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

## Clues:
1. The coffee drinker is not in the first house.
2. The person who loves jazz music is the Desert smoothie lover.
3. The Watermelon smoothie lover is in the fourth house.
4. The coffee drinker is directly left of Peter.
5. The one who only drinks water is the person who loves hip-hop music.
6. The Dragonfruit smoothie lover is directly left of the one who only drinks water.
7. The Desert smoothie lover is in the fifth house.
8. The person who loves stir fry is somewhere to the right of Bob.
9. The person who is a pizza lover is the person who likes Cherry smoothies.
10. The person who loves classical music is the person who loves the stew.
11. The one who only drinks water is Bob.
12. The root beer lover is the Dragonfruit smoothie lover.
13. There is one house between Arnold and the person who likes Cherry smoothies.
14. The person who loves the spaghetti eater is in the fifth house.
15. The person who likes milk is in the fifth house.
16. The person who likes Cherry smoothies is Bob.
17. The Dragonfruit smoothie lover is the person who loves the stew.
18. Eric and Peter are next to each other.
19. The person who loves rock music is not in the first house.
 What is the value of attribute Food for the person whose attribute Smoothie is watermelon? 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: `Peter`, `Eric`, `Alice`, `Arnold`, `Carol`, `Bob`
 - People have unique heights: `very tall`, `very short`, `tall`, `super tall`, `short`, `average`
 - Each person prefers a unique type of vacation: `cruise`, `mountain`, `camping`, `beach`, `city`, `cultural`
 - People have unique hair colors: `blonde`, `red`, `gray`, `brown`, `auburn`, `black`
 - Each person has a unique favorite drink: `tea`, `boba tea`, `milk`, `water`, `coffee`, `root beer`
 - They all have a unique favorite flower: `lilies`, `carnations`, `iris`, `tulips`, `roses`, `daffodils`

## Clues:
1. The person who loves the vase of tulips is the person who has auburn hair.
2. The person who is short is somewhere to the right of Arnold.
3. The person who goes on cultural tours is Arnold.
4. The person who has auburn hair is Bob.
5. The person who is very short is the person who loves a bouquet of daffodils.
6. The person who loves the boquet of iris is the person who is very tall.
7. Peter is somewhere to the right of the root beer lover.
8. The person who loves beach vacations is directly left of the person who prefers city breaks.
9. Bob is directly left of the coffee drinker.
10. The person who is very short is the person who has gray hair.
11. Eric is somewhere to the right of the tea drinker.
12. The person who has red hair is the person who is tall.
13. The person who is tall is somewhere to the right of the person who is short.
14. The person who has black hair is the person who enjoys mountain retreats.
15. The one who only drinks water is the person who prefers city breaks.
16. The person who has brown hair is Carol.
17. The person who has an average height is Arnold.
18. The person who has brown hair is directly left of the person who likes milk.
19. The root beer lover is the person who is very short.
20. The person who is short is somewhere to the left of the coffee drinker.
21. The person who enjoys camping trips is the person who loves a carnations arrangement.
22. The person who is super tall and Arnold are next to each other.
23. There is one house between the person who loves the boquet of lilies and Peter.
24. The person who loves the vase of tulips is directly left of the person who has black hair.
 What is the value of attribute House for the person whose attribute Vacation is cultural? 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: `Arnold`, `Peter`, `Eric`
 - Each mother is accompanied by their child: `Meredith`, `Fred`, `Bella`
 - Each person has a unique birthday month: `jan`, `april`, `sept`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`
 - Everyone has something unique for lunch: `spaghetti`, `grilled cheese`, `pizza`
 - The mothers' names in different houses are unique: `Janelle`, `Aniya`, `Holly`

## Clues:
1. Arnold is the person who loves eating grilled cheese.
2. The person who loves the spaghetti eater and The person whose mother's name is Holly are next to each other.
3. Peter is directly left of the person whose birthday is in September.
4. The person who loves the spaghetti eater is the person whose birthday is in January.
5. The person who likes Cherry smoothies is not in the first house.
6. The person's child is named Bella is the Desert smoothie lover.
7. The person whose birthday is in September is Eric.
8. The person whose mother's name is Aniya is in the first house.
9. The person's child is named Fred is somewhere to the right of the person who likes Cherry smoothies.
10. The person who likes Cherry smoothies is the person whose birthday is in January.
 What is the value of attribute Name for the person whose attribute Birthday is jan? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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

## Clues:
1. The person who has red hair is the person who keeps a pet bird.
2. The person whose mother's name is Kailyn is Eric.
3. There is one house between the person who has gray hair and the person who has a cat.
4. The person whose mother's name is Janelle is Peter.
5. The person who is tall is directly left of The person whose mother's name is Janelle.
6. The person who has an average height is directly left of the person who has brown hair.
7. The person whose mother's name is Kailyn is the person who is very short.
8. The person who is very tall is somewhere to the right of The person whose mother's name is Janelle.
9. The person whose mother's name is Penny is in the first house.
10. The person with a pet hamster is in the first house.
11. The person who is tall is the person with an aquarium of fish.
12. The person whose mother's name is Kailyn is somewhere to the left of The person whose mother's name is Aniya.
13. The person who owns a dog is in the fifth house.
14. The person whose mother's name is Kailyn is directly left of the person who has black hair.
15. Bob is somewhere to the left of the person with an aquarium of fish.
16. The person who keeps a pet bird is directly left of Arnold.
 What is the value of attribute Mother for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: Holly


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`
 - Each person has a unique birthday month: `sept`, `april`, `feb`, `jan`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`, `stew`
 - Each person has a unique hobby: `cooking`, `gardening`, `painting`, `photography`
 - Each person has an occupation: `teacher`, `artist`, `engineer`, `doctor`
 - Each mother is accompanied by their child: `Bella`, `Samantha`, `Fred`, `Meredith`

## Clues:
1. The person who is a teacher is the person who loves cooking.
2. The person whose birthday is in January is directly left of Eric.
3. The person who is an artist is not in the fourth house.
4. The person who loves the spaghetti eater is not in the third house.
5. The person's child is named Bella is the person who paints as a hobby.
6. The person whose birthday is in January is the person who is a teacher.
7. The person who paints as a hobby is the person who is a doctor.
8. The person whose birthday is in September is directly left of Arnold.
9. The person who is a pizza lover is Peter.
10. The person who loves the stew is somewhere to the left of the photography enthusiast.
11. Arnold is the person's child is named Meredith.
12. The person whose birthday is in January is directly left of the person whose birthday is in September.
13. The person whose birthday is in April is in the fourth house.
14. The person's child is named Fred is directly left of the person whose birthday is in April.
15. Alice is in the first house.
16. The person whose birthday is in April is the person who enjoys gardening.
 What is the value of attribute Hobby for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{}

Answer: photography


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

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

Answer: jan


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Arnold`, `Peter`, `Eric`
 - Each person has a unique level of education: `doctorate`, `bachelor`, `associate`, `master`, `high school`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `google pixel 6`, `iphone 13`, `huawei p50`
 - People have unique favorite music genres: `rock`, `classical`, `jazz`, `pop`, `hip hop`
 - The people keep unique animals: `bird`, `dog`, `horse`, `cat`, `fish`
 - Each person has a unique hobby: `painting`, `cooking`, `knitting`, `gardening`, `photography`

## Clues:
1. The person who loves hip-hop music is in the second house.
2. The person who loves classical music is not in the fifth house.
3. The person who loves classical music is the fish enthusiast.
4. The person who uses an iPhone 13 is the person with a doctorate.
5. The person with a master's degree is not in the second house.
6. The person who enjoys knitting is not in the first house.
7. The person with a high school diploma is somewhere to the left of the photography enthusiast.
8. The person who paints as a hobby is somewhere to the right of the person who loves rock music.
9. Arnold is the bird keeper.
10. The person with a bachelor's degree is the person who uses a Google Pixel 6.
11. The person with a bachelor's degree is somewhere to the right of the person who paints as a hobby.
12. Bob is not in the fourth house.
13. The person who loves cooking is the person who loves classical music.
14. The person who loves cooking is the person who uses a Huawei P50.
15. The person who keeps horses is somewhere to the right of Arnold.
16. The person who paints as a hobby is the person who uses a OnePlus 9.
17. The person who uses a OnePlus 9 is somewhere to the left of the person with an associate's degree.
18. The person who uses an iPhone 13 is the dog owner.
19. The person with a high school diploma is Alice.
20. Peter is the person who enjoys gardening.
21. The person who paints as a hobby is the person who loves jazz music.
 What is the value of attribute PhoneModel for the person whose attribute Animal is fish? Please reason step by step, and put your final answer within \boxed{}

Answer: huawei p50


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

## Clues:
1. The person who enjoys mountain retreats is Alice.
2. The Norwegian is somewhere to the left of the person whose birthday is in September.
3. The person who prefers city breaks is the British person.
4. Alice and Eric are next to each other.
5. The person who likes going on cruises is not in the fourth house.
6. Alice is somewhere to the left of the person whose birthday is in January.
7. The person who enjoys mountain retreats is the Norwegian.
8. Alice is directly left of the Swedish person.
9. Eric is in the third house.
10. The person whose birthday is in April is Peter.
11. The Dane is the person whose birthday is in September.
 What is the value of attribute House for the person whose attribute 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: `Peter`, `Bob`, `Alice`, `Arnold`, `Carol`, `Eric`
 - The people are of nationalities: `swede`, `chinese`, `norwegian`, `german`, `brit`, `dane`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`, `chevrolet silverado`, `honda civic`, `bmw 3 series`
 - Each person has a unique favorite drink: `root beer`, `milk`, `tea`, `water`, `coffee`, `boba tea`

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

Answer: honda civic


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Peter`, `Eric`
 - Each person lives in a unique style of house: `victorian`, `craftsman`, `ranch`, `colonial`
 - The people keep unique animals: `fish`, `cat`, `horse`, `bird`
 - Each person has a unique hobby: `cooking`, `painting`, `gardening`, `photography`

## Clues:
1. The person who loves cooking is in the second house.
2. The fish enthusiast is not in the first house.
3. Alice is in the second house.
4. The cat lover is the person residing in a Victorian house.
5. The person who enjoys gardening is the person residing in a Victorian house.
6. The person in a ranch-style home is not in the first house.
7. The person who keeps horses is somewhere to the right of the person residing in a Victorian house.
8. Peter is somewhere to the left of the person who paints as a hobby.
9. The person in a Craftsman-style house is Eric.
10. The person residing in a Victorian house is in the third house.
11. Peter is directly left of the person living in a colonial-style house.
 What is the value of attribute Animal for the person whose attribute HouseStyle is ranch? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Bob`, `Arnold`, `Peter`, `Alice`, `Eric`
 - Each person has a unique favorite drink: `boba tea`, `coffee`, `root beer`, `water`, `milk`, `tea`
 - Each person lives in a unique style of house: `ranch`, `victorian`, `craftsman`, `mediterranean`, `colonial`, `modern`
 - Everyone has a unique favorite cigar: `dunhill`, `prince`, `yellow monster`, `blue master`, `blends`, `pall mall`

## Clues:
1. Eric is the Dunhill smoker.
2. Peter and the root beer lover are next to each other.
3. The Prince smoker is Bob.
4. The person in a modern-style house is somewhere to the left of Arnold.
5. The one who only drinks water is somewhere to the right of Bob.
6. The person in a modern-style house is the person who smokes Blue Master.
7. The person in a ranch-style home is directly left of the person in a modern-style house.
8. The person who smokes Yellow Monster is somewhere to the right of the coffee drinker.
9. The root beer lover is in the sixth house.
10. The person who likes milk is somewhere to the left of the tea drinker.
11. Arnold is the person living in a colonial-style house.
12. The person in a Craftsman-style house is the coffee drinker.
13. The boba tea drinker is somewhere to the left of the person living in a colonial-style house.
14. The boba tea drinker is the person partial to Pall Mall.
15. The person residing in a Victorian house is not in the fifth house.
16. The person living in a colonial-style house is in the third house.
17. Alice is not in the first house.
 What is the value of attribute Cigar for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}

Answer: pall mall


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`
 - People have unique favorite book genres: `science fiction`, `mystery`, `romance`
 - Each mother is accompanied by their child: `Bella`, `Fred`, `Meredith`
 - People have unique hair colors: `black`, `blonde`, `brown`
 - Each person has a unique level of education: `high school`, `associate`, `bachelor`
 - They all have a unique favorite flower: `daffodils`, `lilies`, `carnations`

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

Answer: romance


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Eric`, `Arnold`
 - People use unique phone models: `samsung galaxy s21`, `oneplus 9`, `google pixel 6`, `iphone 13`
 - Everyone has a unique favorite cigar: `dunhill`, `prince`, `blue master`, `pall mall`
 - People have unique hair colors: `red`, `black`, `blonde`, `brown`
 - Each person has a favorite color: `white`, `yellow`, `red`, `green`

## Clues:
1. The person who uses an iPhone 13 is the person who has brown hair.
2. The person who has black hair is somewhere to the left of the Prince smoker.
3. The person who loves yellow is the person who smokes Blue Master.
4. The Dunhill smoker is the person who uses a OnePlus 9.
5. The person who uses a OnePlus 9 is directly left of the person who has red hair.
6. Alice is not in the second house.
7. The Prince smoker is somewhere to the left of the person who uses a OnePlus 9.
8. The person who smokes Blue Master is the person who uses a Samsung Galaxy S21.
9. Eric is the person who loves white.
10. The person who smokes Blue Master is directly left of the person whose favorite color is red.
11. The person partial to Pall Mall is Arnold.
 What is the value of attribute House for the person whose attribute Color is white? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


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

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

Answer: swimming


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 PhoneModel for the person whose attribute CarModel is honda civic? Please reason step by step, and put your final answer within \boxed{}

Answer: oneplus 9


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Bob`, `Alice`
 - Each person has a unique favorite drink: `milk`, `tea`, `root beer`, `water`, `coffee`
 - Each person has a favorite color: `blue`, `white`, `red`, `green`, `yellow`
 - Each mother is accompanied by their child: `Samantha`, `Meredith`, `Timothy`, `Bella`, `Fred`
 - Each person lives in a unique style of house: `modern`, `ranch`, `victorian`, `colonial`, `craftsman`
 - People use unique phone models: `samsung galaxy s21`, `iphone 13`, `huawei p50`, `oneplus 9`, `google pixel 6`

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

Answer: huawei p50


There are 2 houses, numbered 1 to 2 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`
 - People have unique favorite music genres: `pop`, `rock`
 - Each person has a unique type of pet: `dog`, `cat`
 - Everyone has something unique for lunch: `pizza`, `grilled cheese`
 - The people are of nationalities: `brit`, `dane`
 - Each person has a unique birthday month: `april`, `sept`

## Clues:
1. Arnold is the person who loves eating grilled cheese.
2. The person whose birthday is in April is in the first house.
3. Eric is the person who owns a dog.
4. The person who loves rock music is the person who owns a dog.
5. The British person is in the second house.
6. The person who is a pizza lover is directly left of the person who loves eating grilled cheese.
 What is the value of attribute Birthday for the person whose attribute Pet is cat? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Arnold`, `Eric`
 - 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 CarModel for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: ford f150


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Peter`, `Bob`
 - The people are of nationalities: `german`, `norwegian`, `brit`, `dane`, `swede`
 - People have unique favorite sports: `swimming`, `soccer`, `baseball`, `basketball`, `tennis`
 - People have unique hair colors: `blonde`, `brown`, `red`, `black`, `gray`

## Clues:
1. Arnold is the person who loves baseball.
2. The Swedish person is somewhere to the left of Arnold.
3. There are two houses between the person who loves basketball and the person who has black hair.
4. The Norwegian is in the fourth house.
5. Peter is the person who has black hair.
6. The person who has red hair is in the second house.
7. Bob is the German.
8. The person who has blonde hair is not in the third house.
9. The person who loves swimming is the person who has red hair.
10. The German is not in the first house.
11. The person who loves basketball is directly left of the Dane.
12. The person who loves soccer is directly left of Arnold.
13. Alice is the person who has gray hair.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


8,9 |

Four spheres of radius 1 touch each other pairwise. Find the height of the cone that contains these spheres such that all of them touch the lateral surface and three of them touch the base of the cone.

Answer: 1 + \sqrt{3} + \dfrac{2\sqrt{6}}{3}


Vasya has $n{}$ candies of several types, where $n>145$. It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of $n{}$.

[i]Proposed by A. Antropov[/i]

Answer: 160


87. Among all natural numbers up to 2007, at most $\qquad$ numbers can be selected, such that none of them is a multiple of another, and no palindromic numbers (such as 33, 202, 1001) are included.

Answer: 994


7. (10 points) There are 11, 12, and 17 balls of red, yellow, and blue colors respectively. Each operation can replace 2 balls of different colors with 2 balls of the third color. During the operation process, the maximum number of red balls can be $\qquad$.

Answer: 39


The products of the cosines of opposite angles of a quadrilateral are equal. Prove that the quadrilateral is either a trapezoid or a parallelogram.

Answer: \text{The quadrilateral is either a trapezoid or a parallelogram.}


## 107.

On another island of knights, liars, and normal people, the king held opposite views and gave his daughter different paternal advice: “My dear, I don’t want you to marry any knight or liar. I would like your husband to be a solid, reputable normal person. You should not marry a knight because all knights are hypocrites. You should not marry a liar either because all liars are treacherous. No, no matter what, a decent normal person would be just right for you!”

Suppose you are a resident of this island and a normal person. Your task is to convince the king that you are a normal person.
a) How many true statements will you need for this?

b) How many false statements will you need for the same purpose?

(In both cases, we are talking about the minimum number of statements.)

Answer: 1


Example 16. Find the function that maps the upper half of the circle $|z|<1$, $\text{Im } z>0$, onto the upper half-plane $\text{Im } w>0$.

Answer: \left( \frac{z + 1}{1 - z} \right)^2


8.1. Pasha and Sasha made three identical toy cars. Sasha did one-fifth of the total work. After selling the cars, they divided the proceeds in proportion to the work done. Pasha noticed that if he gave Sasha 400 rubles, and Sasha made another such car and sold it, they would have the same amount of money. How much does one car cost?

Answer: 1000


[ Re-grouping areas $]$
[ Properties of symmetries and axes of symmetry ]

Authors: Galperin G.A., Vasilyev N.B.

$F$ - a convex figure with two mutually perpendicular axes of symmetry. Through a point $M$, lying inside the figure and at distances $a$ and $b$ from the axes, lines parallel to the axes are drawn. These lines divide $F$ into four regions. Find the difference between the sum of the areas of the largest and smallest regions and the sum of the areas of the other two.

Answer: 4ab


Example 7: Xiao Li and his brother attended a gathering, along with two other pairs of brothers. After meeting, some people greeted each other with handshakes, but no one shook hands with their own brother, and no one shook hands with the same person twice. At this point, Xiao Li noticed that, apart from himself, everyone had a different number of handshakes. How many times did Xiao Li shake hands? How many times did Xiao Li's brother shake hands?

Answer: 2


A state issues license plates consisting of 6 digits (each digit ranging from 0 to 9) and stipulates that any two license plates must differ by at least two digits (thus, plate numbers 027592 and 020592 cannot be used simultaneously). Determine the maximum number of license plates that can be issued.

Answer: 100000


There are finitely many points on the plane such that no three of them are on the same line. For at most how many points can the segments connecting the points be colored with three colors so that the following conditions are satisfied:

a) no point can have three segments of different colors starting from it;

b) among the points, it is not possible to find three such that the three segments they determine have the same color;

c) all three colors are used.

Answer: 4


Problem 4. Let $\triangle A^{\prime} B^{\prime} C$ be the image of $\triangle A B C$ under a rotation with center $C$. Denote by $M, E$ and $F$ the midpoints of the segments $B A^{\prime}, A C$ and $B^{\prime} C$, respectively. If $A C \neq B C$ and $E M=F M$, find $\Varangle E M F$.

Ivailo Kortezov

Answer: 60^\circ


Question 106, Let the complex number $\mathrm{z}$ be such that the real and imaginary parts of 1 are both greater than 1. If the set of points corresponding to $\mathrm{z}$ on the complex plane is the figure $C$, find the area of figure $C$.

Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.

Answer: \frac{\pi}{8} - \frac{1}{4}


6. Three equal cylinders are pairwise tangent, and their axes are mutually perpendicular. If the radius of the base of each cylinder is equal to $r$, then the radius of the smallest sphere that is tangent to all three cylindrical surfaces is $\qquad$ .

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


11. The smallest positive integer $n$ for which $n^{2}-n+11$ has four prime factors (not necessarily distinct) is $\qquad$

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

Answer: 132


13.9 Given several sets with identical contents, each set consists of 4 objects with integer gram weights. Now, we want to use these objects as weights to measure every integer gram weight from 1 gram to 1985 grams (including 1985 grams). To minimize the total weight of the objects used as weights, how many different ways are there to select the weights of the 4 objects?

Answer: 1


Ex. 140. A circle is inscribed in a trapezoid with integer sides. The midline of the trapezoid divides it into two parts, the areas of which are 15 and 30. Find the radius of the inscribed circle.

## Several problems

Answer: \dfrac{5}{2}


We inscribed the regular hexagon $A B C D E F$ in a circle, then drew semicircles outward over the chords $A B, B D, D E$, and $E A$. Calculate the ratio of the total area of the 4 lens-shaped (bounded by two circular arcs) figures that formed to the area of the hexagon.

Let's start by calculating the area of the regular hexagon. The area of a regular hexagon with side length $s$ is given by:
\[ T_{\text{hexagon}} = \frac{3 \sqrt{3}}{2} s^2 \]

Next, we need to calculate the area of the lens-shaped figures. Each lens is formed by the intersection of two semicircles. The radius of each semicircle is equal to the side length of the hexagon, $s$. The area of each semicircle is:
\[ T_{\text{semicircle}} = \frac{1}{2} \pi s^2 \]

The area of the lens-shaped figure can be found by subtracting the area of the equilateral triangle formed by the two radii and the chord from the area of the semicircle. The area of an equilateral triangle with side length $s$ is:
\[ T_{\text{triangle}} = \frac{\sqrt{3}}{4} s^2 \]

Thus, the area of one lens-shaped figure is:
\[ T_{\text{lens}} = T_{\text{semicircle}} - T_{\text{triangle}} = \frac{1}{2} \pi s^2 - \frac{\sqrt{3}}{4} s^2 = \left( \frac{\pi}{2} - \frac{\sqrt{3}}{4} \right) s^2 \]

Since there are 4 such lens-shaped figures, the total area of the lens-shaped figures is:
\[ T_{\text{total lenses}} = 4 \left( \frac{\pi}{2} - \frac{\sqrt{3}}{4} \right) s^2 = \left( 2\pi - \sqrt{3} \right) s^2 \]

Finally, the ratio of the total area of the lens-shaped figures to the area of the hexagon is:
\[ \text{Ratio} = \frac{T_{\text{total lenses}}}{T_{\text{hexagon}}} = \frac{\left( 2\pi - \sqrt{3} \right) s^2}{\frac{3 \sqrt{3}}{2} s^2} = \frac{2\pi - \sqrt{3}}{\frac{3 \sqrt{3}}{2}} = \frac{2(2\pi - \sqrt{3})}{3 \sqrt{3}} = \frac{4\pi - 2\sqrt{3}}{3 \sqrt{3}} = \frac{4\pi - 2\sqrt{3}}{3 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\pi \sqrt{3} - 6}{9} = \frac{4\pi \sqrt{3} - 6}{9} \]

Thus, the ratio is:
\[ \boxed{\frac{4\pi \sqrt{3} - 6}{9}} \]

Answer: \dfrac{4\pi \sqrt{3} - 6}{9}


7. Given the parabola $x^{2}=-8 y$ whose directrix intersects the $y$-axis at point $A$, a line through $A$ intersects the parabola at points $M$ and $N$. Point $B$ lies on the axis of symmetry of the parabola, and $\left(\overrightarrow{B M}+\frac{1}{2} \overrightarrow{M N}\right) \perp \overrightarrow{M N}$. Then the range of $|\overrightarrow{O B}|$ is $\qquad$

Answer: (6, +\infty)


Problem 3. Find the least positive integer $n$ such that there exists a group of $n$ people such that:

1. There is no group of four every two of which are friends;
2. For any choice of $k \geq 1$ people among which there are no friends there exists a group of three among the remaining $n-k$ every two of which are friends.

Answer: 7


We write 1s at the two endpoints of a segment. Then, step by step, we bisect the resulting segments and write the sum of the numbers at the two ends of the segments to the midpoint. Repeating the bisection of the segments one million times, how many times will the number $1978$ appear among the written numbers?

Answer: 924


Given a function \( f \) defined on the set of integers such that \( f(1)=1, f(2)=2 \), and \( f(n+2)=f[n+2-f(n+1)]+f[n+1-f(n)] \) for \( n \geq 1 \), prove:
1. \( 0 \leq f(n+1) - f(n) \leq 1 \);
2. If \( f(n) \) is odd, then \( f(n+1)=f(n)+1 \);
3. Find all values of \( n \) such that \( f(n)=2^{10}+1 \).

Answer: 2048


15. Given the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ passes through the point $M(0,2)$, and the right focus is $F(2,0)$.
(1) Write the equation of the ellipse $C$;
(2) A line $l$ passing through point $F$ intersects the ellipse $C$ at points $A, B$, and intersects the $y$-axis at point $P$; if $\overrightarrow{P A}=m \overrightarrow{A F}, \overrightarrow{P B}=n \overrightarrow{B F}$. Prove that $m+n$ is a constant;
(3) Under the condition in (2), if point $P$ is not inside the ellipse $C$, and point $Q$ is the point symmetric to $P$ with respect to the origin $O$, find the minimum value of the area of triangle $Q A B$.

Answer: \dfrac{x^2}{8} + \dfrac{y^2}{4} = 1


2. Let $A_{1}, A_{2}, A_{3}, A_{4}, A_{5}$ and $A_{6}$ be six points on a circle in this order such that $\overline{A_{1} A_{2}}=\overline{A_{2} A_{3}}$, $\widetilde{A_{3} A_{4}}=\widehat{A_{4} A_{5}}$ and $\widehat{A_{5} A_{6}}=\widehat{A_{6} A_{1}}$, where $\widehat{A_{1} A_{2}}$ denotes the arc length of the arc $A_{1} A_{2}$ etc. It is also known that $\angle A_{1} A_{3} A_{5}=72^{\circ}$. Find the size of $\angle A_{4} A_{6} A_{2}$ in degrees.

Answer: 54


A square-based pyramid has all edges 2 units long. We attach a pyramid to one of the side faces of the pyramid, where the side edges of the new pyramid are of equal length. The total length of the edges of the resulting solid is 18 units. What is the volume of the solid?

Answer: 2\sqrt{2}


Let $f(x) =x^4+14x^3+52x^2+56x+16.$  Let $z_1, z_2, z_3, z_4$ be the four roots of $f$.  Find the smallest possible value of $|z_az_b+z_cz_d|$ where $\{a,b,c,d\}=\{1,2,3,4\}$.

Answer: 8


Answer: -\frac{1}{2} + \frac{1}{2}i


S is a finite collection of integers, not necessarily distinct. If any element of S is removed, then the remaining integers can be divided into two collections with the same size and the same sum. Show that all elements of S are equal.

Answer: \text{All elements of } S \text{ are equal.}


7. Huanhuan and Lele set off for the market at the same time, traveling at different constant speeds along the same straight road. Initially, they were 200 meters apart, and after 2 hours, they were still 200 meters apart. After another 6 hours, neither of them had reached the market, at which point they were $\qquad$ meters apart.

Answer: 1400


Four copies of an acute scalene triangle $\mathcal T$, one of whose sides has length $3$, are joined to form a tetrahedron with volume $4$ and surface area $24$.  Compute the largest possible value for the circumradius of $\mathcal T$.

Answer: \sqrt{4 + \sqrt{3}}


Problem 8. For the tetrahedron $ABCD$, it is known that $AB=7, DC=8, \angle BCD=\angle ADC=75^{\circ}, \angle BDC=$ $\angle ACD=15^{\circ}$. A sphere is circumscribed around the tetrahedron. Consider the set of all points on this sphere for which the sum of the spherical distances to the points $A, B, C, D$ is no more than $8\pi$. What is the area of this set? The spherical distance between two points on the sphere is the length of the shortest arc of a great circle connecting these points.

Answer: 32\pi


Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$ . Circle $\Omega$ lies tangent to $\overline{AB}$ at a point $P$ and intersects $\Gamma$ at points $Q$ and $R$ . If $QR=3\sqrt3$ and $\angle QPR=60^\circ$ , then the area of $\triangle PQR$ equals $\tfrac{a\sqrt{b}}{c}$ , where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. What is $a+b+c$

Answer: 122


Since guts has 36 questions, they will be combined into posts.

1.**[5]** Farmer Yang has a  $2015$  ×  $2015$  square grid of corn plants. One day, the plant in the very center
of the grid becomes diseased. Every day, every plant adjacent to a diseased plant becomes diseased.
After how many days will all of Yang's corn plants be diseased?

2. **[5]** The three sides of a right triangle form a geometric sequence. Determine the ratio of the length of
the hypotenuse to the length of the shorter leg.

3. **[5]** A parallelogram has  $2$  sides of length  $20$  and  $15$ . Given that its area is a positive integer, ﬁnd the
minimum possible area of the parallelogram.

4. **[6]** Eric is taking a biology class. His problem sets are worth  $100$  points in total, his three midterms are
worth  $100$  points each, and his ﬁnal is worth  $300$  points. If he gets a perfect score on his problem sets
and scores  $60\%$ , $70\%$ , and  $80\%$  on his midterms respectively, what is the minimum possible percentage
he can get on his ﬁnal to ensure a passing grade? (Eric passes if and only if his overall percentage is
at least  $70\%$ ).

5. **[6]** James writes down three integers. Alex picks some two of those integers, takes the average of them,
and adds the result to the third integer. If the possible ﬁnal results Alex could get are  $42$ ,  $13$ , and  $37$ ,
what are the three integers James originally chose?

6. **[6]** Let  $AB$  be a segment of length  $2$  with midpoint  $M$ . Consider the circle with center  $O$  and radius $r$  that is externally tangent to the circles with diameters  $AM$  and  $BM$  and internally tangent to the
circle with diameter  $AB$ . Determine the value of  $r$ .

7. **[7]** Let n be the smallest positive integer with exactly  $2015$  positive factors. What is the sum of
the (not necessarily distinct) prime factors of n? For example, the sum of the prime factors of  $72$  is $2 + 2 + 2 + 3 + 3 = 14$ .

8. **[7]** For how many pairs of nonzero integers  $(c, d)$  with  $-2015 \le c,d \le 2015$  do the equations  $cx = d$ and  $dx = c$  both have an integer solution?

9. **[7]** Find the smallest positive integer n such that there exists a complex number z, with positive real
and imaginary part, satisfying  $z^n = (\overline{z})^n$ .

Answer: 3


In some foreign country's government, there are 12 ministers. Each minister has 5 friends and 6 enemies in the government (friendship/enemyship is a symmetric relation). A triplet of ministers is called **uniform** if all three of them are friends with each other, or all three of them are enemies. How many uniform triplets are there?

Answer: 40


On a chessboard  $8\times 8$ ,  $n>6$  Knights are placed so that for any 6 Knights there are two Knights that attack each other. Find the greatest possible value of  $n$ .

Answer: 10


A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is  $15000$ . What are the page numbers on the torn leaf?

Answer: (112,113)


A square grid  $100 \times 100$  is tiled in two ways - only with dominoes and only with squares  $2 \times 2$ . What is the least number of dominoes that are entirely inside some square  $2 \times 2$ ?

Answer: 100


The graph  $G$  with 2014 vertices doesn’t contain any 3-cliques. If the set of the degrees of the vertices of  $G$  is  $\{1,2,...,k\}$ , find the greatest possible value of  $k$ .

Answer: 1342


Let  $S$  be a square of side length  $1$ .  Two points are chosen independently at random on the sides of  $S$ .  The probability that the straight-line distance between the points is at least  $\tfrac12$  is  $\tfrac{a-b\pi}c$ , where  $a$ ,  $b$ , and  $c$  are positive integers and  $\gcd(a,b,c)=1$ .  What is  $a+b+c$ ? $\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

Answer: 59


Given are 100 different positive integers. We call a pair of numbers *good* if the ratio of these numbers is either 2 or 3. What is the maximum number of good pairs that these 100 numbers can form? (A number can be used in several pairs.)

*Proposed by Alexander S. Golovanov, Russia*

Answer: 180


A set  $A$  of positive integers is called *uniform* if, after any of its elements removed, the remaining ones can be partitioned into two subsets with equal sum of their elements. Find the least positive integer  $n>1$  such that there exist a uniform set  $A$  with  $n$  elements.

Answer: 7


The set  $M= \{1;2;3;\ldots ; 29;30\}$  is  divided in  $k$ subsets such that if  $a+b=n^2, (a,b \in M, a\neq b, n$  is an
integer number  $)$ , then  $a$  and  $b$  belong different subsets.
Determine the minimum value of  $k$ .

Answer: 3


Given a grid rectangle of size  $2010 \times  1340$ . A grid point is called *fair* if the  $2$  axis-parallel lines passing through it from the upper left and lower right corners of the large rectangle cut out a rectangle of equal area (such a point is shown in the figure). How many fair grid points lie inside the rectangle?
![Image](https://cdn.artofproblemsolving.com/attachments/1/b/21d4fb47c94b774994ac1c3aae7690bb98c7ae.png)

Answer: 669


Let the curve  $ C: y \equal{} |x^2 \plus{} 2x \minus{} 3|$  and the line  $ l$  passing through the point  $ ( \minus{} 3,\ 0)$  with a slope  $ m$  in  $ x\minus{}y$  plane. Suppose that  $ C$  intersects to  $ l$  at distinct two points other than the point  $ ( \minus{} 3,\ 0)$ , find the value of  $ m$  for which the area of the figure bounded by  $ C$  and  $ l$  is minimized.

Answer: 12-8\sqrt{2}


A rope of length 10 *m* is tied tautly from the top of a flagpole to the ground 6 *m* away from the base of the pole. An ant crawls up the rope and its shadow moves at a rate of 30 *cm/min*. How many meters above the ground is the ant after 5 minutes? (This takes place on the summer solstice on the Tropic of Cancer so that the sun is directly overhead.)

Answer: 2


There are $20$ people in a certain community. $10$ of them speak English, $10$ of them speak German, and $10$ of them speak French. We call a committee a $3$-subset of this community if there is at least one who speaks English, at least one who speaks German, and at least one who speaks French in this subset. At most, how many committees are there in this community?

Answer: 1020


We have \( n \) objects to place in \( n \) positions. However, some objects are identical. There are \( k \) different types of objects, and there are \( n_{i} \) objects of the \( i^{\text{th}} \) type. Thus, \( \sum_{i=1}^{k} n_{i} = n \).

How many arrangements of these objects are there?

Answer: \dfrac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!}


Problem 1. Find all primary numbers $p$ and $q$, for which $\frac{\left(5^{p}-2^{p}\right)\left(5^{q}-2^{q}\right)}{p q}$ is an integer.

Answer: (3, 3)


Problem 2. The numbers $d$ and $e$ are the roots of the quadratic trinomial $a x^{2}+b x+c$. Could it be that $a, b, c, d, e$ are consecutive integers in some order?

Answer: Yes


Dima needed to arrive at the station at 18:00, where his father was supposed to pick him up by car. However, Dima caught an earlier train and arrived at the station at 17:05. Instead of waiting for his father, he started walking towards him. They met on the way, Dima got into the car, and they arrived home 10 minutes earlier than the calculated time. What was Dima's walking speed before he met his father, if the car's speed was $60 \ \text{km/h}$?

Answer: 6


9.2. Vasya has thought of 8 cells on a chessboard, none of which lie in the same row or column. On his turn, Petya places 8 rooks on the board, none of which attack each other, and then Vasya indicates all the rooks standing on the thought-of cells. If the number of rooks indicated by Vasya on this turn is even (i.e., 0, 2, 4, 6, or 8), then Petya wins; otherwise, all the pieces are removed from the board and Petya makes the next move. In the least number of moves, how many moves will Petya need to guarantee a win?

(I. Bogdanov)

Answer: 2


8.3. In trapezoid $A B C D$ with base $A D$, it is known that $A B=B C, A C=C D$ and $B C+C D=A D$. Find the angles of the trapezoid.

Answer: 72^\circ


We select a real number $\alpha$ uniformly and at random from the interval $(0,500)$.  Define \[ S = \frac{1}{\alpha} \sum_{m=1}^{1000} \sum_{n=m}^{1000} \left\lfloor \frac{m+\alpha}{n} \right\rfloor. \] Let $p$ denote the probability that $S \ge 1200$.  Compute $1000p$.

[i]Proposed by Evan Chen[/i]

Answer: 5


8. (3 points) In $17 \square 17 \square 17 \square 17$, fill in the four $\square$ with the operation symbols “+”, “-”, “$\times$”, “$\div$” each once. The maximum value of the resulting expression is $\qquad$ .

Answer: 305


Problem 9.5. Let $A B C D$ be a cyclic quadrilateral with circumcircle $k$. The rays $\overrightarrow{D A}$ and $\overrightarrow{C B}$ meet at point $N$ and the line $N T$ is tangent to $k, T \in k$. The diagonals $A C$ and $B D$ meet at the centroid $P$ of $\triangle N T D$. Find the ratio $N T: A P$.

Ivailo Kortezov

Answer: 3


11.2. On the sides $A B$ and $A C$ of an equilateral triangle $A B C$ with side length 10, points $P$ and $Q$ are taken such that the segment $P Q$ is tangent to the inscribed circle of the triangle and its length is 4. Find the area of triangle $A P Q$.

Answer: \dfrac{5\sqrt{3}}{3}


6.

when transferring money abroad

Correct answers: when using the Fast Payment System for amounts up to 100 thousand rubles per month, when transferring funds between one's own accounts in the same bank

Question 17

Score: 7.00

Last year, a beauty salon offered a $20 \%$ discount on facial massage when purchasing a membership for 30000 rubles. This year, it was decided to change the loyalty program and, when purchasing a membership for 30000 rubles, an additional $20 \%$ of this amount would be credited to the client's account. How will the number of facial massages that a client can now attend with a membership for 30000 rubles change, if the cost of a massage is 1500 rubles?

In the answer, write a non-negative integer without units of measurement. If the number of massages has not changed, put 0.

Answer: 1


6. In the Cartesian coordinate system, $O$ is the origin, points $A(3, a) 、 B(3, b)$ make $\angle A O B=45^{\circ}$, where $a 、 b$ are integers, and $a>b$. Then the number of pairs $(a, b)$ that satisfy the condition is.

Answer: 6


1. The first three figures in a pattern are shown below.

Figure 1 is formed by three identical squares of side length $1 \mathrm{~cm}$ arranged in two rows. The perimeter of Figure 1 is $8 \mathrm{~cm}$. Given a figure in the pattern, the next figure is obtained by adding a square of side length $1 \mathrm{~cm}$ on the right-hand end of each of the two rows.
(a) How many squares of side length $1 \mathrm{~cm}$ are used to form Figure 8?
(b) Determine the perimeter of Figure 12.
(c) Determine the positive integer $C$ for which the perimeter of Figure $C$ is $38 \mathrm{~cm}$.
(d) Determine the positive integer $D$ for which the ratio of the perimeter of Figure 29 to the perimeter of Figure $D$ is equal to $\frac{4}{11}$.

Answer: 17


$F$ is a convex figure with two mutually perpendicular axes of symmetry. Through a point $M$ located inside the figure and at distances $a$ and $b$ from the axes, lines parallel to the axes are drawn. These lines divide $F$ into four regions. Find the difference between the sum of the areas of the larger and smaller regions and the sum of the areas of the other two regions.

Answer: 4ab


(*) We call $I$ the set of points in the plane such that their abscissa and ordinate are irrational numbers, and $R$ the set of points whose both coordinates are rational. How many points of $R$ at most can lie on a circle of irrational radius whose center belongs to $I$?

Answer: 2


Problem 9.2. At a party, 24 people gathered. A guest is considered an introvert if they have no more than three acquaintances among the other guests. It turned out that each guest has at least three acquaintances who are introverts. How many introverts could there have been at the party? Provide all possible answers and prove that there are no others.

#

Answer: 24


218. Without solving the quadratic equation $x^{2}+p x+q=0$, form a new equation whose roots are the squares of the roots of the given equation.

Answer: x^2 + (2q - p^2)x + q^2 = 0


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`, `Alice`, `Bob`
 - Each person has a unique hobby: `painting`, `cooking`, `knitting`, `gardening`, `photography`
 - People have unique heights: `very tall`, `tall`, `very short`, `average`, `short`
 - Everyone has something unique for lunch: `stew`, `grilled cheese`, `stir fry`, `spaghetti`, `pizza`

## Clues:
1. Bob is the photography enthusiast.
2. The person who loves eating grilled cheese is the person who is tall.
3. Peter is not in the second house.
4. The person who is tall is directly left of the person who loves stir fry.
5. The person who loves cooking is the person who has an average height.
6. Alice is directly left of the person who is a pizza lover.
7. The person who loves the spaghetti eater is not in the second house.
8. Eric is not in the fifth house.
9. The person who is short is Peter.
10. The person who has an average height and the person who enjoys gardening are next to each other.
11. The person who paints as a hobby is directly left of the person who loves eating grilled cheese.
12. The person who is very short is in the fifth house.
13. The person who is tall is in the third house.
14. Alice is somewhere to the right of the photography enthusiast.
 What is the value of attribute Height for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: tall


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - Each person prefers a unique type of vacation: `city`, `beach`, `mountain`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `cherry`
 - People own unique car models: `toyota camry`, `tesla model 3`, `ford f150`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `lilies`

## Clues:
1. The person who likes Cherry smoothies is directly left of the person who loves a bouquet of daffodils.
2. Eric is the person who loves beach vacations.
3. The person who loves a carnations arrangement is somewhere to the left of the person who loves a bouquet of daffodils.
4. The person who owns a Toyota Camry is not in the first house.
5. The Watermelon smoothie lover is directly left of the person who is a pizza lover.
6. Peter is somewhere to the left of Arnold.
7. Arnold is not in the third house.
8. The person who loves beach vacations is the person who owns a Ford F-150.
9. The person who loves the spaghetti eater is the person who prefers city breaks.
10. Peter and the person who loves the spaghetti eater are next to each other.
 What is the value of attribute Food for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: grilled cheese


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`
 - Everyone has something unique for lunch: `pizza`, `stew`, `spaghetti`, `grilled cheese`
 - Each person lives in a unique style of house: `colonial`, `ranch`, `victorian`, `craftsman`
 - The people keep unique animals: `cat`, `horse`, `fish`, `bird`
 - Each mother is accompanied by their child: `Meredith`, `Samantha`, `Fred`, `Bella`

## Clues:
1. The cat lover is not in the third house.
2. There is one house between the person residing in a Victorian house and the person who loves the stew.
3. The person residing in a Victorian house and the bird keeper are next to each other.
4. The person who loves eating grilled cheese is the person in a ranch-style home.
5. The person who loves the spaghetti eater is Peter.
6. Alice is the person in a ranch-style home.
7. The cat lover is Arnold.
8. The person in a Craftsman-style house is the person's child is named Samantha.
9. The person's child is named Bella is directly left of the person who loves the spaghetti eater.
10. The person who is a pizza lover is the person's child is named Fred.
11. The person who loves eating grilled cheese is in the second house.
12. The person's child is named Samantha is somewhere to the right of the fish enthusiast.
 What is the value of attribute House for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


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 House for the person whose attribute PhoneModel is oneplus 9? 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: `Arnold`, `Eric`, `Peter`
 - Each person prefers a unique type of vacation: `beach`, `city`, `mountain`
 - People have unique favorite sports: `soccer`, `basketball`, `tennis`
 - Each person has a unique birthday month: `april`, `jan`, `sept`

## Clues:
1. The person whose birthday is in April is not in the third house.
2. There is one house between the person who enjoys mountain retreats and the person who loves beach vacations.
3. The person who loves soccer is Peter.
4. The person who loves tennis is the person who prefers city breaks.
5. Eric is directly left of the person who loves tennis.
6. The person who loves beach vacations is the person who loves soccer.
7. The person who prefers city breaks is the person whose birthday is in January.
 What is the value of attribute Name for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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

## Clues:
1. Peter is in the first house.
2. The person who loves soccer is the person who paints as a hobby.
3. The person partial to Pall Mall is the person who enjoys knitting.
4. The person who uses a Google Pixel 6 is not in the fifth house.
5. The person who enjoys knitting is somewhere to the left of Alice.
6. Peter is directly left of the person who smokes Blue Master.
7. The Prince smoker is the person who uses a OnePlus 9.
8. The person who loves basketball is directly left of the photography enthusiast.
9. The Dunhill smoker is somewhere to the left of the photography enthusiast.
10. There is one house between the person who drinks Lime smoothies and the person who loves basketball.
11. The person who enjoys gardening is somewhere to the right of the person who enjoys knitting.
12. The Dragonfruit smoothie lover is in the third house.
13. The person who loves baseball is the person partial to Pall Mall.
14. Eric is the person who loves tennis.
15. The person who uses a OnePlus 9 is Peter.
16. The Dunhill smoker is the person who uses a Huawei P50.
17. The Desert smoothie lover is the person who loves soccer.
18. The person who uses a Samsung Galaxy S21 is directly left of Arnold.
19. The person who loves tennis is somewhere to the right of the Watermelon smoothie lover.
 What is the value of attribute Smoothie for the person whose attribute Hobby is gardening? Please reason step by step, and put your final answer within \boxed{}

Answer: watermelon


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Arnold`, `Eric`, `Alice`, `Carol`
 - People have unique hair colors: `red`, `gray`, `auburn`, `blonde`, `brown`, `black`
 - People have unique favorite book genres: `historical fiction`, `biography`, `romance`, `fantasy`, `mystery`, `science fiction`
 - People have unique heights: `very short`, `very tall`, `tall`, `short`, `super tall`, `average`
 - Each person has a unique level of education: `associate`, `doctorate`, `high school`, `trade school`, `bachelor`, `master`
 - Each person has a unique favorite drink: `boba tea`, `milk`, `water`, `root beer`, `coffee`, `tea`

## Clues:
1. The person with a doctorate and Bob are next to each other.
2. There is one house between Carol and the person who has black hair.
3. Peter is the person with a doctorate.
4. The person who loves science fiction books is Eric.
5. The one who only drinks water is the person who has an average height.
6. Alice is the person who loves mystery books.
7. The person who loves romance books is Bob.
8. The person who has brown hair is in the second house.
9. The person who has an average height is in the fourth house.
10. The boba tea drinker is the person who is very tall.
11. The person with a high school diploma is Bob.
12. The person who is short is somewhere to the left of the person who is super tall.
13. The person who loves historical fiction books and the person with a bachelor's degree are next to each other.
14. The person who attended trade school is the person who has red hair.
15. Carol is the person who has auburn hair.
16. The person who is very short is somewhere to the right of the person who has gray hair.
17. There are two houses between Eric and the person who loves fantasy books.
18. There are two houses between the tea drinker and the person who is very short.
19. Eric is somewhere to the left of the person with a bachelor's degree.
20. The root beer lover is Alice.
21. The person with an associate's degree is Arnold.
22. The person who is tall is not in the second house.
23. The person who loves science fiction books is the boba tea drinker.
24. The person who has black hair is not in the fourth house.
25. The person who is short is the coffee drinker.
26. The coffee drinker and the person with a high school diploma are next to each other.
27. The coffee drinker is not in the second house.
 What is the value of attribute Height for the person whose attribute HairColor is blonde? Please reason step by step, and put your final answer within \boxed{}

Answer: very short


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Peter`, `Alice`, `Arnold`, `Carol`
 - People own unique car models: `ford f150`, `honda civic`, `toyota camry`, `tesla model 3`, `chevrolet silverado`, `bmw 3 series`
 - The mothers' names in different houses are unique: `Sarah`, `Penny`, `Holly`, `Aniya`, `Kailyn`, `Janelle`
 - Each person has a unique hobby: `photography`, `cooking`, `knitting`, `gardening`, `woodworking`, `painting`

## Clues:
1. The person who owns a Toyota Camry is in the sixth house.
2. Carol is the photography enthusiast.
3. The person who owns a Chevrolet Silverado is The person whose mother's name is Aniya.
4. The person who owns a Chevrolet Silverado is not in the second house.
5. The person who owns a Ford F-150 is The person whose mother's name is Sarah.
6. The person who owns a BMW 3 Series is Bob.
7. The person whose mother's name is Kailyn is in the sixth house.
8. Eric is directly left of the person who enjoys knitting.
9. There is one house between The person whose mother's name is Sarah and the person who owns a Toyota Camry.
10. The person whose mother's name is Penny is somewhere to the right of the person who enjoys knitting.
11. The person whose mother's name is Aniya is somewhere to the right of the person who owns a Honda Civic.
12. Alice is somewhere to the right of the person who owns a Ford F-150.
13. Eric is the person who enjoys gardening.
14. The woodworking hobbyist is somewhere to the left of the person who enjoys knitting.
15. There is one house between The person whose mother's name is Sarah and the person who loves cooking.
16. The person who owns a Honda Civic is Arnold.
17. The person whose mother's name is Holly is directly left of the person who enjoys knitting.
 What is the value of attribute CarModel for the person whose attribute Hobby is photography? Please reason step by step, and put your final answer within \boxed{}

Answer: ford f150


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`, `Bob`, `Alice`, `Eric`, `Peter`
 - People have unique favorite sports: `basketball`, `soccer`, `tennis`, `baseball`, `swimming`
 - The people keep unique animals: `dog`, `fish`, `horse`, `bird`, `cat`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `samsung galaxy s21`, `iphone 13`, `huawei p50`
 - People have unique hair colors: `black`, `red`, `brown`, `blonde`, `gray`
 - Each person lives in a unique style of house: `modern`, `craftsman`, `victorian`, `ranch`, `colonial`

## Clues:
1. The person who has gray hair is the person who loves tennis.
2. The person in a ranch-style home is in the fourth house.
3. The person living in a colonial-style house is the person who loves swimming.
4. The dog owner is somewhere to the right of the person who has gray hair.
5. The fish enthusiast is the person who has gray hair.
6. The person who has black hair is not in the fifth house.
7. The person who uses a Samsung Galaxy S21 is the bird keeper.
8. There is one house between the fish enthusiast and the person who uses a Samsung Galaxy S21.
9. The person residing in a Victorian house is somewhere to the left of the person in a Craftsman-style house.
10. The person who has red hair is not in the fifth house.
11. There is one house between the person who uses a Samsung Galaxy S21 and the person who uses a Huawei P50.
12. The person who uses an iPhone 13 is the person in a Craftsman-style house.
13. Eric is the person who has red hair.
14. The person who has red hair is not in the fourth house.
15. The person who uses a OnePlus 9 is in the first house.
16. The person who loves soccer is the dog owner.
17. The person who keeps horses is Bob.
18. Arnold is the fish enthusiast.
19. The person who loves baseball is Peter.
20. The person who loves soccer is in the fourth house.
21. The person who has blonde hair is directly left of the person who has black hair.
 What is the value of attribute House for the person whose attribute PhoneModel is google pixel 6? 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`, `Peter`, `Eric`, `Bob`, `Carol`, `Arnold`
 - People have unique favorite sports: `baseball`, `basketball`, `swimming`, `tennis`, `soccer`, `volleyball`
 - Each person has a favorite color: `purple`, `green`, `white`, `red`, `yellow`, `blue`
 - People have unique hair colors: `red`, `auburn`, `blonde`, `black`, `gray`, `brown`
 - Everyone has something unique for lunch: `stew`, `spaghetti`, `stir fry`, `grilled cheese`, `soup`, `pizza`

## Clues:
1. Arnold is somewhere to the left of the person who loves blue.
2. There are two houses between the person whose favorite color is green and Bob.
3. The person who loves white is in the fifth house.
4. The person who loves baseball is directly left of the person who has brown hair.
5. The person who loves tennis is not in the fifth house.
6. The person who has auburn hair is the person who loves the stew.
7. The person whose favorite color is red and the person who loves baseball are next to each other.
8. The person who is a pizza lover is Eric.
9. The person who loves purple is Carol.
10. Peter is in the third house.
11. Carol is the person who loves soccer.
12. The person who loves volleyball is somewhere to the left of the person who has auburn hair.
13. Eric is in the first house.
14. There is one house between the person who has gray hair and the person who loves blue.
15. The person who loves volleyball is directly left of the person who loves baseball.
16. The person who loves the spaghetti eater is in the fifth house.
17. The person who loves eating grilled cheese is the person who has brown hair.
18. The person who loves the soup is in the third house.
19. The person who has blonde hair is somewhere to the left of the person who has gray hair.
20. The person who has red hair is the person who loves white.
21. The person who loves blue is somewhere to the right of the person who loves swimming.
 What is the value of attribute FavoriteSport for the person whose attribute Food is stew? Please reason step by step, and put your final answer within \boxed{}

Answer: soccer


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Bob`, `Carol`, `Eric`, `Alice`, `Arnold`
 - Each person has a unique type of pet: `bird`, `dog`, `cat`, `rabbit`, `fish`, `hamster`
 - Each person lives in a unique style of house: `victorian`, `ranch`, `modern`, `mediterranean`, `colonial`, `craftsman`
 - Each person has a unique birthday month: `mar`, `sept`, `may`, `feb`, `jan`, `april`

## Clues:
1. The person with a pet hamster is somewhere to the right of the person whose birthday is in March.
2. The person whose birthday is in January is somewhere to the left of the person whose birthday is in September.
3. The person whose birthday is in May is in the second house.
4. The person living in a colonial-style house is in the second house.
5. Carol is in the third house.
6. The person in a Mediterranean-style villa is not in the sixth house.
7. The person with an aquarium of fish is somewhere to the right of Bob.
8. Eric is in the sixth house.
9. There is one house between the person who has a cat and the person residing in a Victorian house.
10. There are two houses between the person residing in a Victorian house and the person with a pet hamster.
11. The person in a Craftsman-style house is Arnold.
12. The person living in a colonial-style house is somewhere to the left of the person in a modern-style house.
13. The person with an aquarium of fish is not in the second house.
14. Peter is the person living in a colonial-style house.
15. The person whose birthday is in January is directly left of the person whose birthday is in April.
16. There is one house between the person who keeps a pet bird and the person in a modern-style house.
17. Carol is the person whose birthday is in March.
18. The person in a Craftsman-style house is in the fourth house.
19. The person who owns a dog is in the fourth house.
 What is the value of attribute House for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


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

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

Answer: prince


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Peter`, `Bob`, `Arnold`
 - The mothers' names in different houses are unique: `Penny`, `Holly`, `Janelle`, `Aniya`, `Kailyn`
 - Each mother is accompanied by their child: `Samantha`, `Fred`, `Timothy`, `Meredith`, `Bella`
 - People have unique favorite book genres: `mystery`, `fantasy`, `biography`, `science fiction`, `romance`

## Clues:
1. The person who loves mystery books is somewhere to the left of the person who is the mother of Timothy.
2. The person whose mother's name is Janelle is in the third house.
3. The person who loves science fiction books is directly left of the person's child is named Bella.
4. Arnold is somewhere to the right of the person's child is named Meredith.
5. The person's child is named Fred is Peter.
6. Alice is directly left of Eric.
7. The person whose mother's name is Aniya is in the second house.
8. The person's child is named Fred is not in the fifth house.
9. The person who loves romance books is Alice.
10. The person whose mother's name is Janelle is directly left of the person who is the mother of Timothy.
11. The person whose mother's name is Aniya is somewhere to the left of The person whose mother's name is Kailyn.
12. The person who loves science fiction books is Bob.
13. The person whose mother's name is Janelle is somewhere to the left of the person who loves fantasy books.
14. The person who loves fantasy books is somewhere to the left of The person whose mother's name is Penny.
 What is the value of attribute Mother for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: Janelle


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Peter`, `Eric`
 - People own unique car models: `ford f150`, `tesla model 3`, `toyota camry`, `honda civic`
 - Each person has a unique birthday month: `april`, `sept`, `feb`, `jan`
 - Each person has an occupation: `teacher`, `artist`, `engineer`, `doctor`
 - Everyone has something unique for lunch: `pizza`, `stew`, `spaghetti`, `grilled cheese`

## Clues:
1. The person who owns a Toyota Camry is in the first house.
2. Eric and the person who loves the spaghetti eater are next to each other.
3. The person who is an engineer is the person who loves eating grilled cheese.
4. Peter is the person who owns a Toyota Camry.
5. The person whose birthday is in April is directly left of the person whose birthday is in February.
6. The person who is a doctor is somewhere to the right of the person who is an artist.
7. The person who owns a Ford F-150 is the person who is a doctor.
8. The person whose birthday is in September is somewhere to the left of Alice.
9. Arnold is the person who is an artist.
10. The person who loves the spaghetti eater is the person who owns a Ford F-150.
11. The person who is an engineer is the person whose birthday is in April.
12. The person whose birthday is in February and the person who is a pizza lover are next to each other.
13. The person who owns a Honda Civic is Eric.
 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 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Carol`, `Alice`, `Eric`, `Bob`
 - People have unique favorite book genres: `mystery`, `science fiction`, `fantasy`, `romance`, `biography`, `historical fiction`
 - Everyone has a unique favorite cigar: `dunhill`, `blends`, `yellow monster`, `prince`, `blue master`, `pall mall`
 - The people are of nationalities: `brit`, `swede`, `chinese`, `dane`, `german`, `norwegian`
 - Each person has a unique type of pet: `bird`, `rabbit`, `fish`, `cat`, `hamster`, `dog`

## Clues:
1. The person with an aquarium of fish is in the first house.
2. The person who owns a dog is not in the third house.
3. The British person is the person who loves historical fiction books.
4. Bob is the person who owns a dog.
5. The person with a pet hamster is the British person.
6. Eric is in the sixth house.
7. There is one house between the person who smokes Yellow Monster and the person who owns a dog.
8. Alice is directly left of the person who has a cat.
9. The British person is directly left of the person who smokes Blue Master.
10. Carol is in the second house.
11. The person who loves fantasy books is the Swedish person.
12. The person who loves romance books is directly left of the Dunhill smoker.
13. The person partial to Pall Mall is the Swedish person.
14. The Prince smoker is the Chinese.
15. The person who loves biography books is somewhere to the left of the British person.
16. The person who loves science fiction books is in the fourth house.
17. The British person is Alice.
18. The Norwegian is the person who loves biography books.
19. The Dane and the Dunhill smoker are next to each other.
20. Peter is not in the third house.
21. The person who owns a rabbit is the person who loves romance books.
 What is the value of attribute House for the person whose attribute Cigar is blends? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `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 Education for the person whose attribute Nationality is norwegian? Please reason step by step, and put your final answer within \boxed{}

Answer: high school


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`, `Peter`, `Bob`, `Alice`
 - Each person has an occupation: `engineer`, `doctor`, `artist`, `lawyer`, `teacher`
 - Each person prefers a unique type of vacation: `cruise`, `beach`, `city`, `camping`, `mountain`

## Clues:
1. The person who prefers city breaks is somewhere to the left of the person who is an engineer.
2. There are two houses between the person who likes going on cruises and the person who enjoys camping trips.
3. Bob is in the fourth house.
4. The person who is a doctor is not in the fifth house.
5. The person who likes going on cruises is in the first house.
6. There are two houses between the person who is a doctor and the person who is a lawyer.
7. The person who is a lawyer and Arnold are next to each other.
8. There is one house between Peter and the person who is a teacher.
9. The person who enjoys mountain retreats is somewhere to the left of the person who loves beach vacations.
10. Alice is somewhere to the left of the person who is a teacher.
11. There is one house between the person who prefers city breaks and Arnold.
 What is the value of attribute Occupation for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: teacher


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`
 - Each person has a unique level of education: `associate`, `high school`, `bachelor`
 - Each person prefers a unique type of vacation: `city`, `beach`, `mountain`
 - People have unique favorite music genres: `classical`, `rock`, `pop`
 - Each person has an occupation: `engineer`, `doctor`, `teacher`
 - People have unique favorite sports: `tennis`, `basketball`, `soccer`

## Clues:
1. Peter is in the first house.
2. The person with a high school diploma is directly left of the person who loves beach vacations.
3. The person who is a doctor is not in the third house.
4. The person who loves rock music and the person who loves pop music are next to each other.
5. The person who is a teacher is the person who enjoys mountain retreats.
6. Peter is the person with a bachelor's degree.
7. The person who prefers city breaks is the person who loves rock music.
8. The person who loves basketball is the person who enjoys mountain retreats.
9. Eric is in the third house.
10. The person with a bachelor's degree is the person who loves tennis.
 What is the value of attribute Vacation for the person whose attribute MusicGenre is pop? Please reason step by step, and put your final answer within \boxed{}

Answer: mountain


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`, `Alice`, `Eric`, `Peter`
 - People have unique favorite sports: `swimming`, `basketball`, `volleyball`, `soccer`, `tennis`, `baseball`
 - People have unique favorite music genres: `rock`, `pop`, `jazz`, `country`, `hip hop`, `classical`
 - Each person has a unique level of education: `doctorate`, `trade school`, `bachelor`, `high school`, `associate`, `master`

## Clues:
1. Alice is in the fourth house.
2. Alice is somewhere to the right of the person with a doctorate.
3. The person with a high school diploma is somewhere to the right of the person who loves basketball.
4. The person who loves volleyball is the person with a bachelor's degree.
5. Alice is directly left of the person who loves jazz music.
6. The person with a master's degree is the person who loves swimming.
7. There is one house between the person who loves rock music and the person with a bachelor's degree.
8. Carol is in the second house.
9. The person who loves rock music is somewhere to the left of the person who loves pop music.
10. The person who loves swimming is somewhere to the left of Carol.
11. Eric is directly left of the person who loves hip-hop music.
12. Alice is the person with an associate's degree.
13. Alice is the person who loves classical music.
14. Arnold is the person who loves soccer.
15. The person who loves basketball and the person who loves volleyball are next to each other.
16. Peter is the person who loves tennis.
17. The person with a doctorate is Peter.
 What is the value of attribute MusicGenre for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: jazz