There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Movie-Genre, Music-Genre, Sport.
The attributes have the following possible values:
- Job: chef, journalist, software-developer, teacher
- Movie-Genre: adventure, sports, superhero, zombie
- Music-Genre: folk, gospel, reggae, soul
- Sport: badminton, biathlon, handball, parkour

Given the following premises about the line of people:
- Music-Genre:reggae and Sport:biathlon have different parity positions
- Sport:handball and Job:teacher have different parity positions
- Movie-Genre:adventure is on the right of Movie-Genre:zombie
- Music-Genre:gospel == Sport:biathlon
- Job:journalist is between Movie-Genre:adventure and Sport:badminton
- Sport:biathlon is on the right of Movie-Genre:sports
- Sport:badminton == Music-Genre:soul
- Movie-Genre:zombie is not to the right of Job:chef
- Sport:handball is on the left of Sport:badminton

Answer the following question:

Question: Question: What is the Music-Genre of the person who has/is sports??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: soul


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Movie-Genre, Music-Genre, Pet, Transport.
The attributes have the following possible values:
- Movie-Genre: action, comedy, fantasy, satire, thriller
- Music-Genre: folk, hip-hop, reggae, salsa, techno
- Pet: bird, fish, frog, goldfish, rabbit
- Transport: motorbike, ship, skateboard, snowmobile, trike

Given the following premises about the line of people:
- Pet:fish is somewhere to the right of Music-Genre:techno
- Movie-Genre:action is on the far right
- Pet:frog == Movie-Genre:fantasy
- Transport:trike is somewhere to the right of Transport:skateboard
- Movie-Genre:fantasy is on the right of Music-Genre:hip-hop
- Movie-Genre:satire is somewhere between Movie-Genre:thriller and Music-Genre:hip-hop
- Pet:bird is not to the left of Music-Genre:hip-hop
- Movie-Genre:comedy is somewhere to the left of Transport:snowmobile
- Transport:snowmobile is on the right of Pet:rabbit
- Movie-Genre:fantasy == Music-Genre:folk
- Transport:motorbike and Music-Genre:hip-hop have different parity positions
- Music-Genre:folk is somewhere to the left of Transport:ship
- Music-Genre:reggae is on the left or right of Transport:skateboard

Answer the following question:

Question: Question: What is the Movie-Genre of the person who has/is folk??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: fantasy


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Hobby, Nationality, Pet.
The attributes have the following possible values:
- Job: dressmaker, freelancer, nurse, teacher
- Hobby: card-games, filmmaking, magic-tricks, woodworking
- Nationality: american, argentine, canadian, indonesian
- Pet: cat, hamster, mouse, rat

Given the following premises about the line of people:
- Job:teacher is in an even position
- Hobby:filmmaking is on the far left or far right
- Hobby:magic-tricks is in an even position
- Pet:cat is somewhere to the right of Hobby:woodworking
- Pet:cat is on the left or right of Nationality:argentine
- Pet:hamster != Hobby:filmmaking
- Pet:cat is on the right of Pet:mouse
- Hobby:woodworking is on the left or right of Hobby:card-games
- Job:nurse is somewhere to the right of Pet:cat
- Hobby:magic-tricks is on the right of Job:dressmaker
- Nationality:indonesian is between Nationality:canadian and Job:nurse

Answer the following question:

Question: Question: What is the Nationality of the person who has/is card-games??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: indonesian


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Nationality, Food, Hobby, Beverage, Pet, Transport.
The attributes have the following possible values:
- Nationality: dutch, german, italian, malaysian, mexican
- Food: apple, papaya, radish, spinach, zucchini
- Hobby: baking, filmmaking, hiking, reading, singing
- Beverage: coffee, cola, fanta, lemonade, tea
- Pet: chinchilla, ferret, fish, rat, snake
- Transport: car, helicopter, scooter, taxi, tram

Given the following premises about the line of people:
- Food:spinach is in the middle
- Pet:fish is somewhere to the right of Beverage:cola
- Beverage:cola is between Pet:ferret and Nationality:malaysian
- Food:papaya is somewhere to the right of Beverage:lemonade
- Nationality:mexican is somewhere between Pet:snake and Food:apple
- Hobby:reading is on the left or right of Pet:snake
- Pet:rat == Transport:taxi
- Transport:car is on the right of Hobby:hiking
- Food:apple is on the left of Nationality:dutch
- Hobby:hiking is on the right of Transport:helicopter
- Hobby:baking is somewhere to the left of Hobby:filmmaking
- Beverage:coffee is somewhere to the left of Beverage:fanta
- Beverage:cola is in an odd position
- Nationality:malaysian is on the left or right of Food:zucchini
- Transport:tram is on the right of Food:radish
- Nationality:german is on the left of Hobby:singing
- Pet:fish is in an odd position
- Pet:snake is somewhere to the right of Pet:chinchilla
- Food:apple == Beverage:tea

Answer the following question:

Question: Question: What is the Hobby of the person who has/is fish??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: reading


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Nationality, Movie-Genre, Music-Genre, Hobby, Pet, Sport.
The attributes have the following possible values:
- Nationality: french, italian, russian, spanish
- Movie-Genre: family, satire, superhero, thriller
- Music-Genre: disco, metal, soul, techno
- Hobby: baking, chess, drawing, filmmaking
- Pet: goat, hamster, horse, rat
- Sport: rowing, skateboarding, snowboarding, soccer

Given the following premises about the line of people:
- Pet:hamster is on the right of Music-Genre:soul
- Hobby:chess is between Sport:skateboarding and Movie-Genre:satire
- Music-Genre:metal is somewhere to the left of Music-Genre:techno
- Hobby:chess is between Hobby:drawing and Sport:snowboarding
- Movie-Genre:superhero is somewhere to the right of Nationality:italian
- Music-Genre:techno is on the left or right of Nationality:french
- Sport:snowboarding == Hobby:baking
- Pet:hamster is somewhere to the left of Hobby:chess
- Nationality:french == Hobby:chess
- Pet:goat is on the right of Movie-Genre:family
- Music-Genre:disco is somewhere to the right of Sport:skateboarding
- Nationality:italian is somewhere to the right of Pet:horse
- Sport:rowing is somewhere to the right of Nationality:spanish

Answer the following question:

Question: Question: What is the Pet of the person who is spanish??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: horse


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Food, Movie-Genre, Music-Genre, Pet, Sport.
The attributes have the following possible values:
- Food: kale, orange, radish, raspberry, tomato
- Movie-Genre: horror, scientific, sports, time-travel, zombie
- Music-Genre: blues, gospel, r&b, salsa, techno
- Pet: chinchilla, dog, frog, mouse, pony
- Sport: baseball, climbing, golf, skateboarding, skiing

Given the following premises about the line of people:
- Pet:mouse is not to the left of Sport:skateboarding
- Food:orange is on the far left or far right
- Movie-Genre:sports is on the left or right of Music-Genre:blues
- Food:tomato is not to the left of Food:radish
- Pet:chinchilla == Music-Genre:r&b
- Music-Genre:gospel and Movie-Genre:scientific have the same parity positions
- Movie-Genre:zombie == Sport:baseball or Movie-Genre:zombie == Music-Genre:techno, but not both
- Food:raspberry is on the left or right of Sport:skiing
- Movie-Genre:scientific is somewhere to the left of Movie-Genre:time-travel
- Pet:frog is somewhere between Pet:pony and Pet:mouse
- Sport:skateboarding == Music-Genre:blues
- Sport:climbing == Music-Genre:r&b or Pet:frog == Sport:climbing, but not both
- Movie-Genre:zombie == Food:radish
- Food:radish == Music-Genre:gospel
- Music-Genre:gospel is in an even position
- Pet:dog is not to the right of Food:orange
- Food:raspberry is not to the right of Music-Genre:salsa
- Food:kale is on the left of Food:raspberry

Answer the following question:

Question: Question: What is the Movie-Genre of the person who has/is gospel??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: zombie


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Beverage, Food, Music-Genre, Sport.
The attributes have the following possible values:
- Beverage: 7up, fanta, sprite, water
- Food: peach, spinach, tomato, watermelon
- Music-Genre: d&b, disco, jazz, rock
- Sport: basketball, handball, rowing, surfing

Given the following premises about the line of people:
- Food:spinach is not to the left of Food:tomato
- Food:spinach == Sport:handball or Food:spinach == Music-Genre:disco or both
- Music-Genre:d&b is not to the left of Sport:rowing
- Music-Genre:jazz != Food:watermelon
- Food:spinach is somewhere between Food:peach and Food:tomato
- Music-Genre:disco is somewhere between Beverage:sprite and Music-Genre:rock
- Sport:basketball and Beverage:sprite have different parity positions
- Beverage:water is not to the right of Music-Genre:disco
- Beverage:7up and Food:tomato have the same parity positions
- Sport:rowing != Beverage:7up
- Beverage:7up is not to the right of Music-Genre:rock
- Beverage:fanta is somewhere between Food:watermelon and Beverage:sprite
- Sport:basketball is not to the left of Sport:handball
- Sport:handball and Beverage:7up have different parity positions

Answer the following question:

Question: Question: What is the Food of the person who has/is water??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: watermelon


A cross-pentomino is a shape that consists of a unit square and four other unit squares, each sharing a different edge with the first square. If a cross-pentomino is inscribed in a circle of radius $R$, what is $100R^2$?

Answer: 250


Chip and Dale play the following game. Chip starts by splitting $222$ nuts between two piles, so Dale can see it. In response, Dale chooses some number $N$ from $1$ to $222$. Then Chip moves nuts from the piles he prepared to a new (third) pile until there will be exactly $N$ nuts in any one or two piles. When Chip accomplishes his task, Dale gets an exact amount of nuts that Chip moved. What is the maximal number of nuts that Dale can get for sure, no matter how Chip acts? (Naturally, Dale wants to get as many nuts as possible, while Chip wants to lose as little as possible).

Answer: 111


For positive integers $n$, let $c_n$ be the smallest positive integer for which $n^{c_n} - 1$ is divisible by $210$, if such a positive integer exists, and $c_n = 0$ otherwise. What is $c_1 + c_2 + \dots + c_{210}$?

Answer: 329


Let $n \geq 1$ be a positive integer. We say that an integer $k$ is a fan of $n$ if $0 \leq k \leq n-1$ and there exist integers $x, y, z \in \mathbb{Z}$ such that
\[
\begin{align*}
x^2 + y^2 + z^2 &\equiv 0 \pmod{n}; \\
xyz &\equiv k \pmod{n}.
\end{align*}
\]
Let $f(n)$ be the number of fans of $n$. Determine $f(2020)$.

Answer: 101


For a set $S \subseteq \mathbb{N}$, define $f(S) = \{\lceil \sqrt{s} \rceil \mid s \in S\}$. Find the number of sets $T$ such that $|f(T)| = 2$ and $f(f(T)) = \{2\}$.

Answer: 5043


Let $m$ and $n$ be positive integers. Fuming Zeng gives James a rectangle, such that $m-1$ lines are drawn parallel to one pair of sides and $n-1$ lines are drawn parallel to the other pair of sides, thus dividing the rectangle into an $m \times n$ grid of smaller rectangles. Fuming Zeng chooses $m+n-1$ of the $mn$ smaller rectangles and then tells James the area of each of the smaller rectangles. Of the $\binom{mn}{m+n-1}$ possible combinations of rectangles and their areas Fuming Zeng could have given, let $C_{m,n}$ be the number of combinations which would allow James to determine the area of the whole rectangle. Given that \[A=\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{C_{m,n}\binom{m+n}{m}}{(m+n)^{m+n}},\] then find the greatest integer less than $1000A$.

Answer: 1289


Let $S = \{ A = (a_1, \ldots, a_s) \mid a_i = 0 \text{ or } 1, \ i = 1, \ldots, 8 \}$. For any two elements of $S$, $A = \{ a_1, \ldots, a_8 \}$ and $B = \{ b_1, \ldots, b_8 \}$. Let $d(A,B) = \sum_{i=1}^{8} |a_i - b_i|$. Call $d(A,B)$ the distance between $A$ and $B$. At most, how many elements can $S$ have such that the distance between any two sets is at least 5?

Answer: 4


Two circles have radii $15$ and $95$. If the two external tangents to the circles intersect at $60$ degrees, how far apart are the centers of the circles?

Answer: 160


For a positive integer $n$, let $S(n)$ be the sum of its decimal digits. Determine the smallest positive integer $n$ for which $4 \cdot S(n) = 3 \cdot S(2n)$.

Answer: 14499


Let $\triangle ABC$ be a triangle with $BC = 4$, $CA= 5$, $AB= 6$, and let $O$ be the circumcenter of $\triangle ABC$. Let $O_b$ and $O_c$ be the reflections of $O$ about lines $CA$ and $AB$ respectively. Suppose $BO_b$ and $CO_c$ intersect at $T$, and let $M$ be the midpoint of $BC$. Given that $MT^2 = \frac{p}{q}$ for some coprime positive integers $p$ and $q$, find $p+q$.

Answer: 23


We have  $98$  cards, in each one we will write one of the numbers:  $1, 2, 3, 4,...., 97, 98$ .
We can order the  $98$  cards, in a sequence such that two consecutive numbers  $X$  and  $Y$  and the number  $X - Y$  is greater than  $48$ , determine how and how many ways we can make this sequence!!

Answer: 2


In an isosceles triangle, we drew one of the angle bisectors. At least one of the resulting two smaller ones  triangles is similar to the original. What can be the leg of the original triangle if the length of its base is  $1$  unit?

Answer: \frac{\sqrt{2}}{2}


On an  $n$  ×  $n$  board, the set of all squares that are located on or below the main diagonal of the board is called the $n-ladder$ . For example, the following figure shows a  $3-ladder$ :
[asy]
draw((0,0)--(0,3));
draw((0,0)--(3,0));
draw((0,1)--(3,1));
draw((1,0)--(1,3));
draw((0,2)--(2,2));
draw((2,0)--(2,2));
draw((0,3)--(1,3));
draw((3,0)--(3,1));
[/asy]
In how many ways can a  $99-ladder$  be divided into some rectangles, which have their sides on grid lines, in such a way that all the rectangles have distinct areas?

Answer: 2^{98}


Let  $M=\{1,2,\dots,49\}$  be the set of the first  $49$  positive integers. Determine the maximum integer  $k$  such that the set  $M$  has a subset of  $k$  elements such that there is no  $6$  consecutive integers in such subset. For this value of  $k$ , find the number of subsets of  $M$  with  $k$  elements with the given property.

Answer: 495


A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.

Take five good haikus
Scramble their lines randomly
What are the chances
That you end up with
Five completely good haikus
(With five, seven, five)?
Your answer will be
m over n where m,n
Are numbers such that
m,n positive
Integers where gcd
Of m,n is 1.
Take this answer and
Add the numerator and
Denominator.

*Proposed by Jeff Lin*

Answer: 3004


$n$  coins lies in the circle. If two neighbour coins lies both head up or both tail up, then we can flip both. How many variants of coins are available that can not be obtained from each other by applying such operations?

Answer: 2


The first  $9$  positive integers are placed into the squares of a  $3\times 3$  chessboard. We are taking the smallest number in a column. Let  $a$  be the largest of these three smallest number. Similarly, we are taking the largest number in a row. Let  $b$  be the smallest of these three largest number. How many ways can we distribute the numbers into the chessboard such that  $a=b=4$ ?

Answer: 25920


Let  $M$  be a set of  $2017$  positive integers. For any subset  $A$  of  $M$  we define  $f(A) := \{x\in M\mid \text{ the number of the members of }A\,,\, x \text{ is multiple of, is odd }\}$ .
Find the minimal natural number  $k$ , satisfying the condition:  for any  $M$ , we can color all the subsets of  $M$  with  $k$  colors, such that whenever  $A\neq f(A)$ ,  $A$  and  $f(A)$  are colored with different colors.

Answer: 2


Arrange the natural numbers from 1 to 7 in a row such that each number is either greater than all the numbers before it or less than all the numbers before it. For example, the arrangement 4356271 satisfies this condition: starting from the second position, 3 is less than the first number 4; 5 is greater than the first two numbers 4 and 3; 6 is greater than the first three numbers 4, 3, and 5; 2 is less than the first four numbers 4, 3, 5, and 6; 7 is greater than the first five numbers 4, 3, 5, 6, and 2; 1 is less than the first six numbers 4, 3, 5, 6, 2, and 7.

Determine the number of such arrangements where the number 7 does not occupy the fourth position.

Answer: 60


1. Susie thinks of a positive integer $n$. She notices that, when she divides 2023 by $n$, she is left with a remainder of 43 . Find how many possible values of $n$ there are.

Answer: 19


15. If there exists a positive integer $m$ such that $m!$ ends with exactly $n$ zeros, then the positive integer $n$ is called a "factorial tail number." How many non-"factorial tail number" positive integers are there less than 1992?

Answer: 396


11. (20 points) Given positive real numbers $a, b, c, d$ satisfying $a b c d > 1$. Find the minimum value of $\frac{a^{2}+a b^{2}+a b c^{2}+a b c d^{2}}{a b c d-1}$.

Answer: 4


Example 1. Find $\int\left(x^{2}-x+1\right) \cos 2 x d x$.

Answer: \left( \frac{1}{2}x^2 - \frac{1}{2}x + \frac{1}{4} \right) \sin 2x + \frac{2x - 1}{4} \cos 2x + C


In a football tournament there are n teams, with ${n \ge 4}$, and each pair of teams meets exactly once. Suppose that, at the end of the tournament, the final scores form an arithmetic sequence where each team scores ${1}$ more point than the following team on the scoreboard. Determine the maximum possible score of the lowest scoring team, assuming usual scoring for football games (where the winner of a game gets ${3}$ points, the loser ${0}$ points, and if there is a tie both teams get ${1}$ point).

Answer: n - 2


7.2. A city held three rounds of Go tournaments, with the same participants in each round. It is known that every 2 participants in the three rounds of the tournament won 1 time each, drew 1 time each. A person came in last place in the first two rounds. Question: What place did he get in the third round?

Answer: 1


$[\underline{\text { Properties of Sections }}]$

The edge of the cube $A B C D A 1 B 1 C 1 D 1$ is 12. Point $K$ lies on the extension of edge $B C$ at a distance of 9 from vertex $C$. Point $L$ on edge $A B$ is 5 units away from $A$. Point $M$ divides the segment $A 1 C 1$ in the ratio $1: 3$, counting from $A 1$. Find the area of the section of the cube by the plane passing through points $K, L, M$.

Answer: 156


The center of an equilateral triangle with a side length of 6 cm coincides with the center of a circle with a radius of 2 cm. Determine the area of the part of the triangle that lies outside this circle.

Answer: 6\sqrt{3} - 2\pi


You are playing a game called "Hovse."
Initially you have the number $0$ on a blackboard.
If at any moment the number $x$ is written on the board, you can either:
$\bullet$ replace $x$ with $3x + 1$
$\bullet$ replace $x$ with $9x + 1$
$\bullet$ replace $x$ with $27x + 3$
$\bullet$ or replace $x$ with $\left \lfloor \frac{x}{3} \right \rfloor $.
However, you are not allowed to write a number greater than $2017$ on the board. How many positive numbers can you make with the game of "Hovse?"

Answer: 127


Answer: 5


71*. a) Banach's problem ${ }^{1}$. A person simultaneously bought two boxes of matches and put them in his pocket. After that, every time he needed to light a match, he randomly took one or the other box. After some time, upon emptying one of the boxes, the person discovered that it was empty. What is the probability that at this moment the second box still contained $k$ matches, if the number of matches in the unopened box was $n$?

b) Using the result from part a), find the value of the sum

$$
C_{2 n}^{n}+2 C_{2 n-1}^{n}+4 C_{2 n-2}^{n}+\ldots+2^{n} C_{n}^{n}
$$

Note. Another method for finding this sum is given in the solution to problem 55.

Answer: 4^n


7. In the tetrahedron $P-ABC$, $PB \perp AC$, $PH$ $\perp$ plane $ABC$ at point $H$, $H$ is inside $\triangle ABC$, $PB$ makes a $30^{\circ}$ angle with plane $ABC$, the area of $\triangle PAC$ is 1. When the dihedral angle $P-AC-B$ is $\qquad$, $S_{\triangle ABC}$ is maximized.

Answer: 60^\circ


3. The angle $A C B$ of triangle $A B C$ has a size of $140^{\circ}$. The bisector of angle $A B C$ intersects side $A C$ at point $X$. Point $Y$ lies on side $A B$ such that angle $Y C B$ has a size of $100^{\circ}$. Determine the size of angle $Y X B$.

Answer: 50^\circ


6. In $\triangle A B C$, $\angle B=\frac{\pi}{3}, A C=\sqrt{3}$, point $D$ is on side $A B$, $B D=1$, and $D A=D C$. Then $\angle D C A=$ $\qquad$ .

Answer: \dfrac{\pi}{6}


7. The external angles of a triangle are in the ratio $9: 16: 20$. From the vertex of the largest internal angle, the angle bisector and the altitude to the opposite side are drawn. What is the measure of the angle between the bisector and the altitude of the given triangle?

MINISTRY OF SCIENCE, EDUCATION AND SPORT OF THE REPUBLIC OF CROATIA AGENCY FOR EDUCATION AND UPBRINGING

CROATIAN MATHEMATICAL SOCIETY

## SCHOOL/CITY COMPETITION IN MATHEMATICS

Answer: 16


10. Given that $m$ and $n$ are positive integers. If $1 \leqslant m \leqslant n$ $\leqslant 30$, and $mn$ is divisible by 21, then the number of pairs $(m, n)$ that satisfy the condition is $\qquad$ .

Answer: 57


7. In the Magic Land, there is a magic stone that grows uniformly upwards. To prevent it from piercing the sky, the Elders of the Immortal World decided to send plant warriors to consume the magic stone and suppress its growth. Each plant warrior consumes the same amount every day. If 14 plant warriors are dispatched, the magic stone will pierce the sky after 16 days; if 15 plant warriors are dispatched, the magic stone will pierce the sky after 24 days. At least $\qquad$ plant warriors need to be dispatched to ensure the sky is not pierced.

Answer: 17


6. Two brothers sold a flock of sheep, receiving as many rubles for each sheep as there were sheep in the flock. Wishing to divide the proceeds equally, they began to take 10 rubles from the total sum in turns, starting with the older brother. After the older brother took 10 rubles again, the younger brother was left with less than 10 rubles. To ensure an equal division, the older brother gave the younger brother his knife. For how many rubles was the knife valued?

Answer: 2


8. Given the sequence $\left\{a_{n}\right\}$ with the first term being 2, and satisfying
$$
6 S_{n}=3 a_{n+1}+4^{n}-1 \text {. }
$$

Then the maximum value of $S_{n}$ is $\qquad$.

Answer: 35


4. In a box, there is a large batch of flowers of six types mixed together. Vasya randomly takes flowers one by one from the box. As soon as he collects 5 flowers of the same type, he makes a bouquet and sells it. What is the minimum number of flowers he needs to take to guarantee selling 10 bouquets?

Answer: 70


2. Consider the number $\mathrm{a}_{\mathrm{n}}=18 \underbrace{77 \ldots 7}_{\text {n times }} 889$, where $n$ is a natural number, and $\mathrm{C}_{\mathrm{n}}$ is the quotient of the division of the number $\mathrm{a}_{\mathrm{n}}$ by 13.

a) Show that $\mathrm{a}_{\mathrm{n}}$ is divisible by 13 for any $n$.

b) Determine $n$ for which $s\left(a_{n}\right)=2 s\left(c_{n}\right)$, where $s(m)$ represents the sum of the digits of the number $m$.

Answer: 8


Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as
\[
(x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z)
\]
with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

Answer: 4


An event occurs periodically over $x$ consecutive years, followed by a break of $y$ consecutive years. The event occurred in the years 1964, 1986, 1996, and 2008, and did not occur in 1976, 1993, 2006, or 2013. Determine the first year in which the event will occur again.

Answer: 2018


Benji has a $2 \times 2$ grid, which he proceeds to place chips on. One by one, he places a chip on one of the unit squares of the grid at random. However, if at any point there is more than one chip on the same square, Benji moves two chips on that square to the two adjacent squares, which he calls a chip-fire. He keeps adding chips until there is an infinite loop of chip-fires. What is the expected number of chips that will be added to the board?

Answer: 5


There is a right triangle $\triangle ABC$ where $\angle A$ is the right angle. On side $AB$, there are three points $X$, $Y$, and $Z$ such that $\angle ACX = \angle XCY = \angle YCZ = \angle ZCB$ and $BZ = 2AX$. The smallest angle of $\triangle ABC$ is $\frac{a}{b}$ degrees, where $a$ and $b$ are positive integers such that $\gcd(a, b) = 1$. Find $a + b$.

Answer: 277


In trapezoid $ABCD$, where $AD \parallel BC$ and $\angle ABC + \angle CDA = 270^{\circ}$, compute $AB^2$. Given that $AB \cdot \tan(\angle BCD) = 20$ and $CD = 13$.

Answer: 260


For a positive integer $n$, let $p(n)$ denote the number of distinct prime numbers that divide evenly into $n$. Determine the number of solutions, in positive integers $n$, to the inequality $\log_4 n \leq p(n)$.

Answer: 13


Six chairs are arranged in a row, and six people randomly choose a seat among these chairs. Each person then decides one of three positions for their legs:
1. Feet on the floor
2. Legs crossed to the right
3. Legs crossed to the left

A problem arises only if two adjacent people have the following leg positions:
- The person on the right crosses their legs to the left.
- The person on the left crosses their legs to the right.

Determine the probability that this situation does **not** occur. Express the probability as a fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the sum $m+n$.

Answer: 1106


Consider pairs of functions $(f, g)$ from the set of nonnegative integers to itself such that:
- $f(0) + f(1) + f(2) + \cdots + f(42) \le 2022$;
- For any integers $a \ge b \ge 0$, we have $g(a+b) \le f(a) + f(b)$.

Determine the maximum possible value of $g(0) + g(1) + g(2) + \cdots + g(84)$ over all such pairs of functions.

Answer: 7993


Let $\phi$ be the Euler totient function. Let $\phi^k (n) = (\underbrace{\phi \circ  ... \circ  \phi}_{k})(n)$ be $\phi$ composed with itself $k$ times. Define $\theta (n) = \min \{k \in \mathbb{N} \mid \phi^k (n)=1 \}$.

For example:
- $\phi^1 (13) = \phi(13) = 12$
- $\phi^2 (13) = \phi (\phi (13)) = 4$
- $\phi^3 (13) = \phi(\phi(\phi(13))) = 2$
- $\phi^4 (13) = \phi(\phi(\phi(\phi(13)))) = 1$

Thus, $\theta (13) = 4$.

Let $f(r) = \theta (13^r)$. Determine $f(2012)$.

Answer: 6037


A sequence $\{a_n\}$ is defined such that $a_i = i$ for $i = 1, 2, 3, \ldots, 2020$ and for $i > 2020$, $a_i$ is the average of the previous 2020 terms. What is the largest integer less than or equal to $\displaystyle\lim_{n \to \infty} a_n$?

Answer: 1347


Points $E$ and $F$ lie on diagonal $\overline{AC}$ of square $ABCD$ with side length $24$, such that $AE = CF = 3\sqrt{2}$. An ellipse with foci at $E$ and $F$ is tangent to the sides of the square. Find the sum of the distances from any point on the ellipse to the two foci.

Answer: 30


\( \triangle ABC \) has side lengths \( AB=20 \), \( BC=15 \), and \( CA=7 \). Let the altitudes of \( \triangle ABC \) be \( AD \), \( BE \), and \( CF \). What is the distance between the orthocenter (intersection of the altitudes) of \( \triangle ABC \) and the incenter of \( \triangle DEF \)?

Answer: 15


For any set $S$, let $P(S)$ be its power set, the set of all its subsets. Consider all sets $A$ of 2015 arbitrary finite sets. Let $N$ be the maximum possible number of ordered pairs $(S,T)$ such that $S \in P(A)$, $T \in P(P(A))$, $S \in T$, and $S \subseteq T$. Note that by convention, a set may never contain itself. Find the remainder when $N$ is divided by 1000.

Answer: 872


We are given a $5 \times 5$ square grid, divided into $1 \times 1$ tiles. Two tiles are called **linked** if they lie in the same row or column, and the distance between their centers is 2 or 3. For example, in the picture, the gray tiles are the ones linked to the red tile.

![Grid Image](https://i.imgur.com/JVTQ9wB.png)

Sammy wants to mark as many tiles in the grid as possible, such that no two of them are linked. What is the maximal number of tiles he can mark?

Answer: 10


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`, `Eric`, `Arnold`
 - Each person has a unique hobby: `cooking`, `painting`, `gardening`, `photography`
 - The people keep unique animals: `horse`, `fish`, `cat`, `bird`
 - People have unique favorite book genres: `fantasy`, `mystery`, `romance`, `science fiction`
 - Each person has a unique birthday month: `april`, `jan`, `sept`, `feb`
 - People have unique favorite music genres: `pop`, `rock`, `classical`, `jazz`

## Clues:
1. The person who loves cooking is the person who loves romance books.
2. The person whose birthday is in February is the person who loves pop music.
3. Eric is not in the second house.
4. The person who loves romance books is not in the fourth house.
5. The person whose birthday is in February is the fish enthusiast.
6. Alice is somewhere to the right of the person who loves fantasy books.
7. The person who keeps horses is the person who loves rock music.
8. The person who enjoys gardening is the person whose birthday is in April.
9. The person who loves jazz music is the person who loves cooking.
10. The person who loves rock music is the person who loves mystery books.
11. The person who paints as a hobby is directly left of the person who loves romance books.
12. Peter is the person who loves pop music.
13. The person who enjoys gardening is Arnold.
14. The person who loves rock music is directly left of the person whose birthday is in January.
15. The person who loves cooking is not in the third house.
16. The cat lover is somewhere to the right of the person who keeps horses.
 What is the value of attribute MusicGenre for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{}

Answer: jazz


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`, `Alice`, `Peter`, `Arnold`, `Bob`
 - Each person has a unique type of pet: `cat`, `dog`, `bird`, `fish`, `hamster`
 - Everyone has a unique favorite cigar: `pall mall`, `blends`, `prince`, `dunhill`, `blue master`
 - Everyone has a favorite smoothie: `watermelon`, `desert`, `dragonfruit`, `cherry`, `lime`
 - People use unique phone models: `google pixel 6`, `oneplus 9`, `samsung galaxy s21`, `iphone 13`, `huawei p50`
 - People have unique favorite sports: `baseball`, `soccer`, `basketball`, `swimming`, `tennis`

## Clues:
1. The person who loves swimming is directly left of the Prince smoker.
2. The person who smokes Blue Master is somewhere to the right of Eric.
3. The person who uses a OnePlus 9 is Arnold.
4. Arnold is somewhere to the right of the person with a pet hamster.
5. The person who owns a dog is Peter.
6. Eric is the Desert smoothie lover.
7. The person who loves basketball is not in the fifth house.
8. The person who has a cat is the person who uses a Samsung Galaxy S21.
9. The person who loves baseball is directly left of Peter.
10. There is one house between the Desert smoothie lover and the Dunhill smoker.
11. The person who loves baseball and the person who uses a Google Pixel 6 are next to each other.
12. The person who keeps a pet bird is in the second house.
13. The person who uses a Huawei P50 is in the fourth house.
14. The person who drinks Lime smoothies is in the fourth house.
15. The person partial to Pall Mall is in the fifth house.
16. The Dragonfruit smoothie lover is in the second house.
17. The person who loves tennis is the person with a pet hamster.
18. Bob is somewhere to the right of the person who owns a dog.
19. The Watermelon smoothie lover is Arnold.
20. The Prince smoker is somewhere to the left of the Dunhill smoker.
 What is the value of attribute Name for the person whose attribute PhoneModel is google pixel 6? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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`, `Peter`, `Arnold`, `Alice`, `Bob`
 - Each person has a unique level of education: `associate`, `bachelor`, `high school`, `master`, `doctorate`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `baseball`, `soccer`
 - Each person prefers a unique type of vacation: `camping`, `cruise`, `city`, `mountain`, `beach`
 - People use unique phone models: `google pixel 6`, `huawei p50`, `oneplus 9`, `samsung galaxy s21`, `iphone 13`

## Clues:
1. The person who uses a Huawei P50 is not in the second house.
2. The person with a master's degree is the person who uses a Google Pixel 6.
3. The person who loves swimming is the person with a high school diploma.
4. The person with a bachelor's degree is directly left of the person who loves tennis.
5. The person who enjoys camping trips is the person who loves tennis.
6. Bob is the person who uses a Samsung Galaxy S21.
7. The person with a high school diploma is not in the third house.
8. The person who loves swimming is directly left of the person who loves beach vacations.
9. The person who likes going on cruises is Arnold.
10. Bob is the person with an associate's degree.
11. The person who uses a OnePlus 9 is directly left of the person who loves basketball.
12. The person who enjoys camping trips and the person who prefers city breaks are next to each other.
13. Eric is the person who loves swimming.
14. The person who loves basketball is the person who enjoys mountain retreats.
15. The person who loves tennis is somewhere to the left of Alice.
16. Arnold is the person who loves soccer.
 What is the value of attribute PhoneModel for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: huawei p50


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`, `Alice`, `Eric`, `Peter`, `Arnold`
 - People have unique heights: `average`, `short`, `very tall`, `very short`, `tall`
 - Each person has an occupation: `engineer`, `artist`, `doctor`, `lawyer`, `teacher`
 - The mothers' names in different houses are unique: `Janelle`, `Aniya`, `Holly`, `Penny`, `Kailyn`
 - Each person has a unique birthday month: `mar`, `april`, `jan`, `feb`, `sept`
 - Each person has a unique type of pet: `hamster`, `fish`, `bird`, `dog`, `cat`

## Clues:
1. The person who is a doctor is somewhere to the left of the person with a pet hamster.
2. Arnold is in the first house.
3. The person who is an engineer is the person who owns a dog.
4. The person whose birthday is in March is somewhere to the left of Peter.
5. The person who is an artist is in the third house.
6. The person who is a lawyer is in the fifth house.
7. The person who has a cat is The person whose mother's name is Kailyn.
8. Alice is directly left of Eric.
9. The person whose birthday is in February is directly left of The person whose mother's name is Holly.
10. The person whose birthday is in April is somewhere to the right of the person who is a teacher.
11. The person whose birthday is in February is The person whose mother's name is Kailyn.
12. Peter is in the fourth house.
13. The person with an aquarium of fish is the person who is very short.
14. The person whose mother's name is Kailyn and the person with an aquarium of fish are next to each other.
15. The person who is short is directly left of the person who is a lawyer.
16. The person whose mother's name is Aniya and the person who is a lawyer are next to each other.
17. The person whose birthday is in January is The person whose mother's name is Penny.
18. The person who has an average height is the person who keeps a pet bird.
19. The person who is tall is the person who is an engineer.
 What is the value of attribute Height for the person whose attribute Pet is fish? Please reason step by step, and put your final answer within \boxed{}

Answer: very 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: `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 Height for the person whose attribute Pet is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: short


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`, `Peter`, `Alice`, `Bob`, `Arnold`
 - The people are of nationalities: `norwegian`, `brit`, `swede`, `dane`, `german`
 - Each person prefers a unique type of vacation: `cruise`, `mountain`, `camping`, `beach`, `city`
 - Each person has a unique level of education: `bachelor`, `master`, `associate`, `doctorate`, `high school`
 - Each person has an occupation: `artist`, `doctor`, `engineer`, `teacher`, `lawyer`

## Clues:
1. The person who likes going on cruises is the person who is a lawyer.
2. The person who loves beach vacations is directly left of Arnold.
3. The person with a doctorate is somewhere to the left of Bob.
4. The person with an associate's degree is the person who likes going on cruises.
5. Peter is not in the first house.
6. The person who is an artist is Peter.
7. The person who enjoys camping trips is the person with a master's degree.
8. The Dane is somewhere to the right of the person who is a doctor.
9. The person with an associate's degree is directly left of the person who is an engineer.
10. The person who enjoys camping trips is the British person.
11. The Norwegian and the person with a bachelor's degree are next to each other.
12. The person who is an artist is the Swedish person.
13. Bob is not in the fourth house.
14. The person who enjoys camping trips is Eric.
15. Alice is the German.
16. The person who loves beach vacations is somewhere to the left of the person who prefers city breaks.
17. The person who enjoys mountain retreats is in the fifth house.
18. The person who likes going on cruises is somewhere to the right of the person who loves beach vacations.
19. The person with a bachelor's degree is in the third house.
 What is the value of attribute Nationality for the person whose attribute Occupation is artist? Please reason step by step, and put your final answer within \boxed{}

Answer: swede


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`, `Alice`, `Arnold`
 - Each person has a unique birthday month: `sept`, `feb`, `april`, `jan`
 - People have unique favorite book genres: `romance`, `fantasy`, `mystery`, `science fiction`
 - They all have a unique favorite flower: `roses`, `daffodils`, `lilies`, `carnations`
 - Each person prefers a unique type of vacation: `cruise`, `city`, `mountain`, `beach`

## Clues:
1. There are two houses between the person who loves fantasy books and the person whose birthday is in April.
2. The person whose birthday is in February is directly left of the person who loves a bouquet of daffodils.
3. The person who likes going on cruises is the person who loves a carnations arrangement.
4. The person whose birthday is in September is directly left of the person who loves science fiction books.
5. The person who loves a bouquet of daffodils is the person who prefers city breaks.
6. The person whose birthday is in April is Arnold.
7. The person whose birthday is in February is the person who loves a carnations arrangement.
8. Alice is the person who likes going on cruises.
9. The person who loves the boquet of lilies is the person who enjoys mountain retreats.
10. Eric is not in the third house.
11. The person who loves science fiction books is the person who loves the boquet of lilies.
12. The person who loves mystery books is Eric.
13. Eric and the person whose birthday is in January are next to each other.
 What is the value of attribute Birthday for the person whose attribute Vacation is city? 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: `Eric`, `Arnold`, `Alice`, `Peter`
 - People have unique heights: `tall`, `very short`, `average`, `short`
 - Each person has a unique favorite drink: `tea`, `coffee`, `water`, `milk`
 - Each mother is accompanied by their child: `Fred`, `Meredith`, `Bella`, `Samantha`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`, `Kailyn`

## Clues:
1. The person who is tall is Arnold.
2. The person's child is named Samantha is the tea drinker.
3. The person whose mother's name is Holly is the coffee drinker.
4. The person's child is named Samantha and The person whose mother's name is Holly are next to each other.
5. The person's child is named Meredith and the person who is very short are next to each other.
6. The person's child is named Samantha is The person whose mother's name is Aniya.
7. The person who likes milk is the person who is short.
8. Alice is the person's child is named Fred.
9. Peter and the person who is very short are next to each other.
10. The coffee drinker is somewhere to the right of the person's child is named Samantha.
11. Eric is the person who is short.
12. The person whose mother's name is Janelle is somewhere to the left of The person whose mother's name is Kailyn.
13. Alice is The person whose mother's name is Kailyn.
14. The person's child is named Meredith is not in the first house.
 What is the value of attribute Height for the person whose attribute Mother is Aniya? Please reason step by step, and put your final answer within \boxed{}

Answer: tall


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: `Carol`, `Bob`, `Alice`, `Eric`, `Arnold`, `Peter`
 - Each person has a unique type of pet: `hamster`, `cat`, `fish`, `rabbit`, `dog`, `bird`
 - Each person has a favorite color: `blue`, `purple`, `yellow`, `white`, `red`, `green`

## Clues:
1. The person who has a cat is not in the third house.
2. Arnold is not in the first house.
3. The person who owns a dog is the person whose favorite color is green.
4. The person who loves yellow is somewhere to the left of Bob.
5. The person who loves blue is the person who has a cat.
6. There are two houses between the person with an aquarium of fish and Eric.
7. The person who owns a dog is Carol.
8. The person who loves yellow is the person with a pet hamster.
9. The person who loves white and the person who keeps a pet bird are next to each other.
10. The person who owns a rabbit is somewhere to the left of the person who loves white.
11. The person who owns a dog is directly left of the person who loves blue.
12. The person who loves purple is directly left of Carol.
13. Bob is not in the third house.
14. Alice is the person who loves purple.
 What is the value of attribute Pet for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: hamster


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`, `Bob`, `Eric`, `Arnold`
 - Each person has a unique type of pet: `hamster`, `bird`, `fish`, `cat`, `dog`
 - Each person has an occupation: `doctor`, `engineer`, `lawyer`, `artist`, `teacher`
 - Each person lives in a unique style of house: `colonial`, `modern`, `ranch`, `craftsman`, `victorian`
 - Each person has a unique level of education: `high school`, `bachelor`, `master`, `doctorate`, `associate`

## Clues:
1. There are two houses between the person with a master's degree and the person in a modern-style house.
2. The person with an aquarium of fish is the person who is an artist.
3. The person who has a cat is the person who is an engineer.
4. The person with a bachelor's degree is the person with an aquarium of fish.
5. The person who has a cat is somewhere to the right of the person with an associate's degree.
6. The person with a master's degree is the person who is a lawyer.
7. The person who is a teacher is somewhere to the right of the person who keeps a pet bird.
8. The person with an associate's degree is the person living in a colonial-style house.
9. Peter is the person in a Craftsman-style house.
10. The person residing in a Victorian house is in the second house.
11. The person who is a teacher is Bob.
12. The person with an associate's degree is in the fourth house.
13. The person with a bachelor's degree is somewhere to the right of the person with a high school diploma.
14. There is one house between the person who owns a dog and the person in a ranch-style home.
15. Bob is somewhere to the right of Alice.
16. Eric is the person with a bachelor's degree.
 What is the value of attribute HouseStyle for the person whose attribute Pet is cat? Please reason step by step, and put your final answer within \boxed{}

Answer: modern


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 House for the person whose attribute FavoriteSport is baseball? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


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`, `Arnold`, `Alice`, `Peter`, `Eric`
 - Each person prefers a unique type of vacation: `camping`, `city`, `beach`, `cruise`, `mountain`
 - People have unique heights: `very tall`, `average`, `very short`, `short`, `tall`
 - They all have a unique favorite flower: `tulips`, `roses`, `carnations`, `lilies`, `daffodils`
 - Each person has a unique level of education: `bachelor`, `master`, `associate`, `high school`, `doctorate`
 - People own unique car models: `bmw 3 series`, `tesla model 3`, `toyota camry`, `ford f150`, `honda civic`

## Clues:
1. Eric is the person who prefers city breaks.
2. Arnold is the person who is tall.
3. The person who loves beach vacations is the person who loves the boquet of lilies.
4. The person who owns a BMW 3 Series is the person who is tall.
5. The person with an associate's degree is the person who loves the boquet of lilies.
6. The person who has an average height is Eric.
7. Peter is the person who owns a Toyota Camry.
8. The person who has an average height is somewhere to the right of the person who loves a bouquet of daffodils.
9. The person who owns a Honda Civic is somewhere to the left of the person who loves the boquet of lilies.
10. Eric is the person who owns a Ford F-150.
11. The person who loves the vase of tulips is in the fourth house.
12. The person who is very short is not in the fourth house.
13. Eric is the person with a high school diploma.
14. The person who enjoys mountain retreats is somewhere to the right of the person with a master's degree.
15. The person who likes going on cruises is not in the third house.
16. Bob is the person who is short.
17. The person with an associate's degree is directly left of Arnold.
18. The person with a bachelor's degree is the person who is very tall.
19. The person who loves a carnations arrangement is directly left of the person with a doctorate.
 What is the value of attribute Height for the person whose attribute Name is Bob? 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: `Alice`, `Arnold`, `Carol`, `Peter`, `Bob`, `Eric`
 - People use unique phone models: `huawei p50`, `samsung galaxy s21`, `iphone 13`, `xiaomi mi 11`, `google pixel 6`, `oneplus 9`
 - Each person has a unique hobby: `photography`, `painting`, `gardening`, `woodworking`, `cooking`, `knitting`

## Clues:
1. The woodworking hobbyist is somewhere to the left of Bob.
2. The person who paints as a hobby is in the fifth house.
3. Arnold is directly left of the person who uses a Google Pixel 6.
4. The person who uses a Samsung Galaxy S21 is Arnold.
5. Carol is somewhere to the right of the person who uses a Samsung Galaxy S21.
6. The person who uses a Google Pixel 6 is directly left of the person who uses a OnePlus 9.
7. The person who uses a OnePlus 9 is somewhere to the right of Eric.
8. There are two houses between the person who loves cooking and Peter.
9. The person who enjoys knitting is Eric.
10. Peter is the person who paints as a hobby.
11. The person who enjoys gardening is somewhere to the right of Bob.
12. There is one house between Bob and the person who uses a Huawei P50.
13. The person who uses a Xiaomi Mi 11 is not in the first house.
14. Alice is in the sixth house.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


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`, `Eric`, `Arnold`, `Carol`, `Alice`, `Peter`
 - Each person has a unique hobby: `cooking`, `painting`, `knitting`, `photography`, `woodworking`, `gardening`
 - People have unique hair colors: `brown`, `auburn`, `black`, `red`, `gray`, `blonde`
 - People have unique favorite sports: `swimming`, `volleyball`, `basketball`, `baseball`, `tennis`, `soccer`
 - The mothers' names in different houses are unique: `Sarah`, `Holly`, `Penny`, `Janelle`, `Kailyn`, `Aniya`
 - Everyone has a unique favorite cigar: `dunhill`, `blends`, `pall mall`, `prince`, `yellow monster`, `blue master`

## Clues:
1. The person who has gray hair is directly left of The person whose mother's name is Janelle.
2. The person who loves soccer is in the sixth house.
3. Arnold is the person who has blonde hair.
4. The photography enthusiast is the person who smokes Yellow Monster.
5. There is one house between the Dunhill smoker and the person who loves cooking.
6. The person who paints as a hobby is the person who has blonde hair.
7. The Dunhill smoker is directly left of Alice.
8. The person who paints as a hobby is directly left of the person who loves volleyball.
9. The person who loves soccer is The person whose mother's name is Aniya.
10. The person partial to Pall Mall is somewhere to the left of Carol.
11. The person who enjoys knitting and the person who loves basketball are next to each other.
12. The person partial to Pall Mall is directly left of The person whose mother's name is Holly.
13. The person who loves soccer is the Prince smoker.
14. The person who loves swimming is The person whose mother's name is Penny.
15. The person whose mother's name is Kailyn is the person who loves volleyball.
16. The person who loves basketball is The person whose mother's name is Janelle.
17. The person who enjoys knitting is the person who loves volleyball.
18. The person who enjoys knitting is not in the third house.
19. The person who has auburn hair is somewhere to the left of the photography enthusiast.
20. The person whose mother's name is Sarah is not in the fifth house.
21. Bob is directly left of the person who has red hair.
22. The Dunhill smoker is the person who enjoys gardening.
23. The person who loves baseball is the person who has auburn hair.
24. Eric is the person who smokes many unique blends.
25. The person who has brown hair is somewhere to the right of The person whose mother's name is Penny.
 What is the value of attribute Cigar for the person whose attribute Hobby is woodworking? Please reason step by step, and put your final answer within \boxed{}

Answer: pall mall


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: `Carol`, `Peter`, `Arnold`, `Bob`, `Eric`, `Alice`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `oneplus 9`, `huawei p50`, `samsung galaxy s21`, `xiaomi mi 11`
 - People have unique hair colors: `gray`, `auburn`, `red`, `brown`, `black`, `blonde`
 - Each person has a unique favorite drink: `coffee`, `water`, `root beer`, `tea`, `milk`, `boba tea`
 - People have unique favorite music genres: `classical`, `jazz`, `rock`, `country`, `hip hop`, `pop`

## Clues:
1. Alice is the person who uses a Huawei P50.
2. The person who uses a Xiaomi Mi 11 is somewhere to the left of the one who only drinks water.
3. The one who only drinks water is somewhere to the left of Bob.
4. Carol is the person who has gray hair.
5. Eric is the tea drinker.
6. The person who has black hair is not in the sixth house.
7. The person who uses a Samsung Galaxy S21 is directly left of the person who loves classical music.
8. The person who uses a OnePlus 9 is not in the third house.
9. Alice is somewhere to the right of Carol.
10. The person who loves jazz music is Alice.
11. The person who loves jazz music is the boba tea drinker.
12. There is one house between the person who uses a OnePlus 9 and Eric.
13. The person who has brown hair is the person who loves hip-hop music.
14. The person who loves pop music is not in the first house.
15. The root beer lover is in the second house.
16. Eric is the person who uses an iPhone 13.
17. The person who has red hair is the person who uses an iPhone 13.
18. The person who has red hair and the coffee drinker are next to each other.
19. Peter is in the first house.
20. Arnold is somewhere to the right of the person who has auburn hair.
21. The person who has gray hair is not in the second house.
22. The person who loves country music is in the fifth house.
 What is the value of attribute PhoneModel for the person whose attribute Drink is milk? Please reason step by step, and put your final answer within \boxed{}

Answer: xiaomi mi 11


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`, `Carol`, `Peter`, `Bob`, `Arnold`, `Eric`
 - They all have a unique favorite flower: `carnations`, `iris`, `tulips`, `daffodils`, `lilies`, `roses`
 - Each person prefers a unique type of vacation: `camping`, `mountain`, `city`, `beach`, `cultural`, `cruise`
 - Each person lives in a unique style of house: `colonial`, `modern`, `mediterranean`, `craftsman`, `ranch`, `victorian`
 - People have unique heights: `short`, `very short`, `average`, `tall`, `super tall`, `very tall`
 - People have unique hair colors: `red`, `black`, `gray`, `brown`, `auburn`, `blonde`

## Clues:
1. The person who loves the boquet of lilies is not in the fifth house.
2. The person who is very short is the person who has auburn hair.
3. Arnold is the person who enjoys camping trips.
4. The person who has auburn hair is Eric.
5. The person who goes on cultural tours is somewhere to the left of the person who is super tall.
6. There are two houses between the person who has gray hair and the person who is short.
7. The person who is tall is somewhere to the left of the person who has brown hair.
8. The person in a Craftsman-style house is Bob.
9. The person who is very short is somewhere to the left of the person who has blonde hair.
10. The person who is very tall is Bob.
11. There are two houses between the person who loves the rose bouquet and Peter.
12. The person in a modern-style house is not in the fourth house.
13. The person who prefers city breaks is the person who is short.
14. The person living in a colonial-style house is somewhere to the left of the person who loves the vase of tulips.
15. The person who loves a carnations arrangement is the person who has brown hair.
16. The person who has red hair is the person who loves the boquet of lilies.
17. The person who enjoys mountain retreats is the person residing in a Victorian house.
18. The person who loves a bouquet of daffodils is the person in a modern-style house.
19. Arnold is the person who has an average height.
20. The person in a Mediterranean-style villa is not in the first house.
21. The person who is tall is not in the fifth house.
22. The person who loves the vase of tulips is not in the fourth house.
23. Carol is directly left of Bob.
24. The person who is very tall is the person who goes on cultural tours.
25. The person in a Mediterranean-style villa is the person who has red hair.
26. The person who is super tall is the person who has black hair.
27. Eric is the person who loves the rose bouquet.
28. The person who enjoys mountain retreats is directly left of the person who likes going on cruises.
 What is the value of attribute House for the person whose attribute Flower is tulips? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


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: `Arnold`, `Peter`, `Eric`
 - People have unique favorite book genres: `romance`, `mystery`, `science fiction`
 - Each person has a unique birthday month: `sept`, `april`, `jan`
 - The people are of nationalities: `brit`, `dane`, `swede`

## Clues:
1. Arnold is the person who loves mystery books.
2. Peter is directly left of the person who loves romance books.
3. Peter is somewhere to the left of the person who loves mystery books.
4. The person whose birthday is in April is somewhere to the right of the person whose birthday is in January.
5. The person whose birthday is in September is the Swedish person.
6. The Dane is in the third house.
7. The person who loves science fiction books is the person whose birthday is in September.
 What is the value of attribute BookGenre for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: romance


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`, `Eric`, `Carol`, `Alice`, `Bob`
 - The people are of nationalities: `swede`, `brit`, `chinese`, `dane`, `german`, `norwegian`
 - People have unique favorite book genres: `science fiction`, `mystery`, `fantasy`, `biography`, `historical fiction`, `romance`
 - Everyone has a favorite smoothie: `blueberry`, `dragonfruit`, `desert`, `watermelon`, `cherry`, `lime`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `huawei p50`, `xiaomi mi 11`, `iphone 13`, `google pixel 6`

## Clues:
1. Alice is the British person.
2. There is one house between Eric and the Dragonfruit smoothie lover.
3. The Chinese is somewhere to the left of Alice.
4. The person who loves historical fiction books is directly left of the person who loves biography books.
5. The person who loves biography books is somewhere to the right of Carol.
6. The person who uses an iPhone 13 is somewhere to the right of the Desert smoothie lover.
7. The person who loves mystery books is the British person.
8. Bob is somewhere to the right of the person who uses a OnePlus 9.
9. Carol is the person who drinks Blueberry smoothies.
10. There are two houses between the person who loves romance books and the Watermelon smoothie lover.
11. The person who likes Cherry smoothies is the Swedish person.
12. The person who loves science fiction books is the person who uses a Google Pixel 6.
13. The person who drinks Lime smoothies is Eric.
14. Bob is in the fifth house.
15. Carol is in the second house.
16. The Norwegian and Alice are next to each other.
17. The Swedish person is not in the fifth house.
18. The German and the Norwegian are next to each other.
19. The person who loves fantasy books is the Dragonfruit smoothie lover.
20. The British person is somewhere to the left of Peter.
21. The person who loves romance books is the person who uses a Samsung Galaxy S21.
22. Alice is the person who uses a Xiaomi Mi 11.
23. The person who loves romance books is Eric.
 What is the value of attribute Smoothie for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: watermelon


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`, `Carol`, `Eric`, `Alice`, `Peter`, `Bob`
 - Each person has a favorite color: `red`, `yellow`, `blue`, `white`, `green`, `purple`
 - Each person has a unique birthday month: `sept`, `jan`, `april`, `feb`, `may`, `mar`

## Clues:
1. The person whose favorite color is green is the person whose birthday is in May.
2. The person whose birthday is in September is directly left of Arnold.
3. The person who loves blue is Carol.
4. The person whose birthday is in January is somewhere to the left of Arnold.
5. Eric is not in the sixth house.
6. Eric is not in the first house.
7. The person who loves yellow is not in the third house.
8. The person whose birthday is in May and the person whose favorite color is red are next to each other.
9. Alice is the person whose favorite color is red.
10. Alice is directly left of the person who loves white.
11. The person who loves blue is not in the sixth house.
12. The person who loves purple is the person whose birthday is in April.
13. Arnold is directly left of the person whose birthday is in March.
14. Peter is in the third house.
 What is the value of attribute Birthday for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


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`, `Eric`, `Alice`, `Bob`, `Arnold`
 - The mothers' names in different houses are unique: `Janelle`, `Penny`, `Kailyn`, `Holly`, `Aniya`
 - Each mother is accompanied by their child: `Meredith`, `Bella`, `Timothy`, `Fred`, `Samantha`
 - People have unique favorite sports: `tennis`, `soccer`, `basketball`, `swimming`, `baseball`

## Clues:
1. The person whose mother's name is Holly is the person who loves tennis.
2. The person's child is named Samantha is somewhere to the left of The person whose mother's name is Aniya.
3. The person who loves swimming is directly left of Arnold.
4. The person who loves basketball is Arnold.
5. The person who is the mother of Timothy is the person who loves swimming.
6. Bob is the person's child is named Bella.
7. The person whose mother's name is Kailyn is the person who loves baseball.
8. The person's child is named Meredith is in the second house.
9. The person's child is named Meredith is Arnold.
10. Alice is the person who is the mother of Timothy.
11. The person's child is named Samantha is Eric.
12. The person who loves soccer is not in the fourth house.
13. The person whose mother's name is Penny is the person who loves swimming.
14. The person's child is named Fred is in the fourth house.
15. The person who loves basketball and the person who loves baseball are next to each other.
 What is the value of attribute Name for the person whose attribute FavoriteSport is tennis? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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`, `Arnold`, `Peter`, `Alice`
 - People have unique favorite book genres: `mystery`, `fantasy`, `romance`, `science fiction`
 - Each person prefers a unique type of vacation: `cruise`, `mountain`, `city`, `beach`
 - They all have a unique favorite flower: `roses`, `daffodils`, `carnations`, `lilies`
 - People have unique favorite music genres: `pop`, `rock`, `jazz`, `classical`

## Clues:
1. Alice is the person who loves jazz music.
2. Eric is the person who loves romance books.
3. The person who loves a carnations arrangement is in the first house.
4. The person who loves a bouquet of daffodils is the person who loves science fiction books.
5. The person who enjoys mountain retreats is in the third house.
6. The person who enjoys mountain retreats is the person who loves mystery books.
7. The person who loves science fiction books is Peter.
8. The person who prefers city breaks is in the second house.
9. The person who prefers city breaks and the person who loves classical music are next to each other.
10. The person who loves beach vacations is not in the first house.
11. The person who loves romance books is the person who loves pop music.
12. The person who loves romance books is somewhere to the left of Alice.
13. The person who loves the boquet of lilies is the person who loves mystery books.
14. Peter is somewhere to the left of Alice.
 What is the value of attribute BookGenre for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: mystery


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`, `Eric`, `Arnold`, `Peter`
 - Everyone has a unique favorite cigar: `prince`, `dunhill`, `blue master`, `pall mall`
 - Each person lives in a unique style of house: `victorian`, `colonial`, `ranch`, `craftsman`
 - Each person has a unique birthday month: `april`, `sept`, `feb`, `jan`
 - Each person has an occupation: `teacher`, `artist`, `doctor`, `engineer`
 - Each mother is accompanied by their child: `Samantha`, `Bella`, `Meredith`, `Fred`

## Clues:
1. The person who is a doctor is the person's child is named Fred.
2. The person's child is named Samantha and the person whose birthday is in January are next to each other.
3. Arnold is the person whose birthday is in September.
4. The person in a Craftsman-style house is the person who is an engineer.
5. The Dunhill smoker is the person's child is named Samantha.
6. Alice is directly left of Arnold.
7. The person whose birthday is in April and the Prince smoker are next to each other.
8. Arnold is the person who is a teacher.
9. The person living in a colonial-style house is not in the first house.
10. The person who smokes Blue Master is somewhere to the left of Eric.
11. The person who smokes Blue Master is somewhere to the right of the person's child is named Fred.
12. The person whose birthday is in April is somewhere to the left of the person who is an engineer.
13. Eric is not in the fourth house.
14. The person's child is named Bella is somewhere to the left of the person whose birthday is in April.
15. The person in a ranch-style home is Arnold.
 What is the value of attribute Occupation for the person whose attribute Children is Bella? Please reason step by step, and put your final answer within \boxed{}

Answer: teacher


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`, `Alice`, `Peter`
 - People own unique car models: `bmw 3 series`, `toyota camry`, `honda civic`, `tesla model 3`, `ford f150`
 - People use unique phone models: `google pixel 6`, `huawei p50`, `oneplus 9`, `iphone 13`, `samsung galaxy s21`
 - Each person has a unique hobby: `painting`, `gardening`, `knitting`, `photography`, `cooking`
 - Each person has an occupation: `lawyer`, `artist`, `teacher`, `engineer`, `doctor`
 - Each person has a favorite color: `white`, `blue`, `green`, `red`, `yellow`

## Clues:
1. The person who owns a Ford F-150 is directly left of the person who is a teacher.
2. The photography enthusiast is Arnold.
3. The person who uses a Huawei P50 is Eric.
4. The person who enjoys gardening is somewhere to the right of the person who loves blue.
5. The person who enjoys knitting is the person who loves white.
6. The person who owns a Ford F-150 is the person whose favorite color is red.
7. There is one house between the person who owns a BMW 3 Series and the person who loves yellow.
8. The person who is an artist is the person who uses a OnePlus 9.
9. The person who is a doctor is Arnold.
10. The person who enjoys gardening is the person who uses a Huawei P50.
11. There is one house between the person who loves blue and the person who loves yellow.
12. The person who owns a BMW 3 Series and the person who uses a Google Pixel 6 are next to each other.
13. The person who uses a Google Pixel 6 is directly left of the person who owns a Honda Civic.
14. Peter is the person who is an engineer.
15. The person who uses a Samsung Galaxy S21 is the person who owns a Toyota Camry.
16. The person who owns a Honda Civic is Alice.
17. The person who owns a Tesla Model 3 is the person who paints as a hobby.
 What is the value of attribute Hobby for the person whose attribute House is 4? 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: `Alice`, `Eric`, `Bob`, `Peter`, `Arnold`
 - People have unique favorite book genres: `fantasy`, `mystery`, `science fiction`, `biography`, `romance`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Aniya`, `Kailyn`, `Janelle`
 - People use unique phone models: `oneplus 9`, `iphone 13`, `samsung galaxy s21`, `google pixel 6`, `huawei p50`
 - The people are of nationalities: `dane`, `brit`, `german`, `swede`, `norwegian`
 - Each person lives in a unique style of house: `ranch`, `modern`, `craftsman`, `victorian`, `colonial`

## Clues:
1. The person in a ranch-style home is somewhere to the right of the person residing in a Victorian house.
2. The Dane and the person who uses a Samsung Galaxy S21 are next to each other.
3. The person who uses a Google Pixel 6 is not in the second house.
4. There are two houses between the person who loves fantasy books and the person in a modern-style house.
5. The person whose mother's name is Kailyn is the Swedish person.
6. The Swedish person is Arnold.
7. The British person is somewhere to the left of the person who uses an iPhone 13.
8. The Dane is The person whose mother's name is Penny.
9. The person who uses a Samsung Galaxy S21 is the Norwegian.
10. The Norwegian is The person whose mother's name is Holly.
11. The person who loves fantasy books is Bob.
12. Alice is directly left of the Swedish person.
13. The person who loves romance books is Arnold.
14. The person who loves science fiction books is Peter.
15. The person whose mother's name is Aniya is directly left of the person who loves science fiction books.
16. The person who loves science fiction books is the person who uses a OnePlus 9.
17. The person residing in a Victorian house is the person who loves biography books.
18. The person who loves fantasy books is somewhere to the left of the German.
19. The person who uses an iPhone 13 is the person in a Craftsman-style house.
 What is the value of attribute BookGenre for the person whose attribute PhoneModel is oneplus 9? 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: `Arnold`, `Carol`, `Eric`, `Alice`, `Peter`, `Bob`
 - Each person has a favorite color: `red`, `yellow`, `blue`, `white`, `green`, `purple`
 - Each person has a unique birthday month: `sept`, `jan`, `april`, `feb`, `may`, `mar`

## Clues:
1. The person whose favorite color is green is the person whose birthday is in May.
2. The person whose birthday is in September is directly left of Arnold.
3. The person who loves blue is Carol.
4. The person whose birthday is in January is somewhere to the left of Arnold.
5. Eric is not in the sixth house.
6. Eric is not in the first house.
7. The person who loves yellow is not in the third house.
8. The person whose birthday is in May and the person whose favorite color is red are next to each other.
9. Alice is the person whose favorite color is red.
10. Alice is directly left of the person who loves white.
11. The person who loves blue is not in the sixth house.
12. The person who loves purple is the person whose birthday is in April.
13. Arnold is directly left of the person whose birthday is in March.
14. Peter is in the third house.
 What is the value of attribute Birthday for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


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`, `Eric`, `Arnold`, `Carol`, `Alice`, `Peter`
 - Each person has a unique hobby: `cooking`, `painting`, `knitting`, `photography`, `woodworking`, `gardening`
 - People have unique hair colors: `brown`, `auburn`, `black`, `red`, `gray`, `blonde`
 - People have unique favorite sports: `swimming`, `volleyball`, `basketball`, `baseball`, `tennis`, `soccer`
 - The mothers' names in different houses are unique: `Sarah`, `Holly`, `Penny`, `Janelle`, `Kailyn`, `Aniya`
 - Everyone has a unique favorite cigar: `dunhill`, `blends`, `pall mall`, `prince`, `yellow monster`, `blue master`

## Clues:
1. The person who has gray hair is directly left of The person whose mother's name is Janelle.
2. The person who loves soccer is in the sixth house.
3. Arnold is the person who has blonde hair.
4. The photography enthusiast is the person who smokes Yellow Monster.
5. There is one house between the Dunhill smoker and the person who loves cooking.
6. The person who paints as a hobby is the person who has blonde hair.
7. The Dunhill smoker is directly left of Alice.
8. The person who paints as a hobby is directly left of the person who loves volleyball.
9. The person who loves soccer is The person whose mother's name is Aniya.
10. The person partial to Pall Mall is somewhere to the left of Carol.
11. The person who enjoys knitting and the person who loves basketball are next to each other.
12. The person partial to Pall Mall is directly left of The person whose mother's name is Holly.
13. The person who loves soccer is the Prince smoker.
14. The person who loves swimming is The person whose mother's name is Penny.
15. The person whose mother's name is Kailyn is the person who loves volleyball.
16. The person who loves basketball is The person whose mother's name is Janelle.
17. The person who enjoys knitting is the person who loves volleyball.
18. The person who enjoys knitting is not in the third house.
19. The person who has auburn hair is somewhere to the left of the photography enthusiast.
20. The person whose mother's name is Sarah is not in the fifth house.
21. Bob is directly left of the person who has red hair.
22. The Dunhill smoker is the person who enjoys gardening.
23. The person who loves baseball is the person who has auburn hair.
24. Eric is the person who smokes many unique blends.
25. The person who has brown hair is somewhere to the right of The person whose mother's name is Penny.
 What is the value of attribute Cigar for the person whose attribute Hobby is photography? Please reason step by step, and put your final answer within \boxed{}

Answer: yellow monster


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`, `Peter`, `Eric`, `Bob`, `Arnold`, `Carol`
 - Everyone has a unique favorite cigar: `pall mall`, `yellow monster`, `dunhill`, `blue master`, `prince`, `blends`
 - People have unique favorite music genres: `hip hop`, `jazz`, `country`, `pop`, `classical`, `rock`
 - Each person has a unique favorite drink: `water`, `milk`, `boba tea`, `tea`, `root beer`, `coffee`
 - The mothers' names in different houses are unique: `Kailyn`, `Penny`, `Janelle`, `Holly`, `Sarah`, `Aniya`
 - Everyone has something unique for lunch: `soup`, `pizza`, `spaghetti`, `stir fry`, `stew`, `grilled cheese`

## Clues:
1. Carol is directly left of the person who loves eating grilled cheese.
2. Eric is not in the second house.
3. The person whose mother's name is Holly is somewhere to the right of Carol.
4. The person who loves eating grilled cheese is somewhere to the right of the person who loves rock music.
5. Eric is directly left of Carol.
6. The person who loves pop music is not in the third house.
7. Eric is the person who loves country music.
8. The person who loves classical music is in the sixth house.
9. The coffee drinker is Bob.
10. The person who smokes many unique blends is Peter.
11. The person who loves the stew is not in the fifth house.
12. The root beer lover is directly left of The person whose mother's name is Janelle.
13. There are two houses between The person whose mother's name is Sarah and the person who smokes Yellow Monster.
14. Eric is the tea drinker.
15. The person partial to Pall Mall is somewhere to the right of the person who loves stir fry.
16. The person who loves the soup is Bob.
17. The person who loves hip-hop music is directly left of The person whose mother's name is Kailyn.
18. Arnold is somewhere to the right of The person whose mother's name is Kailyn.
19. The one who only drinks water is directly left of the person who smokes Blue Master.
20. The person who loves the spaghetti eater is somewhere to the left of the person who smokes many unique blends.
21. The person whose mother's name is Sarah is directly left of the person who loves jazz music.
22. The person who loves hip-hop music is directly left of the root beer lover.
23. The one who only drinks water is the person who loves the stew.
24. The Dunhill smoker is not in the second house.
25. The person who likes milk is The person whose mother's name is Janelle.
26. Eric is The person whose mother's name is Aniya.
 What is the value of attribute House for the person whose attribute Cigar is yellow monster? Please reason step by step, and put your final answer within \boxed{}

Answer: 6


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`, `Peter`, `Arnold`, `Bob`, `Eric`, `Carol`
 - Each person has a unique hobby: `cooking`, `gardening`, `painting`, `knitting`, `photography`, `woodworking`
 - Each person has a unique birthday month: `april`, `sept`, `mar`, `jan`, `feb`, `may`
 - Everyone has a unique favorite cigar: `blends`, `prince`, `pall mall`, `dunhill`, `yellow monster`, `blue master`

## Clues:
1. The person whose birthday is in April is Eric.
2. Arnold is in the first house.
3. The person whose birthday is in January is the person who smokes many unique blends.
4. The Dunhill smoker is Alice.
5. Bob is the person whose birthday is in May.
6. The person whose birthday is in September and the photography enthusiast are next to each other.
7. Bob is the person who smokes Blue Master.
8. The person who loves cooking is the person whose birthday is in May.
9. The woodworking hobbyist and the person whose birthday is in March are next to each other.
10. The person who enjoys knitting is directly left of the person who smokes Yellow Monster.
11. Peter and the person who paints as a hobby are next to each other.
12. The person whose birthday is in March is somewhere to the left of the Prince smoker.
13. The person whose birthday is in February is somewhere to the left of the person who smokes many unique blends.
14. The person whose birthday is in March is somewhere to the right of the person whose birthday is in September.
15. The person who smokes many unique blends is directly left of the person whose birthday is in May.
16. The photography enthusiast and Peter are next to each other.
 What is the value of attribute Birthday for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


## Task A-3.2. (4 points)

Legoplus is a body consisting of seven equal cubes joined in such a way that there is one cube that shares a common face with each of the remaining six cubes.

Each face of the legoplus must be painted with one color. How many colors are minimally needed to do this so that no two adjacent faces are painted the same color?

Answer: 3


$\left.\begin{array}{cc}{\left[\begin{array}{l}\text { Law of Cosines } \\ {[\text { Similar triangles (other) }}\end{array}\right]}\end{array}\right]$

In triangle $ABC$, $AC=2\sqrt{3}$, $AB=\sqrt{7}$, $BC=1$. Outside the triangle, a point $K$ is taken such that segment $KC$ intersects segment $AB$ at a point different from $B$, and the triangle with vertices $K, A$, and $C$ is similar to the original triangle. Find the angle $AKC$, given that angle $KAC$ is obtuse.

Answer: 30^\circ


8.6. What is the maximum number of sides a coin can have as a projection of a convex polyhedron with $n$ faces?

Answer: 2n - 4


Example 5 A unit spent 500,000 yuan to purchase a piece of high-tech equipment. According to the tracking survey of this model of equipment, after the equipment is put into use, if the maintenance and repair costs are averaged to each day, the conclusion is: the maintenance and repair cost on the $x$-th day is $\left[\frac{1}{4}(x-1)+500\right]$ yuan.

Answer: 2000


5.

A custodian stores securities, clients' money, and other material assets

Correct answers: A paid investment advisor consults and provides recommendations to the client on investment management, A trustee manages the client's property in their own name

Find the correspondence between the term and the statement so that all 5 pairs are correct. All 5 terms must be used.

The service may involve changing the terms of an existing loan

According to federal law, this service is available only once during the entire loan period

The service provides options for reducing the term or the amount of loan payments

During the term of this service, the bank does not charge the borrower penalties and cannot demand early repayment of the loan

This service may include

refinancing; consolidation of debts; loan holidays; mortgage holidays; early repayment; restructuring;

refinancing; loan holidays; mortgage holidays; early repayment; restructuring;

refinancing; loan holidays; mortgage holidays; early repayment; restructuring;

refinancing; loan holidays; mortgage holidays; early repayment; restructuring;

refinancing; loan holidays; mortgage holidays; early repayment; restructuring;

Correct answer:

The service may involve changing the terms of an existing loan $\rightarrow$ restructuring,

According to federal law, this service is available only once during the entire loan period $\rightarrow$ mortgage holidays,

The service provides options for reducing the term or the amount of loan payments $\rightarrow$ early repayment,

During the term of this service, the bank does not charge the borrower penalties and cannot demand early repayment of the loan $\rightarrow$ loan holidays, This service may include consolidation of debts $\rightarrow$ refinancing

Question 8

Score: 7.00

The Ivanov family carefully plans their budget. Lidia Nikolaevna works as a doctor and earns 1,000,000 rubles per year (before income tax). Arkady Petrovich is an entrepreneur, and his annual profit from the business is 2,000,000 rubles, which is taxed at a rate of $15 \%$ under the simplified taxation system (USN).

On average, the family's expenses amount to 205,000 rubles per month, excluding vacation expenses.

The Ivanovs had been saving money for a trip to Paris for a year, but due to the pandemic, they decided to postpone the trip for a year and temporarily place the accumulated savings in a bank deposit at an annual interest rate of $12 \%$ with interest paid at the end of each quarter. The deposit term is 12 months, and interest income is reinvested. Calculate the Ivanov family's income from the deposit.

Answer: 13806


7. Given $z \in \mathbf{C}$. If the equation $x^{2}-2 z x+\frac{3}{4}+\mathrm{i}=0$ (where $\mathrm{i}$ is the imaginary unit) has real roots, then the minimum value of $|z|$ is . $\qquad$

Answer: 1


7. The inscribed circle of a triangle divides one of its sides into segments equal to 3 and 4. Find the area of the triangle if the radius of the circumscribed circle around it is $7 / \sqrt{3}$.

Answer: 12\sqrt{3}


6.4. A five-digit number is a multiple of 54, and none of its digits are 0. After deleting one of its digits, the resulting four-digit number is still a multiple of 54; after deleting one of the digits of this four-digit number, the resulting three-digit number is still a multiple of 54; after deleting one of the digits of this three-digit number, the resulting two-digit number is still a multiple of 54. Find the original five-digit number.

Answer: 59994


3. The sum of the upper base radius $r$ and the lower base radius $R (R > r)$ of a frustum is 6 times the slant height $l$, and the areas of the upper base, lateral surface, and lower base form a geometric sequence. The height of this frustum is $20 - 3r^2$, then the maximum volume of the frustum is $\qquad$

Answer: \dfrac{500\pi(2 + \sqrt{3})}{9}


## Task $6 / 83$

In every rectangle, the angle bisectors intersect at four points that span a square (if the rectangle is a square, these four points coincide).

The area $A_{Q}$ of this square is to be represented as a function of the side ratio $x=a: b$ (where $a>b, b$ is constant). For which side ratio is the square area $A_{Q}$ equal to the rectangle area $A_{R}$?

Answer: 2 + \sqrt{3}


Given points $A, B, C$, and $D$ such that segments $A C$ and $B D$ intersect at point $E$. Segment $A E$ is 1 cm shorter than segment $A B$, $A E = D C$, $A D = B E$,

$\angle A D C = \angle D E C$. Find the length of $E C$.

Answer: 1


Fix an integer $n \geq 2$. An $n \times n$ sieve is an $n \times n$ array with $n$ cells removed so that exactly one cell is removed from each row and column. For any positive integer $k$, a stick is a $1 \times k$ or $k \times 1$ array. For any sieve $A$, let $m(A)$ denote the minimum number of sticks required to partition $A$. Find all possible values of $m(A)$, since $A$ is different in all possible $n \times n$ sieves, for $n=100$.

Answer: 198


Olja writes down $n$ positive integers $a_{1}, a_{2}, \ldots, a_{n}$ smaller than $p_{n}$ where $p_{n}$ denotes the $n$-th prime number. Oleg can choose two (not necessarily different) numbers $x$ and $y$ and replace one of them with their product $x y$. If there are two equal numbers Oleg wins. Can Oleg guarantee a win? Please provide the smallest integer n for which Oleg can guarantee a win.

Answer: 2


Let $n$ be a positive integer. Let $B_{n}$ be the set of all binary strings of length $n$. For a binary string $s_{1} s_{2} \ldots s_{n}$, we define its twist in the following way. First, we count how many blocks of consecutive digits it has. Denote this number by $b$. Then, we replace $s_{b}$ with $1-s_{b}$. A string $a$ is said to be a descendant of $b$ if $a$ can be obtained from $b$ through a finite number of twists. A subset of $B_{n}$ is called divided if no two of its members have a common descendant. Find the largest possible cardinality of a divided subset of $B_{n}$. Please provide the value when $n = 4$. The answer should be an integer.

Answer: 4


A friendly football match lasts 90 minutes. In this problem, we consider one of the teams, coached by Sir Alex, which plays with 11 players at all times.\na) Sir Alex wants for each of his players to play the same integer number of minutes, but each player has to play less than 60 minutes in total. What is the minimum number of players required?\nb) For the number of players found in a), what is the minimum number of substitutions required, so that each player plays the same number of minutes?\nRemark: Substitutions can only take place after a positive integer number of minutes, and players who have come off earlier can return to the game as many times as needed. There is no limit to the number of substitutions allowed.\nPlease provide the sum of the answers for parts a) and b).

Answer: 35


A positive integer $k \geqslant 3$ is called fibby if there exists a positive integer $n$ and positive integers $d_{1}<d_{2}<\ldots<d_{k}$ with the following properties:  - $d_{j+2}=d_{j+1}+d_{j}$ for every $j$ satisfying $1 \leqslant j \leqslant k-2$,  - $d_{1}, d_{2}, \ldots, d_{k}$ are divisors of $n$,  - any other divisor of $n$ is either less than $d_{1}$ or greater than $d_{k}$.  Find the largest fibby number.

Answer: 4


Let $k$ be a positive integer. Lexi has a dictionary $\mathcal{D}$ consisting of some $k$-letter strings containing only the letters $A$ and $B$. Lexi would like to write either the letter $A$ or the letter $B$ in each cell of a $k \times k$ grid so that each column contains a string from $\mathcal{D}$ when read from top-to-bottom and each row contains a string from $\mathcal{D}$ when read from left-to-right. What is the smallest integer $m$ such that if $\mathcal{D}$ contains at least $m$ different strings, then Lexi can fill her grid in this manner, no matter what strings are in $\mathcal{D}$? Please provide the value of m when k = 2.

Answer: 2


For a sequence $a_{1}<a_{2}<\cdots<a_{n}$ of integers, a pair ( $a_{i}, a_{j}$ ) with $1 \leq i<$ $j \leq n$ is called \underline{interesting} if there exists a pair ( $a_{k}, a_{l}$ ) of integers with $1 \leq k<l \leq n$ such that
$$\frac{a_{l}-a_{k}}{a_{j}-a_{i}}=2$$
For each $n \geq 3$, find the largest possible number of interesting pairs in a sequence of length $n$. Please provide the value of this expression when $n = 5$.

Answer: 7


A domino is a $2 \times 1$ or $1 \times 2$ tile. Determine in how many ways exactly $n^{2}$ dominoes can be placed without overlapping on a $2 n \times 2 n$ chessboard so that every $2 \times 2$ square contains at least two uncovered unit squares which lie in the same row or column. Please provide the value when n = 1.

Answer: 4


Let $m$ be a positive integer. Consider a $4 m \times 4 m$ array of square unit cells. Two different cells are related to each other if they are in either the same row or in the same column. No cell is related to itself. Some cells are coloured blue, such that every cell is related to at least two blue cells. Determine the minimum number of blue cells. Please provide the answer when $m=1$.

Answer: 6