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`, `Bob`, `Arnold`
 - Each person has a favorite color: `white`, `red`, `green`, `blue`, `yellow`
 - Each person lives in a unique style of house: `craftsman`, `modern`, `colonial`, `victorian`, `ranch`
 - People have unique favorite book genres: `fantasy`, `science fiction`, `biography`, `romance`, `mystery`

## Clues:
1. Alice is not in the fourth house.
2. Arnold is not in the fourth house.
3. The person who loves blue is Bob.
4. The person who loves white is in the fifth house.
5. The person who loves romance books is not in the fifth house.
6. The person who loves science fiction books is not in the fifth house.
7. The person whose favorite color is green is not in the fourth house.
8. The person who loves romance books is somewhere to the right of the person in a ranch-style home.
9. The person residing in a Victorian house is in the first house.
10. The person whose favorite color is red is in the second house.
11. The person in a modern-style house is somewhere to the right of Arnold.
12. Eric is the person living in a colonial-style house.
13. The person who loves science fiction books is somewhere to the right of Bob.
14. The person who loves fantasy books is the person residing in a Victorian house.
15. The person in a Craftsman-style house is not in the fifth house.
16. The person who loves biography books is the person whose favorite color is red.
17. Eric is in the third house.
18. The person who loves romance books is somewhere to the right of Eric.
 What is the value of attribute House for the person whose attribute Color is white? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


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`, `Peter`, `Alice`, `Eric`, `Arnold`
 - They all have a unique favorite flower: `daffodils`, `iris`, `lilies`, `roses`, `tulips`, `carnations`
 - People have unique hair colors: `gray`, `red`, `blonde`, `auburn`, `brown`, `black`
 - Each person has a favorite color: `white`, `blue`, `green`, `purple`, `red`, `yellow`
 - People own unique car models: `tesla model 3`, `ford f150`, `honda civic`, `toyota camry`, `chevrolet silverado`, `bmw 3 series`
 - The mothers' names in different houses are unique: `Janelle`, `Holly`, `Kailyn`, `Sarah`, `Penny`, `Aniya`

## Clues:
1. There is one house between Arnold and the person who loves blue.
2. The person whose mother's name is Janelle is directly left of the person who has auburn hair.
3. The person who owns a Chevrolet Silverado and the person who owns a BMW 3 Series are next to each other.
4. Carol is the person who loves a carnations arrangement.
5. The person who loves the boquet of iris is the person who has gray hair.
6. The person who loves the rose bouquet is The person whose mother's name is Kailyn.
7. Bob is the person who loves the boquet of lilies.
8. The person who loves a bouquet of daffodils is somewhere to the left of the person who owns a Chevrolet Silverado.
9. Bob is in the third house.
10. The person who owns a Honda Civic is in the sixth house.
11. The person who loves the vase of tulips is somewhere to the left of The person whose mother's name is Penny.
12. The person who has brown hair is in the fourth house.
13. The person who has brown hair is somewhere to the left of the person who has blonde hair.
14. The person whose mother's name is Holly is Bob.
15. The person who owns a Ford F-150 is in the third house.
16. Eric is directly left of the person who has auburn hair.
17. The person who owns a Tesla Model 3 is the person who loves yellow.
18. There are two houses between the person who loves purple and The person whose mother's name is Aniya.
19. The person who loves purple is in the first house.
20. The person whose favorite color is green is not in the fourth house.
21. The person who has gray hair is the person who owns a BMW 3 Series.
22. The person who has gray hair is somewhere to the left of Alice.
23. The person whose mother's name is Holly is somewhere to the left of the person who loves the vase of tulips.
24. The person who has red hair is the person who loves white.
 What is the value of attribute Color for the person whose attribute Mother is Janelle? Please reason step by step, and put your final answer within \boxed{}

Answer: purple


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Eric`, `Arnold`, `Peter`
 - Each mother is accompanied by their child: `Samantha`, `Meredith`, `Timothy`, `Bella`, `Fred`
 - People have unique hair colors: `red`, `blonde`, `black`, `brown`, `gray`
 - Each person has a unique hobby: `gardening`, `cooking`, `painting`, `knitting`, `photography`
 - Each person has a unique type of pet: `hamster`, `bird`, `cat`, `fish`, `dog`
 - They all have a unique favorite flower: `tulips`, `roses`, `carnations`, `lilies`, `daffodils`

## Clues:
1. The person's child is named Samantha is somewhere to the right of the person's child is named Meredith.
2. The person who loves the boquet of lilies is the person who keeps a pet bird.
3. The person who has blonde hair is the person who loves the vase of tulips.
4. Bob is the person who has brown hair.
5. The person who paints as a hobby is the person with an aquarium of fish.
6. The person's child is named Fred is Peter.
7. The person with an aquarium of fish is somewhere to the left of the person who has a cat.
8. The person with a pet hamster is the person's child is named Samantha.
9. The person's child is named Bella is Arnold.
10. The person's child is named Bella is in the third house.
11. Peter is not in the first house.
12. Alice is the person who has blonde hair.
13. The person who loves the rose bouquet is not in the third house.
14. The person who owns a dog is the person's child is named Fred.
15. The person who has red hair is directly left of the person who has black hair.
16. The photography enthusiast is directly left of Eric.
17. The person who has black hair is the person who loves the rose bouquet.
18. The person who loves a carnations arrangement is not in the third house.
19. The person who enjoys knitting is somewhere to the right of the person's child is named Bella.
20. The person who loves the boquet of lilies is the person who enjoys gardening.
21. The person who keeps a pet bird is in the fifth house.
 What is the value of attribute Pet for the person whose attribute Children is Meredith? 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`
 - People have unique hair colors: `brown`, `black`, `blonde`, `red`
 - People have unique favorite sports: `tennis`, `basketball`, `soccer`, `swimming`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `dragonfruit`, `desert`
 - Each person has a unique birthday month: `april`, `sept`, `feb`, `jan`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `iphone 13`, `google pixel 6`

## Clues:
1. The person whose birthday is in January is somewhere to the left of the person who loves basketball.
2. The person whose birthday is in February is directly left of the person who has red hair.
3. Eric is the person who uses a OnePlus 9.
4. The person whose birthday is in April is Arnold.
5. The person who uses a Samsung Galaxy S21 and the person who likes Cherry smoothies are next to each other.
6. The person who has blonde hair and Peter are next to each other.
7. The person who has black hair is somewhere to the right of the person who has brown hair.
8. The person who has black hair is the person whose birthday is in February.
9. The Dragonfruit smoothie lover is in the second house.
10. The person who loves soccer is somewhere to the right of the person who uses a OnePlus 9.
11. There are two houses between the person who loves tennis and the person who loves basketball.
12. The person who has black hair is the person who uses a Samsung Galaxy S21.
13. Alice is the person who likes Cherry smoothies.
14. The Desert smoothie lover is the person who uses a Google Pixel 6.
 What is the value of attribute House for the person whose attribute HairColor is red? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Bob`, `Arnold`, `Carol`, `Alice`
 - Everyone has a favorite smoothie: `cherry`, `blueberry`, `dragonfruit`, `watermelon`, `lime`, `desert`
 - The people keep unique animals: `bird`, `cat`, `dog`, `rabbit`, `horse`, `fish`
 - The mothers' names in different houses are unique: `Sarah`, `Holly`, `Janelle`, `Aniya`, `Kailyn`, `Penny`
 - They all have a unique favorite flower: `lilies`, `iris`, `carnations`, `tulips`, `roses`, `daffodils`
 - Each person has an occupation: `lawyer`, `doctor`, `teacher`, `engineer`, `artist`, `nurse`

## Clues:
1. The person who loves a bouquet of daffodils is The person whose mother's name is Holly.
2. The person who likes Cherry smoothies is not in the third house.
3. The person who is a nurse is directly left of Peter.
4. The person who is an artist is the person who loves the rose bouquet.
5. The person who drinks Lime smoothies is not in the fifth house.
6. The cat lover is The person whose mother's name is Sarah.
7. Bob is the Desert smoothie lover.
8. The Dragonfruit smoothie lover is in the sixth house.
9. The person who loves the boquet of lilies and The person whose mother's name is Penny are next to each other.
10. Peter is the bird keeper.
11. The person whose mother's name is Kailyn is the person who loves the boquet of iris.
12. Carol is somewhere to the left of the person who likes Cherry smoothies.
13. The person who is a doctor is directly left of The person whose mother's name is Holly.
14. Bob is The person whose mother's name is Janelle.
15. Arnold is in the third house.
16. Alice is the person who is an engineer.
17. The Dragonfruit smoothie lover is the person who is an artist.
18. The dog owner is directly left of the person who is a teacher.
19. The person who keeps horses is The person whose mother's name is Aniya.
20. The person who loves the boquet of lilies is directly left of the person who loves a carnations arrangement.
21. There is one house between the Desert smoothie lover and the person who loves a carnations arrangement.
22. The person who drinks Lime smoothies is somewhere to the right of The person whose mother's name is Aniya.
23. The person whose mother's name is Kailyn is directly left of the fish enthusiast.
24. The Watermelon smoothie lover is the rabbit owner.
 What is the value of attribute Flower for the person whose attribute Animal is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: iris


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Carol`, `Bob`, `Eric`
 - Each person has an occupation: `artist`, `nurse`, `engineer`, `lawyer`, `teacher`, `doctor`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Aniya`, `Janelle`, `Sarah`, `Kailyn`
 - People have unique favorite music genres: `pop`, `rock`, `classical`, `jazz`, `hip hop`, `country`

## Clues:
1. The person who is an engineer is directly left of the person who is an artist.
2. The person whose mother's name is Janelle is the person who is an artist.
3. The person who loves classical music is directly left of Peter.
4. The person whose mother's name is Janelle and the person who loves pop music are next to each other.
5. The person whose mother's name is Sarah is somewhere to the left of Arnold.
6. The person who loves country music is The person whose mother's name is Aniya.
7. The person whose mother's name is Penny is not in the second house.
8. Bob is the person who loves jazz music.
9. Carol is not in the sixth house.
10. The person who is an engineer is the person who loves classical music.
11. The person who is a nurse is somewhere to the right of the person who loves rock music.
12. There are two houses between Bob and The person whose mother's name is Penny.
13. The person who is a lawyer is directly left of the person who is a doctor.
14. The person whose mother's name is Penny is not in the third house.
15. The person who is an engineer is not in the fourth house.
16. The person who is a doctor is The person whose mother's name is Kailyn.
17. Carol is somewhere to the right of the person who loves country music.
18. The person who loves classical music is Alice.
19. The person who is a nurse is The person whose mother's name is Holly.
 What is the value of attribute House for the person whose attribute Mother is Penny? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Bob`, `Arnold`, `Peter`
 - The people keep unique animals: `horse`, `bird`, `dog`, `cat`, `fish`
 - Each person has an occupation: `lawyer`, `teacher`, `doctor`, `engineer`, `artist`

## Clues:
1. The dog owner is not in the first house.
2. The bird keeper is somewhere to the right of the person who is an engineer.
3. Arnold is the fish enthusiast.
4. The bird keeper is directly left of the cat lover.
5. The person who is a teacher is not in the second house.
6. The fish enthusiast is the person who is a lawyer.
7. The person who is an engineer is Alice.
8. The fish enthusiast is in the fourth house.
9. The person who is a doctor is Peter.
10. Bob is the person who is a teacher.
11. Peter is somewhere to the right of Arnold.
 What is the value of attribute Name for the person whose attribute Occupation is doctor? 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`, `Peter`, `Eric`
 - People have unique favorite sports: `soccer`, `tennis`, `basketball`
 - Each person has a unique birthday month: `april`, `sept`, `jan`
 - Each person prefers a unique type of vacation: `city`, `beach`, `mountain`
 - People have unique heights: `average`, `short`, `very short`
 - They all have a unique favorite flower: `lilies`, `carnations`, `daffodils`

## Clues:
1. The person who is very short is the person who prefers city breaks.
2. The person whose birthday is in January is Peter.
3. The person whose birthday is in January is directly left of the person who loves a carnations arrangement.
4. The person who loves beach vacations is somewhere to the right of the person who loves a bouquet of daffodils.
5. Eric is the person who loves basketball.
6. The person who is short is the person whose birthday is in April.
7. The person who enjoys mountain retreats is the person who loves a carnations arrangement.
8. Eric is the person who is short.
9. The person who loves tennis is the person who loves a carnations arrangement.
 What is the value of attribute FavoriteSport for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: tennis


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`, `Bob`, `Carol`, `Arnold`, `Eric`
 - Each person has an occupation: `engineer`, `doctor`, `lawyer`, `artist`, `nurse`, `teacher`
 - Everyone has something unique for lunch: `soup`, `stew`, `grilled cheese`, `spaghetti`, `pizza`, `stir fry`
 - Everyone has a favorite smoothie: `cherry`, `dragonfruit`, `watermelon`, `lime`, `blueberry`, `desert`
 - Each person prefers a unique type of vacation: `camping`, `cultural`, `city`, `cruise`, `beach`, `mountain`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `oneplus 9`, `samsung galaxy s21`, `xiaomi mi 11`, `huawei p50`

## Clues:
1. The person who likes Cherry smoothies is directly left of the person who enjoys camping trips.
2. The person who is a lawyer is the person who uses a Xiaomi Mi 11.
3. The person who is a doctor is Eric.
4. The person who is a doctor and the Desert smoothie lover are next to each other.
5. The person who is an artist is the person who prefers city breaks.
6. The person who loves beach vacations is Alice.
7. There are two houses between the person who is a pizza lover and the person who uses a Huawei P50.
8. The person who uses a Xiaomi Mi 11 is the person who enjoys camping trips.
9. The person who is an artist is directly left of the person who loves eating grilled cheese.
10. The person who uses a Google Pixel 6 is in the first house.
11. There is one house between the person who is a teacher and the person who loves stir fry.
12. There are two houses between the person who is a pizza lover and the person who is an engineer.
13. The person who is a pizza lover is somewhere to the left of the person who loves eating grilled cheese.
14. The Watermelon smoothie lover is the person who uses a Xiaomi Mi 11.
15. The person who loves eating grilled cheese is Carol.
16. Alice is the person who drinks Blueberry smoothies.
17. The person who likes Cherry smoothies is the person who loves the stew.
18. The person who loves the soup is the person who uses a Samsung Galaxy S21.
19. The person who uses an iPhone 13 is directly left of the Dragonfruit smoothie lover.
20. The person who loves the stew is somewhere to the left of Peter.
21. Arnold is somewhere to the right of the Desert smoothie lover.
22. The person who likes going on cruises is the person who uses a Google Pixel 6.
23. The person who goes on cultural tours is not in the fifth house.
24. The person who is a teacher is Alice.
 What is the value of attribute PhoneModel for the person whose attribute Vacation is beach? Please reason step by step, and put your final answer within \boxed{}

Answer: samsung galaxy s21


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`
 - Each person has a unique favorite drink: `water`, `milk`, `tea`
 - They all have a unique favorite flower: `carnations`, `lilies`, `daffodils`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`, `blue master`

## Clues:
1. The person who smokes Blue Master is the person who loves a bouquet of daffodils.
2. Arnold is somewhere to the right of the person who loves a carnations arrangement.
3. Eric is somewhere to the right of the Prince smoker.
4. The one who only drinks water is the person partial to Pall Mall.
5. The person who loves the boquet of lilies is Peter.
6. The tea drinker is not in the third house.
 What is the value of attribute Cigar for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: pall mall


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`, `Bob`, `Eric`, `Arnold`
 - The people keep unique animals: `fish`, `cat`, `bird`, `horse`, `dog`
 - Each person has a unique level of education: `doctorate`, `associate`, `bachelor`, `master`, `high school`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `lilies`, `roses`, `tulips`

## Clues:
1. The person with an associate's degree is not in the first house.
2. Arnold is directly left of Alice.
3. The bird keeper is somewhere to the right of Peter.
4. Eric is in the second house.
5. The fish enthusiast is the person who loves the boquet of lilies.
6. The fish enthusiast is somewhere to the left of the person with a master's degree.
7. The person with a high school diploma is directly left of the person who loves the vase of tulips.
8. The person with a bachelor's degree is not in the fifth house.
9. The person who keeps horses is somewhere to the left of the person with a high school diploma.
10. The person with a bachelor's degree is the bird keeper.
11. The dog owner is the person who loves the rose bouquet.
12. The person who loves a carnations arrangement is in the third house.
13. The person with an associate's degree is not in the second house.
14. Peter is in the third house.
 What is the value of attribute House for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`
 - Each person has a favorite color: `white`, `yellow`, `red`
 - Each person prefers a unique type of vacation: `city`, `mountain`, `beach`
 - Each person has a unique birthday month: `sept`, `april`, `jan`
 - Each mother is accompanied by their child: `Meredith`, `Fred`, `Bella`

## Clues:
1. The person whose birthday is in January is the person's child is named Bella.
2. The person who loves white is Arnold.
3. The person who loves beach vacations is Arnold.
4. The person whose birthday is in September is somewhere to the left of Arnold.
5. The person whose favorite color is red is directly left of the person who enjoys mountain retreats.
6. Peter is somewhere to the right of the person's child is named Fred.
7. The person who loves yellow is the person's child is named Bella.
 What is the value of attribute Color for the person whose attribute Children is Bella? Please reason step by step, and put your final answer within \boxed{}

Answer: yellow


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 HairColor for the person whose attribute HouseStyle is victorian? Please reason step by step, and put your final answer within \boxed{}

Answer: red


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Bob`, `Peter`, `Alice`
 - Each person lives in a unique style of house: `modern`, `victorian`, `ranch`, `craftsman`, `colonial`
 - Each person prefers a unique type of vacation: `mountain`, `cruise`, `camping`, `city`, `beach`
 - Everyone has a favorite smoothie: `watermelon`, `desert`, `lime`, `dragonfruit`, `cherry`

## Clues:
1. Arnold is the person who enjoys mountain retreats.
2. Eric is the person residing in a Victorian house.
3. The Desert smoothie lover is the person who enjoys mountain retreats.
4. Bob is the person who likes going on cruises.
5. The person who prefers city breaks is Peter.
6. The person who likes going on cruises is somewhere to the left of the person who enjoys camping trips.
7. The person living in a colonial-style house is somewhere to the right of Eric.
8. There is one house between Bob and the person who drinks Lime smoothies.
9. The person who enjoys camping trips is not in the third house.
10. The person who likes Cherry smoothies is the person in a ranch-style home.
11. Eric is not in the first house.
12. The person who likes Cherry smoothies is not in the fifth house.
13. The person living in a colonial-style house is not in the fourth house.
14. The person who likes going on cruises is not in the fourth house.
15. There is one house between the person living in a colonial-style house and the person who enjoys camping trips.
16. The Dragonfruit smoothie lover is directly left of the person in a Craftsman-style house.
 What is the value of attribute HouseStyle for the person whose attribute House is 5? Please reason step by step, and put your final answer within \boxed{}

Answer: modern


Example 5. Find $\int e^{x} \sin x d x$.

Answer: \frac{e^{x}}{2}(\sin x - \cos x) + C


8,9

Points $K, L, M$ divide the sides of the convex quadrilateral $A B C D$ in the ratio $A K: B K=C L: B L=C M: D M=1$ : 2. The radius of the circle circumscribed around triangle $K L M$ is $\frac{5}{2}, K L=4, L M=3$. What is the area of quadrilateral $A B C D$, given that $K M<K L$?

Answer: \dfrac{189}{25}


Example 7 A certain consumer good is priced at 120 yuan per piece. Without the collection of additional taxes, approximately 800,000 pieces are sold annually. If the government levies an additional tax, collecting \( r \) yuan for every 100 yuan sold (called a tax rate of \( r \% \)), then the annual sales volume will decrease by \( \frac{20}{3} r \) ten thousand pieces. To ensure that the tax collected annually from this business is no less than 2.56 million yuan, what should \( r \) be?

Answer: 4


4. Misha, over the course of a week, picked an apple each day and weighed it. Each apple weighed a different amount, but the weight of each apple was a whole number of grams and ranged from 221 grams to 230 grams (inclusive). Misha also calculated the average weight of all the apples he picked, and it was always a whole number. The apple picked on the seventh day weighed 225 grams. How much did the apple picked on the sixth day weigh?

Answer: 230


3. Boy Zhenya lives in a building with a total of 100 floors. The elevator takes 1 second to travel between adjacent floors. Unfortunately, only two buttons in the elevator work: “+13 floors” and “-7 floors”. The buttons are pressed instantly.

How many seconds will it take Zhenya to get home from the 1st floor to the 24th floor? (The elevator can only go to existing floors)

Answer: 107


4. Let $A B C D E F$ be a regular hexagon. A frog starts at vertex $A$ and can randomly jump to one of the two adjacent vertices each time. If it reaches point $D$ within 5 jumps, it stops jumping; if it does not reach point $D$ within 5 jumps, it stops after 5 jumps. How many different possible jumping sequences are there from the start to the stop?

The number of different possible jumping sequences is $\qquad$.

Answer: 26


Authors: Dumanov I., Krotovsky R.

Discussing winter holidays in class, Sasha said: "Now, after I flew to Addis Ababa, I have celebrated New Year's in all possible hemispheres of the Earth, except one!"

What is the minimum number of places where Sasha has celebrated New Year's?

Consider the places where Sasha celebrated New Year's as points on a sphere. Points on the boundary of a hemisphere do not belong to that hemisphere.

#

Answer: 4


6. Two circles of radii $R$ and $R^{\prime}$ touch each other externally at point $P$ and touch the line $l$ at points $A$ and $B$, respectively. Let $Q$ be the point of intersection of the line $B P$ with the first circle. Determine the distance from the line $l$ to the point $Q$.

![](https://cdn.mathpix.com/cropped/2024_05_06_bdfe3271f0fede0d79bbg-3.jpg?height=505&width=805&top_left_y=433&top_left_x=274)

Answer: 2R


3. Let $n$ be a given even number, and $a_{1}, a_{2}, \cdots, a_{n}$ be $n$ non-negative real numbers whose sum is 1. Find the maximum possible value of
$$
S=\sum_{1 \leqslant i<j \leqslant n} \min \left\{(j-i)^{2},(n-j+i)^{2}\right\} a_{i} a_{j}
$$

(Fu Yunhao)

Answer: \dfrac{n^2}{16}


Shapovalov A.B.

Thieves Hapok and Glazok are dividing a pile of 100 coins. Hapok grabs a handful of coins from the pile, and Glazok, looking at the handful, decides who of the two will get it. This continues until one of them gets nine handfuls, after which the other takes all the remaining coins (the division may also end if the coins are divided before anyone gets nine handfuls). Hapok can grab any number of coins in a handful. What is the maximum number of coins he can guarantee himself regardless of Glazok's actions?

Answer: 46


1. We say that a polynomial $p(x)$, with real coefficients, is Almeriense if it has the form

$$
p(x)=x^{3}+a x^{2}+b x+a
$$

and its three roots are positive real numbers in arithmetic progression. Find all Almeriense polynomials such that $p(7 / 4)=0$.

Answer: x^3 - \dfrac{21}{4}x^2 + \dfrac{73}{8}x - \dfrac{21}{4}


6. For $x \in \mathbf{R}, n \in \mathbf{N}_{+}$, define
$$
\mathrm{C}_{x}^{n}=\frac{x(x-1) \cdots(x-n+1)}{n!} .
$$

Let $P(x)$ be a polynomial of degree 6 that satisfies
$$
P(0)=1, P(k)=2^{k-1}(k=1,2, \cdots, 6) \text {. }
$$

Express $P(x)=$
$\qquad$

using $\mathrm{C}_{x}^{k}(k=1,2, \cdots, 6)$.

Answer: 1 + \mathrm{C}_{x}^{2} + \mathrm{C}_{x}^{4} + \mathrm{C}_{x}^{6}


Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions:

(i) the set of prime divisors of the elements in $A$ consists of $p-1$ elements;

(ii) for any nonempty subset of $A$, the product of its elements is not a perfect $p$-th power.

What is the largest possible number of elements in $A$ ?

Answer: (p-1)^2


4. There are 85 pebbles on the table, Shu Ke and Bei Ta take turns to take pebbles, each person can only take 2, 4, 5, 7 pebbles or all at a time, and the one who takes the last pebble loses. Shu Ke goes first, his winning strategy is to take $\qquad$ pebbles the first time.

Answer: 2


B2. An integer $n$ is called a combi-number if every pair of different digits from all possible digits $0 \mathrm{t} / \mathrm{m} 9$ appear next to each other at least once in the number. Thus, in a combi-number, the digits 3 and 5 appear next to each other somewhere. It does not matter whether they appear in the order 35 or 53. We agree that a combi-number does not start with the digit 0.

What is the smallest number of digits a combi-number can consist of?

Answer: 50


116. How many lines can divide a plane into 5 parts?

Answer: 4


2. Given are $n$ light bulbs ($n>13$), each with its own switch. It is allowed to change the state of exactly 13 light bulbs simultaneously. Initially, some of the light bulbs are on, and some are off.

a) Is it possible to turn off all the light bulbs?

b) What is the minimum number of steps to turn off all the light bulbs if $n=111$ and all the light bulbs are initially on?

Answer: Yes


10. Now there are 50 drivers and 40 cars, and the lock on each car is different. If you want to ensure that any 40 drivers can work with 40 cars, assuming all the keys are in the hands of the drivers, then at least ( ) keys are needed.

Answer: 440


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 person has a unique hobby: `cooking`, `gardening`, `painting`, `photography`
 - People own unique car models: `honda civic`, `toyota camry`, `tesla model 3`, `ford f150`
 - Everyone has something unique for lunch: `grilled cheese`, `pizza`, `spaghetti`, `stew`
 - Each person has an occupation: `doctor`, `artist`, `engineer`, `teacher`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`

## Clues:
1. The person who loves the stew is Eric.
2. The person who is a teacher is directly left of Peter.
3. The person who loves a carnations arrangement is the person who is an artist.
4. The person who paints as a hobby is the person who loves the spaghetti eater.
5. There is one house between the person who is an artist and the photography enthusiast.
6. The person who enjoys gardening is in the first house.
7. The person who loves the spaghetti eater is somewhere to the right of the person who owns a Honda Civic.
8. The person who is a pizza lover is directly left of the photography enthusiast.
9. The person who loves a carnations arrangement is Alice.
10. The person who owns a Tesla Model 3 is directly left of Eric.
11. The person who is an engineer is the person who loves a bouquet of daffodils.
12. The person who loves the boquet of lilies is somewhere to the left of the person who loves the stew.
13. There is one house between the person who enjoys gardening and the person who owns a Toyota Camry.
14. The person who is a teacher is in the third house.
 What is the value of attribute Hobby for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: photography


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`
 - People have unique favorite sports: `soccer`, `tennis`, `basketball`
 - Each person has a unique birthday month: `april`, `sept`, `jan`
 - Each person prefers a unique type of vacation: `city`, `beach`, `mountain`
 - People have unique heights: `average`, `short`, `very short`
 - They all have a unique favorite flower: `lilies`, `carnations`, `daffodils`

## Clues:
1. The person who is very short is the person who prefers city breaks.
2. The person whose birthday is in January is Peter.
3. The person whose birthday is in January is directly left of the person who loves a carnations arrangement.
4. The person who loves beach vacations is somewhere to the right of the person who loves a bouquet of daffodils.
5. Eric is the person who loves basketball.
6. The person who is short is the person whose birthday is in April.
7. The person who enjoys mountain retreats is the person who loves a carnations arrangement.
8. Eric is the person who is short.
9. The person who loves tennis is the person who loves a carnations arrangement.
 What is the value of attribute Flower for the person whose attribute FavoriteSport is basketball? Please reason step by step, and put your final answer within \boxed{}

Answer: lilies


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 BookGenre for the person whose attribute CarModel is toyota camry? Please reason step by step, and put your final answer within \boxed{}

Answer: science fiction


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`, `Bob`, `Peter`, `Eric`, `Arnold`
 - People have unique favorite sports: `swimming`, `basketball`, `soccer`, `tennis`, `baseball`
 - Each mother is accompanied by their child: `Meredith`, `Bella`, `Fred`, `Samantha`, `Timothy`
 - Each person has a unique birthday month: `april`, `feb`, `sept`, `jan`, `mar`

## Clues:
1. The person who loves soccer is the person's child is named Fred.
2. The person whose birthday is in April is somewhere to the left of the person who loves soccer.
3. The person whose birthday is in January is Peter.
4. The person who loves baseball is the person whose birthday is in September.
5. The person who is the mother of Timothy is the person who loves baseball.
6. There are two houses between the person whose birthday is in February and Peter.
7. Eric is directly left of the person who loves basketball.
8. Eric is the person who loves tennis.
9. The person's child is named Meredith is directly left of Peter.
10. Alice is directly left of the person whose birthday is in September.
11. The person's child is named Bella is somewhere to the left of the person's child is named Samantha.
12. Eric is in the third house.
13. The person who loves soccer is Arnold.
 What is the value of attribute House for the person whose attribute Children is Fred? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Eric`, `Peter`, `Arnold`
 - People use unique phone models: `iphone 13`, `samsung galaxy s21`, `huawei p50`, `google pixel 6`, `oneplus 9`
 - People have unique favorite sports: `soccer`, `swimming`, `baseball`, `basketball`, `tennis`
 - Everyone has a favorite smoothie: `cherry`, `dragonfruit`, `lime`, `desert`, `watermelon`
 - Each person prefers a unique type of vacation: `city`, `cruise`, `beach`, `mountain`, `camping`
 - The people keep unique animals: `horse`, `fish`, `cat`, `dog`, `bird`

## Clues:
1. Peter is somewhere to the right of the person who uses an iPhone 13.
2. The person who loves tennis is directly left of the person who uses a Google Pixel 6.
3. There is one house between the Watermelon smoothie lover and the person who loves beach vacations.
4. The person who loves baseball is in the fourth house.
5. The dog owner is Peter.
6. There is one house between the person who prefers city breaks and the person who keeps horses.
7. The fish enthusiast is directly left of the cat lover.
8. Arnold is the Desert smoothie lover.
9. The person who likes Cherry smoothies is the person who enjoys mountain retreats.
10. The person who loves swimming is not in the first house.
11. Bob is the person who prefers city breaks.
12. The person who uses a Huawei P50 is the fish enthusiast.
13. The person who loves tennis is the person who enjoys mountain retreats.
14. The person who loves baseball is somewhere to the left of the person who loves soccer.
15. The person who uses a OnePlus 9 is Peter.
16. There is one house between the person who prefers city breaks and Alice.
17. The person who enjoys camping trips is the Dragonfruit smoothie lover.
18. The person who loves baseball is the person who enjoys camping trips.
19. The person who loves beach vacations is directly left of the person who enjoys camping trips.
20. Bob is the fish enthusiast.
 What is the value of attribute House for the person whose attribute Animal is dog? 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: `Alice`, `Arnold`, `Eric`, `Peter`
 - Each person has a unique favorite drink: `water`, `tea`, `coffee`, `milk`
 - Each person has a unique level of education: `master`, `associate`, `high school`, `bachelor`
 - The mothers' names in different houses are unique: `Kailyn`, `Holly`, `Aniya`, `Janelle`
 - Each person has a favorite color: `yellow`, `white`, `green`, `red`

## Clues:
1. Eric is the person with an associate's degree.
2. The person whose mother's name is Kailyn is the person who loves yellow.
3. The person who likes milk is in the third house.
4. The person who likes milk is somewhere to the right of the person who loves white.
5. The person with a bachelor's degree is in the first house.
6. The person whose mother's name is Aniya is not in the second house.
7. The one who only drinks water is Arnold.
8. The person whose favorite color is green is Peter.
9. The person whose favorite color is red is somewhere to the left of the person who loves white.
10. The person with a high school diploma is directly left of The person whose mother's name is Janelle.
11. The person with a high school diploma is the tea drinker.
 What is the value of attribute Drink for the person whose attribute Mother is Janelle? Please reason step by step, and put your final answer within \boxed{}

Answer: milk


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`, `Eric`, `Carol`, `Alice`, `Bob`
 - People have unique favorite music genres: `country`, `classical`, `pop`, `hip hop`, `jazz`, `rock`
 - Each person lives in a unique style of house: `modern`, `mediterranean`, `victorian`, `craftsman`, `ranch`, `colonial`
 - People use unique phone models: `huawei p50`, `iphone 13`, `google pixel 6`, `oneplus 9`, `xiaomi mi 11`, `samsung galaxy s21`
 - Each person has a unique type of pet: `hamster`, `dog`, `cat`, `rabbit`, `fish`, `bird`

## Clues:
1. The person who has a cat is the person in a Mediterranean-style villa.
2. Peter is in the second house.
3. The person in a modern-style house is the person who uses a Samsung Galaxy S21.
4. The person who keeps a pet bird is the person who uses a Google Pixel 6.
5. The person who keeps a pet bird is directly left of the person who uses a Xiaomi Mi 11.
6. Arnold is the person residing in a Victorian house.
7. The person who loves rock music is the person living in a colonial-style house.
8. The person in a Craftsman-style house is Carol.
9. The person who owns a rabbit is in the sixth house.
10. The person who loves pop music is somewhere to the left of the person living in a colonial-style house.
11. The person who uses a Huawei P50 and the person who loves country music are next to each other.
12. The person who loves hip-hop music is the person who uses a Samsung Galaxy S21.
13. Carol and Peter are next to each other.
14. The person who uses a OnePlus 9 is Peter.
15. The person with a pet hamster is not in the first house.
16. The person in a ranch-style home is somewhere to the left of the person who loves classical music.
17. There are two houses between Carol and Alice.
18. The person who owns a dog is in the fifth house.
19. The person in a modern-style house is Eric.
20. Peter is the person living in a colonial-style house.
21. The person who has a cat is not in the fourth house.
 What is the value of attribute Pet for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: hamster


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Bob`, `Eric`, `Arnold`
 - Everyone has a favorite smoothie: `lime`, `dragonfruit`, `desert`, `watermelon`, `cherry`
 - The people keep unique animals: `horse`, `dog`, `bird`, `fish`, `cat`
 - The people are of nationalities: `german`, `swede`, `norwegian`, `brit`, `dane`

## Clues:
1. The Swedish person is directly left of the dog owner.
2. There are two houses between the dog owner and the British person.
3. The Dane is the person who keeps horses.
4. The bird keeper is somewhere to the right of the cat lover.
5. The dog owner is directly left of the person who drinks Lime smoothies.
6. Eric is the cat lover.
7. Bob is the bird keeper.
8. The person who likes Cherry smoothies is directly left of Peter.
9. The bird keeper is the Watermelon smoothie lover.
10. The Desert smoothie lover is the dog owner.
11. The person who keeps horses is in the third house.
12. The Norwegian is Alice.
 What is the value of attribute Name for the person whose attribute House is 2? 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`, `Peter`, `Alice`, `Arnold`, `Bob`, `Carol`
 - Each person has a unique hobby: `cooking`, `knitting`, `painting`, `gardening`, `woodworking`, `photography`
 - Everyone has a favorite smoothie: `lime`, `cherry`, `desert`, `blueberry`, `watermelon`, `dragonfruit`
 - The people are of nationalities: `chinese`, `dane`, `german`, `norwegian`, `swede`, `brit`
 - People have unique favorite music genres: `hip hop`, `country`, `rock`, `jazz`, `classical`, `pop`

## Clues:
1. The person who loves rock music is Bob.
2. The person who enjoys knitting is the German.
3. The woodworking hobbyist is in the sixth house.
4. The person who loves rock music is directly left of the woodworking hobbyist.
5. There is one house between the Watermelon smoothie lover and the person who loves hip-hop music.
6. The Chinese is the Watermelon smoothie lover.
7. The Desert smoothie lover is the Norwegian.
8. The person who drinks Blueberry smoothies is not in the sixth house.
9. The person who loves pop music is somewhere to the left of the person who enjoys gardening.
10. Carol is somewhere to the left of the Swedish person.
11. Peter is somewhere to the right of the person who loves pop music.
12. The Norwegian is somewhere to the right of the Dragonfruit smoothie lover.
13. The German is the person who loves country music.
14. The person who paints as a hobby is directly left of the person who loves hip-hop music.
15. The Dane is Arnold.
16. The person who loves jazz music is the Chinese.
17. The British person is not in the third house.
18. The Swedish person is the person who loves pop music.
19. The person who drinks Lime smoothies is the photography enthusiast.
20. The person who loves hip-hop music is directly left of Alice.
21. The Dane is the woodworking hobbyist.
 What is the value of attribute Nationality for the person whose attribute Smoothie is blueberry? 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: `Arnold`, `Peter`, `Eric`, `Alice`, `Bob`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `cruise`, `city`, `camping`
 - Each person lives in a unique style of house: `ranch`, `modern`, `craftsman`, `colonial`, `victorian`
 - Each person has a unique level of education: `bachelor`, `doctorate`, `master`, `associate`, `high school`
 - The mothers' names in different houses are unique: `Aniya`, `Kailyn`, `Janelle`, `Penny`, `Holly`

## Clues:
1. The person whose mother's name is Aniya is the person in a ranch-style home.
2. The person in a Craftsman-style house is Bob.
3. The person with a high school diploma is not in the fourth house.
4. Arnold and the person in a Craftsman-style house are next to each other.
5. Arnold is The person whose mother's name is Penny.
6. The person with an associate's degree is somewhere to the left of the person with a master's degree.
7. Bob and Peter are next to each other.
8. The person living in a colonial-style house is The person whose mother's name is Penny.
9. The person whose mother's name is Holly is the person who enjoys camping trips.
10. The person with a doctorate is the person who prefers city breaks.
11. The person with a bachelor's degree is The person whose mother's name is Aniya.
12. The person with an associate's degree is somewhere to the right of Eric.
13. Arnold is directly left of the person who likes going on cruises.
14. The person who enjoys camping trips is somewhere to the right of the person residing in a Victorian house.
15. The person who enjoys mountain retreats is in the first house.
16. The person whose mother's name is Janelle is the person with an associate's degree.
17. The person with an associate's degree is directly left of Arnold.
 What is the value of attribute House for the person whose attribute Mother is Holly? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


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`
 - People own unique car models: `honda civic`, `tesla model 3`, `ford f150`, `toyota camry`
 - Each person has a unique type of pet: `bird`, `cat`, `fish`, `dog`
 - People use unique phone models: `oneplus 9`, `iphone 13`, `google pixel 6`, `samsung galaxy s21`

## Clues:
1. The person who uses an iPhone 13 is the person who owns a dog.
2. Alice is not in the second house.
3. The person who uses an iPhone 13 is the person who owns a Toyota Camry.
4. The person who uses a Google Pixel 6 is not in the fourth house.
5. The person who owns a Ford F-150 is not in the second house.
6. The person with an aquarium of fish is directly left of the person who keeps a pet bird.
7. Peter is the person who uses a Google Pixel 6.
8. The person who owns a Tesla Model 3 is directly left of the person who uses an iPhone 13.
9. Eric is not in the second house.
10. Eric is not in the first house.
11. The person who uses a Google Pixel 6 and the person who owns a Ford F-150 are next to each other.
12. The person who uses a OnePlus 9 is somewhere to the left of the person who owns a Ford F-150.
 What is the value of attribute PhoneModel for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: google pixel 6


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 has a unique type of pet: `dog`, `fish`, `cat`
 - People have unique favorite book genres: `mystery`, `romance`, `science fiction`
 - People own unique car models: `ford f150`, `tesla model 3`, `toyota camry`
 - Each person has a unique birthday month: `april`, `jan`, `sept`
 - People have unique favorite music genres: `rock`, `classical`, `pop`

## Clues:
1. The person who loves romance books is directly left of the person who owns a Ford F-150.
2. The person whose birthday is in September is the person who owns a Toyota Camry.
3. The person whose birthday is in September is Peter.
4. The person who owns a Tesla Model 3 is the person whose birthday is in January.
5. The person with an aquarium of fish is not in the second house.
6. Arnold is the person who has a cat.
7. The person who owns a Tesla Model 3 is the person who loves science fiction books.
8. The person who loves science fiction books is not in the third house.
9. The person who owns a Ford F-150 is the person who loves pop music.
10. The person who loves rock music is directly left of Eric.
 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: sept


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Bob`, `Eric`, `Arnold`
 - The people are of nationalities: `dane`, `swede`, `norwegian`, `brit`, `german`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`, `dragonfruit`, `lime`
 - They all have a unique favorite flower: `tulips`, `roses`, `daffodils`, `carnations`, `lilies`
 - Each person lives in a unique style of house: `ranch`, `modern`, `victorian`, `colonial`, `craftsman`
 - The people keep unique animals: `dog`, `horse`, `bird`, `fish`, `cat`

## Clues:
1. The person living in a colonial-style house is Peter.
2. The person who keeps horses is the person who drinks Lime smoothies.
3. Alice is the British person.
4. The Swedish person is in the first house.
5. The person who keeps horses and the dog owner are next to each other.
6. There is one house between the person in a Craftsman-style house and the Dane.
7. The person who loves the boquet of lilies is directly left of the fish enthusiast.
8. The person who loves the boquet of lilies is the person in a modern-style house.
9. The Desert smoothie lover is in the first house.
10. The person residing in a Victorian house is somewhere to the right of the Watermelon smoothie lover.
11. The person who loves a bouquet of daffodils is Alice.
12. The Dragonfruit smoothie lover is not in the third house.
13. The person who loves a carnations arrangement is the German.
14. The Dragonfruit smoothie lover is Eric.
15. Bob is somewhere to the right of the person who loves the rose bouquet.
16. The Watermelon smoothie lover is the Norwegian.
17. Bob is the person in a modern-style house.
18. The person residing in a Victorian house is not in the fifth house.
19. The person who loves a carnations arrangement is somewhere to the left of the person in a modern-style house.
20. The Desert smoothie lover is the bird keeper.
 What is the value of attribute Animal for the person whose attribute Smoothie is dragonfruit? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Arnold`, `Peter`, `Alice`
 - People own unique car models: `ford f150`, `honda civic`, `toyota camry`, `bmw 3 series`, `tesla model 3`

## Clues:
1. The person who owns a BMW 3 Series is Arnold.
2. The person who owns a BMW 3 Series is not in the fifth house.
3. Eric is directly left of Peter.
4. Bob is directly left of Alice.
5. Eric is the person who owns a Tesla Model 3.
6. Peter and Arnold are next to each other.
7. Bob and the person who owns a Honda Civic are next to each other.
8. The person who owns a BMW 3 Series is somewhere to the right of the person who owns a Ford F-150.
 What is the value of attribute CarModel for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: tesla model 3


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`, `Carol`, `Eric`, `Arnold`, `Bob`
 - People have unique heights: `very tall`, `very short`, `average`, `tall`, `short`, `super tall`

## Clues:
1. Eric is in the second house.
2. Arnold is the person who is short.
3. Alice is somewhere to the left of the person who has an average height.
4. The person who is tall is somewhere to the right of Peter.
5. Arnold is not in the fourth house.
6. The person who is short is somewhere to the right of Eric.
7. There is one house between Peter and Bob.
8. Arnold is somewhere to the left of the person who is very tall.
9. The person who has an average height is Carol.
10. The person who is very short is Eric.
 What is the value of attribute Height for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: super tall


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`, `Bob`, `Alice`, `Arnold`
 - People have unique favorite book genres: `mystery`, `fantasy`, `science fiction`, `romance`, `biography`
 - Everyone has a favorite smoothie: `lime`, `watermelon`, `dragonfruit`, `cherry`, `desert`

## Clues:
1. There is one house between Eric and the person who loves science fiction books.
2. The person who loves fantasy books is directly left of Alice.
3. The person who loves fantasy books is the Watermelon smoothie lover.
4. Arnold and the Dragonfruit smoothie lover are next to each other.
5. The person who loves biography books is the person who likes Cherry smoothies.
6. The person who loves biography books is in the second house.
7. Eric is in the third house.
8. The Desert smoothie lover is in the third house.
9. The person who loves mystery books is not in the third house.
10. Arnold is directly left of Bob.
 What is the value of attribute House for the person whose attribute BookGenre is romance? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Eric`, `Arnold`, `Peter`
 - Each mother is accompanied by their child: `Samantha`, `Meredith`, `Timothy`, `Bella`, `Fred`
 - People have unique hair colors: `red`, `blonde`, `black`, `brown`, `gray`
 - Each person has a unique hobby: `gardening`, `cooking`, `painting`, `knitting`, `photography`
 - Each person has a unique type of pet: `hamster`, `bird`, `cat`, `fish`, `dog`
 - They all have a unique favorite flower: `tulips`, `roses`, `carnations`, `lilies`, `daffodils`

## Clues:
1. The person's child is named Samantha is somewhere to the right of the person's child is named Meredith.
2. The person who loves the boquet of lilies is the person who keeps a pet bird.
3. The person who has blonde hair is the person who loves the vase of tulips.
4. Bob is the person who has brown hair.
5. The person who paints as a hobby is the person with an aquarium of fish.
6. The person's child is named Fred is Peter.
7. The person with an aquarium of fish is somewhere to the left of the person who has a cat.
8. The person with a pet hamster is the person's child is named Samantha.
9. The person's child is named Bella is Arnold.
10. The person's child is named Bella is in the third house.
11. Peter is not in the first house.
12. Alice is the person who has blonde hair.
13. The person who loves the rose bouquet is not in the third house.
14. The person who owns a dog is the person's child is named Fred.
15. The person who has red hair is directly left of the person who has black hair.
16. The photography enthusiast is directly left of Eric.
17. The person who has black hair is the person who loves the rose bouquet.
18. The person who loves a carnations arrangement is not in the third house.
19. The person who enjoys knitting is somewhere to the right of the person's child is named Bella.
20. The person who loves the boquet of lilies is the person who enjoys gardening.
21. The person who keeps a pet bird is in the fifth house.
 What is the value of attribute House for the person whose attribute HairColor is black? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Bob`, `Peter`, `Carol`, `Alice`
 - Each mother is accompanied by their child: `Fred`, `Bella`, `Timothy`, `Meredith`, `Samantha`, `Alice`
 - Everyone has something unique for lunch: `pizza`, `stir fry`, `grilled cheese`, `spaghetti`, `soup`, `stew`
 - Each person has a unique type of pet: `rabbit`, `fish`, `cat`, `dog`, `bird`, `hamster`
 - Each person has a unique birthday month: `may`, `sept`, `mar`, `jan`, `april`, `feb`
 - Each person has a unique favorite drink: `water`, `milk`, `coffee`, `tea`, `boba tea`, `root beer`

## Clues:
1. The person whose birthday is in May is the person who loves the spaghetti eater.
2. The person's child is named Fred is the person who loves the stew.
3. The person who likes milk is the person with a pet hamster.
4. There is one house between the person who loves the spaghetti eater and the person who has a cat.
5. The person who is the mother of Timothy is Bob.
6. The person whose birthday is in January is directly left of the one who only drinks water.
7. Carol is the coffee drinker.
8. Peter is not in the fifth house.
9. The person whose birthday is in April is the person's child is named Alice.
10. The person who loves eating grilled cheese is somewhere to the right of the person whose birthday is in September.
11. The person's child is named Bella and Bob are next to each other.
12. The person who owns a rabbit is Peter.
13. The person who keeps a pet bird is the person who loves the soup.
14. The person who owns a rabbit is somewhere to the right of the person who loves stir fry.
15. The person who loves eating grilled cheese is somewhere to the left of the person who owns a rabbit.
16. The person whose birthday is in March is the tea drinker.
17. Peter is the person's child is named Bella.
18. The person who is the mother of Timothy is somewhere to the right of the person's child is named Bella.
19. The person with an aquarium of fish and the person's child is named Bella are next to each other.
20. The one who only drinks water is the person's child is named Fred.
21. The person whose birthday is in May is somewhere to the right of the person with an aquarium of fish.
22. The person with a pet hamster is the person's child is named Meredith.
23. The root beer lover is the person who loves the spaghetti eater.
24. Carol is directly left of Alice.
25. Eric is in the third house.
26. The person who is a pizza lover is Bob.
 What is the value of attribute Birthday for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: mar


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
 - People own unique car models: `toyota camry`, `bmw 3 series`, `ford f150`, `honda civic`, `tesla model 3`
 - Each person has a unique favorite drink: `tea`, `milk`, `water`, `root beer`, `coffee`
 - The people are of nationalities: `dane`, `brit`, `german`, `norwegian`, `swede`
 - People use unique phone models: `oneplus 9`, `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `huawei p50`

## Clues:
1. The person who uses a Google Pixel 6 is somewhere to the right of the Dane.
2. The person who uses a Google Pixel 6 is somewhere to the right of the root beer lover.
3. The person who owns a Tesla Model 3 is the British person.
4. The person who owns a Ford F-150 is the Dane.
5. Eric is the tea drinker.
6. The British person is directly left of the person who owns a Honda Civic.
7. Peter is somewhere to the right of the person who likes milk.
8. Bob is directly left of the Swedish person.
9. The person who uses an iPhone 13 is somewhere to the right of the person who uses a Huawei P50.
10. Arnold is the person who uses a Google Pixel 6.
11. Eric is not in the third house.
12. The person who owns a BMW 3 Series is somewhere to the right of the person who uses a Samsung Galaxy S21.
13. The person who owns a Tesla Model 3 is not in the first house.
14. Alice is somewhere to the left of the person who owns a Ford F-150.
15. The person who uses a OnePlus 9 is somewhere to the right of the person who uses a Samsung Galaxy S21.
16. The German is somewhere to the right of the person who owns a Tesla Model 3.
17. Eric is not in the fourth house.
18. Peter is directly left of the person who uses a Samsung Galaxy S21.
19. The coffee drinker is Bob.
 What is the value of attribute Drink for the person whose attribute PhoneModel is huawei p50? Please reason step by step, and put your final answer within \boxed{}

Answer: milk


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`, `Alice`, `Bob`, `Peter`, `Eric`
 - People have unique hair colors: `blonde`, `black`, `gray`, `brown`, `red`
 - They all have a unique favorite flower: `tulips`, `carnations`, `daffodils`, `lilies`, `roses`
 - Each person has a favorite color: `white`, `red`, `green`, `blue`, `yellow`

## Clues:
1. The person who loves the rose bouquet and the person who loves yellow are next to each other.
2. The person who loves a carnations arrangement is Peter.
3. The person who loves the rose bouquet is not in the third house.
4. The person whose favorite color is green is not in the fifth house.
5. The person who loves a bouquet of daffodils is Alice.
6. The person who loves yellow is Eric.
7. The person who has blonde hair is in the fifth house.
8. The person who has black hair is directly left of Arnold.
9. The person who loves yellow is the person who has brown hair.
10. Alice is the person whose favorite color is green.
11. The person who has red hair is the person who loves blue.
12. The person whose favorite color is red is somewhere to the right of the person who loves blue.
13. The person who has black hair is the person who loves the vase of tulips.
 What is the value of attribute Flower for the person whose attribute HairColor is red? Please reason step by step, and put your final answer within \boxed{}

Answer: roses


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Eric`, `Peter`, `Arnold`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`, `blends`, `blue master`, `dunhill`
 - The people are of nationalities: `brit`, `norwegian`, `dane`, `swede`, `german`
 - Each person has a unique birthday month: `april`, `jan`, `sept`, `feb`, `mar`
 - Each person has a unique type of pet: `fish`, `bird`, `hamster`, `cat`, `dog`
 - The people keep unique animals: `cat`, `dog`, `bird`, `fish`, `horse`

## Clues:
1. The British person is Bob.
2. Peter is the bird keeper.
3. The person who smokes Blue Master is the Swedish person.
4. The fish enthusiast is Bob.
5. The Dunhill smoker is the German.
6. The cat lover is the person partial to Pall Mall.
7. The person with a pet hamster is the person whose birthday is in March.
8. The person who keeps horses is somewhere to the right of the person whose birthday is in April.
9. The fish enthusiast is the person whose birthday is in January.
10. Alice is the dog owner.
11. The person whose birthday is in February is in the first house.
12. The person who owns a dog is somewhere to the left of Peter.
13. The Dane is the person who smokes many unique blends.
14. Peter and Arnold are next to each other.
15. The person with an aquarium of fish is Peter.
16. The Dunhill smoker is the person who has a cat.
17. The Swedish person is directly left of the person whose birthday is in January.
18. The person who owns a dog is in the third house.
 What is the value of attribute Pet for the person whose attribute Cigar is blue master? Please reason step by step, and put your final answer within \boxed{}

Answer: hamster


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`, `Peter`, `Eric`, `Arnold`, `Carol`
 - They all have a unique favorite flower: `roses`, `lilies`, `iris`, `tulips`, `daffodils`, `carnations`

## Clues:
1. Bob is somewhere to the right of the person who loves a carnations arrangement.
2. The person who loves the boquet of iris is in the fifth house.
3. Peter is directly left of the person who loves the boquet of lilies.
4. There is one house between the person who loves the rose bouquet and Bob.
5. Alice is in the fourth house.
6. The person who loves the vase of tulips is Peter.
7. Arnold is in the second house.
8. Alice is directly left of Carol.
 What is the value of attribute House for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: 6


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
 - People own unique car models: `toyota camry`, `bmw 3 series`, `ford f150`, `honda civic`, `tesla model 3`
 - Each person has a unique favorite drink: `tea`, `milk`, `water`, `root beer`, `coffee`
 - The people are of nationalities: `dane`, `brit`, `german`, `norwegian`, `swede`
 - People use unique phone models: `oneplus 9`, `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `huawei p50`

## Clues:
1. The person who uses a Google Pixel 6 is somewhere to the right of the Dane.
2. The person who uses a Google Pixel 6 is somewhere to the right of the root beer lover.
3. The person who owns a Tesla Model 3 is the British person.
4. The person who owns a Ford F-150 is the Dane.
5. Eric is the tea drinker.
6. The British person is directly left of the person who owns a Honda Civic.
7. Peter is somewhere to the right of the person who likes milk.
8. Bob is directly left of the Swedish person.
9. The person who uses an iPhone 13 is somewhere to the right of the person who uses a Huawei P50.
10. Arnold is the person who uses a Google Pixel 6.
11. Eric is not in the third house.
12. The person who owns a BMW 3 Series is somewhere to the right of the person who uses a Samsung Galaxy S21.
13. The person who owns a Tesla Model 3 is not in the first house.
14. Alice is somewhere to the left of the person who owns a Ford F-150.
15. The person who uses a OnePlus 9 is somewhere to the right of the person who uses a Samsung Galaxy S21.
16. The German is somewhere to the right of the person who owns a Tesla Model 3.
17. Eric is not in the fourth house.
18. Peter is directly left of the person who uses a Samsung Galaxy S21.
19. The coffee drinker is Bob.
 What is the value of attribute CarModel for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: ford f150


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Bob`, `Arnold`, `Peter`, `Carol`
 - People have unique favorite music genres: `classical`, `rock`, `jazz`, `pop`, `country`, `hip hop`
 - Each person has an occupation: `teacher`, `artist`, `nurse`, `engineer`, `lawyer`, `doctor`
 - Each person prefers a unique type of vacation: `cruise`, `camping`, `city`, `cultural`, `mountain`, `beach`
 - Each person has a favorite color: `blue`, `white`, `red`, `yellow`, `purple`, `green`

## Clues:
1. The person who loves beach vacations is Arnold.
2. The person who is an engineer is the person who enjoys mountain retreats.
3. Peter is not in the second house.
4. The person whose favorite color is red is in the third house.
5. The person who loves beach vacations and the person who is a teacher are next to each other.
6. The person who is a nurse is Carol.
7. The person who goes on cultural tours is the person who loves pop music.
8. The person who loves rock music and the person who loves country music are next to each other.
9. The person whose favorite color is green is in the second house.
10. The person whose favorite color is red is the person who enjoys camping trips.
11. Peter is the person who loves pop music.
12. The person who loves classical music is the person who loves purple.
13. The person who loves blue is directly left of Alice.
14. The person who loves rock music is in the fourth house.
15. The person who loves white is the person who is a doctor.
16. The person who loves jazz music is the person whose favorite color is red.
17. Carol is somewhere to the right of the person who prefers city breaks.
18. The person who loves rock music is the person who is an artist.
19. The person who enjoys mountain retreats is in the first house.
20. Bob is the person who loves classical music.
 What is the value of attribute House for the person whose attribute Color is purple? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Alice`, `Arnold`
 - People have unique hair colors: `brown`, `blonde`, `red`, `black`
 - People have unique heights: `short`, `average`, `tall`, `very short`
 - Each person has a favorite color: `green`, `yellow`, `white`, `red`

## Clues:
1. Eric is the person who has an average height.
2. The person whose favorite color is red is the person who is tall.
3. The person whose favorite color is green is somewhere to the left of Peter.
4. Arnold is in the fourth house.
5. The person who has an average height is not in the first house.
6. The person who is very short is in the fourth house.
7. The person who loves white is the person who has black hair.
8. The person who is tall is directly left of the person who has brown hair.
9. The person who has blonde hair is the person who has an average height.
 What is the value of attribute Height for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: tall


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Peter`
 - People have unique heights: `tall`, `very short`, `average`, `short`
 - Each person has a unique favorite drink: `tea`, `coffee`, `water`, `milk`
 - Each mother is accompanied by their child: `Fred`, `Meredith`, `Bella`, `Samantha`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`, `Kailyn`

## Clues:
1. The person who is tall is Arnold.
2. The person's child is named Samantha is the tea drinker.
3. The person whose mother's name is Holly is the coffee drinker.
4. The person's child is named Samantha and The person whose mother's name is Holly are next to each other.
5. The person's child is named Meredith and the person who is very short are next to each other.
6. The person's child is named Samantha is The person whose mother's name is Aniya.
7. The person who likes milk is the person who is short.
8. Alice is the person's child is named Fred.
9. Peter and the person who is very short are next to each other.
10. The coffee drinker is somewhere to the right of the person's child is named Samantha.
11. Eric is the person who is short.
12. The person whose mother's name is Janelle is somewhere to the left of The person whose mother's name is Kailyn.
13. Alice is The person whose mother's name is Kailyn.
14. The person's child is named Meredith is not in the first house.
 What is the value of attribute House for the person whose attribute Height is very short? 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`, `Alice`, `Arnold`, `Peter`
 - The people keep unique animals: `cat`, `fish`, `horse`, `bird`
 - Each person has a favorite color: `yellow`, `green`, `white`, `red`
 - Each person prefers a unique type of vacation: `cruise`, `city`, `beach`, `mountain`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `stew`, `grilled cheese`
 - People have unique favorite sports: `soccer`, `swimming`, `basketball`, `tennis`

## Clues:
1. The person whose favorite color is red is the person who loves the spaghetti eater.
2. The person who loves the stew is the cat lover.
3. The person who loves yellow is not in the fourth house.
4. Alice is somewhere to the right of Arnold.
5. The person who loves the spaghetti eater is somewhere to the right of the person who loves soccer.
6. The person who loves white is the person who loves tennis.
7. The fish enthusiast is in the second house.
8. Arnold is the person who loves the stew.
9. The person who prefers city breaks is the person who is a pizza lover.
10. The person who prefers city breaks is directly left of the person who likes going on cruises.
11. The person who loves swimming is the person who loves beach vacations.
12. The person who keeps horses and the person whose favorite color is red are next to each other.
13. The person whose favorite color is green is directly left of Eric.
14. The person who prefers city breaks is somewhere to the left of the cat lover.
 What is the value of attribute Name for the person whose attribute Color is green? 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: `Arnold`, `Alice`, `Peter`, `Eric`
 - Everyone has something unique for lunch: `grilled cheese`, `stew`, `pizza`, `spaghetti`
 - People have unique hair colors: `blonde`, `black`, `red`, `brown`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`
 - The mothers' names in different houses are unique: `Holly`, `Aniya`, `Janelle`, `Kailyn`

## Clues:
1. Alice is the person whose favorite color is red.
2. The person who has black hair is the person who loves the stew.
3. The person who loves white is Arnold.
4. The person whose favorite color is red and The person whose mother's name is Kailyn are next to each other.
5. The person who loves yellow and the person who has black hair are next to each other.
6. The person who is a pizza lover is somewhere to the left of the person whose favorite color is green.
7. Peter is The person whose mother's name is Janelle.
8. The person who has blonde hair is the person who loves white.
9. The person who has red hair is Alice.
10. The person who loves white is somewhere to the right of the person who loves eating grilled cheese.
11. The person who loves white is not in the second house.
12. The person whose mother's name is Kailyn is the person who is a pizza lover.
13. The person whose mother's name is Holly is the person who has red hair.
 What is the value of attribute Name for the person whose attribute Color is green? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


Let $N$ be the number of positive integers that are less than or equal to $2003$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the [remainder](https://artofproblemsolving.com/wiki/index.php/Remainder) when $N$ is divided by $1000$.

Answer: 155


8. Suppose $A=\{1,2, \ldots, 20\}$. Call $B$ a visionary set of $A$ if $B \subseteq A, B$ contains at least one even integer, and $|B| \in B$, where $|B|$ is the cardinality of set $B$. How many visionary sets does $A$ have?

Answer: 524032


Example 1. If the total surface area of a cone is $n$ times the area of its inscribed sphere, find the ratio of the volume of the sphere to the volume of the cone.

Answer: \dfrac{1}{n}


The volume of the pyramid $ABCD$ is 1. Points $K, L$, and $M$ are taken on the edges $AD, BD, CD$ respectively, such that $2AK = KD, BL = 2LD$, and $2CM = 3MD$. Find the volume of the polyhedron $ABCKLM$.

#

Answer: \dfrac{41}{45}


Problem 10.2. There are 36 balls lying in a circle, each of which is either red or blue (balls of each of these colors are present). It is known that:

- for any red ball, there is exactly one red ball such that there is exactly one ball between them;
- for any red ball, there is exactly one red ball such that there are exactly three balls between them.

(a) (2 points) Suppose there are no two adjacent red balls. How many red balls can lie in a circle? Indicate all possible options.

(b) (2 points) Suppose there are two adjacent red balls. How many red balls can lie in a circle? Indicate all possible options.

Answer: 12


1. The number $a_{n}$ is formed by writing down the first $n$ squares of consecutive natural numbers in sequence. For example, $a_{11}=149162536496481100$ 121. Determine how many numbers divisible by twelve are among the numbers $a_{1}, a_{2}, \ldots, a_{100000}$.

Answer: 16667


35.2. Two motorcyclists set out simultaneously from points $A$ and $B$ towards each other and met 50 km from $B$. Upon arriving at points $A$ and $B$, the motorcyclists immediately turned around and met again 25 km from $A$. How many kilometers is it from $A$ to $B$?

Answer: 125


Problem 5. Let $p(x)$ and $q(x)$ be polynomials with $m \geq 2$ non-zero coefficients. If $\frac{p(x)}{q(x)}$ is not a constant function, find the least possible number of the non-zero coefficients of the polynomial $f(u, v)=p(u) q(v)-p(v) q(u)$.

Answer: 2(m - 1)


Problem 4. A set of at least three positive integers is called uniform if removing any of its elements the remaining set can be disjoint into two subsets with equal sums of elements. Find the minimal cardinality of a uniform set.

Peter Boyvalenkov, Emil Kolev

Answer: 7


779. At least how many points of a body must be fixed in order for the entire body to be immobile?

Answer: 3


[b]p1.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the square $ABCD$ is $14$ cm.
[img]https://cdn.artofproblemsolving.com/attachments/1/1/0f80fc5f0505fa9752b5c9e1c646c49091b4ca.png[/img]

[b]p2.[/b] If $a, b$, and $c$ are numbers so that $a + b + c = 0$ and $a^2 + b^2 + c^2 = 1$. Compute $a^4 + b^4 + c^4$.

[b]p3.[/b] A given fraction $\frac{a}{b}$ ($a, b$ are positive integers, $a \ne b$) is transformed by the following rule: first, $1$ is added to both the numerator and the denominator, and then the numerator and the denominator of the new fraction are each divided by their greatest common divisor (in other words, the new fraction is put in simplest form). Then the same transformation is applied again and again. Show that after some number of steps the denominator and the numerator differ exactly by $1$.

[b]p4.[/b] A goat uses horns to make the holes in a new $30\times 60$ cm large towel. Each time it makes two new holes. Show that after the goat repeats this $61$ times the towel will have at least two holes whose distance apart is less than $6$ cm.

[b]p5.[/b] You are given $555$ weights weighing $1$ g, $2$ g, $3$ g, $...$ , $555$ g. Divide these weights into three groups whose total weights are equal.

[b]p6.[/b] Draw on the regular $8\times 8$ chessboard a circle of the maximal possible radius that intersects only black squares (and does not cross white squares). Explain why no larger circle can satisfy the condition.

PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Answer: \dfrac{1}{2}


2B. On Monday, three bananas cost as much as a lemon and an orange together. On Tuesday, the prices of all fruits were reduced by the same amount of money, two oranges cost as much as three bananas and one lemon, and the price of half a lemon was 5 denars.

What was the price of one orange on Monday?

Answer: 20


6.8 In the school football championship, each team must play a match against every other team. Each match's winning team gets 2 points, a draw gives each team 1 point, and the losing team gets 0 points. It is known that one team scored the most points (every other team scored fewer points than this team), but it won fewer matches than any other team. How many teams participated at the minimum?

Answer: 6


699. Form the equation for the coordinates of points $(x, y)$ that are equidistant from $(-2,2)$ and $(2,0)$.

Answer: y = 2x + 1


4. In space, four lines satisfy that the angle between any two lines is $\theta$. Then $\theta=$ $\qquad$ .

Answer: \arccos \frac{1}{3}


2. A trading company purchases a certain type of fruit at a cost of 20 yuan per kilogram. After market research, it is found that the selling price $p$ (yuan per kilogram) of this fruit over the next 48 days is related to time $t$ (days) by the function
$$
p=\frac{1}{4} t+30(1 \leqslant t \leqslant 48, t \in \mathbf{Z}),
$$

and the daily sales volume $y$ (kilograms) is related to time $t$ (days) by the function
$$
y=120-2 t(1 \leqslant t \leqslant 48, t \in \mathbf{Z}) \text {. }
$$

In the first 30 days of actual sales, the company decides to donate $n$ yuan of profit for every kilogram of fruit sold to a "targeted poverty alleviation" program. It is observed that: in the sales process over the first 30 days, only in the first 28 days, the daily sales profit $W$ (yuan) after deducting the donation increases with the increase in time $t$ (days). Then the range of values for $n$ is

Answer: (8.75,9.25]


10.081. The area of a rectangle is 9 cm $^{2}$, and the measure of one of the angles formed by the diagonals is $120^{\circ}$ (Fig. 10.79). Find the sides of the rectangle.

Answer: 3^{\frac{5}{4}}


3. Fill the numbers $1,2,3, \ldots, 9,10$ into 10 circles that form a rectangle, such that the sum of the numbers on each side of the rectangle is equal. The maximum sum is ( ).

Answer: 22


5. For the Day of the Russian Flag, the seller decided to decorate the store window with 10 horizontal strips of fabric in three colors. At the same time, he follows two conditions:
1) strips of the same color should not hang next to each other;
2) each blue strip must hang between a white and a red one.

In how many ways can he do this?

Answer: 110


3. Zhenya drew several different lines on the plane. How many lines did Zhenya draw if it is known that they divided the plane into 466 parts? In your answer, indicate the smallest and largest possible values in any order, separated by a semicolon.

Answer: 30


8. Triangle $A B C$ with $\angle B A C>90^{\circ}$ has $A B=5$ and $A C=7$. Points $D$ and $E$ lie on segment $B C$ such that $B D=D E=E C$. If $\angle B A C+\angle D A E=180^{\circ}$, compute $B C$.

Answer: \sqrt{111}


2. (2 points) In the quarry, there was a pile of 20160000 sand grains. A truck carried away from the quarry a quantity of sand grains per trip, which was some power of 9 (including possibly $9^{0}=1$). Could it have carried away the entire pile of sand in exactly 2000 trips?

Answer: Yes


Example 5 Given an arithmetic sequence $\left\{a_{n}\right\}$. Try to find the sum of the first $n$ terms of the sequence: $a_{1}, a_{1}+a_{2}, a_{1}+a_{2}+a_{3}, \cdots, a_{1}+a_{2}+\cdots$ $+a_{n}$.

Answer: \dfrac{n(n + 1)(2a_1 + a_n)}{6}


16. In an addition equation, each addend is composed of the digits 0 and 6, and the result of the calculation is 20222022. The addends in this equation must contain at least $\qquad$ digit 6.

Answer: 26


There is a safe that can be opened by entering a secret code consisting of $n$ digits, each of them is $0$ or $1$. Initially, $n$ zeros were entered, and the safe is closed (so, all zeros is not the secret code).

In one attempt, you can enter an arbitrary sequence of $n$ digits, each of them is $0$ or $1$. If the entered sequence matches the secret code, the safe will open. If the entered sequence matches the secret code in more positions than the previously entered sequence, you will hear a click. In any other cases the safe will remain locked and there will be no click.

Find the smallest number of attempts that is sufficient to open the safe in all cases.

Answer: n


Gálnierin $\Gamma$.

In each cell of a secret $n \times n$ table, one of the digits from 1 to 9 is written. From these, $n$-digit numbers are formed, written from left to right in the rows and from top to bottom in the columns. Petya wants to write an $n$-digit number without zeros in its representation, such that neither this number nor its reverse matches any of the $2n$ numbers in the rows and columns of the table. What is the minimum number of cells Petya needs to know the digits of to achieve this?

Answer: n


Bbotin D.A.

Mosmetrostroy hired two excavators to dig a tunnel. One of them can dig twice as much as the other in an hour, and they are paid the same amount per hour of work according to the contract. Which would be cheaper -
joint work of the excavators from both sides until they meet, or digging half of the tunnel by each excavator in turn?

Answer: \text{joint work}


Initially, a natural number $n$ is written on the blackboard. Then, at each minute, [i]Neymar[/i] chooses a divisor $d>1$ of $n$, erases $n$, and writes $n+d$. If the initial number on the board is $2022$, what is the largest composite number that [i]Neymar[/i] will never be able to write on the blackboard?

Answer: 2033


5. Two equal cones are arranged so that the axis of each is the generatrix of the other. The angles at the vertices in the axial sections of these cones are each $90^{\circ}$. Find the angle between the two generatrices along which these cones intersect.

Answer: \arccos(3 - 2\sqrt{2})


Problem 5.2. During a physical education class, 25 students from 5B class lined up. Each of the students is either an excellent student who always tells the truth, or a troublemaker who always lies.

Excellent student Vlad stood in the 13th place. Everyone except Vlad stated: "There are exactly 6 troublemakers between me and Vlad." How many troublemakers are there in the line?

Answer: 12


5.2. Write the numbers $1,2,3,4,5,6,8,9$ (that is, all numbers from 1 to 9 except 7) in a row so that in any pair of adjacent numbers, one divides the other.

Answer: 5,1,9,3,6,2,4,8


8.1. A dandelion blooms in the morning, remains yellow for this and the next day, turns white on the third morning, and by evening of the third day, it has lost its petals. Yesterday afternoon, there were 20 yellow and 14 white dandelions on the meadow, and today there are 15 yellow and 11 white. How many yellow dandelions were there on the meadow the day before yesterday? Provide all possible answers and prove that there are no others.

Answer: 25


9. (This question is worth 16 points) Let $O$ be the circumcenter of acute $\triangle A B C$, and the areas of $\triangle B O C, \triangle C O A, \triangle A O B$ form an arithmetic sequence. Find the minimum value of $\tan A+2 \tan C$.

---

The translation is provided as requested, maintaining the original format and line breaks.

Answer: 2\sqrt{6}


In a six-page homework assignment, we need to place four figures. The order of the figures is fixed, and a maximum of two figures can be on one page. In how many ways can this be done? (We do not consider the position of the figures on a page.)

Answer: 90


Draw a regular hexagon. Then make a square from each edge of the hexagon. Then form equilateral triangles by drawing an edge between every pair of neighboring squares. If this figure is continued symmetrically off to infinity, what is the ratio between the number of triangles and the number of squares?

Answer: 1


7. Let $a, b, c$ be three distinct real numbers such that the equations $x^{2}+a x+1=0$ and $x^{2}+b x+c=0$ have a common real root, and the equations $x^{2}+x+a=0$ and $x^{2}+c x+b=0$ also have a common real root, then the value of $a+b+c$ is $\qquad$.

Answer: -3


B4. Let $k$ be a circle with radius $R$. On the diameter $AB$ are points $C$ and $D$ such that $|AC|=|CD|=|DB|$. Over $AC$ and $AD$ are semicircles drawn on one side of the segment $AB$, and over $DB$ and $CB$ on the other side (see figure). Express the area of the shaded figure in terms of $R$.

![](https://cdn.mathpix.com/cropped/2024_06_07_ce887fced8b2a2bf0894g-09.jpg?height=377&width=463&top_left_y=1382&top_left_x=1456)

## Solutions to Problems and Scoring

A contestant who arrives at the solution by any correct method (even if the scoring does not provide for it) receives all possible points.

For a correct method, each procedure that

- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete.

A contestant who has only partially solved the problem, from otherwise correct solving procedures but does not show the way to the final solution of the problem, cannot receive more than half of the possible points.

## First Year

I. PART

| Problem | A1 | A2 | A3 | A4 | A5 | A6 |
| :--- | ---: | ---: | ---: | ---: | ---: | :---: |
| Answer | D | E | B | C | B | B |

Answer: \dfrac{\pi R^2}{3}


6.1. Find the largest six-digit number, all digits of which are different, and each of the digits, except for the extreme ones, is either the sum or the difference of the adjacent digits.

Answer: 972538


2. The two positive integers $a, b$ with $a>b$ are such that $a \%$ of $b \%$ of $a$ and $b \%$ of $a \%$ of $b$ differ by 0.003 . Find all possible pairs $(a, b)$.

Answer: (5, 2)


Determine which digit is in the 1000th place after the decimal point in the decimal expansion of the number $\frac{9}{28}$.

(M. Krejčová)

Hint. What does the decimal expansion of the given number look like?

Answer: 4


Eight, (Full marks 10 points) A student buys a certain type of pencil. When he buys $x$ pencils, he pays $y$ yuan ($x, y$ are both integers). The clerk says: "If you buy 10 more, I will charge you a total of 2 yuan. This way, for every 30 pencils you buy, you can save 2 yuan." Find $x$ and $y$.

---

The above text is translated into English, preserving the original text's line breaks and format.

Answer: x = 5


All sides of a convex polygon \( m \) with perimeter \( p = 12 \) and area \( s \) are shifted outward by a distance of 1. The perimeter of the larger polygon \( M \) formed by the shifted sides is denoted by \( P \), and its area by \( S \). Prove that

a) \( P - p > 6 \)

b) \( S - s > 15 \)

c) If the polygon \( m \) is a quadrilateral, then \( P - p \geq 8 \) and \( S - s \geq 16 \)

d) If all angles of the polygon \( m \) are obtuse, then \( P - p < 8 \) and \( S - s < 16 \)

Answer: P - p > 6


Suppose that $a$ is real number. Sequence $a_1,a_2,a_3,....$ satisfies
$$a_1=a, a_{n+1} =
\begin{cases}
a_n - \frac{1}{a_n}, & a_n\ne 0 \\
0, & a_n=0
\end{cases}
(n=1,2,3,..)$$
Find all possible values of $a$ such that $|a_n|<1$ for all positive integer $n$.

Answer: 0


A sequence \( U_{1}, U_{2}, U_{3}, \ldots \) is defined as follows:
- \( U_{1} = 2 \);
- if \( U_{n} \) is prime then \( U_{n+1} \) is the smallest positive integer not yet in the sequence;
- if \( U_{n} \) is not prime then \( U_{n+1} \) is the smallest prime not yet in the sequence.
The integer \( k \) is the smallest such that \( U_{k+1} - U_{k} > 10 \).
What is the value of \( k \times U_{k} \)?

Answer: 270