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`, `Alice`, `Arnold`, `Carol`, `Peter`, `Bob`
 - Each person has a unique level of education: `high school`, `trade school`, `bachelor`, `doctorate`, `master`, `associate`
 - Each person has a favorite color: `purple`, `blue`, `white`, `red`, `green`, `yellow`
 - Each person prefers a unique type of vacation: `cultural`, `cruise`, `beach`, `mountain`, `camping`, `city`
 - People have unique favorite book genres: `science fiction`, `biography`, `historical fiction`, `fantasy`, `romance`, `mystery`

## Clues:
1. Alice is the person with a master's degree.
2. There is one house between the person with an associate's degree and Peter.
3. The person who loves historical fiction books is somewhere to the left of the person who loves biography books.
4. The person who loves historical fiction books is the person with a high school diploma.
5. Bob is the person with a bachelor's degree.
6. Carol is somewhere to the right of Eric.
7. The person with a master's degree is somewhere to the right of the person who enjoys camping trips.
8. The person who loves science fiction books is somewhere to the left of the person who loves yellow.
9. The person who loves historical fiction books is somewhere to the left of the person whose favorite color is green.
10. The person with a high school diploma is not in the second house.
11. The person who loves beach vacations is the person with a doctorate.
12. The person who loves mystery books is somewhere to the left of the person with a bachelor's degree.
13. The person with a doctorate is in the first house.
14. The person with a high school diploma is directly left of the person who enjoys camping trips.
15. The person who loves romance books is not in the third house.
16. There is one house between the person who loves purple and the person with a bachelor's degree.
17. The person who loves biography books is the person whose favorite color is red.
18. There is one house between the person who loves fantasy books and the person who loves blue.
19. The person who goes on cultural tours is the person who loves blue.
20. Carol is not in the fifth house.
21. The person who loves purple is the person with a doctorate.
22. The person who enjoys mountain retreats is somewhere to the right of the person who loves science fiction books.
23. The person who prefers city breaks is the person whose favorite color is green.
 What is the value of attribute Name for the person whose attribute Vacation is mountain? 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: `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 House for the person whose attribute Animal is cat? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`, `Alice`
 - People own unique car models: `ford f150`, `toyota camry`, `honda civic`, `tesla model 3`
 - Everyone has a unique favorite cigar: `prince`, `dunhill`, `blue master`, `pall mall`
 - People have unique favorite sports: `soccer`, `swimming`, `tennis`, `basketball`

## Clues:
1. The person who owns a Tesla Model 3 is not in the third house.
2. The person partial to Pall Mall is not in the third house.
3. The person who owns a Honda Civic is the Prince smoker.
4. Peter is the person who loves soccer.
5. The person who smokes Blue Master is the person who loves swimming.
6. The person who owns a Ford F-150 is not in the fourth house.
7. The person who smokes Blue Master is not in the first house.
8. The person who owns a Ford F-150 is not in the third house.
9. The person who loves soccer is in the second house.
10. The person who loves tennis is directly left of Arnold.
11. Alice is directly left of the person who owns a Honda Civic.
 What is the value of attribute FavoriteSport for the person whose attribute Cigar is prince? Please reason step by step, and put your final answer within \boxed{}

Answer: soccer


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`, `Arnold`, `Peter`, `Eric`
 - The mothers' names in different houses are unique: `Penny`, `Aniya`, `Kailyn`, `Janelle`, `Holly`
 - The people keep unique animals: `cat`, `bird`, `dog`, `fish`, `horse`
 - Everyone has a favorite smoothie: `desert`, `dragonfruit`, `cherry`, `watermelon`, `lime`
 - Each person has an occupation: `doctor`, `engineer`, `artist`, `teacher`, `lawyer`

## Clues:
1. Bob is the Desert smoothie lover.
2. The cat lover is the person who is an engineer.
3. The person whose mother's name is Penny is the bird keeper.
4. The person who is a lawyer is the fish enthusiast.
5. The person who likes Cherry smoothies is directly left of Alice.
6. The Watermelon smoothie lover and the person who is an engineer are next to each other.
7. The person whose mother's name is Holly is the dog owner.
8. The person who is a teacher is in the first house.
9. The person whose mother's name is Penny is in the second house.
10. The person who drinks Lime smoothies is the person who is a doctor.
11. The person whose mother's name is Janelle is the person who is a lawyer.
12. Arnold is not in the third house.
13. The Dragonfruit smoothie lover is directly left of Peter.
14. The person whose mother's name is Kailyn is the Dragonfruit smoothie lover.
15. Arnold is somewhere to the right of Peter.
16. The dog owner is not in the fifth house.
17. The Watermelon smoothie lover is somewhere to the left of the Desert smoothie lover.
 What is the value of attribute Animal for the person whose attribute Occupation is teacher? Please reason step by step, and put your final answer within \boxed{}

Answer: horse


Points \(A\), \(B\), \(C\), and \(D\) are on a circle with diameter \(1\), and \(X\) is on diameter \(\overline{AD}\). If \(BX = CX\) and \(3 \angle BAC = \angle BXC = 36^{\circ}\), then find \(AX\) in the form of a trigonometric expression \(\cos x^\circ \sin y^\circ \csc z^\circ\). Please give the value of x + y + z.

Answer: 36


A cyclist travels at a constant speed of $22.0\,\text{km/hr}$ except for a $20$ minute stop. The cyclist’s average speed was $17.5\,\text{km/hr}$. How far did the cyclist travel? If the answer is $x\,\text{km}$, what is the value of $x \times 10$?

Answer: 285


Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3? Express your answer as a fraction in simplest form, in the format \(\frac{k}{m}\). Please give the value of k + m.

Answer: 265


Let set $T = \{1, 2, 3, 4, 5, 6, 7, 8\}$. Find the number of all nonempty subsets $A$ of $T$ such that $3 \mid S(A)$ and $5 \nmid S(A)$, where $S(A)$ is the sum of all the elements in $A$.

Answer: 70


A positive integer $k$ is said to be \textit{good} if there exists a partition of $\{1, 2, 3, \ldots, 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many \textit{good} numbers are there?

Answer: 6


For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is \textit{good} if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2016$ good partitions.

Answer: 11


Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be six parabolas in the plane, each congruent to the parabola $y = \frac{x^2}{16}$. The vertices of the six parabolas are evenly spaced around a circle. The parabolas open outward with their axes being extensions of six of the circle's radii. Parabola $P_1$ is tangent to $P_2$, which is tangent to $P_3$, which is tangent to $P_4$, which is tangent to $P_5$, which is tangent to $P_6$, which is tangent to $P_1$. What is the diameter of the circle?

Answer: 24


Michelle is drawing segments in the plane. She begins from the origin facing up the $y$-axis and draws a segment of length $1$. Now, she rotates her direction by $120^\circ$, with equal probability clockwise or counterclockwise, and draws another segment of length $1$ beginning from the end of the previous segment. She then continues this until she hits an already drawn segment. What is the expected number of segments she has drawn when this happens?

Answer: 4


Nick has a $3 \times 3$ grid and wants to color each square in the grid one of three colors such that no two squares that are adjacent horizontally or vertically are the same color. Compute the number of distinct grids that Nick can create.

Answer: 246


Given a right triangle $ABC$ with $AC = 3$, $BC = 4$, and $AB = 5$. Squares are constructed externally on each side of the triangle. Calculate the area of the hexagon $PQRSTU$.

Answer: 74


In parallelogram $ABCD$, angles $B$ and $D$ are acute while angles $A$ and $C$ are obtuse. The perpendicular from $C$ to $AB$ and the perpendicular from $A$ to $BC$ intersect at a point $P$ inside the parallelogram. If $PB = 700$ and $PD = 821$, what is $AC$?

Answer: 429


In the figure, a semicircle is folded along the $AN$ string and intersects the $MN$ diameter at $B$. It is given that $MB:BN = 2:3$ and $MN = 10$. If $AN = x$, what is the value of $x^2$?

Answer: 80


In a triangle $ABC$, the median $AD$ divides $\angle BAC$ in the ratio $1:2$. Extend $AD$ to $E$ such that $EB$ is perpendicular to $AB$. Given that $BE=3$ and $BA=4$, find the integer nearest to $BC^2$.

Answer: 29


Towers grow at points along a line. All towers start with height $0$ and grow at the rate of $1$ meter per second. As soon as any two adjacent towers are each at least $1$ meter tall, a new tower begins to grow at a point along the line exactly halfway between those two adjacent towers. Before time $0$, there are no towers, but at time $0$, the first two towers begin to grow at two points along the line. Find the total of all the heights of all towers at time $10$ seconds.

Answer: 1033


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`
 - Each person lives in a unique style of house: `craftsman`, `ranch`, `victorian`, `colonial`
 - Each person has a unique favorite drink: `water`, `tea`, `milk`, `coffee`

## Clues:
1. Eric is the one who only drinks water.
2. The person who likes milk is the person residing in a Victorian house.
3. The person in a ranch-style home is somewhere to the left of the person in a Craftsman-style house.
4. Eric is in the third house.
5. Arnold is somewhere to the right of Alice.
6. The person in a ranch-style home is Arnold.
7. Eric and the person residing in a Victorian house are next to each other.
8. The tea drinker is not in the first house.
 What is the value of attribute Name for the person whose attribute Drink is milk? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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`, `Alice`, `Arnold`, `Bob`, `Peter`, `Carol`
 - Each person has a unique hobby: `woodworking`, `knitting`, `cooking`, `gardening`, `painting`, `photography`
 - Everyone has a unique favorite cigar: `pall mall`, `blue master`, `blends`, `prince`, `dunhill`, `yellow monster`

## Clues:
1. Alice is the person who loves cooking.
2. Bob is not in the second house.
3. The person who enjoys gardening is the person who smokes Blue Master.
4. The person partial to Pall Mall is the person who enjoys knitting.
5. The person who enjoys gardening is directly left of the Dunhill smoker.
6. The person who smokes many unique blends is directly left of Carol.
7. The person partial to Pall Mall is directly left of Bob.
8. The Dunhill smoker is the person who paints as a hobby.
9. The person who smokes Yellow Monster is in the second house.
10. The woodworking hobbyist is somewhere to the right of the Dunhill smoker.
11. Eric is somewhere to the left of the person who enjoys gardening.
12. There are two houses between the photography enthusiast and Peter.
 What is the value of attribute Name for the person whose attribute Hobby is painting? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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`, `Eric`, `Peter`
 - People have unique favorite book genres: `science fiction`, `mystery`, `romance`
 - Each mother is accompanied by their child: `Bella`, `Fred`, `Meredith`
 - People have unique hair colors: `black`, `blonde`, `brown`
 - Each person has a unique level of education: `high school`, `associate`, `bachelor`
 - They all have a unique favorite flower: `daffodils`, `lilies`, `carnations`

## Clues:
1. The person who loves the boquet of lilies is directly left of the person who loves romance books.
2. Peter is the person with a high school diploma.
3. The person who loves the boquet of lilies is the person who has brown hair.
4. The person's child is named Bella is Arnold.
5. The person who loves science fiction books is the person's child is named Fred.
6. The person's child is named Fred is the person who loves the boquet of lilies.
7. The person who has black hair is the person with an associate's degree.
8. The person's child is named Meredith is not in the first house.
9. The person with a high school diploma is the person who loves a carnations arrangement.
10. Peter is not in the third house.
 What is the value of attribute Education for the person whose attribute HairColor is blonde? Please reason step by step, and put your final answer within \boxed{}

Answer: high school


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: `Arnold`, `Peter`, `Alice`, `Eric`
 - People have unique hair colors: `black`, `red`, `blonde`, `brown`
 - Each person has an occupation: `engineer`, `artist`, `doctor`, `teacher`
 - Each person has a unique favorite drink: `coffee`, `water`, `tea`, `milk`

## Clues:
1. The person who has brown hair is not in the second house.
2. The person who is a doctor is Peter.
3. The person who is a doctor is the one who only drinks water.
4. The person who is an artist is somewhere to the left of the person who is an engineer.
5. The person who likes milk is the person who has blonde hair.
6. The person who has red hair is Alice.
7. The person who is an artist and Arnold are next to each other.
8. The person who is an artist is in the third house.
9. The person who likes milk is somewhere to the left of the tea drinker.
10. Arnold is somewhere to the right of Eric.
11. The person who has black hair is the coffee drinker.
 What is the value of attribute Name for the person whose attribute Drink is water? 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: `Peter`, `Eric`, `Arnold`, `Bob`, `Alice`
 - Each person has a unique favorite drink: `milk`, `tea`, `root beer`, `water`, `coffee`
 - Each person has a favorite color: `blue`, `white`, `red`, `green`, `yellow`
 - Each mother is accompanied by their child: `Samantha`, `Meredith`, `Timothy`, `Bella`, `Fred`
 - Each person lives in a unique style of house: `modern`, `ranch`, `victorian`, `colonial`, `craftsman`
 - People use unique phone models: `samsung galaxy s21`, `iphone 13`, `huawei p50`, `oneplus 9`, `google pixel 6`

## Clues:
1. The person in a Craftsman-style house is directly left of the person living in a colonial-style house.
2. The person living in a colonial-style house is somewhere to the left of the person who uses a Samsung Galaxy S21.
3. The person who uses a Huawei P50 is the person in a Craftsman-style house.
4. The root beer lover is somewhere to the left of the person residing in a Victorian house.
5. There is one house between the person in a ranch-style home and the root beer lover.
6. The person who uses a OnePlus 9 is the person whose favorite color is red.
7. The tea drinker is not in the fourth house.
8. The person living in a colonial-style house and Peter are next to each other.
9. The person in a ranch-style home is directly left of the person whose favorite color is green.
10. The person who is the mother of Timothy and the person's child is named Meredith are next to each other.
11. The person who loves blue is the person residing in a Victorian house.
12. Bob is the person residing in a Victorian house.
13. The person's child is named Bella is the person who uses a Google Pixel 6.
14. The person who loves blue is the one who only drinks water.
15. The person who likes milk is the person who is the mother of Timothy.
16. The person who loves white is Arnold.
17. The root beer lover is directly left of the person who uses a Samsung Galaxy S21.
18. The person who likes milk is Alice.
19. The person who is the mother of Timothy is the person whose favorite color is green.
20. The person who loves white is not in the first house.
21. The person's child is named Samantha is the person whose favorite color is red.
 What is the value of attribute House for the person whose attribute HouseStyle is modern? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Alice`
 - People have unique favorite sports: `basketball`, `swimming`, `soccer`, `tennis`
 - People have unique hair colors: `black`, `brown`, `blonde`, `red`
 - Each person has a favorite color: `yellow`, `red`, `white`, `green`
 - The people are of nationalities: `dane`, `norwegian`, `brit`, `swede`
 - Each person has a unique birthday month: `april`, `feb`, `jan`, `sept`

## Clues:
1. The person who has brown hair is in the first house.
2. The person who loves soccer is directly left of the person who loves tennis.
3. The person who has brown hair is directly left of Eric.
4. The person who has blonde hair is somewhere to the left of the British person.
5. The person who loves white is directly left of the Norwegian.
6. The person who has blonde hair and Arnold are next to each other.
7. The person whose birthday is in January is the Swedish person.
8. The person whose birthday is in April is the British person.
9. The person who loves basketball is somewhere to the right of the person who has black hair.
10. The person who has brown hair and the person who loves yellow are next to each other.
11. The person who loves swimming is Peter.
12. The person whose favorite color is red is Peter.
13. The person who has blonde hair is the Swedish person.
14. Eric is the person whose birthday is in February.
 What is the value of attribute Birthday for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: feb


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`, `Alice`, `Eric`, `Arnold`, `Bob`, `Carol`
 - The people are of nationalities: `dane`, `brit`, `norwegian`, `swede`, `german`, `chinese`
 - People have unique favorite book genres: `historical fiction`, `mystery`, `romance`, `fantasy`, `science fiction`, `biography`
 - The mothers' names in different houses are unique: `Holly`, `Aniya`, `Penny`, `Kailyn`, `Sarah`, `Janelle`

## Clues:
1. The person whose mother's name is Janelle is the Swedish person.
2. Alice is the person who loves science fiction books.
3. The person who loves romance books is not in the fifth house.
4. Carol and Peter are next to each other.
5. The person whose mother's name is Sarah is the Norwegian.
6. There is one house between Arnold and the British person.
7. The German and the person who loves historical fiction books are next to each other.
8. There is one house between the German and Carol.
9. The Swedish person is the person who loves fantasy books.
10. Arnold is in the second house.
11. The person whose mother's name is Sarah is in the third house.
12. The person who loves historical fiction books is somewhere to the right of The person whose mother's name is Penny.
13. The Chinese is not in the sixth house.
14. There are two houses between Alice and the person who loves mystery books.
15. Carol is somewhere to the right of Alice.
16. Arnold is the Dane.
17. The person whose mother's name is Aniya is in the fourth house.
18. The person whose mother's name is Penny is somewhere to the left of the Swedish person.
19. The person whose mother's name is Holly and Eric are next to each other.
 What is the value of attribute Mother for the person whose attribute BookGenre is historical fiction? Please reason step by step, and put your final answer within \boxed{}

Answer: Kailyn


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`, `Carol`, `Eric`, `Peter`, `Alice`, `Arnold`
 - Each person lives in a unique style of house: `victorian`, `ranch`, `modern`, `craftsman`, `mediterranean`, `colonial`
 - Everyone has a unique favorite cigar: `yellow monster`, `blue master`, `blends`, `pall mall`, `dunhill`, `prince`

## Clues:
1. The person living in a colonial-style house is not in the second house.
2. The person partial to Pall Mall is somewhere to the left of the person in a ranch-style home.
3. The person in a ranch-style home is not in the fourth house.
4. The Prince smoker is not in the fourth house.
5. The person in a Mediterranean-style villa is not in the fourth house.
6. The person who smokes many unique blends is in the third house.
7. The person in a modern-style house is directly left of Carol.
8. The person residing in a Victorian house is somewhere to the left of the person in a ranch-style home.
9. Eric is not in the sixth house.
10. Alice and the person in a Craftsman-style house are next to each other.
11. There are two houses between the person partial to Pall Mall and Bob.
12. The person in a ranch-style home is not in the third house.
13. The Prince smoker is somewhere to the right of the person in a Craftsman-style house.
14. The person who smokes Yellow Monster is not in the second house.
15. There are two houses between Arnold and Bob.
16. The person living in a colonial-style house is directly left of the person in a Mediterranean-style villa.
17. The person in a Mediterranean-style villa is not in the second house.
18. The person who smokes Blue Master is somewhere to the right of the person living in a colonial-style house.
19. The person living in a colonial-style house is directly left of Peter.
 What is the value of attribute Name for the person whose attribute HouseStyle is mediterranean? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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 Hobby for the person whose attribute Cigar is dunhill? Please reason step by step, and put your final answer within \boxed{}

Answer: gardening


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`
 - The people keep unique animals: `horse`, `cat`, `fish`, `bird`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `iphone 13`, `samsung galaxy s21`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`
 - Each person lives in a unique style of house: `colonial`, `victorian`, `ranch`, `craftsman`

## Clues:
1. The fish enthusiast is the person who uses an iPhone 13.
2. Alice is in the fourth house.
3. Peter is in the first house.
4. The person residing in a Victorian house is the person whose favorite color is red.
5. The person who loves white is the person in a Craftsman-style house.
6. The person whose favorite color is green is in the fourth house.
7. The person living in a colonial-style house is Eric.
8. The bird keeper is Peter.
9. The person who uses a Google Pixel 6 is in the fourth house.
10. The person who loves white is the person who uses a Samsung Galaxy S21.
11. The person living in a colonial-style house is the cat lover.
12. Eric is not in the third house.
 What is the value of attribute PhoneModel for the person whose attribute Color is red? Please reason step by step, and put your final answer within \boxed{}

Answer: iphone 13


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`
 - They all have a unique favorite flower: `lilies`, `carnations`, `roses`, `daffodils`
 - Each person has a favorite color: `white`, `green`, `yellow`, `red`
 - Each person has a unique level of education: `bachelor`, `associate`, `master`, `high school`
 - Each person has a unique hobby: `painting`, `photography`, `gardening`, `cooking`
 - Everyone has a unique favorite cigar: `dunhill`, `blue master`, `prince`, `pall mall`

## Clues:
1. The person partial to Pall Mall is the person with an associate's degree.
2. The person who loves white is the Prince smoker.
3. The person with a high school diploma is not in the fourth house.
4. The person with a master's degree is not in the second house.
5. Arnold is not in the fourth house.
6. The person whose favorite color is red is not in the third house.
7. The person whose favorite color is red is somewhere to the left of Eric.
8. The person who loves cooking is somewhere to the right of Peter.
9. The person who loves a bouquet of daffodils is somewhere to the left of the person who paints as a hobby.
10. The person who loves white is the person who loves the boquet of lilies.
11. Alice is the photography enthusiast.
12. Arnold is somewhere to the right of the person partial to Pall Mall.
13. The person who loves white is in the second house.
14. The person whose favorite color is green is directly left of the person who enjoys gardening.
15. The Dunhill smoker is the person with a high school diploma.
16. The person who loves the rose bouquet is somewhere to the right of the person whose favorite color is green.
 What is the value of attribute Hobby for the person whose attribute Color is red? Please reason step by step, and put your final answer within \boxed{}

Answer: photography


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Carol`, `Peter`, `Eric`, `Alice`
 - People have unique favorite music genres: `classical`, `hip hop`, `jazz`, `pop`, `rock`, `country`
 - Everyone has something unique for lunch: `soup`, `stir fry`, `grilled cheese`, `spaghetti`, `stew`, `pizza`
 - Each person lives in a unique style of house: `craftsman`, `ranch`, `mediterranean`, `modern`, `colonial`, `victorian`
 - Each person has a unique birthday month: `feb`, `sept`, `april`, `may`, `jan`, `mar`
 - People have unique hair colors: `red`, `black`, `blonde`, `brown`, `gray`, `auburn`

## Clues:
1. The person whose birthday is in April is Bob.
2. The person residing in a Victorian house is the person who loves stir fry.
3. There are two houses between the person who loves jazz music and the person who has auburn hair.
4. The person who loves the stew is in the second house.
5. The person who loves stir fry is the person whose birthday is in February.
6. The person who loves classical music is in the fifth house.
7. The person who loves jazz music is Alice.
8. The person living in a colonial-style house is in the fifth house.
9. The person who has gray hair is the person who loves the stew.
10. The person whose birthday is in March is the person who has red hair.
11. The person who loves country music is directly left of the person whose birthday is in September.
12. The person whose birthday is in May is somewhere to the left of the person who loves country music.
13. The person in a ranch-style home is the person who loves the soup.
14. The person whose birthday is in January is the person who loves the stew.
15. The person who has blonde hair is the person in a Mediterranean-style villa.
16. Arnold is somewhere to the right of the person whose birthday is in April.
17. The person whose birthday is in February is the person who loves hip-hop music.
18. The person who has blonde hair is somewhere to the left of Eric.
19. The person residing in a Victorian house is Carol.
20. The person who has black hair is the person in a ranch-style home.
21. The person in a modern-style house is not in the second house.
22. The person who loves the spaghetti eater is somewhere to the left of the person who loves country music.
23. The person who is a pizza lover is Bob.
24. The person who loves pop music is somewhere to the left of the person whose birthday is in September.
 What is the value of attribute HouseStyle for the person whose attribute Name is Bob? 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: `Alice`, `Bob`, `Arnold`, `Carol`, `Eric`, `Peter`
 - People use unique phone models: `xiaomi mi 11`, `google pixel 6`, `samsung galaxy s21`, `oneplus 9`, `huawei p50`, `iphone 13`
 - People have unique favorite book genres: `romance`, `historical fiction`, `fantasy`, `mystery`, `science fiction`, `biography`
 - They all have a unique favorite flower: `tulips`, `roses`, `carnations`, `daffodils`, `iris`, `lilies`
 - The people keep unique animals: `dog`, `cat`, `bird`, `fish`, `rabbit`, `horse`
 - Each person has a unique level of education: `trade school`, `high school`, `master`, `associate`, `bachelor`, `doctorate`

## Clues:
1. The bird keeper is the person who uses a Google Pixel 6.
2. The person who loves a bouquet of daffodils is the person with an associate's degree.
3. The person who uses a OnePlus 9 is not in the fourth house.
4. There are two houses between the person with a master's degree and the rabbit owner.
5. The person who loves a carnations arrangement is Arnold.
6. The person who loves the boquet of lilies is not in the sixth house.
7. The person who loves the vase of tulips is Carol.
8. The person with a high school diploma is Bob.
9. The person who attended trade school is somewhere to the left of the person who keeps horses.
10. The person who loves mystery books is Eric.
11. The fish enthusiast is directly left of Carol.
12. Bob is the fish enthusiast.
13. Eric is the dog owner.
14. The person who loves a carnations arrangement is the person who uses a Google Pixel 6.
15. The person who loves fantasy books is in the second house.
16. The person who uses a Samsung Galaxy S21 is the person who loves the boquet of iris.
17. Alice is the person with an associate's degree.
18. The person who uses a Google Pixel 6 is not in the second house.
19. The dog owner is the person with a doctorate.
20. The fish enthusiast is the person who uses a Samsung Galaxy S21.
21. The person who uses an iPhone 13 is somewhere to the right of the person who loves fantasy books.
22. The person who loves mystery books is somewhere to the right of the person who loves the boquet of iris.
23. Alice is the person who uses an iPhone 13.
24. The person who attended trade school is not in the fourth house.
25. The person who loves biography books is directly left of Eric.
26. There is one house between the rabbit owner and the person who loves historical fiction books.
27. The person who loves the boquet of iris is somewhere to the right of the person who uses a Xiaomi Mi 11.
28. The person who loves science fiction books is the person who uses a Samsung Galaxy S21.
29. The person who loves the boquet of lilies is not in the second house.
 What is the value of attribute Name for the person whose attribute Flower is roses? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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`, `Carol`, `Peter`, `Bob`, `Alice`, `Arnold`
 - People have unique favorite sports: `swimming`, `basketball`, `volleyball`, `tennis`, `soccer`, `baseball`
 - Each person has a unique birthday month: `mar`, `feb`, `may`, `sept`, `april`, `jan`
 - Everyone has a favorite smoothie: `cherry`, `lime`, `desert`, `watermelon`, `blueberry`, `dragonfruit`
 - People have unique favorite music genres: `country`, `hip hop`, `jazz`, `rock`, `pop`, `classical`

## Clues:
1. The person whose birthday is in March is not in the fifth house.
2. Peter is not in the sixth house.
3. The person whose birthday is in September is not in the sixth house.
4. The person whose birthday is in March is somewhere to the left of the Desert smoothie lover.
5. The person who loves jazz music is Eric.
6. The person who loves swimming is somewhere to the right of the person who loves basketball.
7. The person who loves swimming is the person who loves jazz music.
8. There is one house between the person whose birthday is in September and the Desert smoothie lover.
9. The Watermelon smoothie lover is the person who loves jazz music.
10. There are two houses between the person whose birthday is in January and the Desert smoothie lover.
11. The person whose birthday is in February and the person who drinks Lime smoothies are next to each other.
12. There is one house between Bob and the person who loves jazz music.
13. The Watermelon smoothie lover is somewhere to the right of the person whose birthday is in April.
14. The person who loves rock music is Carol.
15. The person who loves volleyball is directly left of the Dragonfruit smoothie lover.
16. The person who loves rock music is directly left of the person whose birthday is in March.
17. The Desert smoothie lover is the person who loves soccer.
18. The person whose birthday is in January is the person who loves pop music.
19. There are two houses between the person who drinks Blueberry smoothies and the Desert smoothie lover.
20. The person whose birthday is in January is somewhere to the left of the person whose birthday is in March.
21. The Dragonfruit smoothie lover is directly left of the person who loves baseball.
22. Alice is the person who loves hip-hop music.
23. The person who loves classical music is in the sixth house.
 What is the value of attribute Birthday for the person whose attribute House is 5? Please reason step by step, and put your final answer within \boxed{}

Answer: feb


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`, `Alice`, `Eric`, `Arnold`, `Bob`
 - Everyone has a favorite smoothie: `lime`, `cherry`, `desert`, `dragonfruit`, `watermelon`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `pizza`, `stew`, `stir fry`
 - People have unique hair colors: `gray`, `black`, `red`, `brown`, `blonde`

## Clues:
1. The person who has blonde hair is the person who loves eating grilled cheese.
2. The Desert smoothie lover is somewhere to the left of the person who has red hair.
3. The person who loves the spaghetti eater is the person who has red hair.
4. Bob and the person who has blonde hair are next to each other.
5. The person who likes Cherry smoothies is in the fourth house.
6. Arnold is the person who has black hair.
7. The Desert smoothie lover is in the third house.
8. The Desert smoothie lover is the person who loves eating grilled cheese.
9. Eric is the person who has blonde hair.
10. The person who has gray hair is somewhere to the left of the person who has black hair.
11. The person who has black hair is the Dragonfruit smoothie lover.
12. Bob is the Watermelon smoothie lover.
13. The person who loves stir fry is the person who has black hair.
14. The person who is a pizza lover is the person who has brown hair.
15. Peter is the person who has gray hair.
 What is the value of attribute Food for the person whose attribute Smoothie is dragonfruit? Please reason step by step, and put your final answer within \boxed{}

Answer: stir fry


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`, `Peter`, `Eric`, `Bob`, `Alice`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `modern`, `craftsman`, `mediterranean`, `victorian`
 - Everyone has something unique for lunch: `pizza`, `stew`, `spaghetti`, `grilled cheese`, `stir fry`, `soup`
 - Each person prefers a unique type of vacation: `cultural`, `cruise`, `mountain`, `camping`, `city`, `beach`
 - People have unique heights: `average`, `very tall`, `very short`, `short`, `tall`, `super tall`
 - Everyone has a unique favorite cigar: `yellow monster`, `prince`, `dunhill`, `pall mall`, `blue master`, `blends`

## Clues:
1. Alice is in the fifth house.
2. The person who loves stir fry is the person living in a colonial-style house.
3. Alice is the person who loves the spaghetti eater.
4. Arnold is the person who loves the stew.
5. There is one house between the person who has an average height and Peter.
6. The person in a Craftsman-style house is not in the third house.
7. The person who has an average height is the person who loves stir fry.
8. The person who loves beach vacations is the person in a ranch-style home.
9. Eric is in the fourth house.
10. There is one house between the person living in a colonial-style house and the person who enjoys camping trips.
11. The person who enjoys mountain retreats is the person who smokes Yellow Monster.
12. The person who enjoys mountain retreats is the person who is very tall.
13. The person who enjoys mountain retreats and the Dunhill smoker are next to each other.
14. The person who loves the spaghetti eater is the person residing in a Victorian house.
15. The person who is tall is the person who loves beach vacations.
16. The person who is tall is somewhere to the left of the person residing in a Victorian house.
17. The person who loves stir fry is directly left of Bob.
18. The person in a modern-style house is somewhere to the left of Alice.
19. The person in a Craftsman-style house is somewhere to the left of the person who is short.
20. The person who loves stir fry is somewhere to the left of the Prince smoker.
21. There are two houses between the person who loves eating grilled cheese and the person who is super tall.
22. The person in a ranch-style home is the person who smokes Blue Master.
23. The person who smokes many unique blends is directly left of the person who smokes Blue Master.
24. The person who goes on cultural tours is the person who is a pizza lover.
25. The person who is a pizza lover is somewhere to the left of the person who likes going on cruises.
 What is the value of attribute Height for the person whose attribute HouseStyle is modern? Please reason step by step, and put your final answer within \boxed{}

Answer: short


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`
 - The mothers' names in different houses are unique: `Janelle`, `Aniya`, `Holly`
 - People have unique favorite sports: `basketball`, `tennis`, `soccer`
 - Each person has a unique favorite drink: `milk`, `water`, `tea`
 - They all have a unique favorite flower: `daffodils`, `lilies`, `carnations`

## Clues:
1. The one who only drinks water is in the first house.
2. The person who loves tennis is Peter.
3. The person who loves a bouquet of daffodils is The person whose mother's name is Aniya.
4. The person who loves soccer is Arnold.
5. Arnold is directly left of The person whose mother's name is Janelle.
6. The person who loves basketball is not in the third house.
7. The person whose mother's name is Janelle is the person who likes milk.
8. The person whose mother's name is Aniya is in the second house.
9. There is one house between the person who loves a carnations arrangement and the person who loves tennis.
 What is the value of attribute Mother for the person whose attribute FavoriteSport is tennis? Please reason step by step, and put your final answer within \boxed{}

Answer: Janelle


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`, `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`
 - They all have a unique favorite flower: `carnations`, `tulips`, `lilies`, `roses`, `daffodils`, `iris`
 - People have unique favorite book genres: `science fiction`, `fantasy`, `romance`, `mystery`, `biography`, `historical fiction`
 - People use unique phone models: `huawei p50`, `oneplus 9`, `iphone 13`, `xiaomi mi 11`, `samsung galaxy s21`, `google pixel 6`
 - People have unique hair colors: `black`, `gray`, `red`, `auburn`, `blonde`, `brown`
 - Each person has a favorite color: `yellow`, `green`, `red`, `white`, `blue`, `purple`

## Clues:
1. There is one house between the person who has gray hair and the person who uses an iPhone 13.
2. The person who loves the rose bouquet is the person who has blonde hair.
3. There is one house between the person who has gray hair and the person who loves yellow.
4. Carol is somewhere to the right of the person who loves a bouquet of daffodils.
5. The person who loves white is somewhere to the left of Peter.
6. The person who loves white is somewhere to the left of the person who loves science fiction books.
7. Arnold is directly left of the person who loves historical fiction books.
8. The person who uses a Huawei P50 is the person who loves a bouquet of daffodils.
9. The person who loves purple is somewhere to the left of the person who loves fantasy books.
10. Eric is the person who loves science fiction books.
11. The person who loves a bouquet of daffodils is the person who has auburn hair.
12. Alice is the person whose favorite color is green.
13. The person who loves fantasy books is in the second house.
14. The person who uses a Samsung Galaxy S21 is Carol.
15. The person who uses a Xiaomi Mi 11 is not in the fourth house.
16. Eric is the person who has auburn hair.
17. The person who loves the rose bouquet is somewhere to the left of the person who has red hair.
18. The person who loves historical fiction books is the person who loves the rose bouquet.
19. The person who uses a Xiaomi Mi 11 is somewhere to the right of the person who loves white.
20. The person who has black hair is in the fourth house.
21. The person whose favorite color is red is the person who loves mystery books.
22. The person who loves historical fiction books and the person who loves the vase of tulips are next to each other.
23. The person who has brown hair is the person who uses a OnePlus 9.
24. The person who loves romance books is somewhere to the left of the person who loves fantasy books.
25. The person who loves the boquet of lilies is in the second house.
26. The person who loves the boquet of iris is somewhere to the right of the person whose favorite color is green.
 What is the value of attribute PhoneModel for the person whose attribute Color is purple? Please reason step by step, and put your final answer within \boxed{}

Answer: oneplus 9


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - Each person lives in a unique style of house: `victorian`, `ranch`, `colonial`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `lilies`
 - People have unique favorite book genres: `romance`, `mystery`, `science fiction`
 - People own unique car models: `ford f150`, `toyota camry`, `tesla model 3`
 - Each person has a unique level of education: `associate`, `high school`, `bachelor`

## Clues:
1. The person who loves science fiction books is in the second house.
2. The person with a high school diploma is the person residing in a Victorian house.
3. The person who loves a bouquet of daffodils is in the second house.
4. The person with an associate's degree is the person in a ranch-style home.
5. The person who owns a Tesla Model 3 is not in the third house.
6. The person who loves the boquet of lilies is the person residing in a Victorian house.
7. Arnold is the person in a ranch-style home.
8. The person who owns a Tesla Model 3 and Eric are next to each other.
9. The person who owns a Ford F-150 is the person who loves romance books.
10. The person who owns a Tesla Model 3 is the person residing in a Victorian house.
 What is the value of attribute House for the person whose attribute CarModel is ford f150? 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: `Arnold`, `Eric`, `Alice`, `Peter`, `Bob`
 - Each person has a unique type of pet: `hamster`, `bird`, `cat`, `dog`, `fish`
 - Each person has a unique birthday month: `jan`, `feb`, `mar`, `april`, `sept`

## Clues:
1. The person who has a cat is not in the fifth house.
2. The person whose birthday is in April is somewhere to the left of Peter.
3. Arnold is the person whose birthday is in April.
4. There are two houses between Arnold and Bob.
5. The person whose birthday is in April is somewhere to the right of the person whose birthday is in February.
6. Arnold is the person with a pet hamster.
7. Eric is not in the fourth house.
8. Peter is the person whose birthday is in January.
9. The person whose birthday is in September is the person with an aquarium of fish.
10. Eric is the person whose birthday is in September.
11. The person who keeps a pet bird is somewhere to the left of the person whose birthday is in January.
12. Alice is somewhere to the left of the person who has a cat.
 What is the value of attribute House for the person whose attribute Pet is cat? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`, `Alice`
 - They all have a unique favorite flower: `lilies`, `roses`, `daffodils`, `carnations`
 - People have unique favorite sports: `swimming`, `tennis`, `soccer`, `basketball`
 - People have unique hair colors: `black`, `brown`, `blonde`, `red`
 - Each person has a unique birthday month: `feb`, `jan`, `april`, `sept`
 - Each person prefers a unique type of vacation: `cruise`, `city`, `mountain`, `beach`

## Clues:
1. The person who enjoys mountain retreats is in the fourth house.
2. The person whose birthday is in April and the person who loves a bouquet of daffodils are next to each other.
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 the person who loves soccer.
5. Arnold is the person who loves a bouquet of daffodils.
6. Alice is in the third house.
7. The person who loves basketball is the person whose birthday is in April.
8. The person whose birthday is in February is the person who has brown hair.
9. The person who has black hair is somewhere to the left of the person who likes going on cruises.
10. The person who prefers city breaks is Eric.
11. The person who has brown hair is Arnold.
12. The person who has blonde hair is the person who loves tennis.
13. The person who loves tennis is the person who likes going on cruises.
14. The person who loves beach vacations is not in the third house.
15. The person who loves the boquet of lilies is in the first house.
 What is the value of attribute Birthday for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: feb


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`, `Alice`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `roses`, `lilies`
 - People have unique heights: `very short`, `short`, `tall`, `average`
 - The mothers' names in different houses are unique: `Janelle`, `Kailyn`, `Holly`, `Aniya`
 - Each person has an occupation: `engineer`, `doctor`, `teacher`, `artist`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `soccer`

## Clues:
1. The person who loves swimming is the person who loves the rose bouquet.
2. The person who loves the rose bouquet is Eric.
3. Arnold is the person who is tall.
4. The person who loves a bouquet of daffodils is somewhere to the right of the person who is an engineer.
5. The person who loves soccer is the person who is short.
6. The person who is a teacher is in the first house.
7. The person whose mother's name is Janelle is the person who loves a carnations arrangement.
8. The person who loves basketball is the person who has an average height.
9. Arnold is not in the third house.
10. The person whose mother's name is Holly is somewhere to the right of the person who has an average height.
11. Peter is the person who is a doctor.
12. The person whose mother's name is Aniya is Alice.
13. Arnold is the person who loves the boquet of lilies.
 What is the value of attribute Flower for the person whose attribute Height is tall? Please reason step by step, and put your final answer within \boxed{}

Answer: lilies


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Eric`, `Arnold`, `Bob`, `Carol`
 - The people are of nationalities: `dane`, `brit`, `norwegian`, `swede`, `german`, `chinese`
 - People have unique favorite book genres: `historical fiction`, `mystery`, `romance`, `fantasy`, `science fiction`, `biography`
 - The mothers' names in different houses are unique: `Holly`, `Aniya`, `Penny`, `Kailyn`, `Sarah`, `Janelle`

## Clues:
1. The person whose mother's name is Janelle is the Swedish person.
2. Alice is the person who loves science fiction books.
3. The person who loves romance books is not in the fifth house.
4. Carol and Peter are next to each other.
5. The person whose mother's name is Sarah is the Norwegian.
6. There is one house between Arnold and the British person.
7. The German and the person who loves historical fiction books are next to each other.
8. There is one house between the German and Carol.
9. The Swedish person is the person who loves fantasy books.
10. Arnold is in the second house.
11. The person whose mother's name is Sarah is in the third house.
12. The person who loves historical fiction books is somewhere to the right of The person whose mother's name is Penny.
13. The Chinese is not in the sixth house.
14. There are two houses between Alice and the person who loves mystery books.
15. Carol is somewhere to the right of Alice.
16. Arnold is the Dane.
17. The person whose mother's name is Aniya is in the fourth house.
18. The person whose mother's name is Penny is somewhere to the left of the Swedish person.
19. The person whose mother's name is Holly and Eric are next to each other.
 What is the value of attribute Nationality for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: german


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`, `Bob`, `Alice`
 - Each person lives in a unique style of house: `modern`, `craftsman`, `ranch`, `victorian`, `colonial`
 - The mothers' names in different houses are unique: `Penny`, `Kailyn`, `Holly`, `Janelle`, `Aniya`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `huawei p50`, `iphone 13`, `samsung galaxy s21`
 - Each person has a unique favorite drink: `coffee`, `water`, `root beer`, `tea`, `milk`
 - The people keep unique animals: `fish`, `dog`, `horse`, `bird`, `cat`

## Clues:
1. The person who uses a Google Pixel 6 is not in the first house.
2. The one who only drinks water is Alice.
3. The person living in a colonial-style house is somewhere to the right of the person who uses a Huawei P50.
4. The person who keeps horses is the person who uses a OnePlus 9.
5. The person in a ranch-style home is The person whose mother's name is Kailyn.
6. The root beer lover is the cat lover.
7. The person living in a colonial-style house is not in the fourth house.
8. The bird keeper is in the fourth house.
9. The tea drinker is Bob.
10. The tea drinker is somewhere to the right of The person whose mother's name is Kailyn.
11. The root beer lover is somewhere to the left of The person whose mother's name is Kailyn.
12. The person who keeps horses is the person in a modern-style house.
13. The person who uses an iPhone 13 is the person who likes milk.
14. The dog owner is the person who likes milk.
15. The person who uses a Google Pixel 6 is the person in a Craftsman-style house.
16. Eric is not in the second house.
17. The tea drinker is in the fourth house.
18. The person who keeps horses is in the third house.
19. The person in a modern-style house is The person whose mother's name is Penny.
20. The root beer lover is Peter.
21. The person whose mother's name is Aniya is not in the fourth house.
22. The person whose mother's name is Janelle is the one who only drinks water.
 What is the value of attribute Animal for the person whose attribute Drink is water? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


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 favorite color: `red`, `white`, `yellow`, `green`
 - Each person has a unique birthday month: `feb`, `sept`, `jan`, `april`
 - The mothers' names in different houses are unique: `Aniya`, `Janelle`, `Kailyn`, `Holly`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`, `blue master`, `dunhill`
 - They all have a unique favorite flower: `lilies`, `daffodils`, `carnations`, `roses`

## Clues:
1. The person who loves yellow is directly left of The person whose mother's name is Kailyn.
2. The person whose birthday is in September and Peter are next to each other.
3. The person whose favorite color is red is not in the fourth house.
4. The person who smokes Blue Master and Arnold are next to each other.
5. The person whose mother's name is Holly is the Dunhill smoker.
6. The person whose birthday is in January is the person who loves a bouquet of daffodils.
7. There are two houses between the person who loves the boquet of lilies and the person whose birthday is in January.
8. The person whose mother's name is Janelle is the person who loves a carnations arrangement.
9. The person whose birthday is in September and the person whose favorite color is red are next to each other.
10. The person whose birthday is in February is the person whose favorite color is green.
11. The person whose favorite color is green is directly left of The person whose mother's name is Janelle.
12. There is one house between the person whose birthday is in February and the person partial to Pall Mall.
13. The person who smokes Blue Master is The person whose mother's name is Aniya.
14. Eric is the person whose birthday is in February.
 What is the value of attribute Mother for the person whose attribute Color is red? Please reason step by step, and put your final answer within \boxed{}

Answer: Kailyn


11. (6 points) The average of 6 different non-zero natural numbers is 12. If one of the two-digit numbers $\overline{\mathrm{ab}}$ is replaced by $\overline{\mathrm{ba}}$ (where $a$ and $b$ are non-zero digits), then the average of these 6 numbers becomes 15. Therefore, the number of $\overline{\mathrm{ab}}$ that satisfy the condition is __.

Answer: 5


Let the sequence $(a_{n})$ be deﬁned by $a_{1} = t$ and $a_{n+1} = 4a_{n}(1 - a_{n})$ for $n \geq 1$. How many possible values of t are there, if $a_{1998} = 0$?

Answer: 2^{1996} + 1


3. A regular quadrilateral pyramid is given. The side of the base is 12, the length of the lateral edge is 10. A sphere \( Q_{1} \) is inscribed in the pyramid. A sphere \( Q_{2} \) touches \( Q_{1} \) and all the lateral faces of the pyramid. Find the radius of the sphere \( Q_{2} \).

Answer: \dfrac{6\sqrt{7}}{49}


2. $A B C D E$ - a pentagon inscribed in circle $S$. Lines $D E$ and $A B$ are parallel, $B C=C D$, $A D=D E=20, B C: B D=4: 5$. Find the radius of circle $S$.

Answer: 16


10. For the string of three letters $\hat{\mathrm{i}}$ "aaa" and "bbb", they are transmitted through a circuit, with each string being sent one letter at a time. Due to issues with the line, each of the 6 letters has a $\frac{1}{3}$ chance of being transmitted incorrectly (an $a$ is received as a $b$, or a $b$ is received as an $a$), and the correctness of each letter's reception is independent of the others. Let $S_{\mathrm{a}}$ be the message received when "aaa" is transmitted, and $S_{\mathrm{b}}$ be the message received when "bbb" is transmitted. Let $P$ be the probability that $S_{\mathrm{a}}$ is lexicographically before $S_{\mathrm{b}}$. When $P$ is written as a reduced fraction, what is the numerator?

Answer: 532


1. The midpoints of adjacent sides of a rectangle with a perimeter of 32 were connected by segments. The same operation was performed on the resulting quadrilateral: the midpoints of adjacent sides were connected by segments (see figure). How many times in total does one need to perform such an operation so that the perimeter of the resulting quadrilateral first becomes less than 1?

Answer: 11


3. Given a cyclic quadrilateral $A B C D$ with perpendicular diagonals. On the circle circumscribed around it, a point $E$ is marked, diametrically opposite to $D$, and the segments $A B$ and $D E$ do not intersect. Find the ratio of the areas of triangle $B C D$ and quadrilateral $A B E D$.

Answer: 1


[Example 5.6.4] On a $6 \times 6$ chessboard, $n$ dominoes of size $1 \times 2$ are placed, each domino covering exactly two squares. If no matter how these $n$ dominoes are placed, it is always possible to place one more domino, find the maximum value of $n$.

Answer: 11


Let $a_n$ be a sequence defined by some $a_0$ and the recursion $a_{n+1} = a_n + 2 \cdot  3^n$ for $n \ge 0$.
Determine all rational values of $a_0$ such that $a^j_k / a^k_j$ is an integer for all integers $j$ and $k$ with $0 < j < k$.

Answer: 1


Example 2. If $\lg ^{2} x \lg 10 x<0$, find the value of $\frac{1}{\lg 10 x} \sqrt{\lg ^{2} x+\lg 10 x^{2}}$.

Answer: -1


## Task B-1.3.

Tangents drawn from point $B$ to a circle of radius $r$ touch the circle at points $A$ and $C$. Determine the area of the figure bounded by the tangents $B C, B A$ and the smaller circular arc $\widetilde{A C}$, if the measure of angle $\varangle A B C$ is $60^{\circ}$.

Answer: r^2 \left( \sqrt{3} - \dfrac{\pi}{3} \right)


Answer: \dfrac{(2x - 5)^{24}}{48}


As shown in the figure, the hyperbola function \( y = \frac{k}{x} \) where \( k > 0 \), passes through the midpoint \( D \) of the hypotenuse \( OB \) of the right triangle \( OAB \) and intersects the leg \( AB \) at point \( C \). If the area of \( \triangle OBC \) is 3, then find the value of \( k \).

Answer: 2


4. Let $\triangle A B K$ be a triangle with acute angles at vertices $A$ and $B$. Points $D$ and $N$ are chosen on side $\overline{K A}$, and points $C$ and $M$ on side $\overline{K B}$ such that the lines $C D, M N$, and $A B$ are parallel. The lines $A B$ and $C D$ are $4 \mathrm{~cm}$ away from the line $N M$. The midpoint of segment $\overline{N M}$ is $4 \mathrm{~cm}$ away from line $A K$ and $3 \mathrm{~cm}$ away from line $B K$. If the area of quadrilateral $A B C D$ is $80 \mathrm{~cm}^{2}$, determine the area of triangle $\triangle A B K$.

Answer: \dfrac{242}{3}


11.1. In each of the four volleyball teams, there are six players, including a captain and a setter, and these are different people. In how many ways can a team of six players be formed from these four teams, such that there is at least one player from each team and there must be a pair of captain and setter from at least one team?

Answer: 9720


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Eric`
 - Each mother is accompanied by their child: `Samantha`, `Fred`, `Meredith`, `Bella`
 - Each person lives in a unique style of house: `craftsman`, `ranch`, `victorian`, `colonial`

## Clues:
1. The person residing in a Victorian house is Arnold.
2. The person in a ranch-style home is the person's child is named Samantha.
3. The person's child is named Meredith is directly left of Peter.
4. Arnold and the person's child is named Meredith are next to each other.
5. Alice is directly left of the person's child is named Bella.
6. The person's child is named Samantha is not in the first house.
7. The person's child is named Bella is in the second house.
8. The person's child is named Meredith is the person in a Craftsman-style house.
 What is the value of attribute Children for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: Fred


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

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

Answer: Samantha


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`
 - Each person lives in a unique style of house: `craftsman`, `ranch`, `victorian`, `colonial`
 - Each person has a unique favorite drink: `water`, `tea`, `milk`, `coffee`

## Clues:
1. Eric is the one who only drinks water.
2. The person who likes milk is the person residing in a Victorian house.
3. The person in a ranch-style home is somewhere to the left of the person in a Craftsman-style house.
4. Eric is in the third house.
5. Arnold is somewhere to the right of Alice.
6. The person in a ranch-style home is Arnold.
7. Eric and the person residing in a Victorian house are next to each other.
8. The tea drinker is not in the first house.
 What is the value of attribute House for the person whose attribute HouseStyle is craftsman? 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: `Peter`, `Eric`, `Alice`, `Bob`, `Arnold`
 - People have unique favorite book genres: `mystery`, `fantasy`, `biography`, `science fiction`, `romance`
 - Each person prefers a unique type of vacation: `mountain`, `camping`, `city`, `beach`, `cruise`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Kailyn`, `Penny`, `Aniya`

## Clues:
1. Bob is the person who loves fantasy books.
2. The person whose mother's name is Janelle and Eric are next to each other.
3. The person who enjoys camping trips is the person who loves biography books.
4. Peter is The person whose mother's name is Kailyn.
5. Peter and The person whose mother's name is Janelle are next to each other.
6. The person who loves beach vacations is Bob.
7. Peter is somewhere to the right of the person who loves romance books.
8. The person whose mother's name is Penny is the person who loves biography books.
9. Bob is directly left of the person who prefers city breaks.
10. The person who loves science fiction books is in the fourth house.
11. Arnold and Bob are next to each other.
12. The person whose mother's name is Aniya is the person who enjoys mountain retreats.
 What is the value of attribute Mother for the person whose attribute Vacation is cruise? Please reason step by step, and put your final answer within \boxed{}

Answer: Janelle


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`
 - Each person has a unique birthday month: `sept`, `april`, `jan`
 - People have unique favorite book genres: `mystery`, `science fiction`, `romance`
 - Everyone has a unique favorite cigar: `prince`, `blue master`, `pall mall`
 - The people keep unique animals: `bird`, `cat`, `horse`
 - People own unique car models: `toyota camry`, `ford f150`, `tesla model 3`

## Clues:
1. Arnold is the Prince smoker.
2. Eric is somewhere to the right of the person who loves romance books.
3. The person who keeps horses is the Prince smoker.
4. The person who smokes Blue Master is the person who loves mystery books.
5. The person who owns a Toyota Camry is the person partial to Pall Mall.
6. The person who owns a Tesla Model 3 is the person whose birthday is in September.
7. Peter is the person partial to Pall Mall.
8. The Prince smoker is somewhere to the right of the person who loves mystery books.
9. The person who keeps horses is the person whose birthday is in January.
10. The person who owns a Toyota Camry is the bird keeper.
 What is the value of attribute Cigar for the person whose attribute BookGenre is science fiction? Please reason step by step, and put your final answer within \boxed{}

Answer: prince


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`
 - Everyone has a unique favorite cigar: `blue master`, `prince`, `pall mall`
 - Each person has a unique hobby: `photography`, `gardening`, `cooking`
 - Each person has a unique level of education: `high school`, `associate`, `bachelor`
 - Each person has a unique favorite drink: `tea`, `milk`, `water`

## Clues:
1. The person partial to Pall Mall is Peter.
2. The person who likes milk is directly left of the person with a high school diploma.
3. Eric is the tea drinker.
4. Arnold and the Prince smoker are next to each other.
5. The person who enjoys gardening is somewhere to the left of the Prince smoker.
6. The person who likes milk is the person with an associate's degree.
7. The person with a bachelor's degree is directly left of the photography enthusiast.
 What is the value of attribute Hobby for the person whose attribute Education is bachelor? Please reason step by step, and put your final answer within \boxed{}

Answer: gardening


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: `Arnold`, `Peter`, `Eric`, `Alice`
 - Each person lives in a unique style of house: `craftsman`, `colonial`, `victorian`, `ranch`
 - People have unique hair colors: `red`, `blonde`, `black`, `brown`
 - Each mother is accompanied by their child: `Bella`, `Fred`, `Meredith`, `Samantha`
 - People have unique favorite book genres: `mystery`, `fantasy`, `romance`, `science fiction`

## Clues:
1. The person in a Craftsman-style house is in the third house.
2. Alice is the person who loves romance books.
3. The person who has brown hair is in the fourth house.
4. The person's child is named Samantha is in the fourth house.
5. The person in a ranch-style home is somewhere to the right of the person who has red hair.
6. Peter is the person's child is named Bella.
7. Arnold is the person who has red hair.
8. Alice is the person living in a colonial-style house.
9. The person who has black hair is in the second house.
10. The person who loves fantasy books is Peter.
11. Arnold is the person's child is named Meredith.
12. The person who has black hair is Eric.
13. The person who loves science fiction books is Arnold.
 What is the value of attribute House for the person whose attribute HairColor is blonde? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Peter`
 - Each person has a favorite color: `red`, `green`, `white`, `yellow`
 - The people keep unique animals: `bird`, `fish`, `horse`, `cat`
 - Each person has a unique level of education: `master`, `associate`, `high school`, `bachelor`

## Clues:
1. The person whose favorite color is green is the cat lover.
2. There are two houses between Alice and the person whose favorite color is red.
3. The cat lover is the person with a bachelor's degree.
4. Arnold is the person with a master's degree.
5. The person who loves yellow is the person with a high school diploma.
6. The person with an associate's degree is in the third house.
7. Alice is the person with a high school diploma.
8. The fish enthusiast is somewhere to the left of the person whose favorite color is red.
9. Peter is the bird keeper.
 What is the value of attribute Animal for the person whose attribute Education is high school? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


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`, `Arnold`, `Bob`, `Carol`, `Alice`, `Eric`
 - People have unique heights: `very tall`, `very short`, `super tall`, `average`, `tall`, `short`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `blueberry`, `desert`, `dragonfruit`, `lime`
 - Everyone has a unique favorite cigar: `prince`, `blue master`, `yellow monster`, `dunhill`, `blends`, `pall mall`
 - Each person prefers a unique type of vacation: `camping`, `city`, `beach`, `cultural`, `mountain`, `cruise`

## Clues:
1. The person who is very short is directly left of the person who drinks Lime smoothies.
2. The person who smokes Yellow Monster is directly left of the person who drinks Lime smoothies.
3. The person who is short and the person who goes on cultural tours are next to each other.
4. The person who smokes many unique blends is the person who loves beach vacations.
5. There are two houses between the person who smokes Blue Master and the person who drinks Blueberry smoothies.
6. The Prince smoker is not in the fourth house.
7. The person who is tall is not in the first house.
8. The person who smokes Yellow Monster is the person who enjoys mountain retreats.
9. The person who is very tall is Alice.
10. The person who likes Cherry smoothies is the person who smokes Blue Master.
11. Bob is in the first house.
12. The person who smokes many unique blends is Alice.
13. The person who goes on cultural tours is the person partial to Pall Mall.
14. Eric is the person who drinks Blueberry smoothies.
15. The person who prefers city breaks is not in the sixth house.
16. The Prince smoker is the person who prefers city breaks.
17. The person who enjoys camping trips is somewhere to the left of Carol.
18. The person who has an average height is the person who prefers city breaks.
19. The Desert smoothie lover is in the fifth house.
20. The Watermelon smoothie lover is Arnold.
21. The person who is very tall is the person who drinks Lime smoothies.
22. Carol is the Dunhill smoker.
 What is the value of attribute Name for the person whose attribute Smoothie is desert? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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`
 - Everyone has something unique for lunch: `stir fry`, `grilled cheese`, `spaghetti`, `pizza`, `soup`, `stew`
 - People use unique phone models: `xiaomi mi 11`, `iphone 13`, `oneplus 9`, `google pixel 6`, `samsung galaxy s21`, `huawei p50`
 - People have unique favorite music genres: `pop`, `country`, `classical`, `jazz`, `rock`, `hip hop`
 - Each person has a unique favorite drink: `water`, `boba tea`, `milk`, `root beer`, `tea`, `coffee`
 - Each person has an occupation: `nurse`, `lawyer`, `artist`, `doctor`, `teacher`, `engineer`

## Clues:
1. The tea drinker is somewhere to the right of the person who is a teacher.
2. Eric is the person who uses a Google Pixel 6.
3. The person who loves the spaghetti eater and the person who loves hip-hop music are next to each other.
4. The boba tea drinker is somewhere to the left of the person who uses a Google Pixel 6.
5. The person who uses a Huawei P50 is somewhere to the left of the person who uses a OnePlus 9.
6. The person who is a pizza lover is the person who loves rock music.
7. The person who is a pizza lover is somewhere to the right of the person who loves the soup.
8. The person who likes milk is somewhere to the left of the person who loves hip-hop music.
9. The person who uses a OnePlus 9 is Arnold.
10. The person who is a lawyer is the person who loves stir fry.
11. The person who is an artist is Alice.
12. Bob is the person who loves classical music.
13. The person who loves rock music is directly left of the person who loves classical music.
14. The person who is a doctor is the person who uses a Huawei P50.
15. Bob is the root beer lover.
16. There are two houses between the person who is an engineer and the person who uses a OnePlus 9.
17. Eric is the person who loves jazz music.
18. The person who uses a Xiaomi Mi 11 is the person who is an artist.
19. Carol is directly left of the person who uses a Huawei P50.
20. The person who is a lawyer is the coffee drinker.
21. The person who loves eating grilled cheese is the person who loves pop music.
22. The person who loves hip-hop music is the person who uses a Samsung Galaxy S21.
23. The coffee drinker is somewhere to the left of the person who likes milk.
 What is the value of attribute House for the person whose attribute Drink is root beer? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Bob`, `Eric`, `Arnold`
 - People have unique heights: `very short`, `short`, `tall`, `average`, `very tall`
 - The mothers' names in different houses are unique: `Janelle`, `Kailyn`, `Penny`, `Holly`, `Aniya`
 - People have unique hair colors: `blonde`, `black`, `gray`, `red`, `brown`

## Clues:
1. The person who is tall is The person whose mother's name is Holly.
2. There are two houses between the person who has an average height and the person who is short.
3. The person who has gray hair is directly left of The person whose mother's name is Janelle.
4. The person who has black hair is not in the fourth house.
5. Eric is the person who has black hair.
6. The person who is very short is The person whose mother's name is Penny.
7. Eric and the person who has gray hair are next to each other.
8. Bob is in the fifth house.
9. The person who has red hair is Peter.
10. The person whose mother's name is Kailyn is directly left of the person who is short.
11. Arnold is the person who has brown hair.
12. The person who has brown hair is somewhere to the left of The person whose mother's name is Janelle.
13. The person whose mother's name is Aniya and the person who is very short are next to each other.
14. The person whose mother's name is Kailyn is in the third house.
 What is the value of attribute Height for the person whose attribute Mother is Janelle? Please reason step by step, and put your final answer within \boxed{}

Answer: short


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Carol`, `Eric`, `Alice`, `Bob`, `Arnold`
 - People use unique phone models: `huawei p50`, `google pixel 6`, `xiaomi mi 11`, `iphone 13`, `samsung galaxy s21`, `oneplus 9`
 - Everyone has a unique favorite cigar: `dunhill`, `pall mall`, `blends`, `blue master`, `prince`, `yellow monster`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `roses`, `tulips`, `lilies`, `iris`
 - Each person has a favorite color: `yellow`, `red`, `green`, `blue`, `white`, `purple`
 - People have unique favorite sports: `soccer`, `tennis`, `basketball`, `volleyball`, `swimming`, `baseball`

## Clues:
1. The person who uses a OnePlus 9 is in the second house.
2. The person who uses a Xiaomi Mi 11 is somewhere to the left of the person who uses a Huawei P50.
3. Carol is the person who loves a carnations arrangement.
4. The person who loves purple is directly left of the person partial to Pall Mall.
5. The person whose favorite color is green is the person who smokes Blue Master.
6. The person who loves yellow and the person who loves blue are next to each other.
7. Eric is somewhere to the right of the person who uses a Samsung Galaxy S21.
8. There are two houses between Carol and the person who loves a bouquet of daffodils.
9. The Prince smoker is the person who loves basketball.
10. The Dunhill smoker is the person who loves volleyball.
11. The person who loves swimming is the person who uses a Google Pixel 6.
12. The person who uses a Huawei P50 is directly left of the person who loves white.
13. The person who uses a OnePlus 9 and the person who loves the rose bouquet are next to each other.
14. The person who loves the boquet of iris is somewhere to the left of Eric.
15. The Dunhill smoker is Peter.
16. The person who loves blue is Peter.
17. The person who loves the vase of tulips is Bob.
18. Alice is in the first house.
19. The person who loves baseball is directly left of the person who smokes Blue Master.
20. The person who uses a Google Pixel 6 is somewhere to the right of the person who smokes many unique blends.
21. The person who loves soccer is Carol.
22. The person who loves a carnations arrangement is directly left of the person who smokes many unique blends.
23. Eric is the person who smokes many unique blends.
24. The person who loves volleyball is the person who uses an iPhone 13.
 What is the value of attribute House for the person whose attribute Color is blue? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


Given an acute-angled triangle \( \triangle ABC \), the angle bisector of \( \angle A \) intersects \( BC \) at \( L \) and the circumcircle at \( N \). From point \( L \), perpendicular lines are drawn to \( AB \) and \( AC \), meeting them at \( K \) and \( M \) respectively. Prove that the area of \( \triangle ABC \) is equal to the area of the quadrilateral \( AKNM \).

Answer: \text{The area of } \triangle ABC \text{ is equal to the area of quadrilateral } AKNM


9. A gardener is preparing to plant a row of 20 trees, with two types of trees available: maple trees or sycamore trees. The number of trees between any two maple trees (not including these two maple trees) cannot be equal to 3. How many maple trees can there be at most among the 20 trees? $\qquad$

Answer: 12


The thief Rumcajs is teaching Cipísek to write numbers. They started writing from one and wrote consecutive natural numbers. Cipísek didn't find it very entertaining and pleaded for them to stop. Rumcajs eventually gave in and promised that he would stop writing as soon as Cipísek had a total of 35 zeros in the written numbers. Which number will Cipísek write as the last one? (M. Krejčová)

Answer: 204


6. Eight people participate in a round-robin tournament (i.e., each pair competes in one match). Each match awards 2 points to the winner, 0 points to the loser, and 1 point to each in the case of a draw. After all matches are completed, it is found that:
(1) The eight participants have different scores;
(2) The second-place score is the sum of the scores of the fifth, sixth, seventh, and eighth-place participants;
(3) The first-place participant did not defeat the fourth-place participant.
Based on the above information, the third-place participant scored $\qquad$ points.

Answer: 10


In a square table consisting of 2015 rows and columns, positive numbers are arranged. The product of the numbers in each row and in each column is equal to 1, and the product of the numbers in any 1008 x 1008 square is equal to 2. What is the number in the center of the table?

Answer: 16


B4. On each of the 10,000 fields of a $100 \times 100$ chessboard, there is a number. On the top row, from left to right, the numbers are 0 to 99. In the left column, from top to bottom, the numbers are 0 to 99. The sum of four numbers in a $2 \times 2$ block is always 20. What number is in the bottom right corner of the board?

Answer: -178


6. In the aquarium, there are 1000 neon tetras, clownfish, and guppies. Among them, the neon tetras are the most numerous, and the guppies are the least, with their numbers in the ratio of $5: 3$. Therefore, the maximum number of clownfish is $\qquad$.

Answer: 384


28. Teacher Zhou takes students to Los Angeles, USA, to participate in a math competition. The first match is held at 6 PM on May 31, Los Angeles time, which is 9 AM on June 1, Beijing time. Teacher Zhou's flight departs at 12:00 PM on May 28 (Beijing time) and arrives at 9:00 AM on May 28 (Los Angeles time). The average speed of the flight is 840 kilometers/hour, and the distance between Beijing and Los Angeles is $\qquad$ kilometers.

Answer: 10080


Four, on the ellipse $\frac{x^{2}}{5^{2}}+\frac{y^{2}}{4^{2}}=1$ there are 16 points, sequentially $P_{1}, P_{2}, \cdots, P_{16}, F$ is the left focus, and the angles between each adjacent pair of points and $F$ are equal $\left(\angle P_{1} F P_{2}=\angle P_{2} F P_{3}=\cdots=\angle P_{16} F P_{1}\right)$. Let the distance from $P_{i}$ to the left directrix be $d_{i}(i=1,2, \cdots, 16)$. Find $\sum_{i=1}^{16} \frac{1}{d_{i}}$.

Answer: 3


7.59 Let
$S=\left\{A=\left(a_{1}, a_{2}, \cdots, a_{8}\right) \mid a_{i}=0\right.$ or $\left.1, i=1,2, \cdots, 8\right\}$.
For any two elements $A$ and $B$ in $S$, define
$$
d(A, B)=\sum_{i=1}^{8}\left|a_{i}-b_{i}\right|,
$$

and call it the distance between $A$ and $B$. How many elements can be selected from $S$ at most, such that the distance between any two of them is not less than 5?

Answer: 4


11. Using 120 unit cubes, form a rectangular prism, and paint its entire surface. The maximum number of unit cubes that have only one face painted is $\qquad$ pieces

Constructing the above text in English, please retain the original text's line breaks and format, and output the translation result directly.

Answer: 64


$7 \cdot 117$ Let $S=\{1,2,3,4\} ; n$ terms of the sequence $a_{1}, a_{2}, \cdots a_{n}$ have the following property: for any non-empty subset $B$ of $S$ (the number of elements in set $B$ is denoted as $|B|$ ), there are adjacent $|B|$ terms in the sequence that exactly form the set $B$. Find the minimum value of the number of terms $n$.

Answer: 8


4. Given an equilateral triangle $ABC$ with side $12 \, \text{cm}$. Two circles with the same radius equal to $\sqrt{3} \, \text{cm}$ touch the side $BC$ from the outside of the triangle, touch each other, and one of the circles touches the line $AB$. Find the area of the triangle whose vertices are the centers of the circles and the midpoint of side $AB$.

Answer: 12


68. Find two numbers, the ratio of which is 3, and the ratio of the sum of their squares to their sum is 5.

Answer: 2


Given two positive integers $m,n$, we say that a function $f : [0,m] \to \mathbb{R}$ is $(m,n)$-[i]slippery[/i] if it has the following properties:

i) $f$ is continuous;
ii) $f(0) = 0$, $f(m) = n$;
iii) If $t_1, t_2\in [0,m]$ with $t_1 < t_2$ are such that $t_2-t_1\in \mathbb{Z}$ and $f(t_2)-f(t_1)\in\mathbb{Z}$, then $t_2-t_1 \in \{0,m\}$.

Find all the possible values for $m, n$ such that there is a function $f$ that is $(m,n)$-slippery.

Answer: \gcd(m, n) = 1


Suppose $ P(x) \equal{} a_nx^n\plus{}\cdots\plus{}a_1x\plus{}a_0$ be a real polynomial of degree $ n > 2$ with $ a_n \equal{} 1$, $ a_{n\minus{}1} \equal{} \minus{}n$, $ a_{n\minus{}2} \equal{}\frac{n^2 \minus{} n}{2}$ such that all the roots of $ P$ are real. Determine the coefficients $ a_i$.

Answer: (-1)^{n-i} \binom{n}{i}


Let  $x_1,x_2, ..., x_n$  be real numbers satisfying  $\sum_{k=1}^{n-1} min(x_k; x_{k+1}) = min(x_1; x_n)$ .
Prove that  $\sum_{k=2}^{n-1} x_k \ge 0$ .

Answer: \sum_{k=2}^{n-1} x_k \ge 0


Consider arrangements of the $9$ numbers $1, 2, 3, \dots, 9$ in a $3 \times 3$ array. For each such arrangement, let $a_1$ , $a_2$ , and $a_3$ be the medians of the numbers in rows $1$ , $2$ , and $3$ respectively, and let $m$ be the median of $\{a_1, a_2, a_3\}$ . Let $Q$ be the number of arrangements for which $m = 5$ . Find the remainder when $Q$ is divided by $1000$ .
Please give the answer directly without any intermediate steps.

Answer: 360


6. A space probe, moving in a straight line at a constant speed, flies past Mars and measures the distance to this planet every day at exactly 12:00. It is known that on February 1st, the distance was 5 million km, on February 10th - 2 million km, and on February 13th - 3 million km.

Determine when the probe will pass at the minimum distance from Mars.

In this problem, Mars can be considered a point.

Answer: February\ 9


3. In the right trapezoid $A B C D$, it is known that $\angle A=$ $\angle D=90^{\circ}, D E \perp A C$ at point $E, \angle A C D=$ $\angle E B C=30^{\circ}, A D=\sqrt{3}$. Then $B C=$ $\qquad$

Answer: 3


8. As shown in Figure 5, the area of rectangle $A B C D$ is $35 \mathrm{~cm}^{2}$, the area of shaded $\triangle A B E$ is $5 \mathrm{~cm}^{2}$, and the area of shaded $\triangle A D F$ is $7 \mathrm{~cm}^{2}$. What is the area of $\triangle A E F$ in square centimeters?

Answer: \dfrac{31}{2}


Given a triangle with sides $A B=2, B C=3, A C=4$. A circle is inscribed in it, and the point $M$ where the circle touches side $B C$ is connected to point $A$. Circles are inscribed in triangles $A M B$ and $A M C$. Find the distance between the points where these circles touch the line $A M$.

Answer: 0


2. As shown in Figure 2, in quadrilateral $ABCD$, $AB=10$, $BC=17$, $CD=13$, $DA$ $=20$, $AC=21$. Then $BD=$

Answer: 10\sqrt{5}


Brian starts at the point $\left(1,0\right)$ in the plane. Every second, he performs one of two moves: he can move from $\left(a,b\right)$ to $\left(a-b,a+b\right)$ or from $\left(a,b\right)$ to $\left(2a-b,a+2b\right)$. How many different paths can he take to end up at $\left(28,-96\right)$?

[i]2018 CCA Math Bonanza Individual Round #14[/i]

Answer: 70


10.

The answer to the task should be some integer or a number written as a finite decimal. If the answer contains a fractional number, use a comma when writing it. Enter all symbols (comma, digits) without spaces.

Find the range of the function given below. In your answer, specify the value of the difference between the largest and smallest values of the function.

$y=x \cdot|3-x|-(x-3) \cdot|x|$

Answer: 4,5


4. Equation
$$
|| \cdots|||x|-1|-2| \cdots|-2011|=2011
$$

has $\qquad$ solutions.

Answer: 4


3. As shown in the figure, $A, B, C$ are the midpoints of three sides of a regular hexagon, and $O$ is the center of the hexagon. The area of the regular hexagon is 120, then the area of the shaded part is $\qquad$ .

Answer: 30


In the drawing below, $ABC$ is a triangle with side lengths $AB=4 \, \text{cm}, AC=$ $6 \, \text{cm}$. Additionally, $\angle DAH=\angle HAB, \angle AHB=90^{\circ}$, and $M$ is the midpoint of $BC$. Find the length of the segment $MH$.

![](https://cdn.mathpix.com/cropped/2024_05_01_12f29789e10e6749ebb9g-25.jpg?height=549&width=465&top_left_y=590&top_left_x=867)

Answer: 1


In the sum

$$
\text { П,Я + Т,Ь + Д,Р + О,Б + Е,Й }
$$

all digits are encoded by letters (different letters represent different digits). It turns out that all five addends are not integers, but the sum itself is an integer. What exactly is the sum? For each possible answer, provide one example with these five addends. Explain why other sums are not possible.

Answer: 18


# 7. Clone 1

In the expression OL $*$ IM $* P *$ IA * DA, it is required to replace the asterisks with two plus signs and two minus signs, and to replace the letters with digits according to the rules of a cryptarithm (identical letters with identical digits, and different letters with different digits). What is the maximum value of the expression that can be obtained in this way? The first digit in a two-digit number must be different from zero.

#

Answer: 263


Given $\triangle ABC$ with $\angle A = 15^{\circ}$, let $M$ be the midpoint of $BC$ and let $E$ and $F$ be points on ray $BA$ and $CA$ respectively such that $BE = BM = CF$. Let $R_1$ be the radius of $(MEF)$ and $R_2$ be the radius of $(AEF)$. If $\frac{R_1^2}{R_2^2} = a - \sqrt{b + \sqrt{c}}$ where $a, b, c$ are integers, find $a^{b^{c}}$.

Answer: 256