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`, `Arnold`, `Bob`, `Eric`
 - Each person has a unique hobby: `photography`, `cooking`, `knitting`, `gardening`, `painting`
 - Each person has a unique favorite drink: `root beer`, `milk`, `water`, `coffee`, `tea`

## Clues:
1. Eric is the coffee drinker.
2. The tea drinker is the person who paints as a hobby.
3. The person who enjoys knitting is not in the fourth house.
4. Peter is not in the fourth house.
5. Eric is somewhere to the right of the root beer lover.
6. Arnold is the person who loves cooking.
7. The one who only drinks water is somewhere to the right of the person who enjoys gardening.
8. There is one house between Bob and the person who paints as a hobby.
9. The person who enjoys gardening is directly left of the root beer lover.
10. The photography enthusiast is the one who only drinks water.
 What is the value of attribute House for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: 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: `Bob`, `Carol`, `Alice`, `Eric`, `Arnold`, `Peter`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `soup`, `grilled cheese`, `stew`, `stir fry`
 - Each person has a unique level of education: `associate`, `doctorate`, `trade school`, `high school`, `master`, `bachelor`
 - The people keep unique animals: `horse`, `fish`, `rabbit`, `cat`, `bird`, `dog`
 - Each person lives in a unique style of house: `ranch`, `victorian`, `modern`, `craftsman`, `mediterranean`, `colonial`

## Clues:
1. The person residing in a Victorian house is in the fourth house.
2. The fish enthusiast is in the first house.
3. The person in a ranch-style home is the bird keeper.
4. The person residing in a Victorian house is somewhere to the left of the person who loves the soup.
5. Carol is directly left of the person who keeps horses.
6. The person who loves stir fry is the person with a doctorate.
7. The person with a master's degree is somewhere to the left of the person with a doctorate.
8. There is one house between the person in a Mediterranean-style villa and the person with a bachelor's degree.
9. The person who attended trade school is the cat lover.
10. The person with a master's degree is the person who is a pizza lover.
11. The person who is a pizza lover is somewhere to the right of Alice.
12. The person living in a colonial-style house is the person who loves the stew.
13. Arnold is the person with a master's degree.
14. There is one house between Arnold and the person who loves eating grilled cheese.
15. The person in a modern-style house and the person who attended trade school are next to each other.
16. Eric is directly left of the person with a doctorate.
17. Alice and the person in a Mediterranean-style villa are next to each other.
18. The person with an associate's degree is directly left of Eric.
19. The dog owner is Bob.
 What is the value of attribute House for the person whose attribute Food is pizza? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Carol`, `Arnold`, `Bob`, `Alice`
 - Each person has a favorite color: `blue`, `red`, `green`, `yellow`, `purple`, `white`
 - Each person prefers a unique type of vacation: `city`, `beach`, `cruise`, `cultural`, `mountain`, `camping`
 - People have unique hair colors: `blonde`, `black`, `gray`, `brown`, `red`, `auburn`
 - Each person has an occupation: `artist`, `lawyer`, `engineer`, `teacher`, `doctor`, `nurse`
 - They all have a unique favorite flower: `roses`, `iris`, `daffodils`, `carnations`, `tulips`, `lilies`

## Clues:
1. The person who loves a bouquet of daffodils is the person who is an artist.
2. The person who enjoys camping trips is the person whose favorite color is green.
3. The person who has brown hair is in the first house.
4. The person who loves blue is somewhere to the right of the person who enjoys mountain retreats.
5. The person who loves the vase of tulips is the person who enjoys mountain retreats.
6. Eric is in the second house.
7. The person who likes going on cruises and Arnold are next to each other.
8. The person who loves yellow and the person who enjoys camping trips are next to each other.
9. The person who loves yellow is somewhere to the left of Peter.
10. The person who is a lawyer and Arnold are next to each other.
11. The person who has red hair is Arnold.
12. Alice is the person who is a doctor.
13. The person who loves purple is directly left of the person who loves the vase of tulips.
14. The person who has brown hair is somewhere to the left of the person who is an engineer.
15. The person who has gray hair is somewhere to the left of the person whose favorite color is red.
16. The person who is an artist is somewhere to the right of Arnold.
17. Alice is the person who loves the boquet of lilies.
18. The person whose favorite color is red is directly left of the person who prefers city breaks.
19. The person who prefers city breaks is somewhere to the right of the person who has blonde hair.
20. There is one house between Alice and Bob.
21. The person who loves the rose bouquet is directly left of the person who is a nurse.
22. The person whose favorite color is green is in the fourth house.
23. The person who loves beach vacations is somewhere to the left of the person who has red hair.
24. The person who has black hair is Alice.
25. The person who loves blue is the person who loves a carnations arrangement.
 What is the value of attribute Occupation for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}

Answer: teacher


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Alice`, `Bob`
 - People use unique phone models: `google pixel 6`, `samsung galaxy s21`, `huawei p50`, `oneplus 9`, `iphone 13`

## Clues:
1. Eric is the person who uses a Samsung Galaxy S21.
2. Alice is in the fifth house.
3. The person who uses a Samsung Galaxy S21 is somewhere to the left of the person who uses a Google Pixel 6.
4. Arnold is directly left of the person who uses a OnePlus 9.
5. The person who uses a Huawei P50 is Bob.
6. The person who uses an iPhone 13 is directly left of the person who uses a OnePlus 9.
7. The person who uses a Huawei P50 is in the first house.
8. The person who uses a Google Pixel 6 is somewhere to the left of the person who uses a OnePlus 9.
 What is the value of attribute Name for the person whose attribute PhoneModel is google pixel 6? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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

Answer: stew


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Arnold`, `Alice`, `Peter`
 - People own unique car models: `bmw 3 series`, `toyota camry`, `honda civic`, `tesla model 3`, `ford f150`
 - People use unique phone models: `google pixel 6`, `huawei p50`, `oneplus 9`, `iphone 13`, `samsung galaxy s21`
 - Each person has a unique hobby: `painting`, `gardening`, `knitting`, `photography`, `cooking`
 - Each person has an occupation: `lawyer`, `artist`, `teacher`, `engineer`, `doctor`
 - Each person has a favorite color: `white`, `blue`, `green`, `red`, `yellow`

## Clues:
1. The person who owns a Ford F-150 is directly left of the person who is a teacher.
2. The photography enthusiast is Arnold.
3. The person who uses a Huawei P50 is Eric.
4. The person who enjoys gardening is somewhere to the right of the person who loves blue.
5. The person who enjoys knitting is the person who loves white.
6. The person who owns a Ford F-150 is the person whose favorite color is red.
7. There is one house between the person who owns a BMW 3 Series and the person who loves yellow.
8. The person who is an artist is the person who uses a OnePlus 9.
9. The person who is a doctor is Arnold.
10. The person who enjoys gardening is the person who uses a Huawei P50.
11. There is one house between the person who loves blue and the person who loves yellow.
12. The person who owns a BMW 3 Series and the person who uses a Google Pixel 6 are next to each other.
13. The person who uses a Google Pixel 6 is directly left of the person who owns a Honda Civic.
14. Peter is the person who is an engineer.
15. The person who uses a Samsung Galaxy S21 is the person who owns a Toyota Camry.
16. The person who owns a Honda Civic is Alice.
17. The person who owns a Tesla Model 3 is the person who paints as a hobby.
 What is the value of attribute Occupation for the person whose attribute CarModel is ford f150? Please reason step by step, and put your final answer within \boxed{}

Answer: lawyer


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`, `Bob`, `Alice`
 - People have unique heights: `average`, `very tall`, `very short`, `short`, `tall`
 - Everyone has a unique favorite cigar: `prince`, `dunhill`, `blends`, `pall mall`, `blue master`
 - Everyone has a favorite smoothie: `lime`, `cherry`, `dragonfruit`, `watermelon`, `desert`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `iphone 13`, `huawei p50`, `google pixel 6`

## Clues:
1. The Prince smoker is the Desert smoothie lover.
2. There is one house between Eric and Alice.
3. The person who is short is the person who smokes many unique blends.
4. The person who uses an iPhone 13 is directly left of the person who smokes Blue Master.
5. The person who has an average height is the Dunhill smoker.
6. Eric is the person who is very tall.
7. Arnold is directly left of the person who uses a Huawei P50.
8. Bob is not in the fourth house.
9. Eric is directly left of the person who likes Cherry smoothies.
10. Bob is the Dunhill smoker.
11. The Dragonfruit smoothie lover is Bob.
12. The person who uses an iPhone 13 and the person who uses a OnePlus 9 are next to each other.
13. The person who uses a Samsung Galaxy S21 is the person who is short.
14. There are two houses between the person who is very tall and the Dragonfruit smoothie lover.
15. The person who uses an iPhone 13 is Eric.
16. The Desert smoothie lover is somewhere to the left of the person who drinks Lime smoothies.
17. Arnold and the person who is very short are next to each other.
 What is the value of attribute Height for the person whose attribute PhoneModel is samsung galaxy s21? Please reason step by step, and put your final answer within \boxed{}

Answer: short


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`, `Eric`, `Alice`, `Peter`
 - People own unique car models: `honda civic`, `tesla model 3`, `toyota camry`, `ford f150`
 - They all have a unique favorite flower: `roses`, `daffodils`, `lilies`, `carnations`
 - Each person has a unique favorite drink: `milk`, `water`, `tea`, `coffee`
 - People have unique heights: `very short`, `average`, `tall`, `short`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`

## Clues:
1. The person who owns a Toyota Camry is somewhere to the left of the person who owns a Ford F-150.
2. The coffee drinker is the person who owns a Ford F-150.
3. The person who has an average height is in the third house.
4. The person whose favorite color is green and the person who owns a Tesla Model 3 are next to each other.
5. There are two houses between the person who loves the rose bouquet and the person who owns a Tesla Model 3.
6. The person who is very short is the tea drinker.
7. Alice is the person who loves white.
8. The person who loves the boquet of lilies is the person who owns a Tesla Model 3.
9. Peter is the person whose favorite color is red.
10. The person who owns a Toyota Camry is Eric.
11. The person who is short is somewhere to the left of the person who loves the boquet of lilies.
12. The person who loves a bouquet of daffodils is the person who loves white.
13. The coffee drinker is in the second house.
14. The person who owns a Tesla Model 3 is the person who likes milk.
 What is the value of attribute Height for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: very short


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`
 - People have unique favorite music genres: `classical`, `pop`, `rock`
 - They all have a unique favorite flower: `carnations`, `daffodils`, `lilies`
 - People have unique heights: `average`, `very short`, `short`
 - Everyone has a unique favorite cigar: `prince`, `pall mall`, `blue master`

## Clues:
1. Peter is the person who has an average height.
2. The person who smokes Blue Master is the person who is short.
3. The person who loves a bouquet of daffodils is directly left of the person partial to Pall Mall.
4. The person who loves pop music is the person who loves a bouquet of daffodils.
5. The person whose mother's name is Janelle is the person who loves the boquet of lilies.
6. The person whose mother's name is Aniya is in the first house.
7. The person who has an average height is in the second house.
8. The person who loves pop music and the person who loves classical music are next to each other.
9. Eric is in the third house.
10. The person whose mother's name is Holly is the person who is very short.
 What is the value of attribute Mother for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: Janelle


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Alice`, `Bob`, `Peter`
 - Each person prefers a unique type of vacation: `mountain`, `city`, `cruise`, `beach`, `camping`
 - Each person has a unique level of education: `doctorate`, `high school`, `bachelor`, `associate`, `master`
 - Each person has a favorite color: `blue`, `red`, `white`, `yellow`, `green`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `oneplus 9`, `huawei p50`, `samsung galaxy s21`
 - Everyone has something unique for lunch: `grilled cheese`, `stir fry`, `pizza`, `spaghetti`, `stew`

## Clues:
1. The person who loves the stew is not in the first house.
2. There are two houses between the person who loves stir fry and the person with an associate's degree.
3. The person who enjoys mountain retreats is the person with a bachelor's degree.
4. The person with a doctorate is somewhere to the right of Bob.
5. The person who uses a Samsung Galaxy S21 is in the third house.
6. Eric is the person with a doctorate.
7. The person with a doctorate is in the third house.
8. The person who loves stir fry is the person with a bachelor's degree.
9. The person with a doctorate is the person who is a pizza lover.
10. The person whose favorite color is green is somewhere to the right of Peter.
11. The person who enjoys camping trips is the person who uses an iPhone 13.
12. The person who likes going on cruises is Alice.
13. There is one house between the person with a high school diploma and the person who uses a Samsung Galaxy S21.
14. The person who uses a Google Pixel 6 is Arnold.
15. The person who uses a OnePlus 9 is somewhere to the right of the person who uses a Huawei P50.
16. Arnold is the person who loves eating grilled cheese.
17. The person who loves eating grilled cheese is not in the fourth house.
18. There are two houses between the person with a bachelor's degree and the person whose favorite color is red.
19. The person who loves beach vacations is somewhere to the right of the person who prefers city breaks.
20. The person whose favorite color is green is not in the second house.
21. The person who loves blue is somewhere to the right of Peter.
22. There is one house between the person who enjoys camping trips and the person who loves yellow.
 What is the value of attribute House for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Alice`, `Carol`, `Arnold`, `Eric`
 - Each person has a unique favorite drink: `milk`, `coffee`, `tea`, `root beer`, `boba tea`, `water`
 - They all have a unique favorite flower: `iris`, `roses`, `carnations`, `daffodils`, `lilies`, `tulips`
 - Each person has a unique birthday month: `sept`, `jan`, `feb`, `april`, `may`, `mar`
 - Each person has a unique hobby: `photography`, `cooking`, `woodworking`, `gardening`, `knitting`, `painting`

## Clues:
1. The root beer lover is in the fifth house.
2. The person whose birthday is in February is Peter.
3. The boba tea drinker is not in the fourth house.
4. The person who loves the boquet of iris is somewhere to the right of the person whose birthday is in April.
5. The coffee drinker is Arnold.
6. Eric is the photography enthusiast.
7. The person who paints as a hobby is the person who loves the vase of tulips.
8. The person whose birthday is in April is in the third house.
9. The tea drinker is the person who loves a bouquet of daffodils.
10. There are two houses between the person who loves the boquet of lilies and the person who likes milk.
11. The person whose birthday is in March is somewhere to the left of the person who enjoys gardening.
12. There are two houses between the person who loves a bouquet of daffodils and the person who paints as a hobby.
13. The person whose birthday is in February is in the fourth house.
14. The person who loves cooking is not in the fourth house.
15. The person who enjoys gardening is somewhere to the left of the person who loves a carnations arrangement.
16. The person whose birthday is in January is Bob.
17. Carol is the person whose birthday is in April.
18. The woodworking hobbyist is in the second house.
19. The person who likes milk is in the sixth house.
20. Peter is the person who loves a carnations arrangement.
21. The person whose birthday is in May is the woodworking hobbyist.
 What is the value of attribute House for the person whose attribute Flower is daffodils? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Peter`, `Alice`, `Eric`, `Carol`
 - The people are of nationalities: `swede`, `dane`, `german`, `norwegian`, `chinese`, `brit`
 - Each person has a favorite color: `red`, `green`, `purple`, `blue`, `white`, `yellow`

## Clues:
1. Alice is directly left of Bob.
2. The person whose favorite color is green is the British person.
3. Alice is somewhere to the left of the German.
4. The person who loves blue is the Dane.
5. Eric is the person who loves purple.
6. Carol is in the second house.
7. The Swedish person is not in the second house.
8. The Swedish person is not in the first house.
9. The person who loves white is directly left of the Norwegian.
10. There are two houses between Alice and the person who loves yellow.
11. The Dane is in the fifth house.
12. The person who loves blue is Peter.
 What is the value of attribute Nationality for the person whose attribute Name is Arnold? 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: `Bob`, `Peter`, `Alice`, `Eric`, `Arnold`
 - Each person has a unique type of pet: `hamster`, `fish`, `bird`, `cat`, `dog`
 - Each person lives in a unique style of house: `colonial`, `ranch`, `victorian`, `modern`, `craftsman`
 - Each person has a unique birthday month: `mar`, `sept`, `jan`, `feb`, `april`
 - The people keep unique animals: `dog`, `cat`, `fish`, `bird`, `horse`

## Clues:
1. There is one house between Alice and the fish enthusiast.
2. The person in a ranch-style home is not in the fifth house.
3. The person whose birthday is in March is Alice.
4. The person in a ranch-style home is not in the fourth house.
5. Arnold is somewhere to the right of the person whose birthday is in April.
6. The person who has a cat is the person who keeps horses.
7. There are two houses between the cat lover and the person living in a colonial-style house.
8. Eric is somewhere to the left of the person whose birthday is in September.
9. The person whose birthday is in April is directly left of the person who keeps horses.
10. Alice is the person residing in a Victorian house.
11. The person whose birthday is in March and the cat lover are next to each other.
12. Peter is the person whose birthday is in February.
13. The cat lover is the person who owns a dog.
14. The bird keeper is somewhere to the left of the person who keeps a pet bird.
15. Bob is the person who keeps a pet bird.
16. Eric is in the fourth house.
17. The person who has a cat is somewhere to the left of the person with a pet hamster.
18. The person who keeps a pet bird is the person in a modern-style house.
 What is the value of attribute House for the person whose attribute Pet is fish? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Eric`, `Peter`, `Alice`, `Bob`, `Arnold`
 - Everyone has something unique for lunch: `spaghetti`, `soup`, `grilled cheese`, `stir fry`, `stew`, `pizza`
 - Each person has a favorite color: `green`, `purple`, `blue`, `red`, `white`, `yellow`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Aniya`, `Kailyn`, `Sarah`, `Janelle`
 - Each person lives in a unique style of house: `colonial`, `craftsman`, `modern`, `ranch`, `victorian`, `mediterranean`

## Clues:
1. The person whose mother's name is Janelle is the person living in a colonial-style house.
2. Peter is the person whose favorite color is green.
3. The person who loves blue is directly left of the person who loves the spaghetti eater.
4. The person whose mother's name is Penny is directly left of the person whose favorite color is red.
5. The person who loves blue is The person whose mother's name is Penny.
6. The person whose mother's name is Aniya is directly left of the person whose favorite color is green.
7. The person who loves white is somewhere to the left of The person whose mother's name is Janelle.
8. Eric is the person who loves eating grilled cheese.
9. The person whose mother's name is Holly is directly left of Alice.
10. The person whose mother's name is Holly and the person who loves the stew are next to each other.
11. Alice is the person in a ranch-style home.
12. Carol is not in the sixth house.
13. The person who loves stir fry is the person who loves purple.
14. The person whose favorite color is red and the person living in a colonial-style house are next to each other.
15. The person who loves the soup is Peter.
16. The person who is a pizza lover is somewhere to the left of The person whose mother's name is Sarah.
17. The person residing in a Victorian house is the person who loves eating grilled cheese.
18. The person who loves the spaghetti eater is not in the fifth house.
19. There are two houses between the person who loves eating grilled cheese and The person whose mother's name is Aniya.
20. Arnold is the person whose favorite color is red.
21. The person in a modern-style house is Arnold.
22. The person who loves yellow is the person in a Mediterranean-style villa.
 What is the value of attribute HouseStyle for the person whose attribute Mother is Kailyn? Please reason step by step, and put your final answer within \boxed{}

Answer: victorian


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 House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: oneplus 9


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Eric`, `Carol`, `Alice`, `Peter`
 - Each person has a unique level of education: `doctorate`, `trade school`, `associate`, `high school`, `bachelor`, `master`
 - People own unique car models: `chevrolet silverado`, `ford f150`, `honda civic`, `tesla model 3`, `bmw 3 series`, `toyota camry`
 - Each person has a unique favorite drink: `tea`, `water`, `boba tea`, `coffee`, `root beer`, `milk`
 - They all have a unique favorite flower: `carnations`, `iris`, `lilies`, `tulips`, `daffodils`, `roses`

## Clues:
1. Eric is the one who only drinks water.
2. The person with an associate's degree is the person who owns a Toyota Camry.
3. The boba tea drinker is in the fourth house.
4. The person who loves the vase of tulips is directly left of the person who owns a Tesla Model 3.
5. The person with a master's degree is somewhere to the left of the person who owns a BMW 3 Series.
6. The person who owns a Toyota Camry is directly left of the person who loves a bouquet of daffodils.
7. The person who loves the boquet of iris is somewhere to the left of the root beer lover.
8. The person who likes milk is somewhere to the right of the person who owns a Honda Civic.
9. Carol is the person who owns a Honda Civic.
10. The person with a high school diploma is in the second house.
11. The person with a master's degree is the boba tea drinker.
12. The person who owns a Tesla Model 3 is Arnold.
13. The person who loves the boquet of lilies is the person who owns a Tesla Model 3.
14. The person who loves a carnations arrangement is somewhere to the left of Alice.
15. The person with a bachelor's degree is somewhere to the right of the person who likes milk.
16. The one who only drinks water is the person who owns a Ford F-150.
17. Carol is somewhere to the left of the person with a doctorate.
18. There are two houses between the person who owns a Toyota Camry and the person who loves the vase of tulips.
19. The person who owns a Tesla Model 3 is somewhere to the left of the coffee drinker.
20. Bob is in the third house.
 What is the value of attribute Flower for the person whose attribute CarModel is honda civic? Please reason step by step, and put your final answer within \boxed{}

Answer: tulips


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`
 - Everyone has a unique favorite cigar: `blue master`, `dunhill`, `pall mall`, `prince`
 - Everyone has something unique for lunch: `stew`, `spaghetti`, `pizza`, `grilled cheese`
 - People have unique favorite book genres: `fantasy`, `mystery`, `science fiction`, `romance`

## Clues:
1. Arnold is the person who loves eating grilled cheese.
2. The person who smokes Blue Master is Alice.
3. Arnold is in the second house.
4. The person who is a pizza lover and the person who loves the stew are next to each other.
5. Arnold is the person who loves science fiction books.
6. The person partial to Pall Mall is the person who loves eating grilled cheese.
7. Eric is somewhere to the left of Alice.
8. The person who is a pizza lover is somewhere to the left of the Dunhill smoker.
9. The person who loves fantasy books is the person who loves the stew.
10. The person who loves mystery books is not in the first house.
 What is the value of attribute House for the person whose attribute Food is stew? 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: `Alice`, `Bob`, `Peter`, `Eric`, `Carol`, `Arnold`
 - People own unique car models: `toyota camry`, `ford f150`, `bmw 3 series`, `tesla model 3`, `chevrolet silverado`, `honda civic`
 - Each person has a unique hobby: `knitting`, `gardening`, `woodworking`, `painting`, `photography`, `cooking`
 - Each person has a unique favorite drink: `coffee`, `root beer`, `milk`, `water`, `tea`, `boba tea`

## Clues:
1. There is one house between the coffee drinker and the woodworking hobbyist.
2. The person who enjoys knitting is in the first house.
3. There are two houses between the person who loves cooking and the person who likes milk.
4. The person who paints as a hobby is the person who owns a Toyota Camry.
5. Arnold is the tea drinker.
6. There is one house between the person who owns a Chevrolet Silverado and the person who owns a BMW 3 Series.
7. The woodworking hobbyist is somewhere to the right of Carol.
8. The person who paints as a hobby is not in the fourth house.
9. The boba tea drinker is in the third house.
10. Bob is somewhere to the left of the person who likes milk.
11. The person who owns a Chevrolet Silverado is not in the sixth house.
12. The person who owns a Toyota Camry is the root beer lover.
13. The person who owns a Chevrolet Silverado is Alice.
14. Peter is the person who owns a Toyota Camry.
15. The root beer lover is not in the sixth house.
16. The person who owns a Tesla Model 3 is not in the first house.
17. The person who enjoys gardening is Eric.
18. The tea drinker is somewhere to the left of the one who only drinks water.
19. The person who owns a Honda Civic and the boba tea drinker are next to each other.
20. Bob is in the third house.
 What is the value of attribute House for the person whose attribute Drink is tea? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Bob`, `Alice`, `Peter`, `Eric`, `Arnold`
 - Each person has a unique level of education: `master`, `doctorate`, `high school`, `bachelor`, `trade school`, `associate`
 - People have unique favorite book genres: `romance`, `fantasy`, `historical fiction`, `science fiction`, `mystery`, `biography`
 - Each mother is accompanied by their child: `Alice`, `Samantha`, `Timothy`, `Meredith`, `Bella`, `Fred`
 - Everyone has something unique for lunch: `stew`, `soup`, `stir fry`, `spaghetti`, `pizza`, `grilled cheese`
 - People have unique favorite sports: `volleyball`, `basketball`, `tennis`, `baseball`, `swimming`, `soccer`

## Clues:
1. Alice is the person who loves swimming.
2. The person's child is named Alice is directly left of the person who loves historical fiction books.
3. The person who loves the stew is the person who loves historical fiction books.
4. The person with a doctorate is the person who loves the soup.
5. The person who loves science fiction books is in the sixth house.
6. The person who attended trade school is directly left of the person who loves historical fiction books.
7. The person who loves baseball and the person's child is named Bella are next to each other.
8. The person with a master's degree is somewhere to the left of the person who loves mystery books.
9. The person who loves soccer is somewhere to the left of the person who loves fantasy books.
10. The person who loves stir fry is the person with an associate's degree.
11. The person who is a pizza lover is Peter.
12. Bob is not in the sixth house.
13. The person's child is named Meredith is directly left of the person who loves fantasy books.
14. The person's child is named Samantha is Eric.
15. There is one house between the person who loves eating grilled cheese and Bob.
16. The person who loves eating grilled cheese is the person's child is named Samantha.
17. The person who loves volleyball is the person's child is named Samantha.
18. The person's child is named Fred is somewhere to the left of the person's child is named Samantha.
19. Bob is the person who loves tennis.
20. The person who loves soccer is directly left of Carol.
21. The person who loves eating grilled cheese is somewhere to the left of the person who loves mystery books.
22. The person's child is named Meredith is the person who loves biography books.
23. The person's child is named Meredith is the person who loves the soup.
24. Carol is the person with a high school diploma.
 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: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Carol`, `Bob`, `Peter`, `Eric`, `Alice`
 - People own unique car models: `bmw 3 series`, `honda civic`, `toyota camry`, `chevrolet silverado`, `ford f150`, `tesla model 3`

## Clues:
1. Carol is not in the third house.
2. Arnold is not in the third house.
3. Arnold is not in the fifth house.
4. The person who owns a Toyota Camry is somewhere to the right of the person who owns a Honda Civic.
5. Bob is somewhere to the right of Arnold.
6. Arnold is not in the second house.
7. The person who owns a Tesla Model 3 is not in the second house.
8. Alice is the person who owns a Chevrolet Silverado.
9. The person who owns a Chevrolet Silverado is not in the third house.
10. Alice is not in the sixth house.
11. The person who owns a Chevrolet Silverado is not in the fifth house.
12. Peter is somewhere to the left of Carol.
13. Bob is the person who owns a Tesla Model 3.
14. The person who owns a Chevrolet Silverado is not in the second house.
15. The person who owns a Ford F-150 is Arnold.
16. Carol is the person who owns a Honda Civic.
 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: ford f150


13. Place the addition and subtraction signs between the digits

$$
\begin{array}{lllllllll}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9
\end{array}
$$

so that the result is 100.

Answer: 123 - 45 - 67 + 89 = 100


2. (15 points) The bisectors of the obtuse angles at the base of a trapezoid intersect on the other base of the trapezoid. Find the area of the trapezoid if its height is 12 cm, and the lengths of the bisectors are 15 cm and $13 \mathrm{~cm}$.

Answer: \dfrac{1302}{5}


4. There is a simple pendulum, which has a period of $T=1$ second in summer. In winter, the pendulum length shortens by 0.01 cm. In winter, this pendulum is approximately faster by $\qquad$ seconds in 24 hours (rounded to 1 second).

Note: The formula for the period of a simple pendulum is $T=2 \pi \sqrt{\frac{l}{g}}$, where the period $T$ is in seconds, the pendulum length $l$ is in centimeters, and the gravitational acceleration $g=980$ cm/s${}^2$.

Answer: 17


In the figure, let $\triangle ABC$ have points $D, E, F$ on $BC, AC, AB$ respectively. If quadrilateral $AFDE$ is an inscribed quadrilateral, prove that: $\frac{4 S_{\triangle DEF}}{S_{\triangle ABC}} \leqslant\left(\frac{EF}{AD}\right)^{2}$.

Answer: \frac{4 S_{\triangle DEF}}{S_{\triangle ABC}} \leqslant\left(\frac{EF}{AD}\right)^{2}


A certain consumer product costs 120 yuan per item, and without additional tax, approximately 800,000 items are sold each year. If the government imposes an additional tax, where $r$ yuan is taxed for every 100 yuan of sales (this is called a tax rate of $r\%$), the annual sales volume will decrease by $\frac{20}{3}r$ ten thousand items. To ensure that the tax revenue from this business operation is no less than 2.56 million yuan each year, how should $r$ be determined?

Answer: [4, 8]


Zaslavsky A.A.

In triangle $A B C$, the bisectors $B B_{1}$ and $C C_{1}$ are drawn. It is known that the center of the circumcircle of triangle $B B_{1} C_{1}$ lies on the line $A C$. Find the angle $C$ of the triangle.

Answer: 120^\circ


$f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies the equation
\[f(x)f(y)-af(xy)=x+y\]
, for every real numbers $x,y$. Find all possible real values of $a$.

Answer: 1


6. Let a composition contain $n>1$ wagons with gold coins. There are two types of visually identical gold coins: genuine and fake. In each wagon, there are only coins of one type. Coins of the same type have the same mass, while coins of different types do not have the same mass. The mass of a genuine gold coin is known.

Determine the minimum number of measurements on a digital scale that can determine which wagons contain fake gold coins, as well as the mass of a fake gold coin.

(It is assumed that any number of coins can be taken from each wagon.)

(Miloš Milosavljević)

Answer: 2


Given 10 natural numbers: \(a_{1} < a_{2} < a_{3} < \cdots < a_{10}\). Prove that their least common multiple is not less than \(10 a_{1}\).

Answer: \text{LCM}(a_1, a_2, \ldots, a_{10}) \geq 10a_1


Problem 3. Senya cannot write some letters and always makes mistakes in them. In the word TETRAHEDRON he would make five mistakes, in the word DODECAHEDRON - six, and in the word ICOSAHEDRON - seven. How many mistakes would he make in the word OCTAHEDRON?

![](https://cdn.mathpix.com/cropped/2024_05_06_5f8bb630dfa91fa0065fg-2.jpg?height=41&width=471&top_left_y=799&top_left_x=802)

Answer: 5


Let there be a figure with $9$ disks and $11$ edges, as shown below.

We will write a real number in each and every disk. Then, for each edge, we will write the square of the difference between the two real numbers written in the two disks that the edge connects. We must write $0$ in disk $A$, and $1$ in disk $I$. Find the minimum sum of all real numbers written in $11$ edges.

Answer: \dfrac{20}{33}


3. Let $z_{1}, z_{2}, \cdots, z_{12}$ be the roots of the equation $z^{12}=2^{36}$. Let $w_{j} (j=1,2, \cdots, 12)$ be one of $z_{j}$ and i $z_{j}$. Then the maximum value of $\operatorname{Re} \sum_{j=1}^{12} w_{j}$ is . $\qquad$

Answer: 16(1 + \sqrt{3})


3. A hundred people are sitting in a circle. Some of them are knights, who always tell the truth, and the rest are liars, who always lie. For some natural number $k<100$, each of those sitting said: "The next $k$ people sitting to my right are liars." What could the number $k$ be? (S. Berlov)

Answer: 1


For example, the area of $\triangle ABC$ is $1$, points $D$ and $E$ are on sides $AB$ and $AC$ respectively, $BE$ and $CD$ intersect at point $P$, and the area of quadrilateral $BCED$ is twice the area of $\triangle PBC$. Find the maximum value of the area of $\triangle PDE$.

Answer: 5\sqrt{2} -7


In a plane, there is a set of $n \geq 9$ points. For any 9 points from this set, it is possible to choose two circles such that all these points lie on the chosen circles. Prove that all $n$ points lie on two circles.

Answer: \text{All } n \text{ points lie on two circles.}


Problem 10.1. Find the least natural number $a$ such that the equation $\cos ^{2} \pi(a-x)-2 \cos \pi(a-x)+\cos \frac{3 \pi x}{2 a} \cdot \cos \left(\frac{\pi x}{2 a}+\frac{\pi}{3}\right)+2=0$ has a root.

Answer: 6


360. The sums of the terms of each of the arithmetic progressions, having $n$ terms, are equal to $n^{2}+p n$ and $3 n^{2}-2 n$. Find the condition under which the $n$-th terms of these progressions will be equal.

Answer: p = 4n - 4


Three, (15 points) The three coefficients $a, b, c$ and the two roots $x_{1}, x_{2}$ of the quadratic equation $a x^{2}+b x+c$ are written in some order as five consecutive integers. Find all such quadratic equations, and briefly describe the reasoning process.

Answer: -2x^2 + 2


IMO 1965 Problem A3 The tetrahedron ABCD is divided into two parts by a plane parallel to AB and CD. The distance of the plane from AB is k times its distance from CD. Find the ratio of the volumes of the two parts.

Answer: \dfrac{k^3 + 3k^2}{3k + 1}


Let $ABCD$ be a convex quadrilateral such that the triangle $ABD$ is equilateral and the triangle $BCD$ is isosceles, with $\angle C = 90^o$. If $E$ is the midpoint of the side $AD$, determine the measure of the angle $\angle CED$.

Answer: 45^\circ


Robots Robert and Hubert assemble and disassemble coffee grinders. Each of them assembles a coffee grinder four times faster than they can disassemble it. When they came to the workshop in the morning, several coffee grinders were already assembled there.

At 7:00, Hubert started assembling and Robert started disassembling, and precisely at 12:00, Hubert finished assembling a coffee grinder and Robert finished disassembling another one. In total, 70 coffee grinders were added during this shift.

At 13:00, Robert started assembling and Hubert started disassembling, and precisely at 22:00, Robert finished assembling the last coffee grinder and Hubert finished disassembling another one. In total, 36 coffee grinders were added during this shift.

How long would it take to assemble 360 coffee grinders if Robert and Hubert were assembling together? (K. Pazourek)

Hint. How many coffee grinders will be added per hour in each shift?

Answer: 15


9. Person A and Person B start from points $A$ and $B$ respectively at the same time and walk towards each other. When A reaches the midpoint of $A B$, the two are 5 kilometers apart; when B reaches the midpoint of $A B$, the two are $\frac{45}{8}$ kilometers apart. The distance between $A$ and $B$ is $\qquad$ kilometers.

Answer: 90


Let \( x_{1} > 0 \) and \( a > 0 \), and define the sequence \( x_{n+1} = \left( x_{n} + \frac{a}{x_{n}} \right) \) for \( n \geqslant 1 \). Calculate the limit \( \lim_{n \rightarrow \infty} x_{n} \).

Answer: \sqrt{a}


Example 5. For what values of $B$ and $C$ is the line $\frac{x-3}{1}=\frac{y-2}{B}=$ $=\frac{z+2}{C}$ perpendicular to the plane $3 x-2 y+5 z=0$?

Answer: -\dfrac{2}{3}


15 (12 points) As shown in the figure, plane $m \parallel$ plane $n$, line segment $AD$ intersects $m$ and $n$ at points $B$ and $C$, respectively. Another line through point $A$ intersects $m$ and $n$ at points $M$ and $P$, respectively, and another line through point $D$ intersects $m$ and $n$ at points $N$ and $Q$, respectively. Given $\frac{S_{\triangle BMN}}{S_{\triangle CPQ}}=\frac{3}{2}$, find the minimum value of $\frac{AD}{CD}$.

Answer: 3+\sqrt{3}


Label the vertices of a trapezoid $T$ inscribed in the unit circle as $A,B,C,D$ counterclockwise with $AB\parallel CD$. Let $s_1,s_2,$ and $d$ denote the lengths of $AB$, $CD$, and $OE$, where $E$ is the intersection of the diagonals of $T$, and $O$ is the center of the circle. Determine the least upper bound of $\frac{s_1-s_2}d$ over all $T$ for which $d\ne0$, and describe all cases, if any, in which equality is attained.

Answer: 2


5. In a circle, a diameter $A B$ and a chord $C D$ parallel to it were drawn such that the distance between them is half the radius of this circle (see figure). Find the angle $CAB$.

Answer: 75^\circ


7. (10 points) The Spring Festival is approaching. Starting from January 17, 2011 (Monday), workers in the factory gradually return home for the New Year to reunite with their families. If the number of workers leaving the factory each day is the same, by January 31, there are still 121 workers left in the factory. During this 15-day period, the total workdays (one person working one day counts as 1 workday, and workers do not need to be counted after they leave the factory) recorded in the factory is 2011 workdays, with weekends off and no one absent. How many workers have returned home by January 31? $\qquad$

Answer: 120


12. Let's number from 1 to 28 the squares placed along the edge of a common $8 \times 8$ chessboard (see the figure on the side). Initially, a cubic die is placed on square 1 so that its bottom face exactly overlaps the entire square. We move the die along the squares as follows: the first move consists of rotating the die $90^{\circ}$ around the common side between square 1 and square 2 so that at the end the die is exactly on top of square 2. Similarly, we move the die from square 2 to 3, from 3 to 4, and so on, rotating the die each time around the common side between the square it is on and the next one. From square 28, we return to square 1, and after completing one round, we start over. How many moves are required at a minimum for the die to return to square 1 with the same top face it had at the beginning?

| 1 | 2 3 | 3 4 | 5 | 6 | 7 | 8 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 28 |  |  |  |  |  | 9 |
| 27 |  |  |  |  |  | 10 |
| 26 |  |  |  |  |  | 11 |
| 25 |  |  |  |  |  | 12 |
| 24 |  |  |  |  |  | 13 |
| 23 |  |  |  |  |  | 14 |
| 22 | 21 20 | 20 19 | 18 | 17 | 16 | 15 |

Answer: 84


Let $a_{i}$ and $b_{i}$, where $i \in \{1,2, \dots, 2005 \}$, be real numbers such that the inequality $(a_{i}x-b_{i})^{2} \ge \sum_{j=1, j \not= i}^{2005} (a_{j}x-b_{j})$ holds for all $x \in \mathbb{R}$ and all $i \in \{1,2, \dots, 2005 \}$. Find the maximum possible number of positive numbers amongst $a_{i}$ and $b_{i}$, $i \in \{1,2, \dots, 2005 \}$.

Answer: 4009


[Pairing and Grouping; Bijections] $[$ Evenness and Oddness $]$

Given a 29-digit number $X=\overline{a_{1}} \overline{-} \overline{-} \overline{29}\left(0 \leq a_{k} \leq 9, a_{1} \neq 0\right)$. It is known that for every $k$ the digit $a_{k}$ appears in the representation of this number $a_{30-k}$ times (for example, if $a_{10}=7$, then the digit $a_{20}$ appears seven times). Find the sum of the digits of the number $X$.

Answer: 201


3. In a certain state, only liars and economists live (liars always lie, while economists tell the truth). At a certain moment, the state decided to carry out monetary and credit, as well as budgetary and tax reforms. Since it was unknown what the residents expected, everyone was asked several questions (with only "yes" and "no" answers). The questions were answered affirmatively as follows:

"Will taxes be raised?" $40 \%$,

"Will there be an increase in the money supply?" $30 \%$,

"Will bonds be issued?" $50 \%$,

"Will the gold and foreign exchange reserves be squandered?" $\mathbf{0} \%$.

What percentage of the population actually expects taxes to be raised, if each of them realistically expects only one measure? (14 points).

Answer: 30


Problem 9.7. In front of a sweet-tooth lie five boxes of candies: the first box contains 11 candies, the second - 22 candies, the third - 33 candies, the fourth - 44 candies, the fifth - 55 candies. In one move, the sweet-tooth can take four candies from one box and distribute them, one candy to each of the remaining four boxes.

At any point, the sweet-tooth can take the candies from any box and leave. What is the maximum number of candies he can take?

Answer: 159


5. Solve the equation $x^{2}-x y-6 y^{2}+2 x+19 y=18$ in integers.

Rewrite the equation as

$$
x^{2}-x(y-2)-6 y^{2}+19 y+k=18+k
$$

and we will choose the number $k$ so that the discriminant of the quadratic trinomial on the left side of the equation becomes a perfect square. We have

$$
D=25 y^{2}-80 y+4-4 k
$$

To obtain a perfect square, we need $k=-15$. Now we have the opportunity to factor the left side of our equation

$$
x_{1,2}=\frac{y-2 \pm(5 y-8)}{2} \Rightarrow(x-3 y+5)(x+2 y-3)=3
$$

The factors on the left side are integers, so we get four systems

$$
\left\{\begin{array}{l}
x-3 y+5=1 \\
x+2 y-3=3
\end{array},\left\{\begin{array}{l}
x-3 y+5=-1 \\
x+2 y-3=-3
\end{array},\left\{\begin{array}{l}
x-3 y+5=3 \\
x+2 y-3=1
\end{array},\left\{\begin{array}{l}
x-3 y+5=-3 \\
x+2 y-3=-1
\end{array}\right.\right.\right.\right.
$$

Solve these systems and we get the answer.

Answer: (2, 2)


Let $ABC$ be an acute triangle and $AD$ a height. The angle bissector of $\angle DAC$ intersects $DC$ at $E$. Let $F$ be a point on $AE$ such that $BF$ is perpendicular to $AE$. If $\angle BAE=45º$, find $\angle BFC$.

Answer: 135^\circ


6. [4 points] Two parallel lines $\ell_{1}$ and $\ell_{2}$ touch the circle $\omega_{1}$ with center $O_{1}$ at points $A$ and $B$ respectively. The circle $\omega_{2}$ with center $O_{2}$ touches the line $\ell_{1}$ at point $D$, intersects the line $\ell_{2}$ at points $B$ and $E$, and intersects the circle $\omega_{1}$ again at point $C$ (with point $O_{2}$ lying between the lines $\ell_{1}$ and $\ell_{2}$). It is known that the ratio of the area of quadrilateral $B O_{1} C O_{2}$ to the area of triangle $O_{2} B E$ is $\frac{6}{5}$. Find the ratio of the radii of the circles $\omega_{2}$ and $\omega_{1}$.

Answer: \dfrac{7}{6}


4. On the plane $(x ; y)$, plot the set of points whose coordinates satisfy the system $\left\{\begin{array}{l}(|x|+x)^{2}+(|y|-y)^{2} \leq 16, \\ y-3 x \leq 0\end{array}\right.$, and find the area of the resulting figure.

Answer: \frac{20}{3} + \pi


Example 45. An isosceles trapezoid is circumscribed around a circle. Find the ratio of the area of the trapezoid to the area of the circle, given that the distance between the points of tangency of the circle with the lateral sides of the trapezoid is to the radius of the circle as $\sqrt{3}: 1$.

Given (Fig. 49): $A B C D$-trapezoid, $|A B|=|C D|, \omega(0,|O K|)-$ circle inscribed in the trapezoid, $K$ and $M$ - points of tangency with segments $A B$ and $C D,|K M|:|O K|=\sqrt{3}: 1$.

Find: $S_{A B C D}: S_{\omega}$.

Answer: \dfrac{8\sqrt{3}}{3\pi}


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

Answer: 256


$\mathrm{Az} A B C D$ quadrilateral has all four of its side lines tangent to a circle, the center of which is $O$. Express the sum of the angles $A O B$ and $C O D$ in terms of the angles of the quadrilateral, as well as the sum of the angles $B O C$ and $D O A$, in the following situations:

I. Each of the four side lines touches the circle at a point on the side segment.

II. Two side lines touch the circle at a point outside the side segment.

III. All four side lines touch the circle at a point outside the side segment.

Answer: 180^\circ


There is a quadrilateral ABCD inscribed in a circle with radius \( R \) that has perpendicular diagonals. The distance \( KO \) from the intersection point of the diagonals to the center of the circle is \( t \). Prove that the midpoints of the sides of the quadrilateral and the feet of the perpendiculars dropped from point \( K \) to the sides all lie on one circle. Find the radius of this circle.

Answer: \dfrac{\sqrt{2R^2 - t^2}}{2}


Problem 2. Given 10 equal segments in a plane. All intersection points are colored. Each colored point divides each of the given segments it lies on in the ratio $3: 4$. What is the maximum number of colored points?

Answer: 10


In the diagram below, square $ABCD$ with side length 23 is cut into nine rectangles by two lines parallel to $\overline{AB}$ and two lines parallel to $\overline{BC}$. The areas of four of these rectangles are indicated in the diagram. Compute the largest possible value for the area of the central rectangle.

[asy]
size(250);
 defaultpen (linewidth (0.7) + fontsize (10));
		 draw ((0,0)--(23,0)--(23,23)--(0,23)--cycle);
		 label("$A$", (0,23), NW);
		 label("$B$", (23, 23), NE);
		 label("$C$", (23,0), SE);
		 label("$D$", (0,0), SW);
		 draw((0,6)--(23,6));
		 draw((0,19)--(23,19));
		 draw((5,0)--(5,23));
		 draw((12,0)--(12,23));
		 label("13", (17/2, 21));
		 label("111",(35/2,25/2));
		 label("37",(17/2,3));
		 label("123",(2.5,12.5));[/asy]

[i]Proposed by Lewis Chen[/i]

Answer: 180


70. As shown in the figure, on the two legs $AC$ and $BC$ of the right $\triangle ABC$, construct squares $ACDE$ and $CBFG$, respectively. Connect $DG$, connect $AF$ intersecting $BC$ at $W$, and connect $GW$. If $AC=14, BC=28$, then the area of $\triangle AGW$ is . $\qquad$

Answer: 196


2. Given a regular $n$-sided polygon inscribed in a circle. If the line connecting a vertex to the center of the circle is exactly on the perpendicular bisector of a certain side, then $n \equiv$ $\qquad$ $(\bmod 2)$

Answer: 1


2. The base of the pyramid $M A B C$ is an isosceles $\triangle A B C$ ($A B \cong A C$). The foot of the height from the vertex $M$ is at the midpoint of the height $A A_{0}$ in $\triangle A B C$. A plane is placed through the edge $B C$ which is perpendicular to the edge $M A$ and intersects it at point $S$. It is known that this plane forms an angle of $30^{\circ}$ with the plane $A B C$. If the volume of the pyramid $S A B C$ is 2017, find the volume of the pyramid $M S B C$.

Answer: 2017


Example 4 Given that the base of the quadrilateral pyramid $P-ABCD$ is a parallelogram, with height $h$, a section is made through the base side $BC$ intersecting the side face $PAD$ at $EF$. If the section divides the pyramid into two parts of equal volume, find the distance from $EF$ to the base.

---

The translation maintains the original text's line breaks and format as requested.

Answer: \dfrac{(3 - \sqrt{5})h}{2}


7. Let $z=x+y \mathrm{i} (x, y \in \mathbf{R}, \mathrm{i}$ be the imaginary unit), the imaginary part of $z$ and the real part of $\frac{z-\mathrm{i}}{1-z}$ are both non-negative. Then the area of the region formed by the points $(x, y)$ on the complex plane that satisfy the conditions is $\qquad$ .

Answer: \dfrac{3\pi}{8} + \dfrac{1}{4}


23. King Qi and General Tian Ji had a horse race. Each had three horses, divided into three classes. Tian Ji knew that the fastest of the six horses was King Qi's first-class horse, followed by his own first-class horse, King Qi's second-class horse, his own second-class horse, King Qi's third-class horse, and his own third-class horse. Tian Ji also knew that King Qi's horses would appear in the order of first-class, second-class, and third-class. Therefore, Tian Ji has $\qquad$ ways to arrange the order of his three horses to ensure that he wins at least one race.

Answer: 5


In triangle \( \triangle ABC \), the three internal angles are \( \angle A \), \( \angle B \), and \( \angle C \), where \( \angle C = 2 \angle B \). Point \( P \) is a point inside \( \triangle ABC \) such that \( AP = AC \) and \( PB = PC \). Prove that \( \angle PAB = \frac{1}{3} \angle A \).

Answer: \dfrac{1}{3} \angle A


As shown in the figure, the area of a regular 18-sided polygon is 2016 square centimeters. What is the area of the shaded rectangle in the figure, in square centimeters?

Answer: 448


14. Insert parentheses in the equation: $1: 2: 3: 4: 5: 6: 7: 8: 9: 10=7$, to make it true.

Answer: (1:2:3:4:5):(6:7:8:9:10)


Task 4. Fill in the asterisks with six different digits so that all fractions are irreducible and the equation is correct: $\frac{*}{*}+\frac{*}{*}=\frac{*}{*} . \quad[6$ points] (A.V. Lapovalov)

Answer: \frac{1}{6} + \frac{7}{3} = \frac{5}{2}


53rd Putnam 1992 Problem B3 Let S be the set of points (x, y) in the plane such that the sequence a n defined by a 0 = x, a n+1 = (a n 2 + y 2 )/2 converges. What is the area of S? Solution

Answer: 4 + \pi


[Parallel lines, properties and signs. Transversals]

Points $M$ and $N$ are located on the lateral sides $A B$ and $C D$ of trapezoid $A B C D$, respectively, such that $M N \| A D$. It is known that the area of trapezoid $M B C N$ is to the area of trapezoid $A M N D$ as $2: 3$. Find $M N$, if $B C = a, A D = b$.

Answer: \sqrt{\dfrac{3a^2 + 2b^2}{5}}


3. The king summoned two male wizards to the palace. He asked Wizard A to first write down 100 positive real numbers (allowing duplicates) on a card, without revealing them to Wizard B. Then, B must accurately write down all 100 positive real numbers, or both wizards will be beheaded. A is allowed to provide B with a list of numbers, each of which is either one of the 100 positive real numbers or the sum of several of these 100 positive real numbers, but he cannot tell B which are the numbers on the card and which are the sums of the numbers on the card. In the end, the king decided to pull out the same number of whiskers from each wizard based on the number of numbers in the list. Given that the two wizards cannot communicate beforehand, how many whiskers at least need to be pulled out to ensure they do not lose their lives?

Answer: 101


14. (15 points) Person A and Person B start from locations $A$ and $B$ respectively at the same time, walking towards each other, and meet at location $C$. After meeting, A continues walking towards $B$, while B rests for 14 minutes and then continues walking towards $A$. After A and B reach $B$ and $A$ respectively, they immediately turn back and meet again at $C$. It is known that A walks 60 meters per minute, and B walks 80 meters per minute. How far apart are $A$ and $B$?

Answer: 1680


[ The Sum of Angles in a Triangle. The Exterior Angle Theorem. ] [ Criteria and Properties of an Isosceles Triangle. ]

In an isosceles triangle $A B C$, the heights $A D$ and $C E$, dropped to the lateral sides, form an angle $A M C$ equal to $48^{\circ}$. Find the angles of triangle $A B C$.

Answer: 24^\circ


II. (40 points) Let $n(n \geqslant 3)$ be a given natural number, and for $n$ given real numbers $a_{1}, a_{2}, \cdots, a_{n}$, denote the minimum value of $\left|a_{i}-a_{j}\right|(1 \leqslant i < j \leqslant n)$ as $m$. If $a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=1$, find the maximum value of $m$.

Answer: \sqrt{\dfrac{12}{n(n^2 - 1)}}


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Bob`, `Peter`
 - People have unique heights: `average`, `short`, `tall`, `very tall`, `very short`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Kailyn`, `Janelle`, `Aniya`
 - Each person has a unique type of pet: `cat`, `dog`, `hamster`, `fish`, `bird`
 - The people keep unique animals: `bird`, `dog`, `fish`, `horse`, `cat`
 - People have unique favorite sports: `swimming`, `basketball`, `baseball`, `tennis`, `soccer`

## Clues:
1. The person who loves basketball is the person who keeps a pet bird.
2. The person who loves soccer is directly left of the person who is very short.
3. The person whose mother's name is Aniya is directly left of the person who is tall.
4. The person who loves tennis is Eric.
5. The person whose mother's name is Holly is in the first house.
6. The person who owns a dog is somewhere to the right of Bob.
7. The person who owns a dog is the cat lover.
8. The person whose mother's name is Aniya is Peter.
9. The person who is very short is directly left of Alice.
10. The person whose mother's name is Janelle is the bird keeper.
11. The person who has a cat is directly left of the person who has an average height.
12. The person whose mother's name is Penny is the person with an aquarium of fish.
13. The person with an aquarium of fish is directly left of the dog owner.
14. The person who is very tall is somewhere to the right of the fish enthusiast.
15. The person who loves tennis is in the fourth house.
16. The person who loves basketball is directly left of the person who loves baseball.
 What is the value of attribute House for the person whose attribute Height is tall? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


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`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `samsung galaxy s21`
 - The people keep unique animals: `bird`, `horse`, `cat`
 - Each person has an occupation: `doctor`, `engineer`, `teacher`
 - Each person has a unique birthday month: `jan`, `sept`, `april`
 - Everyone has something unique for lunch: `grilled cheese`, `pizza`, `spaghetti`

## Clues:
1. The person who keeps horses is directly left of the person who is a teacher.
2. The person who loves the spaghetti eater is the person who is an engineer.
3. The person who uses a Samsung Galaxy S21 is not in the third house.
4. The person who is a doctor is the cat lover.
5. The person who uses an iPhone 13 is Peter.
6. The person who loves the spaghetti eater is the person whose birthday is in January.
7. The person who uses a Samsung Galaxy S21 is the person who loves eating grilled cheese.
8. The person who uses a Samsung Galaxy S21 and Eric are next to each other.
9. The person whose birthday is in January is somewhere to the right of the person whose birthday is in September.
 What is the value of attribute Food for the person whose attribute PhoneModel is iphone 13? Please reason step by step, and put your final answer within \boxed{}

Answer: pizza


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`, `Carol`, `Peter`, `Eric`, `Arnold`
 - The mothers' names in different houses are unique: `Holly`, `Sarah`, `Penny`, `Kailyn`, `Aniya`, `Janelle`

## Clues:
1. Carol is somewhere to the right of Bob.
2. The person whose mother's name is Aniya is in the first house.
3. Eric is in the second house.
4. Peter is The person whose mother's name is Penny.
5. There is one house between Peter and Carol.
6. The person whose mother's name is Holly is somewhere to the left of Arnold.
7. The person whose mother's name is Sarah and The person whose mother's name is Penny are next to each other.
8. The person whose mother's name is Sarah is not in the fifth house.
9. The person whose mother's name is Janelle is Bob.
10. The person whose mother's name is Sarah is not in the second house.
 What is the value of attribute House for the person whose attribute Mother is Kailyn? Please reason step by step, and put your final answer within \boxed{}

Answer: 6


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Peter`, `Carol`, `Bob`, `Arnold`
 - Everyone has a favorite smoothie: `watermelon`, `blueberry`, `desert`, `cherry`, `dragonfruit`, `lime`
 - Everyone has something unique for lunch: `stew`, `pizza`, `grilled cheese`, `stir fry`, `soup`, `spaghetti`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `xiaomi mi 11`, `huawei p50`, `samsung galaxy s21`, `oneplus 9`
 - People own unique car models: `tesla model 3`, `honda civic`, `toyota camry`, `ford f150`, `chevrolet silverado`, `bmw 3 series`
 - Each person lives in a unique style of house: `craftsman`, `ranch`, `modern`, `victorian`, `mediterranean`, `colonial`

## Clues:
1. Bob is the person who uses a Xiaomi Mi 11.
2. The person who loves the soup is in the fourth house.
3. The Dragonfruit smoothie lover is somewhere to the left of the person in a ranch-style home.
4. There is one house between the person who owns a Chevrolet Silverado and the person residing in a Victorian house.
5. The person in a Mediterranean-style villa is the person who drinks Lime smoothies.
6. Eric is in the sixth house.
7. The Desert smoothie lover is the person who is a pizza lover.
8. The person living in a colonial-style house is the person who drinks Blueberry smoothies.
9. The Dragonfruit smoothie lover and the person who uses a Google Pixel 6 are next to each other.
10. The person who loves the soup is Peter.
11. Alice is somewhere to the right of the person who owns a BMW 3 Series.
12. The person who loves stir fry is the person in a ranch-style home.
13. The person who owns a Ford F-150 is the person living in a colonial-style house.
14. The person in a Craftsman-style house is somewhere to the right of the person in a modern-style house.
15. The person who loves the stew is directly left of the person in a ranch-style home.
16. The person who owns a Tesla Model 3 is directly left of the person who loves stir fry.
17. The person who loves eating grilled cheese is the person who owns a Honda Civic.
18. The person in a Mediterranean-style villa is the person who uses a Google Pixel 6.
19. The person in a Craftsman-style house is the Watermelon smoothie lover.
20. The person in a modern-style house is Carol.
21. The person who uses an iPhone 13 is in the first house.
22. The person who owns a Honda Civic is somewhere to the left of the person who owns a BMW 3 Series.
23. The person who uses a Xiaomi Mi 11 is the person who is a pizza lover.
24. The person who uses a Samsung Galaxy S21 is the person who loves stir fry.
25. The person residing in a Victorian house is directly left of the person who uses a OnePlus 9.
 What is the value of attribute HouseStyle for the person whose attribute Food is pizza? Please reason step by step, and put your final answer within \boxed{}

Answer: victorian


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 Nationality for the person whose attribute Animal is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: swede


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Alice`, `Arnold`, `Carol`, `Peter`
 - The people are of nationalities: `swede`, `norwegian`, `german`, `chinese`, `dane`, `brit`
 - They all have a unique favorite flower: `daffodils`, `roses`, `carnations`, `lilies`, `tulips`, `iris`
 - Each person has a unique favorite drink: `water`, `tea`, `root beer`, `coffee`, `boba tea`, `milk`
 - Each person has a favorite color: `green`, `blue`, `purple`, `red`, `yellow`, `white`
 - Each person has a unique level of education: `high school`, `associate`, `bachelor`, `doctorate`, `trade school`, `master`

## Clues:
1. Arnold is somewhere to the left of the person who loves a carnations arrangement.
2. The coffee drinker is the person who loves the vase of tulips.
3. The German is directly left of the one who only drinks water.
4. The person whose favorite color is green is Alice.
5. The person who loves white is the coffee drinker.
6. The coffee drinker is the person with a bachelor's degree.
7. The one who only drinks water is the person who loves purple.
8. The person who loves a bouquet of daffodils is the person who loves blue.
9. The tea drinker is directly left of the person who loves a carnations arrangement.
10. The person with a high school diploma is Carol.
11. There is one house between the person who loves the boquet of iris and Peter.
12. The person with an associate's degree is the person whose favorite color is green.
13. Arnold is the Chinese.
14. The person whose favorite color is red is in the third house.
15. The boba tea drinker and the person who loves the vase of tulips are next to each other.
16. Alice is directly left of the Swedish person.
17. There is one house between the British person and the Chinese.
18. The person who loves the rose bouquet is somewhere to the left of the person with a bachelor's degree.
19. The person who loves yellow is not in the first house.
20. There are two houses between the person who likes milk and Peter.
21. Bob is the person with a master's degree.
22. The Swedish person is the person with a master's degree.
23. The person who attended trade school is somewhere to the left of the Chinese.
24. The German is in the fifth house.
25. The Norwegian is the one who only drinks water.
 What is the value of attribute Flower for the person whose attribute Drink is boba tea? Please reason step by step, and put your final answer within \boxed{}

Answer: iris


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`, `Alice`
 - Everyone has something unique for lunch: `spaghetti`, `pizza`, `stew`, `grilled cheese`
 - People own unique car models: `tesla model 3`, `honda civic`, `toyota camry`, `ford f150`
 - The people keep unique animals: `fish`, `cat`, `bird`, `horse`

## Clues:
1. The cat lover is somewhere to the right of the fish enthusiast.
2. Eric is the person who owns a Ford F-150.
3. The person who owns a Honda Civic is in the fourth house.
4. The person who is a pizza lover is the fish enthusiast.
5. The person who keeps horses is somewhere to the left of Arnold.
6. Arnold is the person who loves the spaghetti eater.
7. The person who owns a Toyota Camry is the person who loves the stew.
8. Alice and Arnold are next to each other.
9. Alice is in the first house.
 What is the value of attribute Food for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: grilled cheese


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Carol`, `Bob`, `Peter`
 - Each person has a unique level of education: `master`, `bachelor`, `high school`, `doctorate`, `associate`, `trade school`
 - People use unique phone models: `huawei p50`, `google pixel 6`, `oneplus 9`, `xiaomi mi 11`, `iphone 13`, `samsung galaxy s21`

## Clues:
1. The person with a bachelor's degree is somewhere to the left of the person with a master's degree.
2. The person with an associate's degree is the person who uses a Samsung Galaxy S21.
3. The person who uses a Google Pixel 6 is Alice.
4. The person who uses a Xiaomi Mi 11 is the person with a bachelor's degree.
5. The person who uses a Huawei P50 is in the third house.
6. There is one house between Arnold and Bob.
7. Arnold is in the second house.
8. Peter is somewhere to the left of the person who uses a Huawei P50.
9. There is one house between the person who uses an iPhone 13 and Eric.
10. The person with a doctorate is directly left of the person with an associate's degree.
11. The person with a master's degree is the person who uses a Huawei P50.
12. The person with a high school diploma is Arnold.
 What is the value of attribute PhoneModel for the person whose attribute Education is high school? Please reason step by step, and put your final answer within \boxed{}

Answer: oneplus 9


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Arnold`, `Eric`
 - People own unique car models: `toyota camry`, `tesla model 3`, `honda civic`, `ford f150`
 - People have unique heights: `average`, `very short`, `short`, `tall`
 - The mothers' names in different houses are unique: `Aniya`, `Kailyn`, `Holly`, `Janelle`
 - Each person has a unique level of education: `bachelor`, `high school`, `master`, `associate`
 - The people keep unique animals: `cat`, `fish`, `horse`, `bird`

## Clues:
1. The bird keeper is The person whose mother's name is Kailyn.
2. The person who owns a Ford F-150 is the person who is tall.
3. The person whose mother's name is Kailyn is not in the third house.
4. The person with a master's degree is the fish enthusiast.
5. The person who is very short is somewhere to the left of the cat lover.
6. The fish enthusiast is the person who is short.
7. The person whose mother's name is Janelle is directly left of the person who owns a Toyota Camry.
8. The person whose mother's name is Kailyn is not in the first house.
9. The person who is short is somewhere to the right of Alice.
10. The person with a high school diploma is the person who has an average height.
11. There is one house between the person who is very short and The person whose mother's name is Holly.
12. The person with a master's degree is somewhere to the right of the person with a bachelor's degree.
13. The person with a master's degree and Peter are next to each other.
14. The person who owns a Tesla Model 3 is Eric.
15. The person who has an average height is somewhere to the right of Arnold.
16. The person who keeps horses is directly left of the person with a master's degree.
 What is the value of attribute Education for the person whose attribute CarModel is tesla model 3? Please reason step by step, and put your final answer within \boxed{}

Answer: high school


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 Food for the person whose attribute Occupation is lawyer? Please reason step by step, and put your final answer within \boxed{}

Answer: stir fry


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Eric`, `Carol`, `Alice`, `Peter`
 - Each person has a unique level of education: `doctorate`, `trade school`, `associate`, `high school`, `bachelor`, `master`
 - People own unique car models: `chevrolet silverado`, `ford f150`, `honda civic`, `tesla model 3`, `bmw 3 series`, `toyota camry`
 - Each person has a unique favorite drink: `tea`, `water`, `boba tea`, `coffee`, `root beer`, `milk`
 - They all have a unique favorite flower: `carnations`, `iris`, `lilies`, `tulips`, `daffodils`, `roses`

## Clues:
1. Eric is the one who only drinks water.
2. The person with an associate's degree is the person who owns a Toyota Camry.
3. The boba tea drinker is in the fourth house.
4. The person who loves the vase of tulips is directly left of the person who owns a Tesla Model 3.
5. The person with a master's degree is somewhere to the left of the person who owns a BMW 3 Series.
6. The person who owns a Toyota Camry is directly left of the person who loves a bouquet of daffodils.
7. The person who loves the boquet of iris is somewhere to the left of the root beer lover.
8. The person who likes milk is somewhere to the right of the person who owns a Honda Civic.
9. Carol is the person who owns a Honda Civic.
10. The person with a high school diploma is in the second house.
11. The person with a master's degree is the boba tea drinker.
12. The person who owns a Tesla Model 3 is Arnold.
13. The person who loves the boquet of lilies is the person who owns a Tesla Model 3.
14. The person who loves a carnations arrangement is somewhere to the left of Alice.
15. The person with a bachelor's degree is somewhere to the right of the person who likes milk.
16. The one who only drinks water is the person who owns a Ford F-150.
17. Carol is somewhere to the left of the person with a doctorate.
18. There are two houses between the person who owns a Toyota Camry and the person who loves the vase of tulips.
19. The person who owns a Tesla Model 3 is somewhere to the left of the coffee drinker.
20. Bob is in the third house.
 What is the value of attribute House for the person whose attribute CarModel is bmw 3 series? Please reason step by step, and put your final answer within \boxed{}

Answer: 6


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`
 - Each person has a unique level of education: `bachelor`, `high school`, `associate`, `master`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`, `Kailyn`
 - The people keep unique animals: `fish`, `horse`, `cat`, `bird`
 - Each person has a unique birthday month: `jan`, `feb`, `sept`, `april`
 - People have unique favorite music genres: `rock`, `pop`, `jazz`, `classical`

## Clues:
1. The person with an associate's degree is The person whose mother's name is Aniya.
2. Peter is the person whose birthday is in September.
3. The person with a bachelor's degree is directly left of the person who loves pop music.
4. Arnold is The person whose mother's name is Janelle.
5. The person with a master's degree is Arnold.
6. The person whose mother's name is Janelle is the person who loves classical music.
7. The person whose birthday is in April is somewhere to the right of the person who loves rock music.
8. The cat lover is in the second house.
9. The fish enthusiast is in the third house.
10. Eric is the person whose birthday is in April.
11. The person who loves pop music is the person whose birthday is in January.
12. The person whose mother's name is Holly is the person with a bachelor's degree.
13. The person who keeps horses is the person with a master's degree.
14. The person whose mother's name is Aniya is the fish enthusiast.
15. Arnold is somewhere to the left of the fish enthusiast.
 What is the value of attribute House for the person whose attribute Animal is horse? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Eric`, `Alice`, `Arnold`
 - Each mother is accompanied by their child: `Timothy`, `Meredith`, `Fred`, `Bella`, `Samantha`

## Clues:
1. The person who is the mother of Timothy is directly left of the person's child is named Meredith.
2. Eric is the person who is the mother of Timothy.
3. The person's child is named Samantha is somewhere to the right of the person's child is named Meredith.
4. Bob is directly left of Arnold.
5. The person's child is named Meredith is somewhere to the right of the person's child is named Fred.
6. Bob is the person's child is named Fred.
7. Peter is the person's child is named Meredith.
 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: 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`, `Eric`, `Arnold`, `Peter`, `Alice`
 - Everyone has a unique favorite cigar: `blends`, `prince`, `pall mall`, `blue master`, `dunhill`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `stew`, `pizza`, `stir fry`
 - The people are of nationalities: `norwegian`, `swede`, `brit`, `dane`, `german`
 - Each person has an occupation: `teacher`, `lawyer`, `doctor`, `artist`, `engineer`
 - People have unique heights: `average`, `very tall`, `short`, `very short`, `tall`

## Clues:
1. The person who smokes Blue Master is the person who is a lawyer.
2. Alice is the person partial to Pall Mall.
3. The person partial to Pall Mall is somewhere to the left of the Dunhill smoker.
4. The German is the person who loves stir fry.
5. The person who is tall is the person who smokes many unique blends.
6. The person who is an artist is the person who loves the spaghetti eater.
7. Peter is the person who is short.
8. The Dunhill smoker is not in the fourth house.
9. The person who is very short is the person who smokes Blue Master.
10. There are two houses between the person who loves the spaghetti eater and the person who loves stir fry.
11. The person who has an average height is in the second house.
12. The Prince smoker is the person who is a doctor.
13. The Swedish person is the person who is an engineer.
14. Eric is the person who loves stir fry.
15. There is one house between the British person and the person who loves the stew.
16. The person who is very tall is the Norwegian.
17. The Dane is the person who is short.
18. The person who is very tall is Arnold.
19. There are two houses between the person who is a pizza lover and the person who is very tall.
20. The person who is a teacher is not in the third house.
 What is the value of attribute House for the person whose attribute Nationality is german? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
 - Each person prefers a unique type of vacation: `camping`, `city`, `beach`, `cruise`, `mountain`
 - People have unique heights: `very tall`, `average`, `very short`, `short`, `tall`
 - They all have a unique favorite flower: `tulips`, `roses`, `carnations`, `lilies`, `daffodils`
 - Each person has a unique level of education: `bachelor`, `master`, `associate`, `high school`, `doctorate`
 - People own unique car models: `bmw 3 series`, `tesla model 3`, `toyota camry`, `ford f150`, `honda civic`

## Clues:
1. Eric is the person who prefers city breaks.
2. Arnold is the person who is tall.
3. The person who loves beach vacations is the person who loves the boquet of lilies.
4. The person who owns a BMW 3 Series is the person who is tall.
5. The person with an associate's degree is the person who loves the boquet of lilies.
6. The person who has an average height is Eric.
7. Peter is the person who owns a Toyota Camry.
8. The person who has an average height is somewhere to the right of the person who loves a bouquet of daffodils.
9. The person who owns a Honda Civic is somewhere to the left of the person who loves the boquet of lilies.
10. Eric is the person who owns a Ford F-150.
11. The person who loves the vase of tulips is in the fourth house.
12. The person who is very short is not in the fourth house.
13. Eric is the person with a high school diploma.
14. The person who enjoys mountain retreats is somewhere to the right of the person with a master's degree.
15. The person who likes going on cruises is not in the third house.
16. Bob is the person who is short.
17. The person with an associate's degree is directly left of Arnold.
18. The person with a bachelor's degree is the person who is very tall.
19. The person who loves a carnations arrangement is directly left of the person with a doctorate.
 What is the value of attribute Flower for the person whose attribute Education is high school? Please reason step by step, and put your final answer within \boxed{}

Answer: roses


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Bob`, `Peter`, `Alice`, `Eric`, `Arnold`
 - Each person has a unique birthday month: `feb`, `may`, `mar`, `april`, `sept`, `jan`
 - Each person prefers a unique type of vacation: `city`, `beach`, `cultural`, `mountain`, `camping`, `cruise`
 - People have unique hair colors: `blonde`, `brown`, `black`, `gray`, `auburn`, `red`

## Clues:
1. Arnold is directly left of the person whose birthday is in April.
2. The person who loves beach vacations is not in the second house.
3. The person whose birthday is in January is in the second house.
4. The person who has red hair is not in the fourth house.
5. The person who likes going on cruises and the person who enjoys camping trips are next to each other.
6. Eric is the person whose birthday is in February.
7. Arnold is somewhere to the left of the person whose birthday is in February.
8. The person who prefers city breaks is not in the second house.
9. The person whose birthday is in February is the person who enjoys mountain retreats.
10. Alice and Carol are next to each other.
11. The person whose birthday is in September is the person who likes going on cruises.
12. The person who has brown hair is directly left of the person who has blonde hair.
13. The person whose birthday is in March is in the sixth house.
14. The person who has black hair is Alice.
15. The person who has gray hair is in the third house.
16. The person who has black hair is the person whose birthday is in September.
17. Peter is not in the third house.
18. There are two houses between the person who loves beach vacations and the person whose birthday is in February.
 What is the value of attribute Birthday for the person whose attribute Vacation is cultural? Please reason step by step, and put your final answer within \boxed{}

Answer: jan


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

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

Answer: soccer