There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Bob`, `Eric`
 - People have unique heights: `very tall`, `average`, `tall`, `very short`, `short`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `cruise`, `city`, `camping`
 - Each person has a unique level of education: `master`, `doctorate`, `associate`, `high school`, `bachelor`

## Clues:
1. The person who is very tall is directly left of the person with an associate's degree.
2. The person who has an average height and the person who enjoys mountain retreats are next to each other.
3. Eric is the person who is short.
4. The person who prefers city breaks is in the fourth house.
5. The person who loves beach vacations is the person who is very tall.
6. The person with a master's degree is Bob.
7. The person who enjoys camping trips is not in the third house.
8. The person who has an average height is not in the second house.
9. The person with a doctorate is directly left of Alice.
10. The person with a master's degree is the person who enjoys camping trips.
11. The person who enjoys camping trips is not in the first house.
12. The person with a doctorate is the person who is tall.
13. The person with a high school diploma is somewhere to the right of the person who has an average height.
14. Arnold is the person with a doctorate.
 What is the value of attribute House for the person whose attribute Vacation is mountain? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Carol`, `Bob`, `Alice`, `Eric`, `Peter`
 - People have unique favorite sports: `swimming`, `basketball`, `volleyball`, `soccer`, `tennis`, `baseball`
 - People have unique favorite music genres: `rock`, `pop`, `jazz`, `country`, `hip hop`, `classical`
 - Each person has a unique level of education: `doctorate`, `trade school`, `bachelor`, `high school`, `associate`, `master`

## Clues:
1. Alice is in the fourth house.
2. Alice is somewhere to the right of the person with a doctorate.
3. The person with a high school diploma is somewhere to the right of the person who loves basketball.
4. The person who loves volleyball is the person with a bachelor's degree.
5. Alice is directly left of the person who loves jazz music.
6. The person with a master's degree is the person who loves swimming.
7. There is one house between the person who loves rock music and the person with a bachelor's degree.
8. Carol is in the second house.
9. The person who loves rock music is somewhere to the left of the person who loves pop music.
10. The person who loves swimming is somewhere to the left of Carol.
11. Eric is directly left of the person who loves hip-hop music.
12. Alice is the person with an associate's degree.
13. Alice is the person who loves classical music.
14. Arnold is the person who loves soccer.
15. The person who loves basketball and the person who loves volleyball are next to each other.
16. Peter is the person who loves tennis.
17. The person with a doctorate is Peter.
 What is the value of attribute House for the person whose attribute MusicGenre is jazz? 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: `Carol`, `Bob`, `Eric`, `Alice`, `Peter`, `Arnold`
 - People have unique heights: `super tall`, `tall`, `very short`, `average`, `very tall`, `short`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `samsung galaxy s21`, `xiaomi mi 11`, `oneplus 9`, `huawei p50`
 - Each mother is accompanied by their child: `Fred`, `Timothy`, `Alice`, `Bella`, `Meredith`, `Samantha`

## Clues:
1. The person who uses a Xiaomi Mi 11 is the person who is short.
2. Alice is somewhere to the right of Bob.
3. The person who uses a OnePlus 9 is Carol.
4. The person who uses a Xiaomi Mi 11 is the person's child is named Alice.
5. The person who has an average height is not in the fourth house.
6. The person who uses a Samsung Galaxy S21 is in the second house.
7. Alice is directly left of the person's child is named Fred.
8. The person's child is named Meredith and Arnold are next to each other.
9. The person's child is named Samantha is the person who is super tall.
10. The person who is short is somewhere to the right of the person who uses a Samsung Galaxy S21.
11. The person who is very tall is the person who is the mother of Timothy.
12. Bob is not in the fourth house.
13. The person who is very short is the person who uses a Huawei P50.
14. The person who uses a Samsung Galaxy S21 is directly left of the person who is super tall.
15. Peter is somewhere to the right of the person who is tall.
16. Arnold is somewhere to the left of the person's child is named Bella.
17. The person who uses a Google Pixel 6 is directly left of the person who uses an iPhone 13.
18. Eric is in the first house.
 What is the value of attribute House for the person whose attribute Children is Fred? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Bob`, `Arnold`, `Carol`, `Eric`, `Alice`
 - Everyone has something unique for lunch: `spaghetti`, `stir fry`, `stew`, `soup`, `grilled cheese`, `pizza`
 - Everyone has a unique favorite cigar: `blends`, `pall mall`, `dunhill`, `yellow monster`, `prince`, `blue master`
 - Each person has a unique favorite drink: `boba tea`, `water`, `coffee`, `root beer`, `milk`, `tea`
 - Each person has a unique level of education: `associate`, `bachelor`, `trade school`, `doctorate`, `high school`, `master`

## Clues:
1. Bob is not in the second house.
2. The person who smokes many unique blends is somewhere to the left of the person who is a pizza lover.
3. The person partial to Pall Mall is somewhere to the right of the Dunhill smoker.
4. The person who loves the soup is not in the first house.
5. The person who likes milk is directly left of the person who loves the stew.
6. The person who loves the soup is the person with a doctorate.
7. The coffee drinker is the person with a high school diploma.
8. There are two houses between the person who loves the spaghetti eater and the person with an associate's degree.
9. The Prince smoker is not in the third house.
10. The Dunhill smoker and Bob are next to each other.
11. There are two houses between Alice and the boba tea drinker.
12. The person with a high school diploma is Eric.
13. The person who attended trade school is the Prince smoker.
14. There is one house between the Dunhill smoker and the person who loves stir fry.
15. The Dunhill smoker is directly left of the person who is a pizza lover.
16. There is one house between the person partial to Pall Mall and the person who smokes Blue Master.
17. The person who smokes Blue Master is the coffee drinker.
18. The person with a bachelor's degree is the Dunhill smoker.
19. Alice is the tea drinker.
20. Arnold is directly left of the root beer lover.
21. The person with a master's degree is Peter.
22. Arnold is somewhere to the left of the person who likes milk.
 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: soup


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`
 - People have unique favorite sports: `soccer`, `basketball`, `tennis`
 - The people keep unique animals: `cat`, `bird`, `horse`
 - Each person has a unique favorite drink: `water`, `tea`, `milk`
 - The people are of nationalities: `swede`, `dane`, `brit`

## Clues:
1. The person who likes milk is not in the first house.
2. The person who keeps horses is somewhere to the left of Eric.
3. The person who loves basketball is Peter.
4. The Swedish person is somewhere to the left of the one who only drinks water.
5. The person who loves tennis is the tea drinker.
6. The person who loves soccer is the bird keeper.
7. The cat lover is the British person.
8. Peter is not in the third house.
9. The person who likes milk is the person who loves soccer.
 What is the value of attribute Animal for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: horse


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Carol`, `Eric`, `Peter`, `Arnold`
 - Each person has a unique hobby: `knitting`, `woodworking`, `photography`, `cooking`, `gardening`, `painting`
 - Each person lives in a unique style of house: `ranch`, `modern`, `mediterranean`, `colonial`, `craftsman`, `victorian`
 - People have unique favorite music genres: `classical`, `jazz`, `rock`, `hip hop`, `pop`, `country`
 - People have unique hair colors: `blonde`, `red`, `gray`, `black`, `brown`, `auburn`

## Clues:
1. The woodworking hobbyist is somewhere to the right of the person living in a colonial-style house.
2. The person who loves jazz music is in the sixth house.
3. The person who has black hair is directly left of Bob.
4. The person who has blonde hair is the person living in a colonial-style house.
5. Arnold is directly left of the person in a ranch-style home.
6. The person who paints as a hobby is somewhere to the left of the photography enthusiast.
7. Peter is the person who has black hair.
8. Eric is not in the first house.
9. The person who enjoys gardening is the person who has brown hair.
10. The person who has red hair is the person who loves pop music.
11. The person in a Craftsman-style house is in the sixth house.
12. Carol is in the sixth house.
13. The person who has blonde hair is directly left of the person residing in a Victorian house.
14. The person who enjoys knitting is Bob.
15. The person who paints as a hobby is the person who has blonde hair.
16. The person who enjoys knitting is the person who has auburn hair.
17. The photography enthusiast is somewhere to the left of the person in a Craftsman-style house.
18. There is one house between the person who loves hip-hop music and the person who loves country music.
19. The person who has black hair is somewhere to the right of the person who loves hip-hop music.
20. The person who enjoys gardening is the person residing in a Victorian house.
21. The person who loves pop music is somewhere to the left of the person in a Mediterranean-style villa.
22. The person who loves rock music is in the second house.
 What is the value of attribute HairColor for the person whose attribute MusicGenre is jazz? Please reason step by step, and put your final answer within \boxed{}

Answer: gray


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`, `Bob`
 - Everyone has something unique for lunch: `stir fry`, `spaghetti`, `stew`, `grilled cheese`, `pizza`
 - People own unique car models: `ford f150`, `tesla model 3`, `bmw 3 series`, `toyota camry`, `honda civic`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `oneplus 9`, `huawei p50`
 - Each person has an occupation: `teacher`, `lawyer`, `doctor`, `artist`, `engineer`
 - Each person has a unique favorite drink: `tea`, `milk`, `water`, `root beer`, `coffee`

## Clues:
1. The root beer lover is the person who owns a Honda Civic.
2. The person who likes milk is directly left of the person who loves eating grilled cheese.
3. Alice is the person who uses a Samsung Galaxy S21.
4. Alice is the person who loves stir fry.
5. The tea drinker is not in the fifth house.
6. The person who owns a BMW 3 Series is somewhere to the left of the tea drinker.
7. The person who is a doctor is Arnold.
8. The person who uses an iPhone 13 is the coffee drinker.
9. The person who is an engineer is the person who owns a BMW 3 Series.
10. The person who loves the stew is the person who uses an iPhone 13.
11. The person who is a doctor is directly left of the person who uses a OnePlus 9.
12. The person who owns a Honda Civic is directly left of the person who loves the spaghetti eater.
13. The person who uses a Google Pixel 6 is the tea drinker.
14. Alice is the person who is an artist.
15. There is one house between Alice and the person who owns a Ford F-150.
16. Arnold is the person who owns a Toyota Camry.
17. Eric is in the fourth house.
18. The person who uses a OnePlus 9 is the person who is a lawyer.
19. The person who loves eating grilled cheese is Peter.
 What is the value of attribute House for the person whose attribute Food is stir fry? 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: `Eric`, `Peter`, `Arnold`, `Alice`
 - Each person lives in a unique style of house: `victorian`, `ranch`, `colonial`, `craftsman`
 - Each person has an occupation: `teacher`, `engineer`, `doctor`, `artist`
 - Everyone has a unique favorite cigar: `prince`, `dunhill`, `pall mall`, `blue master`
 - People have unique favorite music genres: `pop`, `rock`, `classical`, `jazz`
 - People have unique heights: `very short`, `tall`, `average`, `short`

## Clues:
1. The person who loves classical music is the person living in a colonial-style house.
2. The person who smokes Blue Master is the person in a ranch-style home.
3. The person who loves rock music is the person who smokes Blue Master.
4. The Dunhill smoker is directly left of Peter.
5. Eric is in the first house.
6. The person residing in a Victorian house is in the third house.
7. The person who loves rock music and the person who loves classical music are next to each other.
8. The person who is a teacher is the person who is short.
9. The person who loves classical music is the Prince smoker.
10. The person who is a doctor is somewhere to the left of the person who is very short.
11. Arnold is the person who is an artist.
12. The person who smokes Blue Master is in the first house.
13. The person who is tall is directly left of the person who loves pop music.
14. The person residing in a Victorian house is the person who is a teacher.
 What is the value of attribute Height for the person whose attribute MusicGenre is pop? Please reason step by step, and put your final answer within \boxed{}

Answer: short


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Carol`, `Peter`, `Eric`, `Arnold`
 - The mothers' names in different houses are unique: `Holly`, `Sarah`, `Penny`, `Kailyn`, `Aniya`, `Janelle`

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

Answer: Peter


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Peter`, `Eric`
 - Each person has a unique birthday month: `sept`, `feb`, `april`, `jan`
 - The mothers' names in different houses are unique: `Aniya`, `Kailyn`, `Holly`, `Janelle`
 - The people keep unique animals: `horse`, `bird`, `cat`, `fish`
 - Each person has a favorite color: `green`, `white`, `yellow`, `red`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `soccer`

## Clues:
1. The bird keeper is The person whose mother's name is Kailyn.
2. The fish enthusiast is not in the fourth house.
3. The person who loves white is Arnold.
4. The person who keeps horses is Peter.
5. The person who keeps horses is not in the first house.
6. The person whose mother's name is Kailyn is Arnold.
7. The person whose birthday is in January is in the first house.
8. The person whose birthday is in April is somewhere to the left of the person whose birthday is in September.
9. Alice is the person whose birthday is in September.
10. The person whose mother's name is Aniya and the person whose birthday is in September are next to each other.
11. Peter is directly left of the person whose favorite color is red.
12. There is one house between the person whose birthday is in April and the person who loves soccer.
13. The fish enthusiast is the person whose birthday is in September.
14. The person who loves swimming is The person whose mother's name is Holly.
15. The person whose favorite color is green is the person who loves swimming.
16. The person who loves yellow is the person who loves basketball.
 What is the value of attribute Birthday for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: jan


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Bob`, `Eric`, `Carol`
 - Everyone has something unique for lunch: `stir fry`, `grilled cheese`, `spaghetti`, `pizza`, `soup`, `stew`
 - People use unique phone models: `xiaomi mi 11`, `iphone 13`, `oneplus 9`, `google pixel 6`, `samsung galaxy s21`, `huawei p50`
 - People have unique favorite music genres: `pop`, `country`, `classical`, `jazz`, `rock`, `hip hop`
 - Each person has a unique favorite drink: `water`, `boba tea`, `milk`, `root beer`, `tea`, `coffee`
 - Each person has an occupation: `nurse`, `lawyer`, `artist`, `doctor`, `teacher`, `engineer`

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

Answer: 5


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

## Clues:
1. The person who uses an iPhone 13 is the person who loves white.
2. Peter is the person whose favorite color is red.
3. The Prince smoker is directly left of the person partial to Pall Mall.
4. The person who smokes Blue Master and the Dunhill smoker are next to each other.
5. The person who loves blue is the person who uses a Xiaomi Mi 11.
6. The person who loves white is Alice.
7. The person who uses a OnePlus 9 is somewhere to the left of Arnold.
8. The person who is tall is the person who smokes Blue Master.
9. The person who uses a Samsung Galaxy S21 is the Dunhill smoker.
10. Bob is directly left of the person who is very short.
11. The person who loves a bouquet of daffodils is Eric.
12. The person who loves the boquet of iris is the person who uses a OnePlus 9.
13. The person who uses a Huawei P50 is the person who is very short.
14. The person who smokes many unique blends is directly left of Peter.
15. The Prince smoker is not in the first house.
16. The person who is very tall is the person who loves a bouquet of daffodils.
17. The person who uses a Google Pixel 6 is not in the fifth house.
18. The person who is tall is not in the third house.
19. The person who uses a Huawei P50 is somewhere to the left of the person who uses a Samsung Galaxy S21.
20. Arnold is somewhere to the left of the person who is short.
21. The person who loves the vase of tulips is not in the first house.
22. The person who loves the boquet of lilies is directly left of the person who loves yellow.
23. Arnold is the person who loves purple.
24. The person who uses a OnePlus 9 and the person who uses a Samsung Galaxy S21 are next to each other.
25. The person who has an average height is not in the first house.
26. The person who loves the rose bouquet is Alice.
 What is the value of attribute House for the person whose attribute Cigar is blue master? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Arnold`, `Alice`, `Eric`
 - Everyone has a unique favorite cigar: `blue master`, `dunhill`, `prince`, `blends`, `pall mall`
 - People have unique favorite music genres: `rock`, `classical`, `hip hop`, `pop`, `jazz`
 - Each person has a unique birthday month: `feb`, `april`, `sept`, `jan`, `mar`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`, `biography`, `fantasy`

## Clues:
1. The person who loves pop music is the person who loves romance books.
2. The person who loves fantasy books is in the second house.
3. The person who smokes Blue Master is the person who loves science fiction books.
4. The person who loves hip-hop music is the person whose birthday is in September.
5. The person who loves biography books is Arnold.
6. Alice is the person who loves science fiction books.
7. The person who loves jazz music is somewhere to the left of Eric.
8. The person who smokes Blue Master is the person whose birthday is in April.
9. The person who smokes many unique blends is not in the third house.
10. The Dunhill smoker is the person whose birthday is in February.
11. The person who loves science fiction books is in the fifth house.
12. The person who loves classical music is Bob.
13. The person who loves pop music is Eric.
14. The person who loves mystery books is the person partial to Pall Mall.
15. Peter is the person whose birthday is in February.
16. The person whose birthday is in March is the person who smokes many unique blends.
17. The person who loves classical music is not in the first house.
 What is the value of attribute Name for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Alice`, `Arnold`, `Peter`
 - Each person has an occupation: `engineer`, `doctor`, `artist`, `lawyer`, `teacher`
 - Everyone has a favorite smoothie: `desert`, `dragonfruit`, `lime`, `cherry`, `watermelon`
 - Each mother is accompanied by their child: `Fred`, `Bella`, `Meredith`, `Samantha`, `Timothy`

## Clues:
1. Arnold is somewhere to the right of the person who is a teacher.
2. The person's child is named Bella is in the third house.
3. The person who is the mother of Timothy is the person who is an artist.
4. Alice is in the third house.
5. The person who is a doctor is directly left of the person who is an engineer.
6. The Desert smoothie lover is the person who is an engineer.
7. The person's child is named Fred is the person who is a teacher.
8. The person who is an engineer is not in the third house.
9. Bob is the Dragonfruit smoothie lover.
10. Peter is somewhere to the right of the person who is an engineer.
11. The person's child is named Meredith is the person who drinks Lime smoothies.
12. The person's child is named Meredith is in the first house.
13. The person who likes Cherry smoothies is somewhere to the right of the Desert smoothie lover.
 What is the value of attribute Children for the person whose attribute Occupation is engineer? Please reason step by step, and put your final answer within \boxed{}

Answer: Samantha


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`
 - Everyone has something unique for lunch: `pizza`, `grilled cheese`, `spaghetti`
 - Each person has a unique birthday month: `april`, `jan`, `sept`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `desert`
 - People own unique car models: `toyota camry`, `ford f150`, `tesla model 3`
 - People have unique favorite music genres: `pop`, `classical`, `rock`

## Clues:
1. The person whose birthday is in January is not in the second house.
2. Arnold is directly left of the person who loves eating grilled cheese.
3. The person who owns a Tesla Model 3 is the person who loves pop music.
4. The Watermelon smoothie lover and the person who owns a Tesla Model 3 are next to each other.
5. The person whose birthday is in April is not in the third house.
6. The person who loves eating grilled cheese is the Desert smoothie lover.
7. The person who owns a Toyota Camry is the person who is a pizza lover.
8. The person who owns a Tesla Model 3 is the person who loves eating grilled cheese.
9. The person whose birthday is in January is the person who loves classical music.
10. The person who is a pizza lover is Eric.
11. Eric is not in the third house.
 What is the value of attribute MusicGenre for the person whose attribute Smoothie is watermelon? Please reason step by step, and put your final answer within \boxed{}

Answer: rock


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`
 - They all have a unique favorite flower: `roses`, `carnations`, `daffodils`, `lilies`
 - Each person lives in a unique style of house: `victorian`, `craftsman`, `colonial`, `ranch`
 - Everyone has a unique favorite cigar: `prince`, `dunhill`, `blue master`, `pall mall`

## Clues:
1. The person in a ranch-style home is in the third house.
2. The person who smokes Blue Master is somewhere to the left of the person residing in a Victorian house.
3. There are two houses between Eric and the person who loves the rose bouquet.
4. The Prince smoker is Peter.
5. The person in a Craftsman-style house is the Dunhill smoker.
6. The Dunhill smoker is the person who loves a carnations arrangement.
7. Arnold is in the fourth house.
8. The Prince smoker is in the second house.
9. The person who loves the boquet of lilies is the person who smokes Blue Master.
 What is the value of attribute Flower for the person whose attribute HouseStyle is victorian? Please reason step by step, and put your final answer within \boxed{}

Answer: roses


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Peter`
 - People use unique phone models: `iphone 13`, `samsung galaxy s21`, `google pixel 6`, `oneplus 9`
 - The people are of nationalities: `norwegian`, `swede`, `dane`, `brit`
 - People have unique favorite sports: `tennis`, `swimming`, `soccer`, `basketball`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `cherry`, `dragonfruit`

## Clues:
1. The person who loves basketball is the British person.
2. The person who likes Cherry smoothies is directly left of Arnold.
3. The person who uses an iPhone 13 is somewhere to the left of the person who loves tennis.
4. The person who loves soccer is the person whose favorite color is red.
5. Eric is the person who loves basketball.
6. Alice is the Watermelon smoothie lover.
7. The Norwegian is not in the fourth house.
8. Eric is somewhere to the left of the person who loves yellow.
9. The Desert smoothie lover is not in the second house.
10. The person who loves tennis is the person whose favorite color is green.
11. The person who uses a OnePlus 9 is the Norwegian.
12. Alice is the person who uses a Samsung Galaxy S21.
13. Eric is in the third house.
14. Peter is somewhere to the left of the Dane.
 What is the value of attribute Color for the person whose attribute Smoothie is dragonfruit? Please reason step by step, and put your final answer within \boxed{}

Answer: green


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Eric`, `Peter`
 - Everyone has something unique for lunch: `stew`, `spaghetti`, `grilled cheese`, `pizza`, `stir fry`
 - Each person has a favorite color: `green`, `blue`, `yellow`, `red`, `white`
 - The people are of nationalities: `swede`, `brit`, `dane`, `german`, `norwegian`
 - People have unique favorite book genres: `fantasy`, `biography`, `romance`, `mystery`, `science fiction`
 - Each mother is accompanied by their child: `Timothy`, `Samantha`, `Meredith`, `Fred`, `Bella`

## Clues:
1. The Swedish person is the person's child is named Bella.
2. There is one house between the person who loves science fiction books and the person who loves the stew.
3. The person who is a pizza lover is the person's child is named Fred.
4. The person who loves white is not in the fourth house.
5. The person who loves yellow is Peter.
6. The person who loves eating grilled cheese is directly left of the person's child is named Meredith.
7. The German is the person who loves the spaghetti eater.
8. The person who loves romance books is the person who loves the stew.
9. The person's child is named Samantha is directly left of the German.
10. Alice is the Swedish person.
11. Arnold is the person whose favorite color is green.
12. Arnold is in the third house.
13. The person's child is named Meredith is the person who loves biography books.
14. The Norwegian is the person whose favorite color is red.
15. The person who loves romance books is not in the third house.
16. The person who loves the spaghetti eater is Eric.
17. The Dane is somewhere to the left of the person who loves yellow.
18. The person who loves mystery books is directly left of the person whose favorite color is red.
19. The person's child is named Fred is not in the third house.
 What is the value of attribute Children for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: Samantha


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`, `Peter`, `Arnold`
 - People have unique favorite book genres: `fantasy`, `mystery`, `science fiction`, `biography`, `romance`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Aniya`, `Kailyn`, `Janelle`
 - People use unique phone models: `oneplus 9`, `iphone 13`, `samsung galaxy s21`, `google pixel 6`, `huawei p50`
 - The people are of nationalities: `dane`, `brit`, `german`, `swede`, `norwegian`
 - Each person lives in a unique style of house: `ranch`, `modern`, `craftsman`, `victorian`, `colonial`

## Clues:
1. The person in a ranch-style home is somewhere to the right of the person residing in a Victorian house.
2. The Dane and the person who uses a Samsung Galaxy S21 are next to each other.
3. The person who uses a Google Pixel 6 is not in the second house.
4. There are two houses between the person who loves fantasy books and the person in a modern-style house.
5. The person whose mother's name is Kailyn is the Swedish person.
6. The Swedish person is Arnold.
7. The British person is somewhere to the left of the person who uses an iPhone 13.
8. The Dane is The person whose mother's name is Penny.
9. The person who uses a Samsung Galaxy S21 is the Norwegian.
10. The Norwegian is The person whose mother's name is Holly.
11. The person who loves fantasy books is Bob.
12. Alice is directly left of the Swedish person.
13. The person who loves romance books is Arnold.
14. The person who loves science fiction books is Peter.
15. The person whose mother's name is Aniya is directly left of the person who loves science fiction books.
16. The person who loves science fiction books is the person who uses a OnePlus 9.
17. The person residing in a Victorian house is the person who loves biography books.
18. The person who loves fantasy books is somewhere to the left of the German.
19. The person who uses an iPhone 13 is the person in a Craftsman-style house.
 What is the value of attribute BookGenre for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: science fiction


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

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

Answer: short


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Peter`
 - They all have a unique favorite flower: `lilies`, `roses`, `carnations`, `daffodils`
 - People have unique favorite book genres: `fantasy`, `mystery`, `science fiction`, `romance`
 - Each person prefers a unique type of vacation: `city`, `beach`, `cruise`, `mountain`
 - People have unique hair colors: `black`, `red`, `brown`, `blonde`
 - Everyone has something unique for lunch: `grilled cheese`, `pizza`, `stew`, `spaghetti`

## Clues:
1. The person who likes going on cruises is the person who has brown hair.
2. The person who loves the stew is the person who loves beach vacations.
3. Peter is the person who loves romance books.
4. There is one house between the person who has brown hair and the person who loves the spaghetti eater.
5. The person who is a pizza lover is Arnold.
6. The person who prefers city breaks is somewhere to the right of the person who loves the boquet of lilies.
7. The person who loves the stew is directly left of the person who loves mystery books.
8. The person who has black hair is directly left of the person who loves fantasy books.
9. The person who loves a carnations arrangement is directly left of the person who is a pizza lover.
10. The person who has blonde hair is not in the fourth house.
11. The person who prefers city breaks is directly left of Eric.
12. The person who loves the boquet of lilies is somewhere to the right of Alice.
13. The person who loves a bouquet of daffodils is in the fourth house.
 What is the value of attribute Food for the person whose attribute Vacation is cruise? 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: `Peter`, `Arnold`, `Eric`, `Alice`, `Bob`, `Carol`
 - Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `lime`, `desert`, `blueberry`, `watermelon`
 - Each person has a unique type of pet: `rabbit`, `fish`, `bird`, `hamster`, `cat`, `dog`
 - Each person has a unique level of education: `master`, `bachelor`, `high school`, `trade school`, `associate`, `doctorate`
 - People have unique heights: `short`, `very tall`, `very short`, `tall`, `average`, `super tall`

## Clues:
1. The person with an aquarium of fish is in the first house.
2. The person with a doctorate is somewhere to the left of the person who owns a dog.
3. Peter and the person with a high school diploma are next to each other.
4. There is one house between the person with a pet hamster and the person who owns a rabbit.
5. The person who has a cat is Eric.
6. Bob and the person who owns a dog are next to each other.
7. The person who is tall is Carol.
8. The person with a master's degree is the person with a pet hamster.
9. Peter is the person who is very tall.
10. The person who likes Cherry smoothies is the person who is very tall.
11. The person who is super tall is somewhere to the right of Arnold.
12. The Watermelon smoothie lover is in the sixth house.
13. The person who is super tall is the Desert smoothie lover.
14. The person who is very short is the person with an associate's degree.
15. The person with an associate's degree and the Watermelon smoothie lover are next to each other.
16. The person who drinks Lime smoothies is the person with a pet hamster.
17. The person who is short is in the second house.
18. The person who drinks Blueberry smoothies is the person who attended trade school.
19. The person who is very short is somewhere to the right of Bob.
 What is the value of attribute Pet for the person whose attribute Height is tall? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`, `Alice`
 - People have unique favorite sports: `basketball`, `soccer`, `swimming`, `tennis`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `samsung galaxy s21`, `iphone 13`
 - People own unique car models: `toyota camry`, `ford f150`, `tesla model 3`, `honda civic`
 - They all have a unique favorite flower: `roses`, `daffodils`, `lilies`, `carnations`

## Clues:
1. The person who owns a Honda Civic is the person who uses an iPhone 13.
2. The person who loves tennis is the person who uses an iPhone 13.
3. The person who uses a Google Pixel 6 is somewhere to the right of the person who loves basketball.
4. The person who uses an iPhone 13 is directly left of Eric.
5. Alice is the person who owns a Ford F-150.
6. Arnold is the person who loves a bouquet of daffodils.
7. The person who loves the boquet of lilies is in the third house.
8. The person who loves soccer is Peter.
9. The person who uses a OnePlus 9 is the person who loves the rose bouquet.
10. The person who loves a carnations arrangement is not in the fourth house.
11. Peter is the person who loves the boquet of lilies.
12. The person who owns a Toyota Camry is directly left of the person who owns a Tesla Model 3.
 What is the value of attribute FavoriteSport for the person whose attribute CarModel is tesla model 3? Please reason step by step, and put your final answer within \boxed{}

Answer: soccer


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`, `Alice`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Kailyn`, `Aniya`
 - The people are of nationalities: `dane`, `norwegian`, `brit`, `swede`
 - People have unique favorite book genres: `science fiction`, `romance`, `fantasy`, `mystery`
 - The people keep unique animals: `fish`, `cat`, `bird`, `horse`
 - People have unique hair colors: `black`, `blonde`, `red`, `brown`

## Clues:
1. The Swedish person is the person who has black hair.
2. The bird keeper is in the first house.
3. Arnold is the person who loves fantasy books.
4. The person who loves science fiction books is not in the second house.
5. The British person and Arnold are next to each other.
6. The bird keeper is the person who has red hair.
7. The person who keeps horses is Peter.
8. The person who has black hair is the person who loves romance books.
9. The person whose mother's name is Kailyn is Arnold.
10. The Swedish person is The person whose mother's name is Janelle.
11. The person who has blonde hair is in the third house.
12. Alice is The person whose mother's name is Aniya.
13. The Swedish person is the fish enthusiast.
14. Arnold is directly left of the person who has black hair.
15. The Norwegian is not in the first house.
 What is the value of attribute BookGenre for the person whose attribute Nationality is norwegian? Please reason step by step, and put your final answer within \boxed{}

Answer: fantasy


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Alice`, `Eric`, `Arnold`
 - Each person has a unique type of pet: `hamster`, `fish`, `bird`, `cat`, `dog`
 - Each person lives in a unique style of house: `colonial`, `ranch`, `victorian`, `modern`, `craftsman`
 - Each person has a unique birthday month: `mar`, `sept`, `jan`, `feb`, `april`
 - The people keep unique animals: `dog`, `cat`, `fish`, `bird`, `horse`

## Clues:
1. There is one house between Alice and the fish enthusiast.
2. The person in a ranch-style home is not in the fifth house.
3. The person whose birthday is in March is Alice.
4. The person in a ranch-style home is not in the fourth house.
5. Arnold is somewhere to the right of the person whose birthday is in April.
6. The person who has a cat is the person who keeps horses.
7. There are two houses between the cat lover and the person living in a colonial-style house.
8. Eric is somewhere to the left of the person whose birthday is in September.
9. The person whose birthday is in April is directly left of the person who keeps horses.
10. Alice is the person residing in a Victorian house.
11. The person whose birthday is in March and the cat lover are next to each other.
12. Peter is the person whose birthday is in February.
13. The cat lover is the person who owns a dog.
14. The bird keeper is somewhere to the left of the person who keeps a pet bird.
15. Bob is the person who keeps a pet bird.
16. Eric is in the fourth house.
17. The person who has a cat is somewhere to the left of the person with a pet hamster.
18. The person who keeps a pet bird is the person in a modern-style house.
 What is the value of attribute House for the person whose attribute HouseStyle is craftsman? 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`, `Bob`, `Peter`, `Arnold`, `Eric`
 - People have unique hair colors: `black`, `red`, `brown`, `gray`, `blonde`
 - Each person has a unique level of education: `bachelor`, `associate`, `high school`, `master`, `doctorate`
 - Each mother is accompanied by their child: `Bella`, `Timothy`, `Samantha`, `Meredith`, `Fred`
 - The people keep unique animals: `horse`, `cat`, `fish`, `bird`, `dog`

## Clues:
1. The person who has red hair is directly left of the bird keeper.
2. Peter is the person with a doctorate.
3. The fish enthusiast is somewhere to the right of the person who has brown hair.
4. The cat lover is the person with a bachelor's degree.
5. The person's child is named Samantha is somewhere to the right of the person who has black hair.
6. The fish enthusiast is the person's child is named Bella.
7. Arnold and the person with an associate's degree are next to each other.
8. The person who keeps horses is the person who is the mother of Timothy.
9. The person who has brown hair is somewhere to the right of Bob.
10. The person who has blonde hair is the bird keeper.
11. The person with a master's degree is the fish enthusiast.
12. The person with a high school diploma is the person's child is named Fred.
13. The person with a high school diploma is Eric.
14. The person with a bachelor's degree is the person who has red hair.
15. The person who has gray hair is the person with an associate's degree.
 What is the value of attribute Children for the person whose attribute Animal is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: Fred


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`
 - People have unique favorite book genres: `romance`, `mystery`, `science fiction`
 - Each person has a unique birthday month: `sept`, `april`, `jan`
 - The people are of nationalities: `brit`, `dane`, `swede`

## Clues:
1. Arnold is the person who loves mystery books.
2. Peter is directly left of the person who loves romance books.
3. Peter is somewhere to the left of the person who loves mystery books.
4. The person whose birthday is in April is somewhere to the right of the person whose birthday is in January.
5. The person whose birthday is in September is the Swedish person.
6. The Dane is in the third house.
7. The person who loves science fiction books is the person whose birthday is in September.
 What is the value of attribute Birthday for the person whose attribute BookGenre is science fiction? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


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 Birthday for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Arnold`, `Bob`, `Eric`, `Peter`, `Alice`
 - People use unique phone models: `iphone 13`, `huawei p50`, `oneplus 9`, `google pixel 6`, `xiaomi mi 11`, `samsung galaxy s21`
 - Everyone has a unique favorite cigar: `pall mall`, `yellow monster`, `prince`, `blends`, `dunhill`, `blue master`
 - Each person prefers a unique type of vacation: `cultural`, `beach`, `city`, `camping`, `cruise`, `mountain`
 - People have unique hair colors: `brown`, `red`, `gray`, `black`, `blonde`, `auburn`
 - Each person has a unique favorite drink: `coffee`, `boba tea`, `milk`, `water`, `root beer`, `tea`

## Clues:
1. The person who prefers city breaks is Bob.
2. The person who has brown hair is the one who only drinks water.
3. Eric is directly left of the person who uses a Samsung Galaxy S21.
4. There is one house between the person who loves beach vacations and the person who enjoys camping trips.
5. The one who only drinks water is somewhere to the left of the person who uses a Xiaomi Mi 11.
6. The person who has brown hair is the Dunhill smoker.
7. The person who has red hair is Arnold.
8. The person who has auburn hair is the person who uses an iPhone 13.
9. The person who has gray hair is not in the sixth house.
10. The person who likes milk is the person who has auburn hair.
11. The person who smokes Blue Master is Alice.
12. The person who uses a OnePlus 9 is the person who likes going on cruises.
13. The coffee drinker is the Prince smoker.
14. The coffee drinker is the person who uses a Huawei P50.
15. Eric is the tea drinker.
16. The Prince smoker is in the third house.
17. The person who smokes Yellow Monster is somewhere to the right of the coffee drinker.
18. The person who enjoys mountain retreats is Peter.
19. The person who uses a Huawei P50 is the person who has blonde hair.
20. There are two houses between the person who loves beach vacations and the person who uses a Google Pixel 6.
21. The one who only drinks water is in the fourth house.
22. Eric is the person partial to Pall Mall.
23. Bob is the boba tea drinker.
 What is the value of attribute Drink for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: tea


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Peter`
 - The people keep unique animals: `cat`, `fish`, `horse`, `bird`
 - Each person has a favorite color: `yellow`, `green`, `white`, `red`
 - Each person prefers a unique type of vacation: `cruise`, `city`, `beach`, `mountain`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `stew`, `grilled cheese`
 - People have unique favorite sports: `soccer`, `swimming`, `basketball`, `tennis`

## Clues:
1. The person whose favorite color is red is the person who loves the spaghetti eater.
2. The person who loves the stew is the cat lover.
3. The person who loves yellow is not in the fourth house.
4. Alice is somewhere to the right of Arnold.
5. The person who loves the spaghetti eater is somewhere to the right of the person who loves soccer.
6. The person who loves white is the person who loves tennis.
7. The fish enthusiast is in the second house.
8. Arnold is the person who loves the stew.
9. The person who prefers city breaks is the person who is a pizza lover.
10. The person who prefers city breaks is directly left of the person who likes going on cruises.
11. The person who loves swimming is the person who loves beach vacations.
12. The person who keeps horses and the person whose favorite color is red are next to each other.
13. The person whose favorite color is green is directly left of Eric.
14. The person who prefers city breaks is somewhere to the left of the cat lover.
 What is the value of attribute Food for the person whose attribute FavoriteSport is soccer? 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: `Bob`, `Arnold`, `Alice`, `Peter`, `Carol`, `Eric`
 - Each person prefers a unique type of vacation: `mountain`, `city`, `cultural`, `camping`, `beach`, `cruise`
 - Everyone has a favorite smoothie: `lime`, `desert`, `cherry`, `blueberry`, `dragonfruit`, `watermelon`
 - People have unique favorite sports: `tennis`, `volleyball`, `basketball`, `baseball`, `swimming`, `soccer`
 - Everyone has a unique favorite cigar: `blends`, `yellow monster`, `dunhill`, `pall mall`, `blue master`, `prince`
 - People have unique heights: `tall`, `very tall`, `average`, `super tall`, `very short`, `short`

## Clues:
1. Peter is the Prince smoker.
2. The person who smokes many unique blends is the person who enjoys mountain retreats.
3. The person who loves baseball is the person who is tall.
4. The person who goes on cultural tours is the person who drinks Blueberry smoothies.
5. The person who likes Cherry smoothies is not in the fourth house.
6. The person who loves soccer is the person who enjoys camping trips.
7. The person who is super tall is somewhere to the right of the person who likes going on cruises.
8. The person who drinks Blueberry smoothies is the person who is tall.
9. The person who is very tall is the Dunhill smoker.
10. The person who enjoys mountain retreats is Eric.
11. The person who is super tall is Alice.
12. Bob is the person who smokes Blue Master.
13. The person who loves volleyball is in the first house.
14. Peter and the person who drinks Lime smoothies are next to each other.
15. Carol is the person who likes Cherry smoothies.
16. Peter is the person who is short.
17. The person who smokes many unique blends is somewhere to the right of the Dragonfruit smoothie lover.
18. The person who loves swimming is Carol.
19. The person who is very short is the person who smokes Blue Master.
20. The person who loves tennis is somewhere to the left of the person who is very short.
21. The person who loves swimming is directly left of the person who enjoys camping trips.
22. The person who drinks Blueberry smoothies is not in the third house.
23. Alice is somewhere to the right of the person who drinks Lime smoothies.
24. The person who loves tennis is the person who is super tall.
25. The person who smokes Yellow Monster and the person who prefers city breaks are next to each other.
26. The Watermelon smoothie lover is the person who is very short.
27. The person who loves beach vacations is somewhere to the left of the person who is very tall.
 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 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`
 - Each person has a unique type of pet: `hamster`, `bird`, `fish`, `cat`, `dog`
 - Each person has an occupation: `doctor`, `engineer`, `lawyer`, `artist`, `teacher`
 - Each person lives in a unique style of house: `colonial`, `modern`, `ranch`, `craftsman`, `victorian`
 - Each person has a unique level of education: `high school`, `bachelor`, `master`, `doctorate`, `associate`

## Clues:
1. There are two houses between the person with a master's degree and the person in a modern-style house.
2. The person with an aquarium of fish is the person who is an artist.
3. The person who has a cat is the person who is an engineer.
4. The person with a bachelor's degree is the person with an aquarium of fish.
5. The person who has a cat is somewhere to the right of the person with an associate's degree.
6. The person with a master's degree is the person who is a lawyer.
7. The person who is a teacher is somewhere to the right of the person who keeps a pet bird.
8. The person with an associate's degree is the person living in a colonial-style house.
9. Peter is the person in a Craftsman-style house.
10. The person residing in a Victorian house is in the second house.
11. The person who is a teacher is Bob.
12. The person with an associate's degree is in the fourth house.
13. The person with a bachelor's degree is somewhere to the right of the person with a high school diploma.
14. There is one house between the person who owns a dog and the person in a ranch-style home.
15. Bob is somewhere to the right of Alice.
16. Eric is the person with a bachelor's degree.
 What is the value of attribute House for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Bob`, `Eric`
 - Each person has a favorite color: `white`, `blue`, `yellow`, `red`, `green`
 - Everyone has a unique favorite cigar: `blue master`, `dunhill`, `prince`, `pall mall`, `blends`
 - People have unique favorite book genres: `biography`, `mystery`, `romance`, `fantasy`, `science fiction`

## Clues:
1. There are two houses between the Prince smoker and the person who loves yellow.
2. The person who loves white is Bob.
3. The person who smokes Blue Master is directly left of Bob.
4. Alice is somewhere to the left of the person who loves science fiction books.
5. The person who loves biography books is not in the first house.
6. Arnold is somewhere to the left of the person who smokes Blue Master.
7. The person who loves mystery books and the person who loves biography books are next to each other.
8. The person who loves fantasy books is Bob.
9. The person whose favorite color is green is the person who smokes Blue Master.
10. The person partial to Pall Mall is in the fourth house.
11. The person who loves science fiction books is the Dunhill smoker.
12. The person whose favorite color is red is not in the fifth house.
13. Peter is directly left of the Dunhill smoker.
 What is the value of attribute BookGenre for the person whose attribute Cigar is pall mall? Please reason step by step, and put your final answer within \boxed{}

Answer: romance


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `samsung galaxy s21`
 - The people keep unique animals: `bird`, `horse`, `cat`
 - Each person has an occupation: `doctor`, `engineer`, `teacher`
 - Each person has a unique birthday month: `jan`, `sept`, `april`
 - Everyone has something unique for lunch: `grilled cheese`, `pizza`, `spaghetti`

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

Answer: pizza


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - People own unique car models: `ford f150`, `tesla model 3`, `toyota camry`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `city`
 - The people are of nationalities: `dane`, `swede`, `brit`
 - People have unique hair colors: `blonde`, `black`, `brown`
 - Each person lives in a unique style of house: `colonial`, `ranch`, `victorian`

## Clues:
1. The Swedish person is the person who owns a Ford F-150.
2. The person who has blonde hair is not in the second house.
3. The person who owns a Toyota Camry and Arnold are next to each other.
4. The person who has black hair is somewhere to the right of the person in a ranch-style home.
5. The person who has black hair is the Dane.
6. The person residing in a Victorian house is not in the second house.
7. Peter is directly left of the person who loves beach vacations.
8. The person living in a colonial-style house is the person who has blonde hair.
9. The person who has black hair is Arnold.
10. The Dane is the person who loves beach vacations.
11. Peter is the person who prefers city breaks.
 What is the value of attribute CarModel for the person whose attribute HairColor is brown? Please reason step by step, and put your final answer within \boxed{}

Answer: toyota camry


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`
 - They all have a unique favorite flower: `tulips`, `daffodils`, `lilies`, `roses`, `carnations`
 - Each mother is accompanied by their child: `Fred`, `Bella`, `Timothy`, `Meredith`, `Samantha`
 - Each person has a unique level of education: `high school`, `bachelor`, `doctorate`, `master`, `associate`
 - The people keep unique animals: `cat`, `fish`, `dog`, `horse`, `bird`

## Clues:
1. Peter is somewhere to the right of the dog owner.
2. There is one house between the person who loves a carnations arrangement and the person with a bachelor's degree.
3. There is one house between the person with a doctorate and the cat lover.
4. The person with an associate's degree is directly left of the person who loves the vase of tulips.
5. There is one house between the cat lover and the person who keeps horses.
6. The person's child is named Fred is somewhere to the left of the person with a bachelor's degree.
7. The person's child is named Meredith is not in the third house.
8. The person with a master's degree is the person's child is named Bella.
9. The person with a bachelor's degree is Bob.
10. The person who is the mother of Timothy is the person who loves a bouquet of daffodils.
11. The person's child is named Samantha is Bob.
12. Arnold is directly left of Peter.
13. The bird keeper is directly left of the person who loves a bouquet of daffodils.
14. The person who loves the rose bouquet is the bird keeper.
15. There is one house between the dog owner and the fish enthusiast.
16. The person who loves the vase of tulips is Eric.
 What is the value of attribute Children for the person whose attribute Animal is dog? Please reason step by step, and put your final answer within \boxed{}

Answer: Timothy


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`, `Arnold`, `Peter`, `Bob`
 - People use unique phone models: `samsung galaxy s21`, `huawei p50`, `oneplus 9`, `iphone 13`, `google pixel 6`
 - Everyone has a favorite smoothie: `desert`, `dragonfruit`, `watermelon`, `cherry`, `lime`
 - The mothers' names in different houses are unique: `Janelle`, `Holly`, `Penny`, `Aniya`, `Kailyn`
 - Each person has an occupation: `doctor`, `lawyer`, `engineer`, `teacher`, `artist`

## Clues:
1. The person whose mother's name is Kailyn is Peter.
2. The Dragonfruit smoothie lover is not in the fourth house.
3. The person who uses an iPhone 13 is the person who is an artist.
4. The person whose mother's name is Aniya is Arnold.
5. The person whose mother's name is Aniya is the Desert smoothie lover.
6. The person whose mother's name is Aniya is in the second house.
7. The person who drinks Lime smoothies is in the first house.
8. The person who uses a Huawei P50 is the person who is a doctor.
9. Eric is the person who drinks Lime smoothies.
10. There is one house between the person who is a teacher and Bob.
11. The person whose mother's name is Aniya is the person who is a lawyer.
12. The person who uses an iPhone 13 is the person who likes Cherry smoothies.
13. Arnold is the person who uses a OnePlus 9.
14. The person who is a doctor is The person whose mother's name is Penny.
15. Peter is the person who uses a Samsung Galaxy S21.
16. Bob is not in the third house.
17. The person whose mother's name is Holly is directly left of Peter.
 What is the value of attribute Smoothie for the person whose attribute Mother is Penny? Please reason step by step, and put your final answer within \boxed{}

Answer: lime


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Eric`
 - People have unique hair colors: `black`, `red`, `blonde`, `brown`
 - Each person has an occupation: `engineer`, `artist`, `doctor`, `teacher`
 - Each person has a unique favorite drink: `coffee`, `water`, `tea`, `milk`

## Clues:
1. The person who has brown hair is not in the second house.
2. The person who is a doctor is Peter.
3. The person who is a doctor is the one who only drinks water.
4. The person who is an artist is somewhere to the left of the person who is an engineer.
5. The person who likes milk is the person who has blonde hair.
6. The person who has red hair is Alice.
7. The person who is an artist and Arnold are next to each other.
8. The person who is an artist is in the third house.
9. The person who likes milk is somewhere to the left of the tea drinker.
10. Arnold is somewhere to the right of Eric.
11. The person who has black hair is the coffee drinker.
 What is the value of attribute Drink for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: tea


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

## Clues:
1. The British person is Bob.
2. Peter is the bird keeper.
3. The person who smokes Blue Master is the Swedish person.
4. The fish enthusiast is Bob.
5. The Dunhill smoker is the German.
6. The cat lover is the person partial to Pall Mall.
7. The person with a pet hamster is the person whose birthday is in March.
8. The person who keeps horses is somewhere to the right of the person whose birthday is in April.
9. The fish enthusiast is the person whose birthday is in January.
10. Alice is the dog owner.
11. The person whose birthday is in February is in the first house.
12. The person who owns a dog is somewhere to the left of Peter.
13. The Dane is the person who smokes many unique blends.
14. Peter and Arnold are next to each other.
15. The person with an aquarium of fish is Peter.
16. The Dunhill smoker is the person who has a cat.
17. The Swedish person is directly left of the person whose birthday is in January.
18. The person who owns a dog is in the third house.
 What is the value of attribute Birthday for the person whose attribute House is 5? 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`, `Alice`, `Peter`, `Arnold`
 - Each person has a unique level of education: `bachelor`, `high school`, `associate`, `master`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`, `Kailyn`
 - The people keep unique animals: `fish`, `horse`, `cat`, `bird`
 - Each person has a unique birthday month: `jan`, `feb`, `sept`, `april`
 - People have unique favorite music genres: `rock`, `pop`, `jazz`, `classical`

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

Answer: 4


Given a positive integer \( n \geq 2 \) and non-negative real numbers \( x_{1}, x_{2}, \cdots, x_{n} \) such that \( \sum_{i=1}^{n} x_{i}^{2} = 1 \). Let \( M = \max \left\{ x_{1}, x_{2}, \cdots, x_{n} \right\} \). Find the maximum and minimum values of
\[
S = n M^{2} + 2 \sum_{1 \leq i < j \leq n} x_{i} x_{j}
\]

Answer: n + \sqrt{n} - 1


There is a tunnel between locations $A$ and $B$. A car departs from $B$ towards $A$ at 08:16, and a truck departs from $A$ towards $B$ at 09:00. It is known that the truck and the car arrive at the two ends of the tunnel simultaneously, but the truck leaves the tunnel 2 minutes later than the car. If the car arrives at $A$ at 10:56, and the truck arrives at $B$ at 12:20, then the time they simultaneously arrive at the two ends of the tunnel is $\qquad$.

Answer: 10:00


5. The solution can only exist if $a \in\left\{-\frac{\pi}{12}\right\} \cup\left(0 ; \frac{\pi}{12}\right]$, since otherwise the left side of the equation is either undefined or strictly positive. When $a=-\frac{\pi}{12}$, the equation becomes $2|x-16|=0$. Therefore, when $a=-\frac{\pi}{12}$, $x=16$. If $a \in\left(0 ; \frac{\pi}{12}\right]$, then $2^{\frac{1}{\sin ^{2}(2 a)}}>16$, and $2^{-4 \operatorname{tg}(3 a)}<1$. Therefore, the minimum value of the function $f(x)=\left|x-2^{\frac{1}{\sin ^{2}(2 a)}}\right|+\left|x-2^{-4 \operatorname{tg}(3 a)}\right|$ is not less than 15. On the other hand, the absolute value of the expression $g(a)=a\left(a+\frac{\pi}{12}\right)^{2}\left(a-\frac{\pi}{12}\right)$ on the half-interval $\left(0 ; \frac{\pi}{12}\right]$ is certainly no more than $1:|g(a)|<a\left(a+\frac{\pi}{12}\right)^{3}<1$.
Therefore, when $a \in\left(0 ; \frac{\pi}{12}\right]$, there are no solutions.

Answer: 16


Binkov A, A:

In a football championship, 16 teams participated. Each team played against each of the others once, with 3 points awarded for a win, 1 point for a draw, and 0 points for a loss. We will call a team successful if it scored at least half of the maximum possible number of points. What is the maximum number of successful teams that could have been in the tournament?

Answer: 15


6. As shown in Figure 1, let $P$ be a point inside the convex quadrilateral $A B C D$. Draw perpendiculars from $P$ to $A B$, $B C$, $C D$, and $D A$, with the feet of the perpendiculars being $E$, $F$, $G$, and $H$, respectively. Given that $A H=3$, $H D=4$, $D G=1$, $G C=5$, $C F=6$, $F B=4$, and $B E-A E=1$. Then the perimeter of quadrilateral $A B C D$ is $\qquad$

Answer: 34


8. In Rt $\triangle A C B$, $\angle C=90^{\circ}, D, E$ are on $B C, C A$ respectively, and $\frac{B D}{A C}=\frac{A E}{C D}=K$, let $B E \cap A D=$ $O$, if $\angle B O D=75^{\circ}$, then $K=$ $\qquad$ .

Answer: 2 + \sqrt{3}


9. Let $A B C D$ be a regular tetrahedron with side length 1 . Let $X$ be the point in triangle $B C D$ such that $[X B C]=2[X B D]=4[X C D]$, where $[\varpi]$ denotes the area of figure $\varpi$. Let $Y$ lie on segment $A X$ such that $2 A Y=Y X$. Let $M$ be the midpoint of $B D$. Let $Z$ be a point on segment $A M$ such that the lines $Y Z$ and $B C$ intersect at some point. Find $\frac{A Z}{Z M}$.

Answer: \dfrac{4}{7}


Answer: \dfrac{1991009}{1991011}


Let $ n$ be an integer greater than $ 3.$ Points $ V_{1},V_{2},...,V_{n},$ with no three collinear, lie on a plane. Some of the segments $ V_{i}V_{j},$ with $ 1 \le i < j \le n,$ are constructed. Points  $ V_{i}$ and $ V_{j}$ are [i]neighbors[/i] if  $ V_{i}V_{j}$ is constructed. Initially, chess pieces $ C_{1},C_{2},...,C_{n}$ are placed at points $ V_{1},V_{2},...,V_{n}$ (not necessarily in that order) with exactly one piece at each point. In a move, one can choose some of the $ n$ chess pieces, and simultaneously relocate each of the chosen piece from its current position to one of its neighboring positions such that after the move, exactly one chess piece is at each point and no two chess pieces have exchanged their positions. A set of constructed segments is called [i]harmonic[/i] if for any initial positions of the chess pieces, each chess piece $ C_{i}(1 \le i \le n)$ is at the point $ V_{i}$ after a finite number of moves. Determine the minimum number of segments in a harmonic set.

Answer: n + 1


# Task 1.

Mom told Dima that he should eat 13 spoons of porridge. Dima told his friend that he had eaten 26 spoons of porridge. Each child, telling about Dima's feat, increased the amount of porridge Dima had eaten by 2 or 3 times. As a result, one of the children told Dima's mom about 33696 spoons of porridge. How many times in total did the children, including Dima, tell about the feat?

Answer: 9


11. The product of the two ten-digit numbers 1111111111 and 9999999999 has how many odd digits?

Answer: 10


9.9. On a plane, $N$ points are marked. Any three of them form a triangle, the angles of which in degrees are expressed by natural numbers. For what largest $N$ is this possible

$$
\text { (E. Bakayev) }
$$

Answer: 180


10. In the complex plane, take any three different roots of the equation $z^{100}-1=0$ as vertices to form a triangle, then the number of different acute triangles is $\qquad$ .

Answer: 39200


13.427 After passing through the filter, the liquid is evenly poured into a 40-bucket barrel and can be drained through a tap at the bottom of the barrel. If this tap is open, the inflow and outflow of the liquid are such that every 4 minutes, one bucket is reduced in the barrel. How long will it take for the filtered liquid to fill an empty barrel with the lower tap closed, if it is known that this will take 3 minutes less than the time it takes for the open lower tap to pass 66 buckets?

Answer: 96


9.4. In the country, there are 20 cities. An airline wants to organize two-way flights between them so that from any city, it is possible to reach any other city with no more than $k$ transfers. At the same time, the number of air routes from any city should not exceed four. What is the smallest $k$ for which this is possible?

## 9th grade

## Second day

Answer: 2


Answer: 15


## Task A-3.2.

Four spheres of radius $R$ lie on the base of a cone such that each touches two of the other spheres and the lateral surface of the cone. A fifth sphere of the same radius touches the first four spheres and the lateral surface of the cone. Determine the volume of this cone.

Answer: \dfrac{\pi R^3}{3} (25 + 22\sqrt{2})


10. The base of the pyramid $T A B C D$ is a rectangle $A B C D$. The height of the pyramid coincides with the lateral edge $T A$, and the lateral edge $T C$ is inclined to the base plane at an angle of $45^{\circ}$. The plane passing through the edge $T C$ and parallel to the diagonal of the base $B D$ forms an angle of $60^{\circ}$ with the base plane, and the distance between this plane and the diagonal $B D$ is $a$. What is the smallest area that the section of the pyramid by a plane passing through the diagonal of the base $A C$ can have?

Answer: \dfrac{8\sqrt{13}}{13}a^2


4. As shown in Figure 4, given $\angle A O M=60^{\circ}$, there is a point $B$ on ray $O M$ such that the lengths of $A B$ and $O B$ are both integers, thus $B$ is called an "olympic point". If $O A=8$, then the number of olympic points $B$ in Figure 4 is $\qquad$

Answer: 4


12. Let the foci of an ellipse be $F_{1}(-1,0)$ and $F_{2}(1,0)$ with eccentricity $e$. A parabola with vertex at $F_{1}$ and focus at $F_{2}$ intersects the ellipse at a point $P$. If $\frac{\left|P F_{1}\right|}{\left|P F_{2}\right|}=e$, then the value of $e$ is $\qquad$.

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


1. $n(n \geqslant=2)$ players participate in a competition that lasts for $k$ days, with each player scoring exactly $1,2, \cdots, n$ points each day (no two players have the same score). At the end of $k$ days, each player's total score is 26. Find all possible values of $(n, k)$.
(22nd Canadian Intermediate Mathematics Competition)

Answer: (3, 13)


## Task 2 - 170622

At a school sports festival, Christa, Doris, Elke, Franziska, and Gitta competed in the 60-meter sprint. When asked which places these five students would take at the finish line, some of the watching classmates made the following predictions:

(1) Christa will not come immediately before Elke at the finish line.

(2) Elke will either finish as the second-to-last or even achieve a better position.

(3) It is not true that Doris will not run faster than Gitta.

(4) Franziska will not finish in third place.

After the sprint was over, it was found that all five students had run different times and that all four predictions about the finish were false.

What is the actual order of the finish according to these statements?

Answer: Gitta, Doris, Franziska, Christa, Elke


Anthony writes the $(n+1)^2$ distinct positive integer divisors of $10^n$, each once, on a whiteboard. On a move, he may choose any two distinct numbers $a$ and $b$ on the board, erase them both, and write $\gcd(a, b)$ twice. Anthony keeps making moves until all of the numbers on the board are the same. Find the minimum possible number of moves Anthony could have made.

[i]Proposed by Andrew Wen[/i]

Answer: n(n +1)


B5. Three wheels are pressed against each other so that they do not slip when rotating. The circumferences of the wheels are 14, 10, and \(6 \, \text{cm}\) respectively. Each wheel has an arrow pointing straight down. The largest wheel is set in motion, and the other wheels rotate along with it. They stop at the first moment when all three arrows point down again. A whistle sounds whenever one of the arrows points straight up.
![](https://cdn.mathpix.com/cropped/2024_04_17_88f9acd303931df7c94dg-2.jpg?height=668&width=622&top_left_y=201&top_left_x=1228)
If two or three arrows point up at the same time, only one whistle sounds. How many whistles sound in total?

## C-Problems

For the C-Problems, the answer is not the only thing that matters; a reasoning must also be provided that shows your answer is correct. Each correctly worked-out C-Problem is worth 10 points. Partial solutions can also earn points. Therefore, write everything clearly and submit your scratch paper as well.

NOTE: Solve each C-Problem on a separate sheet of paper and submit the corresponding scratch paper per problem.

Answer: 57


A square piece of paper $A B C D$ is white on one side and grey on the other side. Initially, the paper is flat on a table with the grey side down. Point $E$ is on $B C$ so when the paper is folded along $A E, B$ lands on diagonal $A C$. Similarly, point $F$ is on $D C$ so that when the paper is folded along $A F, D$ lands on $A C$. After these folds, the resulting shape is kite $A E C F$. What fraction of the area of $A E C F$ is grey?

![](https://cdn.mathpix.com/cropped/2024_04_17_b6707dc71c8b7ccd30afg-4.jpg?height=509&width=1000&top_left_y=453&top_left_x=622)

Answer: \dfrac{\sqrt{2}}{2}


For a semicircle with diameter $AB = 2r$ and center $O$, the tangent at point $M$ intersects the tangents at points $A$ and $B$ at points $A'$ and $B'$, respectively. The perpendicular projection of $M$ onto the diameter $AB$ is $P$. Let $x = \frac{AP}{PB} (>0)$.

Rotate the trapezoid $AA'BB'$ and the triangle $A'OB'$ around $AB$, and denote the volumes of the resulting solids of revolution by $V_1$ and $V_2$, respectively. Express these volumes as functions of $r$ and $x$. Determine $x$ such that $\frac{V_2}{V_1} = m$, where $m$ is a given positive number. Discussion.

What is the maximum value that $m$ can take, and what position of $M$ corresponds to this value?

Answer: \dfrac{2}{3}


Problem 2. Determine the smallest natural number $n$ such that, no matter which $n$ vertices of a cube we color red, there exists a vertex of the cube that has all three adjacent vertices red.

Answer: 5


(Mid-term training Valbonne 2019 group B)

Let $n \geqslant 1$ be an integer. We have $n$ cubes with sides $1, 2, \ldots, n$. We want to stack them in a certain order, such that a cube of side $k$ can only be placed on a cube of side $l$ with $l \geqslant k-2$. How many such different stackings exist?

Answer: 2 \times 3^{n-2}


Problem 10.7. At one meal, Karlson can eat no more than 5 kg of jam. If he opens a new jar of jam, he must eat it completely during this meal. (Karlson will not open a new jar if he has to eat more than 5 kg of jam together with what he has just eaten.)

Little Boy has several jars of raspberry jam weighing a total of 50 kg, each weighing no more than 1 kg. In the least number of meals, how many meals will Karlson definitely be able to eat all the jam?

Answer: 12


10.1. The sequences $\left\{x_{n}\right\},\left\{y_{n}\right\}$ are defined by the conditions $x_{1}=11, y_{1}=7, x_{n+1}=3 x_{n}+2 y_{n}$, $y_{n+1}=4 x_{n}+3 y_{n}, n \in \mathbb{N}$. Find the remainder of the division of the number $y_{1855}^{2018}-2 x_{1855}^{2018}$ by 2018.

Answer: 1825


[Completion of a Tetrahedron to a Parallelepiped]

On the faces of a regular tetrahedron with edge $a$, equal regular pyramids are constructed. The plane angles at the vertices of these pyramids, opposite to the faces of the tetrahedron, are right angles. Consider the polyhedron formed by the tetrahedron and the constructed pyramids. How many faces does this polyhedron have? What is it called?

Answer: 6


1. Multiplying the first and fourth, and the second and third factors, we get the equation: $\left(x^{2}+5 x+4\right)\left(x^{2}+5 x+6\right)=360$.

By making the substitution $y=x^{2}+5 x+4$, we get $y^{2}+2 y-360=0$, from which $y_{1}=-20$, $y_{2}=18$. Therefore, we have the equations:

$x^{2}+5 x+24=0, x^{2}+5 x-14=0$

The first equation has no solutions, from the second we get: $x_{1}=-7, x_{2}=2$.

Answer: -7


4. A tangent is drawn from point $C$ to a circle with radius $2 \sqrt{5}$ and center at point $O$. Point $A$ is the point of tangency. Segment $C O$ intersects the circle at point $B$. A perpendicular is dropped from point $B$ to line $B C$ until it intersects line $A C$ at point $F$. Find the radius of the circumscribed circle around triangle $A B C$ if $B F=2$.

points)

#

Answer: \dfrac{\sqrt{30}}{2}


10. Given a sequence $\left\{a_{n}\right\}$ whose terms are all non-negative real numbers, and satisfies: for any integer $n \geqslant 2$, we have $a_{n+1}=$ $a_{n}-a_{n-1}+n$. If $a_{2} a_{2022}=1$, find the maximum possible value of $a_{1}$.

Answer: \dfrac{4051}{2025}


Let the coefficients of the equation \( x^{n} + a_{n-1}x^{n-1} + \cdots + a_{1}x + a_{0} = 0 \) be real numbers that satisfy the condition \( 0 < a_{0} \leq a_{1} \leq \cdots \leq a_{n-1} \leq 1 \). Given that \( \lambda \) is a complex root of this equation and \( |\lambda| \geq 1 \), prove that \( \lambda^{n+1} = 1 \).

Answer: \lambda^{n+1} = 1


Task 4. (20 points) To get from the first to the second building of the university, Sasha took a car-sharing vehicle, while Zhenya rented a scooter. Sasha and Zhenya left the first building for the second at the same time, and at the same time, Professor Vladimir Sergeyevich left the second building for the first in a car. The cars met after 3 minutes, and the scooter and the professor's car met 1 km from the first building. Find the speeds of the cars and the scooter, if the scooter takes 1.25 hours longer to cover a distance of 30 km compared to the car-sharing vehicle, and its speed is 4 times less than the speed of Professor Vladimir Sergeyevich's car. In your answer, indicate the speeds in $\frac{\kappa M}{4}$ in ascending order.

Answer: 15


1. A domino has a left end and a right end, each of a certain color. Alice has four dominos, colored red-red, red-blue, blue-red, and blue-blue. Find the number of ways to arrange the dominos in a row end-to-end such that adjacent ends have the same color. The dominos cannot be rotated.

Answer: 4


25. In the Garden School, 300 students are playing games in groups on the playground, with 3 people in each group. Among them, there are 27 groups with only 1 girl, and 42 groups with 2 or 3 boys, and the number of all-girl groups is the same as all-boy groups. Then, among these 300 students, there are $\qquad$ girls.

Answer: 158


Let \( n \) be a natural number, and let \( \mathcal{E} \) be a set of \( 2n+1 \) non-zero natural numbers. It is assumed that by removing any element from \( \mathcal{E} \), it is always possible to partition the remaining elements into two parts of \( n \) elements each, with the sums of the elements in each part being equal. Show that all the elements of \( \mathcal{E} \) are equal.

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


In an inscribed quadrilateral \(ABCD\), the degree measures of the angles are in the ratio \(\angle A : \angle B : \angle C = 2 : 3 : 4\). Find the length of \(AC\) if \(CD = 18\) and \(BC = 12\sqrt{3} - 9\).

Answer: 30


Let $ABCD$ be a cyclic quadrilateral, and suppose that $BC = CD = 2$. Let $I$ be the incenter of triangle $ABD$. If $AI = 2$ as well, find the minimum value of the length of diagonal $BD$.

Answer: 2\sqrt{3}


1. Find all real numbers $p$ such that the system

$$
\begin{aligned}
& x^{2}+(p-1) x+p \leqslant 0 \\
& x^{2}-(p-1) x+p \leqslant 0
\end{aligned}
$$

of inequalities has at least one solution $x \in \mathbb{R}$.

Answer: (-\infty, 0]


9. A museum has 7 exhibition rooms in a row. The guard starts by walking into the first room, then to the second... and after reaching the seventh room, he turns back, walking to the sixth room, then to the fifth... Each time he enters a room, he toggles the light switch in that room. If all 7 rooms are initially lit, how many rooms will still have their lights on after he has walked into 100 rooms.

Answer: 3


5. a Swiss cross consists of five unit squares, one in the center and four on the sides. Determine the smallest natural number $n$ with the following property: For every $n$ points inside or on the edge of a Swiss cross, there are always two whose distance is less than 1.

Answer: 13


Consider the sequence $a_{1}, a_{2}, a_{3}, \ldots$ defined by $a_{1}=9$ and

$$
a_{n+1}=\frac{(n+5) a_{n}+22}{n+3}
$$

for $n \geqslant 1$.

Find all natural numbers $n$ for which $a_{n}$ is a perfect square of an integer.

Answer: 1


Exercise 16. Suzanne multiplies two integers whose difference is 5, and Martin multiplies two integers whose difference is 8. They get the same result, which we denote as $C$. What are the possible values of $C$?

Answer: 0


## Task 1 - 250711

In a bag, there is $1 \mathrm{~kg}$ of sugar. Using a balance scale with two weighing pans (each large enough for $1 \mathrm{~kg}$ of loose sugar) and exactly one $50 \mathrm{~g}$ weight, $300 \mathrm{~g}$ of sugar should be weighed out.

Show that this is possible with only three weighings!

Answer: 300


Given point \( F(1,0) \), line \( l: x=-1 \), and line \( l' \perp l \) at point \( P \). The perpendicular bisector of segment \( PF \) intersects \( l' \) at point \( Q \).
1. Find the equation of the locus \( C \) of point \( Q \);
2. Given point \( H(1,2) \), a line passing through \( F \) and not perpendicular to the \( x \)-axis intersects curve \( C \) at points \( A \) and \( B \). Lines \( AH \) and \( BH \) intersect \( l \) at points \( M \) and \( N \), respectively. Prove that the circle with diameter \( MN \) passes through a fixed point.

Answer: y^2 = 4x


A $2.9 \, \text{cm}$ diameter measuring cylinder is filled with water to a height of $4 \, \text{cm}$, then dense copper cubes with an edge length of $2 \, \text{cm}$ are placed into the cylinder. What is the maximum number of cubes that the water in the cylinder can submerge?

Answer: 5


The parabola defined by the equation \( y^2 = 2px \) is bounded by the line \( x = a \). Inscribe rectangles within this parabolic segment such that their midlines coincide with the axis of the parabola. Which rectangle will have the maximum area?

Answer: \frac{2a}{3}


Ha az előbbi feladatban azt is tudni akarjuk, hogy a hamis pénz könnyebb-e, vagy nehezebb a többinél, hány pénzdarab a legtöbb, amennyi közül még három mérés elegendő ennek a kérdésnek az eldöntéséhez?

If in the previous task we also want to know whether the counterfeit coin is lighter or heavier than the others, how many coins is the maximum number from which three measurements are still sufficient to determine the answer to this question?

Answer: 12


## 296. Math Puzzle $1 / 90$

Sabine values physical exercise a lot, and so her daily commute to school by bike means more to her than just fulfilling a necessary "evil."

On her route, she has to cross a $100 \mathrm{~m}$ long bridge. One day, she meets a classmate on the bridge after $40 \mathrm{~m}$, who was cycling towards her at the same speed as she was. "Forgot my sports gear," Sabine thought.

A car driver traveling at 70 km/h met one of the girls at the beginning of the bridge and the other at the end of the bridge.

At what speed were the two students cycling?

Answer: 14


$ABCD$ - quadrilateral inscribed in circle, and $AB=BC,AD=3DC$ . Point $R$ is on the $BD$ and $DR=2RB$. Point $Q$ is on $AR$ and $\angle ADQ = \angle BDQ$.  Also $\angle ABQ + \angle CBD = \angle QBD$ . $AB$ intersect line $DQ$ in point $P$.
Find $\angle APD$

Answer: 90^\circ


A square-based frustum has a base edge and every side edge of 4. The edge of its top face is 2. What is the maximum distance between two vertices of the frustum?

Answer: 4\sqrt{2}