$10 \cdot 44$ What is the maximum difference between a three-digit number and the sum of the cubes of its digits? What kind of three-digit number can achieve this maximum difference? What is the smallest positive difference?

(10th All-Russian Mathematical Olympiad, 1984)

Answer: 396


1. As shown in Figure $4, \triangle A B C$ has an incircle $\odot O_{1}$ that touches side $B C$ at point $D, \odot O_{2}$ is the excircle of $\triangle A B C$ inside $\angle A$. If $O_{1} B=6, O_{1} C=3, O_{1} D=2$, then $O_{1} O_{2}=$ $\qquad$

Answer: 9


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Eric`, `Peter`
 - Each person has a unique level of education: `master`, `bachelor`, `high school`, `associate`
 - People have unique heights: `short`, `average`, `tall`, `very short`
 - Each person has a unique favorite drink: `water`, `coffee`, `tea`, `milk`

## Clues:
1. The person who likes milk is somewhere to the left of the person with a bachelor's degree.
2. The person who likes milk and the person with a bachelor's degree are next to each other.
3. Alice is the person who is tall.
4. Peter is not in the second house.
5. Arnold is in the first house.
6. The person who has an average height is not in the second house.
7. Arnold is directly left of the person with a master's degree.
8. The coffee drinker is the person who is short.
9. There is one house between the person with a master's degree and the person with a high school diploma.
10. The one who only drinks water is directly left of Alice.
11. The person who is short is somewhere to the left of the person who is tall.
 What is the value of attribute House for the person whose attribute Height is very short? Please reason step by step, and put your final answer within \boxed{}

Answer: 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: `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 Education is bachelor? Please reason step by step, and put your final answer within \boxed{}

Answer: google pixel 6


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

## Clues:
1. The person who owns a dog is Arnold.
2. The bird keeper is in the fourth house.
3. The person who keeps a pet bird is directly left of the dog owner.
4. The person who loves white is somewhere to the left of the person who is a lawyer.
5. The person who loves yellow is directly left of the person whose favorite color is green.
6. The cat lover is Bob.
7. The cat lover is somewhere to the left of Eric.
8. The person who keeps horses is in the fifth house.
9. The person who is a lawyer is directly left of the person who is a teacher.
10. The person who is a doctor is in the first house.
11. Alice is the person who loves yellow.
12. The person who loves blue is directly left of the person with an aquarium of fish.
13. The person who loves yellow is in the first house.
14. The person with a pet hamster is the person who is an artist.
15. Eric is the dog owner.
 What is the value of attribute Color for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: white


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`
 - The people keep unique animals: `horse`, `bird`, `cat`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `desert`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`
 - People have unique favorite music genres: `pop`, `classical`, `rock`
 - Each person has a favorite color: `yellow`, `white`, `red`

## Clues:
1. The cat lover is in the third house.
2. There is one house between the person who loves yellow and the cat lover.
3. The person who loves classical music is directly left of Eric.
4. Arnold is the Desert smoothie lover.
5. The bird keeper is the Watermelon smoothie lover.
6. The person whose mother's name is Janelle is somewhere to the left of the person who loves white.
7. The person who loves yellow is the person who loves rock music.
8. The person whose mother's name is Holly is directly left of Peter.
 What is the value of attribute MusicGenre for the person whose attribute Animal is cat? Please reason step by step, and put your final answer within \boxed{}

Answer: pop


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`
 - Everyone has something unique for lunch: `pizza`, `grilled cheese`, `spaghetti`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `cherry`
 - Each person has a unique type of pet: `fish`, `dog`, `cat`

## Clues:
1. The person who likes Cherry smoothies is the person with an aquarium of fish.
2. The person who loves the spaghetti eater is directly left of the person who has a cat.
3. The person who has a cat is Eric.
4. The Watermelon smoothie lover is Arnold.
5. The person who owns a dog is directly left of the person who is a pizza lover.
6. The Desert smoothie lover is not in the third house.
 What is the value of attribute Food for the person whose attribute Pet is fish? Please reason step by step, and put your final answer within \boxed{}

Answer: grilled cheese


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Alice`, `Eric`, `Peter`, `Arnold`, `Bob`
 - The mothers' names in different houses are unique: `Penny`, `Janelle`, `Aniya`, `Sarah`, `Holly`, `Kailyn`
 - Everyone has a unique favorite cigar: `blends`, `dunhill`, `blue master`, `prince`, `yellow monster`, `pall mall`
 - Each person has a unique type of pet: `cat`, `dog`, `rabbit`, `hamster`, `bird`, `fish`
 - The people keep unique animals: `bird`, `horse`, `cat`, `rabbit`, `dog`, `fish`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `lime`, `cherry`, `blueberry`, `dragonfruit`

## Clues:
1. The dog owner is not in the third house.
2. The fish enthusiast is not in the fourth house.
3. The person who owns a dog is Eric.
4. The cat lover is not in the second house.
5. The person who smokes Blue Master is somewhere to the right of The person whose mother's name is Sarah.
6. The cat lover is not in the fourth house.
7. There are two houses between Carol and the person who owns a rabbit.
8. The person whose mother's name is Penny is the person with a pet hamster.
9. The person whose mother's name is Kailyn is somewhere to the right of the person with a pet hamster.
10. The dog owner and the person who drinks Lime smoothies are next to each other.
11. The Prince smoker is somewhere to the right of The person whose mother's name is Janelle.
12. There is one house between The person whose mother's name is Penny and Bob.
13. There are two houses between the person partial to Pall Mall and The person whose mother's name is Penny.
14. Peter is somewhere to the left of the person who drinks Lime smoothies.
15. The bird keeper is the Watermelon smoothie lover.
16. The person with a pet hamster is the Prince smoker.
17. The person who smokes Yellow Monster is the person who drinks Lime smoothies.
18. The person whose mother's name is Sarah is the person who smokes Yellow Monster.
19. The Dunhill smoker is somewhere to the left of the Watermelon smoothie lover.
20. The person whose mother's name is Holly is directly left of the person who owns a dog.
21. The bird keeper is The person whose mother's name is Kailyn.
22. The person whose mother's name is Sarah is not in the third house.
23. The person who smokes many unique blends is the rabbit owner.
24. The person who has a cat is somewhere to the right of the person with an aquarium of fish.
25. The person who smokes Yellow Monster is Eric.
26. The person who keeps a pet bird is the Desert smoothie lover.
27. Alice is The person whose mother's name is Kailyn.
28. The Dragonfruit smoothie lover is Carol.
29. The person who keeps horses is the person who likes Cherry smoothies.
 What is the value of attribute Mother for the person whose attribute Animal is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: Kailyn


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Alice`
 - People have unique favorite music genres: `rock`, `jazz`, `classical`, `pop`
 - The people are of nationalities: `dane`, `brit`, `swede`, `norwegian`
 - Each person has a favorite color: `green`, `red`, `yellow`, `white`

## Clues:
1. The British person and Peter are next to each other.
2. Eric is somewhere to the left of the British person.
3. The Swedish person is directly left of the person whose favorite color is red.
4. The person who loves classical music is in the second house.
5. There is one house between the British person and the person whose favorite color is red.
6. The person who loves rock music is the Norwegian.
7. The person who loves yellow is not in the first house.
8. The Norwegian is Arnold.
9. The Dane and the person who loves white are next to each other.
10. There are two houses between the person who loves pop music and the person who loves rock music.
 What is the value of attribute Color for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: red


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

Answer: sept


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Alice`, `Arnold`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`
 - Each person has a unique birthday month: `feb`, `sept`, `jan`, `april`
 - The mothers' names in different houses are unique: `Aniya`, `Janelle`, `Kailyn`, `Holly`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`, `blue master`, `dunhill`
 - They all have a unique favorite flower: `lilies`, `daffodils`, `carnations`, `roses`

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

Answer: Kailyn


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`
 - Each person has an occupation: `doctor`, `teacher`, `engineer`
 - Each person has a unique level of education: `associate`, `high school`, `bachelor`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`
 - Each person has a unique hobby: `gardening`, `cooking`, `photography`

## Clues:
1. The Desert smoothie lover is the person who is a doctor.
2. Arnold is not in the third house.
3. The person who likes Cherry smoothies is somewhere to the right of Peter.
4. The person who loves cooking is in the second house.
5. The person who loves cooking is Peter.
6. The person with an associate's degree is somewhere to the right of the person who enjoys gardening.
7. The person with a bachelor's degree is somewhere to the right of the Desert smoothie lover.
8. The person who loves cooking is the person who is a doctor.
9. The photography enthusiast is the person who is a teacher.
 What is the value of attribute Hobby for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: photography


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - Each person lives in a unique style of house: `colonial`, `victorian`, `ranch`
 - People have unique favorite book genres: `science fiction`, `mystery`, `romance`
 - Each person has a unique type of pet: `fish`, `cat`, `dog`
 - Everyone has a unique favorite cigar: `prince`, `blue master`, `pall mall`
 - Each person has a unique hobby: `cooking`, `photography`, `gardening`

## Clues:
1. The person who loves cooking is in the second house.
2. The person who loves romance books is the person who smokes Blue Master.
3. The person who loves mystery books is the person partial to Pall Mall.
4. The person who smokes Blue Master and the person who loves cooking are next to each other.
5. The person who owns a dog is the person in a ranch-style home.
6. The person residing in a Victorian house is the person with an aquarium of fish.
7. The person who has a cat is the Prince smoker.
8. Arnold and the person who has a cat are next to each other.
9. The photography enthusiast is directly left of Arnold.
10. Eric is not in the third house.
11. Eric is the person with an aquarium of fish.
 What is the value of attribute BookGenre for the person whose attribute Hobby is cooking? Please reason step by step, and put your final answer within \boxed{}

Answer: mystery


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

Answer: spaghetti


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Carol`, `Bob`, `Alice`
 - People have unique heights: `super tall`, `very tall`, `tall`, `very short`, `short`, `average`

## Clues:
1. Carol is not in the sixth house.
2. There is one house between Bob and the person who is very tall.
3. The person who is super tall is not in the sixth house.
4. Eric is the person who is very tall.
5. Peter is not in the sixth house.
6. The person who is short is not in the first house.
7. The person who is short is not in the third house.
8. The person who is short is directly left of Eric.
9. The person who is very short is somewhere to the right of the person who is very tall.
10. The person who is tall is Alice.
11. Carol is not in the second house.
12. There are two houses between the person who is super tall and Eric.
 What is the value of attribute House for the person whose attribute Height is very tall? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `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 Food is pizza? 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: `Peter`, `Eric`, `Arnold`
 - Each person prefers a unique type of vacation: `mountain`, `beach`, `city`
 - Each person has a unique hobby: `gardening`, `cooking`, `photography`
 - People have unique favorite sports: `tennis`, `soccer`, `basketball`
 - They all have a unique favorite flower: `carnations`, `daffodils`, `lilies`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`

## Clues:
1. The person who loves soccer is the person who loves a bouquet of daffodils.
2. Arnold is the person who loves the boquet of lilies.
3. The person who loves beach vacations is directly left of Eric.
4. The Desert smoothie lover is the person who loves basketball.
5. The person who loves the boquet of lilies is directly left of the person who loves a carnations arrangement.
6. Eric is directly left of Peter.
7. The person who loves basketball and the person who loves a bouquet of daffodils are next to each other.
8. The person who loves a bouquet of daffodils is the person who likes Cherry smoothies.
9. The person who enjoys mountain retreats is the person who loves cooking.
10. The photography enthusiast is the person who loves basketball.
 What is the value of attribute House for the person whose attribute Smoothie is cherry? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


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`
 - Each person has a unique type of pet: `cat`, `dog`
 - The people are of nationalities: `dane`, `brit`
 - Each person has an occupation: `doctor`, `engineer`
 - People have unique favorite sports: `soccer`, `basketball`
 - Everyone has a unique favorite cigar: `prince`, `pall mall`

## Clues:
1. The person who has a cat and the Prince smoker are next to each other.
2. Eric is directly left of the person who owns a dog.
3. The Dane is not in the second house.
4. Arnold is the person who is a doctor.
5. The person who loves basketball is not in the second house.
 What is the value of attribute Cigar for the person whose attribute FavoriteSport is basketball? Please reason step by step, and put your final answer within \boxed{}

Answer: pall mall


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

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

Answer: pall mall


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

## Clues:
1. The dog owner is not in the first house.
2. The bird keeper is somewhere to the right of the person who is an engineer.
3. Arnold is the fish enthusiast.
4. The bird keeper is directly left of the cat lover.
5. The person who is a teacher is not in the second house.
6. The fish enthusiast is the person who is a lawyer.
7. The person who is an engineer is Alice.
8. The fish enthusiast is in the fourth house.
9. The person who is a doctor is Peter.
10. Bob is the person who is a teacher.
11. Peter is somewhere to the right of Arnold.
 What is the value of attribute House for the person whose attribute Name is Peter? 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: `Peter`, `Alice`, `Carol`, `Eric`, `Arnold`, `Bob`
 - Each person has a unique birthday month: `mar`, `feb`, `jan`, `sept`, `may`, `april`
 - Each person has a unique hobby: `knitting`, `gardening`, `woodworking`, `painting`, `cooking`, `photography`

## Clues:
1. There is one house between Carol and the woodworking hobbyist.
2. The woodworking hobbyist is somewhere to the right of Bob.
3. The person whose birthday is in March is directly left of the person who loves cooking.
4. Carol is in the third house.
5. The person whose birthday is in March is somewhere to the left of the person who paints as a hobby.
6. Arnold is not in the fifth house.
7. The photography enthusiast is somewhere to the right of the person whose birthday is in May.
8. The person whose birthday is in September is the woodworking hobbyist.
9. There is one house between the person who loves cooking and Eric.
10. The person whose birthday is in May is directly left of the person whose birthday is in September.
11. Bob is somewhere to the right of the person who enjoys knitting.
12. The person whose birthday is in February is the photography enthusiast.
13. The person whose birthday is in April is the person who paints as a hobby.
14. Alice is in the first house.
 What is the value of attribute Hobby for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}

Answer: painting


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Carol`, `Bob`, `Eric`, `Peter`
 - People use unique phone models: `iphone 13`, `samsung galaxy s21`, `xiaomi mi 11`, `oneplus 9`, `google pixel 6`, `huawei p50`
 - Each person has a unique birthday month: `april`, `mar`, `feb`, `may`, `jan`, `sept`
 - Each person has an occupation: `nurse`, `doctor`, `engineer`, `teacher`, `artist`, `lawyer`

## Clues:
1. The person who uses a Google Pixel 6 is directly left of the person whose birthday is in March.
2. The person who is a nurse is Eric.
3. Arnold is directly left of the person whose birthday is in April.
4. Alice and the person whose birthday is in January are next to each other.
5. The person who is an artist is the person whose birthday is in May.
6. The person whose birthday is in September is not in the sixth house.
7. There is one house between the person whose birthday is in March and the person who is a nurse.
8. The person whose birthday is in March is the person who is a lawyer.
9. The person who uses a Xiaomi Mi 11 is directly left of the person who is a nurse.
10. The person who is a teacher is not in the sixth house.
11. Eric is the person who uses an iPhone 13.
12. The person who is an engineer and Peter are next to each other.
13. The person whose birthday is in January is in the third house.
14. The person who uses a Samsung Galaxy S21 is Bob.
15. The person who is a lawyer and the person whose birthday is in April are next to each other.
16. The person who uses a OnePlus 9 is somewhere to the right of the person who is a doctor.
 What is the value of attribute PhoneModel for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: samsung galaxy s21


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Peter`, `Carol`, `Bob`, `Eric`
 - Each person prefers a unique type of vacation: `mountain`, `beach`, `city`, `cruise`, `camping`, `cultural`
 - Everyone has a unique favorite cigar: `yellow monster`, `pall mall`, `prince`, `dunhill`, `blue master`, `blends`
 - People have unique favorite sports: `swimming`, `soccer`, `tennis`, `basketball`, `volleyball`, `baseball`

## Clues:
1. Arnold is the person who prefers city breaks.
2. There is one house between the person who loves volleyball and the person who loves swimming.
3. Alice is in the fourth house.
4. The person who likes going on cruises is the person who smokes many unique blends.
5. The person who enjoys camping trips is somewhere to the left of Eric.
6. The person who loves volleyball is the person who smokes many unique blends.
7. Arnold is the Prince smoker.
8. The person who loves baseball is Peter.
9. The Prince smoker and the person partial to Pall Mall are next to each other.
10. The person who loves beach vacations is in the sixth house.
11. Carol is the person who loves beach vacations.
12. Carol is the person who smokes Yellow Monster.
13. The person who loves tennis is somewhere to the right of the person who goes on cultural tours.
14. The person who goes on cultural tours is the person who smokes Blue Master.
15. The person who loves basketball is directly left of the person who smokes many unique blends.
16. The person who loves volleyball and Eric are next to each other.
17. The person who loves tennis is Arnold.
 What is the value of attribute FavoriteSport for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: volleyball


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Bob`, `Alice`, `Eric`, `Carol`
 - The mothers' names in different houses are unique: `Holly`, `Aniya`, `Kailyn`, `Janelle`, `Penny`, `Sarah`
 - Everyone has something unique for lunch: `stir fry`, `stew`, `spaghetti`, `soup`, `pizza`, `grilled cheese`
 - The people keep unique animals: `dog`, `horse`, `cat`, `fish`, `rabbit`, `bird`
 - Everyone has a favorite smoothie: `watermelon`, `desert`, `lime`, `cherry`, `blueberry`, `dragonfruit`
 - Each person lives in a unique style of house: `mediterranean`, `craftsman`, `victorian`, `colonial`, `modern`, `ranch`

## Clues:
1. The person whose mother's name is Penny is the person residing in a Victorian house.
2. The person in a modern-style house is The person whose mother's name is Kailyn.
3. The person in a modern-style house is somewhere to the right of the person residing in a Victorian house.
4. The person who drinks Blueberry smoothies is not in the sixth house.
5. The person whose mother's name is Sarah is the person who drinks Blueberry smoothies.
6. The person who likes Cherry smoothies is not in the third house.
7. The person who likes Cherry smoothies is Eric.
8. The person whose mother's name is Janelle is not in the third house.
9. The person who loves the spaghetti eater is in the third house.
10. The fish enthusiast is The person whose mother's name is Holly.
11. The person who keeps horses is in the first house.
12. The person in a Mediterranean-style villa is the person who loves the stew.
13. The bird keeper is in the third house.
14. The person living in a colonial-style house is the person who drinks Blueberry smoothies.
15. Carol is the person who keeps horses.
16. There are two houses between the Desert smoothie lover and Peter.
17. The person who loves eating grilled cheese is the Dragonfruit smoothie lover.
18. The person who is a pizza lover is somewhere to the right of the Dragonfruit smoothie lover.
19. The person whose mother's name is Janelle is somewhere to the left of Arnold.
20. Carol is directly left of The person whose mother's name is Aniya.
21. The person in a ranch-style home is The person whose mother's name is Aniya.
22. Arnold is somewhere to the right of the person in a Craftsman-style house.
23. The cat lover is Bob.
24. There is one house between the Watermelon smoothie lover and the person who is a pizza lover.
25. There are two houses between the cat lover and the person who loves stir fry.
26. Arnold is the rabbit owner.
27. Peter is The person whose mother's name is Janelle.
 What is the value of attribute Mother for the person whose attribute HouseStyle is craftsman? Please reason step by step, and put your final answer within \boxed{}

Answer: Holly


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`, `Carol`, `Alice`, `Bob`
 - The people are of nationalities: `swede`, `brit`, `chinese`, `dane`, `german`, `norwegian`
 - People have unique favorite book genres: `science fiction`, `mystery`, `fantasy`, `biography`, `historical fiction`, `romance`
 - Everyone has a favorite smoothie: `blueberry`, `dragonfruit`, `desert`, `watermelon`, `cherry`, `lime`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `huawei p50`, `xiaomi mi 11`, `iphone 13`, `google pixel 6`

## Clues:
1. Alice is the British person.
2. There is one house between Eric and the Dragonfruit smoothie lover.
3. The Chinese is somewhere to the left of Alice.
4. The person who loves historical fiction books is directly left of the person who loves biography books.
5. The person who loves biography books is somewhere to the right of Carol.
6. The person who uses an iPhone 13 is somewhere to the right of the Desert smoothie lover.
7. The person who loves mystery books is the British person.
8. Bob is somewhere to the right of the person who uses a OnePlus 9.
9. Carol is the person who drinks Blueberry smoothies.
10. There are two houses between the person who loves romance books and the Watermelon smoothie lover.
11. The person who likes Cherry smoothies is the Swedish person.
12. The person who loves science fiction books is the person who uses a Google Pixel 6.
13. The person who drinks Lime smoothies is Eric.
14. Bob is in the fifth house.
15. Carol is in the second house.
16. The Norwegian and Alice are next to each other.
17. The Swedish person is not in the fifth house.
18. The German and the Norwegian are next to each other.
19. The person who loves fantasy books is the Dragonfruit smoothie lover.
20. The British person is somewhere to the left of Peter.
21. The person who loves romance books is the person who uses a Samsung Galaxy S21.
22. Alice is the person who uses a Xiaomi Mi 11.
23. The person who loves romance books is Eric.
 What is the value of attribute PhoneModel for the person whose attribute Nationality is norwegian? 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: `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 Name is Arnold? 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: `Bob`, `Peter`, `Arnold`, `Eric`, `Alice`
 - Each person has a unique hobby: `gardening`, `painting`, `knitting`, `cooking`, `photography`

## Clues:
1. The person who enjoys gardening is somewhere to the left of Bob.
2. Peter is in the second house.
3. Alice is not in the fourth house.
4. The person who loves cooking is directly left of the person who enjoys knitting.
5. The person who enjoys knitting is directly left of Bob.
6. The person who paints as a hobby is Eric.
7. Alice is not in the third house.
 What is the value of attribute House for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: 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: `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`, `Carol`
 - People have unique favorite sports: `volleyball`, `soccer`, `basketball`, `baseball`, `swimming`, `tennis`
 - Each person has a unique hobby: `gardening`, `cooking`, `painting`, `woodworking`, `knitting`, `photography`
 - Each mother is accompanied by their child: `Bella`, `Meredith`, `Alice`, `Samantha`, `Fred`, `Timothy`
 - Each person has a unique birthday month: `feb`, `jan`, `mar`, `may`, `sept`, `april`
 - Everyone has something unique for lunch: `stir fry`, `grilled cheese`, `pizza`, `spaghetti`, `stew`, `soup`

## Clues:
1. The person who loves soccer is somewhere to the right of Eric.
2. The person who paints as a hobby is somewhere to the left of the woodworking hobbyist.
3. The person whose birthday is in February is not in the third house.
4. The person who is the mother of Timothy is directly left of Arnold.
5. The person who loves cooking is somewhere to the left of the person's child is named Fred.
6. Alice is the person who paints as a hobby.
7. The person's child is named Alice is the person who loves cooking.
8. The person whose birthday is in March is the person's child is named Samantha.
9. The person who loves the soup is the person whose birthday is in May.
10. Carol is somewhere to the right of the person whose birthday is in May.
11. The person whose birthday is in March and the person who is a pizza lover are next to each other.
12. The person whose birthday is in February is not in the fourth house.
13. There are two houses between the person who loves baseball and the person who loves the stew.
14. The person whose birthday is in May is somewhere to the left of the person who loves volleyball.
15. The person whose birthday is in January is somewhere to the right of Arnold.
16. The person who loves cooking is not in the third house.
17. The person who enjoys knitting is the person whose birthday is in September.
18. The person who enjoys gardening is the person's child is named Meredith.
19. The person who is a pizza lover is the person who loves swimming.
20. There is one house between the person who loves basketball and the person whose birthday is in May.
21. The person who loves the soup is directly left of the person who loves the spaghetti eater.
22. The person who loves the stew is the person's child is named Bella.
23. The person who enjoys knitting is somewhere to the left of the person who loves the soup.
24. Bob is somewhere to the left of the person who loves the stew.
25. Arnold is the person who loves stir fry.
26. The woodworking hobbyist is directly left of the person who loves the soup.
 What is the value of attribute Hobby for the person whose attribute FavoriteSport is tennis? Please reason step by step, and put your final answer within \boxed{}

Answer: woodworking


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`
 - Each person has a unique birthday month: `jan`, `april`, `sept`
 - Each person has a unique type of pet: `cat`, `dog`, `fish`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `city`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`

## Clues:
1. The person whose birthday is in January is somewhere to the left of the person who is a pizza lover.
2. The person who owns a dog is in the second house.
3. The person whose birthday is in September is the person who loves the spaghetti eater.
4. The person whose birthday is in January is the person who enjoys mountain retreats.
5. The Watermelon smoothie lover is the person with an aquarium of fish.
6. The person whose birthday is in September is not in the third house.
7. Eric is somewhere to the left of the person who prefers city breaks.
8. The person who has a cat is Peter.
9. The person who loves beach vacations is in the third house.
10. The person who likes Cherry smoothies is the person whose birthday is in April.
 What is the value of attribute Food for the person whose attribute Pet is cat? Please reason step by step, and put your final answer within \boxed{}

Answer: pizza


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

## Clues:
1. The person whose birthday is in April is Bob.
2. The person residing in a Victorian house is the person who loves stir fry.
3. There are two houses between the person who loves jazz music and the person who has auburn hair.
4. The person who loves the stew is in the second house.
5. The person who loves stir fry is the person whose birthday is in February.
6. The person who loves classical music is in the fifth house.
7. The person who loves jazz music is Alice.
8. The person living in a colonial-style house is in the fifth house.
9. The person who has gray hair is the person who loves the stew.
10. The person whose birthday is in March is the person who has red hair.
11. The person who loves country music is directly left of the person whose birthday is in September.
12. The person whose birthday is in May is somewhere to the left of the person who loves country music.
13. The person in a ranch-style home is the person who loves the soup.
14. The person whose birthday is in January is the person who loves the stew.
15. The person who has blonde hair is the person in a Mediterranean-style villa.
16. Arnold is somewhere to the right of the person whose birthday is in April.
17. The person whose birthday is in February is the person who loves hip-hop music.
18. The person who has blonde hair is somewhere to the left of Eric.
19. The person residing in a Victorian house is Carol.
20. The person who has black hair is the person in a ranch-style home.
21. The person in a modern-style house is not in the second house.
22. The person who loves the spaghetti eater is somewhere to the left of the person who loves country music.
23. The person who is a pizza lover is Bob.
24. The person who loves pop music is somewhere to the left of the person whose birthday is in September.
 What is the value of attribute House for the person whose attribute MusicGenre is hip hop? Please reason step by step, and put your final answer within \boxed{}

Answer: 6


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Bob`, `Eric`, `Arnold`, `Carol`
 - Each person lives in a unique style of house: `mediterranean`, `modern`, `ranch`, `craftsman`, `victorian`, `colonial`

## Clues:
1. Alice is not in the fifth house.
2. The person in a Mediterranean-style villa is directly left of the person in a Craftsman-style house.
3. The person in a Craftsman-style house is in the fifth house.
4. Carol is directly left of Alice.
5. The person in a modern-style house is Eric.
6. The person residing in a Victorian house is not in the third house.
7. Bob is in the first house.
8. Bob and Eric are next to each other.
9. Peter is not in the third house.
10. The person in a ranch-style home is somewhere to the right of the person residing in a Victorian house.
11. The person living in a colonial-style house is somewhere to the left of Carol.
 What is the value of attribute House for the person whose attribute HouseStyle is colonial? 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: `Arnold`, `Eric`, `Peter`, `Alice`, `Carol`, `Bob`
 - People have unique favorite music genres: `jazz`, `pop`, `classical`, `rock`, `hip hop`, `country`

## Clues:
1. Bob is directly left of the person who loves jazz music.
2. Eric is somewhere to the left of the person who loves hip-hop music.
3. Carol is in the sixth house.
4. Eric and the person who loves hip-hop music are next to each other.
5. The person who loves country music is Carol.
6. Arnold is not in the fifth house.
7. Arnold is somewhere to the right of the person who loves pop music.
8. The person who loves pop music is Peter.
9. The person who loves hip-hop music is in the third house.
10. There is one house between Peter and Bob.
11. The person who loves rock music is not in the fifth house.
 What is the value of attribute House for the person whose attribute MusicGenre is hip hop? 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: `Peter`, `Eric`, `Arnold`, `Alice`
 - The mothers' names in different houses are unique: `Kailyn`, `Aniya`, `Holly`, `Janelle`
 - Each person has a unique birthday month: `feb`, `sept`, `jan`, `april`
 - Each person has a unique hobby: `cooking`, `gardening`, `photography`, `painting`

## Clues:
1. Peter is the person whose birthday is in February.
2. Arnold is not in the third house.
3. The person whose mother's name is Aniya is somewhere to the left of the person who enjoys gardening.
4. The person whose birthday is in January is in the second house.
5. The person whose birthday is in February is not in the third house.
6. The person whose birthday is in February is the photography enthusiast.
7. Alice is somewhere to the right of the person whose birthday is in April.
8. The person whose mother's name is Kailyn is somewhere to the left of the person whose birthday is in September.
9. The person who enjoys gardening is not in the fourth house.
10. The person who loves cooking is The person whose mother's name is Janelle.
11. Peter is The person whose mother's name is Kailyn.
 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: Holly


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Eric`, `Bob`, `Arnold`
 - Each person has a favorite color: `white`, `red`, `green`, `blue`, `yellow`
 - Each person lives in a unique style of house: `craftsman`, `modern`, `colonial`, `victorian`, `ranch`
 - People have unique favorite book genres: `fantasy`, `science fiction`, `biography`, `romance`, `mystery`

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

Answer: red


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Peter`, `Bob`
 - People have unique hair colors: `black`, `red`, `gray`, `blonde`, `brown`
 - The people are of nationalities: `dane`, `norwegian`, `swede`, `brit`, `german`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `modern`, `craftsman`, `victorian`
 - Each person has a unique birthday month: `sept`, `jan`, `mar`, `april`, `feb`
 - Each person has an occupation: `engineer`, `doctor`, `lawyer`, `teacher`, `artist`

## Clues:
1. The person who has brown hair is the person whose birthday is in March.
2. The person who is a doctor is the Dane.
3. Eric is the person whose birthday is in March.
4. The person whose birthday is in February is directly left of the person who is an engineer.
5. There is one house between the person in a ranch-style home and the German.
6. Bob is the person whose birthday is in January.
7. The person who is an artist is the Swedish person.
8. The Swedish person is in the second house.
9. Peter is not in the fourth house.
10. The person who is an artist is somewhere to the left of the person who is a teacher.
11. The person who is an artist is directly left of the person who has gray hair.
12. Alice is the person who has red hair.
13. The person whose birthday is in September is not in the third house.
14. Arnold is somewhere to the left of the person living in a colonial-style house.
15. The person in a modern-style house is in the fifth house.
16. The person who has black hair is the Norwegian.
17. There are two houses between the person whose birthday is in January and the British person.
18. The person who has red hair is the person in a Craftsman-style house.
19. The person who is an engineer and the British person are next to each other.
20. There is one house between Arnold and the person who has red hair.
 What is the value of attribute Nationality for the person whose attribute Birthday is feb? Please reason step by step, and put your final answer within \boxed{}

Answer: swede


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`, `Alice`, `Bob`
 - Each person has a unique level of education: `associate`, `doctorate`, `high school`, `bachelor`, `master`
 - Each mother is accompanied by their child: `Fred`, `Meredith`, `Bella`, `Timothy`, `Samantha`
 - Everyone has a unique favorite cigar: `prince`, `pall mall`, `blue master`, `dunhill`, `blends`
 - The people are of nationalities: `swede`, `brit`, `dane`, `german`, `norwegian`
 - People have unique favorite book genres: `romance`, `mystery`, `biography`, `science fiction`, `fantasy`

## Clues:
1. The person with a master's degree is the person partial to Pall Mall.
2. The person with a high school diploma is in the fourth house.
3. The person's child is named Samantha is in the first house.
4. The Norwegian and the person with a master's degree are next to each other.
5. The German is the Dunhill smoker.
6. The person who is the mother of Timothy is directly left of the person with a master's degree.
7. Arnold and the Dane are next to each other.
8. The Swedish person is the person who loves mystery books.
9. The Swedish person is Alice.
10. Arnold is the person with an associate's degree.
11. The person with a doctorate is in the second house.
12. Peter is somewhere to the right of the person who loves biography books.
13. The person who smokes many unique blends is Alice.
14. The person who loves fantasy books is in the first house.
15. Peter is the Norwegian.
16. The person with a doctorate is the person's child is named Meredith.
17. The Prince smoker is somewhere to the left of the person who smokes many unique blends.
18. The person partial to Pall Mall is the person's child is named Bella.
19. The person who loves romance books is Bob.
 What is the value of attribute House for the person whose attribute Nationality is dane? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


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`
 - Everyone has a unique favorite cigar: `pall mall`, `dunhill`, `blue master`, `prince`
 - Each person has a unique birthday month: `sept`, `april`, `feb`, `jan`
 - Each person has a unique hobby: `gardening`, `cooking`, `photography`, `painting`
 - People have unique favorite music genres: `pop`, `classical`, `jazz`, `rock`

## Clues:
1. The person who loves jazz music is the person whose birthday is in April.
2. Alice is the person whose birthday is in April.
3. Peter is directly left of the person who loves cooking.
4. The person whose birthday is in February is Arnold.
5. The Dunhill smoker is not in the third house.
6. The Prince smoker is Arnold.
7. The person who loves jazz music is the person partial to Pall Mall.
8. The person whose birthday is in January is not in the third house.
9. The person who loves rock music is Peter.
10. The person who enjoys gardening is directly left of the photography enthusiast.
11. The person who loves classical music is in the first house.
12. The person who loves pop music is somewhere to the left of the person partial to Pall Mall.
13. The person who smokes Blue Master is the person who loves pop music.
 What is the value of attribute House for the person whose attribute Birthday is april? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - Each person has a favorite color: `white`, `red`, `yellow`
 - People use unique phone models: `google pixel 6`, `samsung galaxy s21`, `iphone 13`
 - They all have a unique favorite flower: `lilies`, `daffodils`, `carnations`
 - Each person has a unique favorite drink: `water`, `tea`, `milk`
 - Each person has a unique level of education: `bachelor`, `associate`, `high school`

## Clues:
1. The person with an associate's degree is Eric.
2. Peter is the person who loves a carnations arrangement.
3. The person with a bachelor's degree is in the second house.
4. The person with a high school diploma is the person who loves yellow.
5. The one who only drinks water is the person with a high school diploma.
6. The person who loves a bouquet of daffodils is the person who uses an iPhone 13.
7. The person who uses a Google Pixel 6 is not in the third house.
8. The tea drinker is in the second house.
9. Eric is somewhere to the left of the person with a high school diploma.
10. The tea drinker is the person who uses an iPhone 13.
11. The person who loves white is not in the first house.
 What is the value of attribute PhoneModel for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: samsung galaxy s21


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - Each person has an occupation: `doctor`, `teacher`, `engineer`
 - Each person has a unique hobby: `cooking`, `photography`, `gardening`

## Clues:
1. The person who is a doctor and Eric are next to each other.
2. The person who loves cooking is directly left of the person who is a teacher.
3. The person who is a doctor is somewhere to the right of the person who enjoys gardening.
4. The photography enthusiast is the person who is a teacher.
5. The person who is an engineer is Peter.
 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 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`
 - Each person lives in a unique style of house: `victorian`, `ranch`, `craftsman`, `colonial`
 - People have unique favorite music genres: `pop`, `classical`, `rock`, `jazz`
 - They all have a unique favorite flower: `roses`, `lilies`, `carnations`, `daffodils`
 - Each person prefers a unique type of vacation: `city`, `beach`, `mountain`, `cruise`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`

## Clues:
1. The person who loves classical music is Peter.
2. The person in a ranch-style home is not in the fourth house.
3. The person who loves a bouquet of daffodils is the person living in a colonial-style house.
4. The person living in a colonial-style house is the person who loves white.
5. The person who prefers city breaks and the person whose favorite color is red are next to each other.
6. The person who loves the rose bouquet is not in the fourth house.
7. The person who likes going on cruises is in the second house.
8. The person who prefers city breaks is the person who loves rock music.
9. The person who loves pop music is Alice.
10. The person who loves the rose bouquet is the person who prefers city breaks.
11. Eric is the person who enjoys mountain retreats.
12. The person who enjoys mountain retreats is directly left of the person who loves a bouquet of daffodils.
13. The person who loves yellow is somewhere to the left of the person who likes going on cruises.
14. The person who enjoys mountain retreats is the person residing in a Victorian house.
15. The person who loves classical music is the person who likes going on cruises.
16. Peter is somewhere to the right of the person who loves the boquet of lilies.
 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: 2


(4) (Total 20 points) In the right triangle $\triangle ABC$, $\angle ACB=90^{\circ}$, $M$ is a point on $AB$, and $AM^2 + BM^2 + CM^2 = 2AM + 2BM + 2CM - 3$. If $P$ is a moving point on the segment $AC$, and $\odot O$ is the circle passing through points $P, M, C$, and a line through $P$ parallel to $AB$ intersects $\odot O$ at point $D$.
(1) (8 points) Prove that $M$ is the midpoint of $AB$;
(2) (12 points) Find the length of $PD$.

Answer: 1


4. Find all real solutions to the system of equations

$$
\left\{\begin{array}{l}
\sqrt{x-997}+\sqrt{y-932}+\sqrt{z-796}=100 \\
\sqrt{x-1237}+\sqrt{y-1121}+\sqrt{3045-z}=90 \\
\sqrt{x-1621}+\sqrt{2805-y}+\sqrt{z-997}=80 \\
\sqrt{2102-x}+\sqrt{y-1237}+\sqrt{z-932}=70
\end{array}\right.
$$

(L. S. Korechkova, A. A. Tessler)

Answer: 2021


The struggle of the land, by 1996, the forest coverage rate of the whole county had reached 30% (becoming an oasis). From 1997 onwards, each year will see such changes: 16% of the original desert area will be converted, and 4% of it will be eroded, turning back into desert.
(1) Assuming the total area of the county is 1, in 1996 the oasis area is $a_{1}=\frac{3}{10}$, after $n$ years the oasis area becomes $a_{n+1}$. Prove:
$$
a_{n+1}=\frac{4}{5} a_{n}+\frac{4}{25} \text {. }
$$
(2) How many years at least are needed to make the forest coverage rate of the whole county exceed 60% (round to the nearest whole number)?

Answer: 5


(19) (This question is worth 12 points) In response to the national strategy of "targeted poverty alleviation and industrial support," Harbin City is recruiting volunteer advocates for the "Poverty Alleviation Policy" from the public. From 500 volunteers aged between $[20,45]$, 100 are randomly selected, and their age frequency distribution histogram is shown in the figure.
(1) Find the value of $x$ in the figure;
(2) From the 100 selected volunteers, 10 are chosen using stratified sampling based on age to participate in the central square publicity campaign. Then, 3 of these 10 volunteers are selected to serve as primary leaders. Let the number of these 3 volunteers who are "under 35 years old" be $X$. Find the distribution of $X$ and its expected value.

Answer: 1.8


## Task 3 - 040813

On a double-track route to the suburb of a large city, a tram departs from the starting station and from the terminal station every 10 minutes and each requires 50 minutes of travel time. The dwell time at these two stations is 10 minutes each.

How many trams are in operation on this route in total?

Answer: 12


A game show offers a contestant three prizes $A, B$ and $C$, each of which is worth a whole number of dollars from $＄$ $1$ to $＄$ $9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order $A, B, C$. As a hint, the digits of the three prices are given. On a particular day, the digits given were $1, 1, 1, 1, 3, 3, 3$ . Find the total number of possible guesses for all three prizes consistent with the hint.

Answer: 420


In triangle $ABC$ , $AB = 10$ , $BC = 14$ , and $CA = 16$ . Let $D$ be a point in the interior of $\overline{BC}$ . Let points $I_B$ and $I_C$ denote the incenters of triangles $ABD$ and $ACD$ , respectively. The circumcircles of triangles $BI_BD$ and $CI_CD$ meet at distinct points $P$ and $D$ . The maximum possible area of $\triangle BPC$ can be expressed in the form $a - b\sqrt {c}$ , where $a$ , $b$ , and $c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a + b + c$ .

Answer: 150


For each positive integer $n$ , let $f(n) = \sum\limits_{k = 1}^{100} \lfloor \log_{10} (kn) \rfloor$ . Find the largest value of $n$ for which $f(n) \le 300$ .
Note: $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$ .

Answer: 109


The probability that a set of three distinct vertices chosen at random from among the vertices of a regular n-gon determine an obtuse triangle is $\frac{93}{125}$ . Find the sum of all possible values of $n$ .

Answer: 503


Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that $p$ does not exceed $1$ percent.

Answer: 594


There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,$ $2,$ $3,$ $...,$ $n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that
 ( $1$ ) for each $k,$ the mathematician who was seated in seat $k$ before the break is seated in seat $ka$ after the break (where seat $i + n$ is seat $i$ );
 ( $2$ ) for every pair of mathematicians, the number of mathematicians sitting between them after the break, counting in both the clockwise and the counterclockwise directions, is different from either of the number of mathematicians sitting between them before the break. Find the number of possible values of $n$ with $1 < n < 1000.$

Answer: 332


Ms. Math's kindergarten class has $16$ registered students. The classroom has a very large number, $N$ , of play blocks which satisfies the conditions:
$(a)$ If $16$ , $15$ , or $14$ students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and $(b)$ There are three integers $0 < x < y < z < 14$ such that when $x$ , $y$ , or $z$ students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.
Find the sum of the distinct prime divisors of the least possible value of $N$ satisfying the above conditions.

Answer: 148


Let $\triangle PQR$ be a triangle with $\angle P = 75^\circ$ and $\angle Q = 60^\circ$ . A regular hexagon $ABCDEF$ with side length $1$ is drawn inside $\triangle PQR$ so that side $\overline{AB}$ lies on $\overline{PQ}$ , side $\overline{CD}$ lies on $\overline{QR}$ , and one of the remaining vertices lies on $\overline{RP}$ . There are positive integers $a, b, c,$ and $d$ such that the area of $\triangle PQR$ can be expressed in the form $\frac{a+b\sqrt{c}}{d}$ , where $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$ .

Answer: 21


Triangle $AB_0C_0$ has side lengths $AB_0 = 12$ , $B_0C_0 = 17$ , and $C_0A = 25$ . For each positive integer $n$ , points $B_n$ and $C_n$ are located on $\overline{AB_{n-1}}$ and $\overline{AC_{n-1}}$ , respectively, creating three similar triangles $\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1}C_{n-1}$ . The area of the union of all triangles $B_{n-1}C_nB_n$ for $n\geq1$ can be expressed as $\tfrac pq$ , where $p$ and $q$ are relatively prime positive integers. Find $q$ .

Answer: 961


$A$ rectangle has sides of length $a$ and $36$. $A$ hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length $36$. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length $a$ parallel and separated by a distance of $24$, the hexagon has the same area as the original rectangle. Find $a^2$.

Answer: 720


For positive integer $n$ , let $s(n)$ denote the sum of the digits of $n$ . Find the smallest positive integer satisfying $s(n) = s(n+864) = 20$ .

Answer: 695


Points $A, B, Q, D,$ and $C$ lie on the circle shown. The measures of arcs $\widehat{BQ}$ and $\widehat{QD}$ are $42^\circ$ and $38^\circ$, respectively. Find the sum of the measures of angles $P$ and $Q$ in degrees.

Answer: 40


Let $AD$ be a median of $\triangle ABC$ such that $m(\widehat{ADB})=45^{\circ}$ and $m(\widehat{ACB})=30^{\circ}$. If the measure of $\widehat{ABC}$ is $x^\circ$, what is the value of $x$?

Answer: 105


For real numbers $x$, let \[P(x)=1+\cos(x)+i \sin(x)-\cos(2x)-i \sin(2x)+\cos(3x)+i \sin(3x)\] where $i=\sqrt{-1}$. Find the number of values of $x$ with $0 \leq x<2 \pi$ such that $P(x)=0$.

Answer: 0


Let $ABC$ be a triangle with $m(\widehat{A}) = 70^\circ$ and the incenter $I$. If $|BC| = |AC| + |AI|$, then what is the value of $m(\widehat{B})$ in degrees? If the answer is $x^\circ$, what is the value of $x$?

Answer: 35


Answer: 10


Triangles $ABC$ and $ABD$ are isosceles with $AB = AC = BD$, and $BD$ intersects $AC$ at $E$. If $BD$ is perpendicular to $AC$, then what is the value of $\angle C + \angle D$ if the answer is in degrees?

Answer: 135


For the function $$ g(a) = \underbrace{\max}_{x\in R} \left\{ \cos x + \cos \left(x + \frac{\pi}{6} \right)+ \cos \left(x + \frac{\pi}{4} \right) + cos(x + a) \right\},$$ let $b \in R$ be the input that maximizes $g$. If $\cos^2 b = \frac{m+\sqrt{n}+\sqrt{p}-\sqrt{q}}{24}$ for positive integers $m, n, p, q$, find $m + n + p + q$.

Answer: 54


Observe the set $S =\{(x, y) \in Z^2 : |x| \le 5$ and $-10 \le y\le 0\}$. Find the number of points $P$ in $S$ such that there exists a tangent line from $P$ to the parabola $y = x^2 + 1$ that can be written in the form $y = mx + b$, where $m$ and $b$ are integers.

Answer: 15


Let $\vartriangle ABC$ be a triangle with $BC = 7$, $CA = 6$, and, $AB = 5$. Let $I$ be the incenter of $\vartriangle ABC$. Let the incircle of $\vartriangle ABC$ touch sides $BC$, $CA$, and $AB$ at points $D,E$, and $F$. Let the circumcircle of $\vartriangle AEF$ meet the circumcircle of $\vartriangle ABC$ for a second time at point $X\ne A$. Let $P$ denote the intersection of $XI$ and $EF$. If the product $XP \cdot IP$ can be written as $\frac{m}{n}$  for relatively prime positive integers $m, n$, find $m + n$.

Answer: 629


An ellipse has foci $A$ and $B$ and has the property that there is some point $C$ on the ellipse such that the area of the circle passing through $A$, $B$, and, $C$ is equal to the area of the ellipse. Let $e$ be the largest possible eccentricity of the ellipse. One may write $e^2$ as $\frac{a+\sqrt{b}}{c}$ , where $a, b$, and $c$ are integers such that $a$ and $c$ are relatively prime, and b is not divisible by the square of any prime. Find $a^2 + b^2 + c^2$.

Answer: 30


Let  $\vartriangle ABC$ be a triangle with $AB = 4$, $BC = 6$, and $CA = 5$. Let the angle bisector of $\angle BAC$ intersect $BC$ at the point $D$ and the circumcircle of $\vartriangle ABC$ again at the point $M\ne A$. The perpendicular bisector of segment $DM$ intersects the circle centered at $M$ passing through $B$ at two points, $X$ and $Y$ . Compute $AX \cdot AY$.

Answer: 36


13. Let $\triangle T B D$ be a triangle with $T B=6, B D=8$, and $D T=7$. Let $I$ be the incenter of $\triangle T B D$, and let $T I$ intersect the circumcircle of $\triangle T B D$ at $M \neq T$. Let lines $T B$ and $M D$ intersect at $Y$, and let lines $T D$ and $M B$ intersect at $X$. Let the circumcircles of $\triangle Y B M$ and $\triangle X D M$ intersect at $Z \neq M$. If the area of $\triangle Y B Z$ is $x$ and the area of $\triangle X D Z$ is $y$, then the ratio $\frac{x}{y}$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Answer: 97


Amir enters Fine Hall and sees the number $2$ written on the blackboard. Amir can perform the following operation: he flips a coin, and if it is heads, he replaces the number $x$ on the blackboard with $3x+1;$ otherwise, he replaces $x$ with $\lfloor x/3\rfloor.$ If Amir performs the operation four times, let $\tfrac{m}{n}$ denote the expected number of times that he writes the digit $1$ on the blackboard, where $m,n$ are relatively prime positive integers. Find $m+n.$

Answer: 27


10. The sum $\sum_{k=1}^{2020} k \cos \left(\frac{4 k \pi}{4041}\right)$ can be written in the form

$$
\frac{a \cos \left(\frac{p \pi}{q}\right)-b}{c \sin ^{2}\left(\frac{p \pi}{q}\right)}
$$

where $a, b, c$ are relatively prime positive integers and $p, q$ are relatively prime positive integers where $p<q$. Determine $a+b+c+p+q$.

Answer: 4049


A spider is walking on the boundary of equilateral triangle $\triangle{ABC}$ (vertices labelled in counterclockwise order), starting at vertex $A$. Each second, she moves to one of her two adjacent vertices with equal probability. The windiness of a path that starts and ends at $A$ is the net number of counterclockwise revolutions made. For example, the windiness of the path $ABCA$ is $1,$ and the windiness of the path $ABCACBACBA$ is $-1$. What is the remainder modulo $1000$ of the sum of the squares of the windiness values taken over all possible paths that end back at vertex $A$ after $2025$ seconds?

Answer: 50


15. Let $a_{n}$ denote the number of ternary strings of length $n$ so that there does not exist a $k<n$ such that the first $k$ digits of the string equals the last $k$ digits. What is the largest integer $m$ such that $3^{m} \mid a_{2023}$ ?

Answer: 9


For a positive integer $n,$ let $P_n$ be the set of sequences of $2n$ elements, each $0$ or $1,$ where there are exactly $n$ $1$’s and $n$ $0$’s. I choose a sequence uniformly at random from $P_n.$ Then, I partition this sequence into maximal blocks of consecutive $0$’s and $1$’s. Define $f(n)$ to be the expected value of the sum of squares of the block lengths of this uniformly random sequence. What is the largest integer value that $f(n)$ can take on?

Answer: 121


Let $\vartriangle ABC$ have $AB = 15$, $AC = 20$, and $BC = 21$. Suppose $\omega$ is a circle passing through $A$ that is tangent to segment $BC$. Let point $D\ne A$ be the second intersection of AB with $\omega$, and let point $E \ne A$ be the second intersection of $AC$ with $\omega$. Suppose $DE$ is parallel to $BC$. If $DE = \frac{a}{b}$ , where $a$, $b$ are relatively prime positive integers, find $a + b$.

Answer: 361


The integers from $1$ to $25,$ inclusive, are randomly placed into a $5$ by $5$ grid such that in each row, the numbers are increasing from left to right. If the columns from left to right are numbered $1,2,3,4,$ and $5,$ then the expected column number of the entry $23$ can be written as $\tfrac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b.$

Answer: 17


A quadratic polynomial $f(x)$ is called sparse if its degree is exactly 2 , if it has integer coefficients, and if there exists a nonzero polynomial $g(x)$ with integer coefficients such that $f(x) g(x)$ has degree at most 3 and $f(x) g(x)$ has at most two nonzero coefficients. Find the number of sparse quadratics whose coefficients lie between 0 and 10, inclusive.

Answer: 228


9. The real quartic $P x^{4}+U x^{3}+M x^{2}+A x+C$ has four different positive real roots. Find the square of the smallest real number $z$ for which the expression $M^{2}-2 U A+z P C$ is always positive, regardless of what the roots of the quartic are.

Answer: 16


14. Kelvin the frog is hopping on the coordinate plane $\mathbb{R}^{2}$. He starts at the origin, and every second, he hops one unit to the right, left, up, or down, such that he always remains in the first quadrant $\{(x, y): x \geq 0, y \geq 0\}$. In how many ways can Kelvin make his first 14 jumps such that his 14 th jump lands at the origin?

Answer: 613470


Let $\vartriangle ABC$  be a triangle with $AB = 4$ and $AC = \frac72$ . Let $\omega$ denote the $A$-excircle of $\vartriangle ABC$. Let $\omega$ touch lines $AB$, $AC$ at the points $D$, $E$, respectively. Let $\Omega$ denote the circumcircle of $\vartriangle ADE$. Consider the line $\ell$ parallel to $BC$ such that $\ell$  is tangent to $\omega$ at a point $F$ and such that $\ell$  does not intersect $\Omega$. Let $\ell$  intersect lines $AB$, $AC$ at the points $X$, $Y$ , respectively, with $XY = 18$ and $AX = 16$. Let the perpendicular bisector of $XY$ meet the circumcircle of $\vartriangle AXY$ at $P$, $Q$, where the distance from $P$ to $F$ is smaller than the distance from $Q$ to$ F$. Let ray $\overrightarrow {PF}$ meet $\Omega$ for the first time at the point $Z$. If $PZ^2 = \frac{m}{n}$ for relatively prime positive integers $m$, $n$, find $m + n$.

Answer: 1159


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

Answer: 4


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Bob`, `Arnold`
 - People have unique favorite music genres: `rock`, `classical`, `hip hop`, `pop`, `jazz`
 - Each person has a unique hobby: `photography`, `knitting`, `cooking`, `painting`, `gardening`
 - Each person prefers a unique type of vacation: `camping`, `city`, `beach`, `mountain`, `cruise`
 - Each mother is accompanied by their child: `Timothy`, `Samantha`, `Fred`, `Bella`, `Meredith`
 - They all have a unique favorite flower: `lilies`, `roses`, `tulips`, `carnations`, `daffodils`

## Clues:
1. The person who is the mother of Timothy is somewhere to the right of the person who likes going on cruises.
2. The photography enthusiast is somewhere to the right of the person who loves hip-hop music.
3. The person who paints as a hobby is the person who enjoys camping trips.
4. The person who loves the boquet of lilies is somewhere to the right of the person who prefers city breaks.
5. The person who loves a carnations arrangement is the person who loves cooking.
6. The person who loves a bouquet of daffodils is the person who loves classical music.
7. The person's child is named Samantha is somewhere to the right of Arnold.
8. The person who loves the rose bouquet is the person who likes going on cruises.
9. The person who loves hip-hop music is the person who loves the vase of tulips.
10. The person who is the mother of Timothy is somewhere to the left of the person who prefers city breaks.
11. The person who loves jazz music is the person's child is named Bella.
12. Eric is the person who likes going on cruises.
13. The person who loves a carnations arrangement is directly left of the person who loves jazz music.
14. The person who enjoys mountain retreats is the person who is the mother of Timothy.
15. The person who loves the boquet of lilies is Bob.
16. The person who enjoys gardening is directly left of the person who is the mother of Timothy.
17. The person's child is named Fred is the person who loves the boquet of lilies.
18. Alice is the person who loves rock music.
 What is the value of attribute MusicGenre for the person whose attribute Children is Fred? Please reason step by step, and put your final answer within \boxed{}

Answer: pop


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

## Clues:
1. The person whose favorite color is green is in the third house.
2. Carol is the person who loves yellow.
3. Alice is not in the fourth house.
4. There are two houses between the person whose favorite color is red and Bob.
5. Alice is directly left of the person who loves yellow.
6. Arnold is directly left of Bob.
7. Bob is somewhere to the left of Carol.
8. Bob is directly left of the person who loves purple.
9. Eric is the person who loves blue.
 What is the value of attribute House for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


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

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

Answer: horse


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`
 - Each person has a unique favorite drink: `milk`, `water`, `tea`
 - Each person prefers a unique type of vacation: `mountain`, `city`, `beach`
 - Each person lives in a unique style of house: `colonial`, `victorian`, `ranch`
 - The people keep unique animals: `cat`, `bird`, `horse`
 - Each person has a unique birthday month: `jan`, `sept`, `april`

## Clues:
1. The person living in a colonial-style house is somewhere to the left of the person who likes milk.
2. The person who prefers city breaks is directly left of the person residing in a Victorian house.
3. The person whose birthday is in January is directly left of the cat lover.
4. The one who only drinks water is the person who enjoys mountain retreats.
5. The person who keeps horses is Peter.
6. The person residing in a Victorian house is somewhere to the right of the person who loves beach vacations.
7. Peter is the person who prefers city breaks.
8. The person who enjoys mountain retreats is the person whose birthday is in April.
9. Eric is the one who only drinks water.
 What is the value of attribute HouseStyle for the person whose attribute Birthday is jan? Please reason step by step, and put your final answer within \boxed{}

Answer: ranch


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Alice`
 - Each person has a unique birthday month: `april`, `sept`, `jan`, `feb`
 - Everyone has a unique favorite cigar: `dunhill`, `prince`, `pall mall`, `blue master`
 - People use unique phone models: `google pixel 6`, `samsung galaxy s21`, `oneplus 9`, `iphone 13`
 - They all have a unique favorite flower: `lilies`, `roses`, `carnations`, `daffodils`
 - People have unique heights: `average`, `short`, `tall`, `very short`

## Clues:
1. The person who loves a bouquet of daffodils is Arnold.
2. The person whose birthday is in April is directly left of the person who smokes Blue Master.
3. The person who is short is the person who uses an iPhone 13.
4. The person who uses a Samsung Galaxy S21 is the person who is tall.
5. Eric is the person who uses a OnePlus 9.
6. The person who uses a Samsung Galaxy S21 is directly left of the person who uses a Google Pixel 6.
7. The person whose birthday is in February is Alice.
8. The Prince smoker is somewhere to the right of the person who is short.
9. The Prince smoker is the person whose birthday is in February.
10. The person who uses a OnePlus 9 is directly left of the person who loves the rose bouquet.
11. The person partial to Pall Mall is somewhere to the left of the person who loves the boquet of lilies.
12. The person who has an average height and the person whose birthday is in September are next to each other.
13. The person who loves a bouquet of daffodils is in the fourth house.
 What is the value of attribute House for the person whose attribute Birthday is feb? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


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

## Clues:
1. The person who is tall is The person whose mother's name is Holly.
2. There are two houses between the person who has an average height and the person who is short.
3. The person who has gray hair is directly left of The person whose mother's name is Janelle.
4. The person who has black hair is not in the fourth house.
5. Eric is the person who has black hair.
6. The person who is very short is The person whose mother's name is Penny.
7. Eric and the person who has gray hair are next to each other.
8. Bob is in the fifth house.
9. The person who has red hair is Peter.
10. The person whose mother's name is Kailyn is directly left of the person who is short.
11. Arnold is the person who has brown hair.
12. The person who has brown hair is somewhere to the left of The person whose mother's name is Janelle.
13. The person whose mother's name is Aniya and the person who is very short are next to each other.
14. The person whose mother's name is Kailyn is in the third house.
 What is the value of attribute Height for the person whose attribute Mother is Penny? 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: `Bob`, `Carol`, `Eric`, `Alice`, `Peter`, `Arnold`
 - Each person prefers a unique type of vacation: `camping`, `cruise`, `mountain`, `cultural`, `city`, `beach`
 - Each person has a unique type of pet: `fish`, `hamster`, `bird`, `rabbit`, `dog`, `cat`
 - The people are of nationalities: `brit`, `swede`, `german`, `norwegian`, `dane`, `chinese`
 - People have unique favorite book genres: `biography`, `romance`, `mystery`, `fantasy`, `science fiction`, `historical fiction`
 - People own unique car models: `tesla model 3`, `bmw 3 series`, `honda civic`, `chevrolet silverado`, `toyota camry`, `ford f150`

## Clues:
1. The person who loves fantasy books is Peter.
2. The Chinese is somewhere to the right of the Dane.
3. The person who owns a Tesla Model 3 is the person with a pet hamster.
4. Carol is the person who loves beach vacations.
5. The person who has a cat is in the second house.
6. There is one house between the person who loves mystery books and the person who prefers city breaks.
7. The person who owns a Honda Civic is in the first house.
8. The person who loves science fiction books is directly left of the person who prefers city breaks.
9. The British person is the person who owns a Honda Civic.
10. There is one house between the person who loves fantasy books and the person who owns a Toyota Camry.
11. The person who keeps a pet bird is the person who owns a Toyota Camry.
12. The Dane is not in the third house.
13. Eric is not in the sixth house.
14. The person who loves romance books is the person who enjoys camping trips.
15. Arnold is the person who likes going on cruises.
16. The person who owns a dog is in the fifth house.
17. The person who enjoys camping trips is directly left of the person who owns a rabbit.
18. The person who owns a Ford F-150 is the person who owns a rabbit.
19. The person who loves fantasy books is directly left of Eric.
20. The German is the person who owns a BMW 3 Series.
21. The person who goes on cultural tours is not in the first house.
22. The person who loves mystery books is not in the sixth house.
23. The person who loves historical fiction books is somewhere to the left of the person who enjoys camping trips.
24. The Dane is Bob.
25. There is one house between the Dane and the Chinese.
26. The Swedish person is somewhere to the right of the person who likes going on cruises.
27. Eric is not in the third house.
 What is the value of attribute Nationality for the person whose attribute Vacation is cultural? Please reason step by step, and put your final answer within \boxed{}

Answer: german