Islamic scholar is organizing a series of weekend classes to teach young Muslims about the principles of Islam. He plans to hold classes every Saturday and Sunday for 4 weeks. Each class can accommodate 15 students. On Saturdays, he teaches about the history of the Quran, and on Sundays, he teaches about the application of Islamic principles in daily life.

If the scholar wants to ensure that each student attends the same number of Saturday and Sunday classes, how many students can fully participate in all the classes over the 4 weeks?

Answer: 60


Alex, an education technology expert, is invited to give a presentation about the future of library technology at a conference. For the presentation, Alex plans to use a total of 5 gadgets: 2 tablets, 1 laptop, and 2 smartboards. Each tablet requires 3 hours to charge fully, the laptop requires 4 hours, and each smartboard needs 5 hours to be ready. If Alex starts charging all devices at the same time and can charge 3 devices simultaneously, how many hours in total will Alex need to charge all the gadgets for the presentation?

Answer: 8


Alex, an aspiring screenwriter, is passionate about crafting scripts that pay homage to the adventure genre of the 80s. In one of his scripts, the heroes must solve a series of puzzle challenges to recover a lost treasure. The treasure is buried under an ancient pyramid that requires entering several numbers into a secret keypad to unlock the chamber.

The first number is the total number of classic 80s adventure films Alex has watched, which is 24. The second number is the number of days Alex has spent writing his script, which is 15. The third number is the sum of the years the two main actors in his favorite 80s adventure movie were born: 1950 and 1955.

To unlock the chamber, Alex needs to calculate the sum of these three numbers and then divide the result by the number of main characters in his script, which is 3. What is the final number Alex should enter into the keypad?

Answer: 1314


A retired Azerbaijani musician from Baku, who was a close friend of the famous pianist Oqtay Radjabov, decides to host a small gathering to celebrate the anniversary of Oqtay's most popular concert. The retired musician plans to play 3 of Oqtay's compositions during the event, each lasting 12 minutes. He also wants to share 15 stories about his experiences with Oqtay, with each story taking about 4 minutes. If the musician wants the entire program to last exactly 90 minutes, how many minutes does he have left for a tea break during the gathering?

Answer: 0


Mr. Thanh, a local resident of Khương Đình, Hanoi, loves to share historical stories with his neighbors. One day, he decided to organize a walking tour to explore some significant historical sites in the area. The tour starts at the Khương Đình Pagoda, built in the year 1010, and includes stops at three other sites. Each site Mr. Thanh shares is 200 years younger than the previous one. If the tour covers a total of 4 sites, what is the average age of all the historical sites visited during the tour?

Answer: 713


Mr. Tanaka, a retiree with a passion for collecting historical data about Japanese financial institutions, decides to analyze the number of banks established in Japan over a century. He discovers that from 1900 to 1950, an average of 5 banks were established per year. From 1951 to 2000, the number of banks established each year increased by 2. How many banks were established from 1900 to 2000?

Answer: 605


You are a tenant who pays $1,200 in rent each month. Recently, you lost your job and have only $3,600 in savings to cover your expenses. You are determined to find a new job within 3 months. Additionally, your monthly expenses for groceries and utilities are $400. If you use your savings to cover only rent and these basic expenses, how much of your savings will be left after 3 months?

Answer: 0


At a retail store, the manager assigns a dedicated sales associate, Alex, to train new employees every month. In January, Alex trained 5 new employees. In February, the manager recognized Alex's dedication and decided to increase the number of new employees trained by 20%. In March, the number of new employees trained was increased by another 20% from February's number. How many new employees did Alex train in March?

Answer: 7


A blogger who is a fan of the Tequila Mockingbird Orchestra attended a three-day music festival to write concert reviews. Over the course of the festival, they attended a total of 12 concerts. On the first day, they attended 4 concerts and wrote 2 reviews. On the second day, they attended 5 concerts and wrote 3 reviews. On the final day, they attended the remaining concerts and wrote enough reviews to have reviewed every concert they attended. How many reviews did the blogger write on the third day?

Answer: 7


An artisan from a small village in India specializes in crafting beautiful wooden chairs. Each chair takes him 4 days to complete. He works 6 days a week, taking one day off for rest. If the artisan starts working on a batch of 5 chairs on the first day of the month, how many days will he need to finish crafting all 5 chairs?

Answer: 23


Alex, a history major with a keen interest in the social and political context of law enforcement, is analyzing the distribution of police precincts in a city over the decades. In 1970, there were 5 precincts. By 1990, the number of precincts had increased by 60% due to urban expansion. From 1990 to 2020, the number of precincts decreased by 10% because of consolidation efforts. How many precincts were there in the city in 2020?

Answer: 7


Alex is an aspiring soccer coach who loves studying the game's tactics and strategies. During a training session, Alex sets up a practice drill for his team. The drill involves passing the ball through a series of cones arranged in a zigzag pattern. There are 5 cones set up, each 3 meters apart from the next. After passing through each cone, players must take a shot at the goal, which is 10 meters away from the last cone. If a player completes the drill and returns to the starting point by running in a straight line from the goal back to the first cone, how many meters in total does the player travel during one complete cycle of the drill?

Answer: 44


In 2012, the Anchuria School Board decided that three questions were too few. Now, one needs to correctly answer six questions out of 40. The question is, if one knows nothing and simply guesses the answers, in which year is the probability of obtaining the Anchurian certificate higher - in 2011 or in 2012?

Answer: 2012


4. Calculate $\int \frac{\sin 2 x-\operatorname{tg}^{2} x}{e^{\operatorname{tg} x}+\sin ^{2} x} d x, x \in\left(\frac{\pi}{24}, \frac{\pi}{3}\right)$.

Narcis Gabriel Turcu, Brăila

Answer: \ln\left(e^{\tan x} + \sin^2 x\right) - \tan x + C


Example 4 Let $f(x)$ be an odd function defined on $\mathbf{R}$, and when $x<0$, $f(x)$ is monotonically increasing, $f(-1)=0$. Let
$$
\begin{array}{l}
\varphi(x)=\sin ^{2} x+m \cos x-2 m, \\
M=\left\{m \mid \text { for any } x \in\left[0, \frac{\pi}{2}\right], \varphi(x)<0\right\}, \\
N=\left\{m \mid \text { for any } x \in\left[0, \frac{\pi}{2}\right], f(\varphi(x))<0\right\} .
\end{array}
$$

Find $M \cap N$.

Answer: (4 - 2\sqrt{2}, +\infty)


6. A circle with diameter $A B$ intersects segments $A C$ and $B C$ at points $M$ and $N$, respectively, and the length of segment $M N$ is equal to the radius of the circle. Find the area of quadrilateral $A B N M$, given that $A C=12$ and $B C=8$.

![](https://cdn.mathpix.com/cropped/2024_05_06_355bebf291524cccadf1g-11.jpg?height=862&width=899&top_left_y=1057&top_left_x=470)

ANSWER: $18 \sqrt{3}$.

Answer: 18\sqrt{3}


5. A $20 \times 19$ rectangle is plotted on the Cartesian plane with one corner at the origin and with sides parallel to the coordinate axes. How many unit squares do the two diagonals of this rectangle pass through?

Answer: 74


In the $xOy$ plane, draw two lines $A_{1}B_{1}$ and $A_{2}B_{2}$ through a point $P$ on the $x$-axis, where the points $A_{1}$, $B_{1}$, $A_{2}$, and $B_{2}$ are all on the parabola $\Gamma: y^{2}=x$. Let $A_{1}B_{2}$ and $A_{2}B_{1}$ respectively pass through points $S$ and $T$ on the $x$-axis. Compare the magnitudes of $\overrightarrow{OS} \cdot \overrightarrow{OT}$ and $|OP|^{2}$.

Answer: \overrightarrow{OS} \cdot \overrightarrow{OT} = |OP|^{2}


1. The number of pairs of skew lines among all the edges of a regular pentagonal prism is $\qquad$ pairs.

Answer: 50


4. Let $E$ be the midpoint of side $CD$ of square $ABCD$. Point $M$ inside the square is such that $\measuredangle MAB = \measuredangle MBC = \measuredangle BME = x$. Determine $x$?

Answer: 75^\circ


Example 7 As shown in Figure 7, there is a fixed point $P$ inside $\angle M A N$. It is known that $\tan \angle M A N=3$, the distance from point $P$ to line $A N$ is $P D=12, A D=$
30, and a line is drawn through $P$ intersecting
$A N$ and $A M$ at points
$B$ and $C$ respectively. Find the minimum
value of the area of $\triangle A B C$. ${ }^{[7]}$
(2010, Taiyuan Junior High School Mathematics Competition)

Answer: 624


4. How many 6-tuples $\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}\right)$ are there such that each of $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ is from the set $\{1,2,3,4\}$ and the six expressions

$$
a_{j}^{2}-a_{j} a_{j+1}+a_{j+1}^{2}
$$

for $j=1,2,3,4,5,6$ (where $a_{7}$ is to be taken as $a_{1}$ ) are all equal to one another?

Answer: 40


7. The equation $\sum_{k=1}^{2 n+1} \cos ^{2} k x=n\left(n \in \mathbf{N}_{+}\right)$ has $\qquad$ solutions in $[0,2 \pi)$.

Answer: 8n +2


3. Given a convex quadrilateral $A B C D$ such that

$$
\measuredangle A B D=50^{\circ}, \measuredangle A D B=80^{\circ}, \measuredangle A C B=40^{\circ}, \measuredangle D B C=\measuredangle B D C+30^{\circ}
$$

Calculate $\Varangle D B C$.

Answer: 70^\circ


Let $\mathbb{Z}_{\geqslant 0}$ be the set of non-negative integers, and let $f: \mathbb{Z}_{\geqslant 0} \times \mathbb{Z}_{\geqslant 0} \rightarrow \mathbb{Z}_{\geqslant 0}$ be a bijection such that whenever $f\left(x_{1}, y_{1}\right)>f\left(x_{2}, y_{2}\right)$, we have $f\left(x_{1}+1, y_{1}\right)>f\left(x_{2}+1, y_{2}\right)$ and $f\left(x_{1}, y_{1}+1\right)>f\left(x_{2}+1, y_{2}\right)$.  Let $N$ be the number of pairs of integers $(x, y)$, with $0 \leqslant x, y<100$, such that $f(x, y)$ is odd. Find the smallest and largest possible value of $N$.  Answer: The optimal bounds are $2500 \leqslant N \leqslant 7500$.

Answer: 2500


# Problem 4. (3 points)

On the side $B C$ of triangle $A B C$, points $A_{1}$ and $A_{2}$ are marked such that $B A_{1}=6, A_{1} A_{2}=8$, $C A_{2}=4$. On the side $A C$, points $B_{1}$ and $B_{2}$ are marked such that $A B_{1}=9, C B_{2}=6$. Segments $A A_{1}$ and $B B_{1}$ intersect at point $K$, and $A A_{2}$ and $B B_{2}$ intersect at point $L$. Points $K, L$ and $C$ lie on the same line. Find $B_{1} B_{2}$.

Answer: 12


Task 4. Let $n \geq 3$ be an integer and consider an $n \times n$ board, divided into $n^{2}$ unit squares. We have for every $m \geq 1$ arbitrarily many $1 \times m$ rectangles (type I) and arbitrarily many $m \times 1$ rectangles (type II) available. We cover the board with $N$ of these rectangles, which do not overlap and do not extend beyond the board. The total number of type I rectangles on the board must be equal to the total number of type II rectangles on the board. (Note that a $1 \times 1$ rectangle is both of type I and type II.) What is the smallest value of $N$ for which this is possible?

Answer: 2n - 1


11. Let $\left\{a_{n}\right\}$ be an integer sequence, and it satisfies: for any $n\left(n \in \mathbf{N}^{*}\right)$, we have $(n-1) a_{n+1}=$ $(n+1) a_{n}-2(n-1)$, and $2008 \mid a_{2007}$, find the smallest positive integer $n(n \geqslant 2)$, such that $2008 \mid a_{n}$.

Answer: 501


5. The sphere touches all the edges of a tetrahedron, in which there is a vertex from which the edges emanate pairwise perpendicularly. Find the radius of this sphere if the radius of the sphere circumscribed around this tetrahedron is $3 \sqrt{3}$.

Answer: 6(\sqrt{2} - 1)


3. In a $100 \times 25$ rectangular table, each cell is filled with a non-negative real number, with the number in the $i$-th row and $j$-th column being $x_{i, j}(i=1,2, \cdots, 100 ; j=1,2, \cdots, 25)$ (Table 1). Then, the numbers in each column of Table 1 are rearranged in descending order from top to bottom as $x_{1, j}^{\prime} \geqslant x_{2, j}^{\prime} \geqslant \cdots \geqslant x_{100, j}^{\prime}(j=1,2, \cdots, 25)$ (Table 2). Find the smallest natural number $k$ such that if the numbers in Table 1 satisfy $\sum_{j=1}^{25} x_{i, j} \leqslant 1(i=1,2, \cdots, 100)$, then when $i \geqslant k$, in Table 2, $\sum_{j=1}^{25} x_{i, j}^{\prime} \leqslant 1(i=1,2, \cdots, 100)$ holds.

Answer: 97


а) A rectangle with an area of 2018 is placed on the coordinate plane such that its sides are parallel to the coordinate axes, and all four vertices lie in different quadrants with integer coordinates. Find the length of its diagonal.

б) Does there exist a rectangle with an area of 2018, whose vertices with integer coordinates lie in different quadrants, but whose sides are not parallel to the coordinate axes?

Answer: \sqrt{1018085}


5. Given a regular 22-gon $M$. Find the number of quadruples of vertices of this 22-gon that are the vertices of trapezoids.

Answer: 990


B5. A sawtooth number is a positive integer with the following property: of each triplet of adjacent digits, the middle digit is either greater than its two neighboring digits or smaller than its two neighboring digits. The numbers 352723 and 314 are, for example, sawtooth numbers, but 3422 and 1243 are not.

How many eight-digit sawtooth numbers are there where each digit is either a 1, a 2, or a 3?

## C-problems

For the C-problems, the answer is not the only thing that matters; there should also be a reasoning that shows that your answer is correct. Solve each C-problem on a separate sheet of paper. Each correctly worked-out C-problem is worth 10 points.

Partial solutions can also earn points. Therefore, write everything clearly and submit your scratch paper (for each problem!) as well.

Answer: 110


3. Let's call a number $x$ semi-integer if the number $2x$ is an integer. The semi-integer part of a number $x$ is defined as the greatest semi-integer number not exceeding $x$, and we will denote it as $] x[$. Solve the equation $x^{2} + 2 \cdot ] x[ = 6$. (20 points)

Answer: \sqrt{3}


Three. (25 points) A positive integer $M$ has 8 positive divisors. Xiao Feng, while calculating the sum of the 8 positive divisors of $M$, forgot to add one of the divisors, resulting in a sum of 2776. Find all possible values of the positive integer $M$.

Answer: 2008


35*. What is the greatest number of rays in space that form pairwise obtuse angles?

Answer: 4


We start with any finite list of distinct positive integers. We may replace any pair $n, n + 1$ (not necessarily adjacent in the list) by the single integer $n-2$, now allowing negatives and repeats in the list. We may also replace any pair $n, n + 4$ by $n - 1$. We may repeat these operations as many times as we wish. Either determine the most negative integer which can appear in a list, or prove that there is no such minimum.

Answer: -3


11. Arrange the following 4 numbers in ascending order: $\sin \left(\cos \frac{3 \pi}{8}\right), \cos \left(\sin \frac{3}{8} \pi\right), \cos \left(\cos \frac{3}{8} \pi\right)$. $\sin \left(\sin \frac{3}{8} \pi\right)$ $\qquad$ .

Answer: \sin \left(\cos \frac{3 \pi}{8}\right) < \cos \left(\sin \frac{3 \pi}{8}\right) < \sin \left(\sin \frac{3 \pi}{8}\right) < \cos \left(\cos \frac{3 \pi}{8}\right)


1. Task: Compare the numbers $\left(\frac{2}{3}\right)^{2016}$ and $\left(\frac{4}{3}\right)^{-1580}$.

Answer: \left(\frac{2}{3}\right)^{2016} < \left(\frac{4}{3}\right)^{-1580}


6. If from the set $S=\{1,2, \cdots, 20\}$, we take a three-element subset $A=\left\{a_{1}, a_{2}, a_{3}\right\}$, such that it simultaneously satisfies: $a_{2}-a_{1} \geqslant 5,4 \leqslant a_{3}-a_{2} \leqslant 9$, then the number of all such subsets $A$ is $\qquad$ (answer with a specific number).

Answer: 251