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`
 - The people keep unique animals: `cat`, `fish`, `bird`, `dog`, `horse`
 - Each person has a unique birthday month: `mar`, `jan`, `feb`, `april`, `sept`

## Clues:
1. The fish enthusiast and Arnold are next to each other.
2. The person whose birthday is in February is not in the fourth house.
3. The person who keeps horses is in the fourth house.
4. The person whose birthday is in March is somewhere to the left of the dog owner.
5. Peter is the person whose birthday is in September.
6. The dog owner is not in the fifth house.
7. The bird keeper is in the third house.
8. Peter and the person whose birthday is in February are next to each other.
9. Bob is not in the third house.
10. The dog owner is somewhere to the right of Alice.
11. The person whose birthday is in January is the bird keeper.
 What is the value of attribute Animal for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{}

Answer: horse


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 Color for the person whose attribute Children is Bella? Please reason step by step, and put your final answer within \boxed{}

Answer: white


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`, `Alice`
 - The mothers' names in different houses are unique: `Holly`, `Kailyn`, `Aniya`, `Janelle`
 - People have unique favorite book genres: `science fiction`, `mystery`, `fantasy`, `romance`
 - Each person has a favorite color: `green`, `red`, `white`, `yellow`
 - Each person prefers a unique type of vacation: `mountain`, `cruise`, `beach`, `city`

## Clues:
1. The person who loves romance books and the person who likes going on cruises are next to each other.
2. Arnold is The person whose mother's name is Janelle.
3. The person who prefers city breaks is the person who loves yellow.
4. The person who loves science fiction books is directly left of Peter.
5. The person who loves science fiction books is somewhere to the left of The person whose mother's name is Kailyn.
6. The person who loves beach vacations is not in the first house.
7. There is one house between Alice and the person who likes going on cruises.
8. The person who loves mystery books is the person who loves white.
9. Peter is the person who prefers city breaks.
10. The person whose favorite color is green is in the third house.
11. The person whose mother's name is Aniya is Peter.
12. The person whose favorite color is green is The person whose mother's name is Janelle.
 What is the value of attribute House for the person whose attribute BookGenre is romance? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


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

## Clues:
1. The person who is very short is the person who loves swimming.
2. The person who loves baseball is the person who is short.
3. Alice is the person who owns a Honda Civic.
4. The person who loves baseball is somewhere to the right of Bob.
5. The person who loves soccer is somewhere to the right of Carol.
6. There are two houses between the person who loves volleyball and the person who has an average height.
7. Arnold is in the third house.
8. The person who loves tennis is not in the fourth house.
9. The person who owns a Chevrolet Silverado and the person who owns a Toyota Camry are next to each other.
10. Alice is directly left of the person who is super tall.
11. The person who owns a Ford F-150 is the person who loves swimming.
12. Eric is in the fourth house.
13. The person who is short is the person who owns a Chevrolet Silverado.
14. The person who loves baseball is in the fifth house.
15. Eric is the person who owns a BMW 3 Series.
16. The person who loves swimming is somewhere to the left of the person who is very tall.
17. The person who has an average height is the person who owns a BMW 3 Series.
 What is the value of attribute House for the person whose attribute Height is tall? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


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

## Clues:
1. There is one house between the person's child is named Bella and the person's child is named Meredith.
2. The person who likes Cherry smoothies is the person who loves beach vacations.
3. The person who prefers city breaks is Arnold.
4. The person whose favorite color is red is the person's child is named Fred.
5. The person's child is named Fred is the Watermelon smoothie lover.
6. Eric is not in the first house.
7. The person's child is named Meredith is the person who loves white.
8. The person who loves white is the Desert smoothie lover.
9. The Desert smoothie lover is Eric.
 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: Fred


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

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

Answer: lilies


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

Answer: 1


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Peter`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `cherry`, `dragonfruit`
 - Each person lives in a unique style of house: `craftsman`, `colonial`, `ranch`, `victorian`
 - Each person has a unique hobby: `cooking`, `photography`, `gardening`, `painting`
 - Each person has a unique favorite drink: `coffee`, `tea`, `milk`, `water`

## Clues:
1. The person in a Craftsman-style house is the Dragonfruit smoothie lover.
2. The tea drinker is the person who loves a bouquet of daffodils.
3. The Desert smoothie lover is the person living in a colonial-style house.
4. There is one house between the Watermelon smoothie lover and the person who loves a bouquet of daffodils.
5. The person who likes Cherry smoothies is directly left of the photography enthusiast.
6. The person who likes milk is the person who loves cooking.
7. The tea drinker is Arnold.
8. The person who loves the boquet of lilies is the person living in a colonial-style house.
9. The person who loves the boquet of lilies is not in the third house.
10. The person living in a colonial-style house is directly left of the person who paints as a hobby.
11. The person in a ranch-style home is the person who likes milk.
12. The person who loves the rose bouquet is the one who only drinks water.
13. Eric is somewhere to the left of the Dragonfruit smoothie lover.
14. The person who loves the rose bouquet is somewhere to the right of the person who loves a carnations arrangement.
15. Peter is the person who loves the boquet of lilies.
 What is the value of attribute Drink for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: water


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

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

Answer: iris


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

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

Answer: 2


There are 2 houses, numbered 1 to 2 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`
 - Each person has a unique type of pet: `cat`, `dog`
 - Each person has a unique favorite drink: `tea`, `water`
 - People have unique favorite music genres: `rock`, `pop`
 - The people keep unique animals: `horse`, `cat`
 - People own unique car models: `ford f150`, `tesla model 3`

## Clues:
1. The tea drinker is the person who owns a Tesla Model 3.
2. The person who owns a dog is the cat lover.
3. Arnold is the tea drinker.
4. The person who loves pop music is directly left of the person who has a cat.
5. The person who loves rock music is the tea drinker.
 What is the value of attribute Drink for the person whose attribute MusicGenre is pop? Please reason step by step, and put your final answer within \boxed{}

Answer: water


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

## Clues:
1. Arnold is not in the first house.
2. Alice is directly left of the person's child is named Samantha.
3. Peter is directly left of Alice.
4. Arnold is the person's child is named Bella.
5. Eric is not in the third house.
6. The person's child is named Meredith is in the fifth house.
7. The person who is the mother of Timothy is not in the first house.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Arnold`, `Eric`, `Alice`
 - Each person has a unique level of education: `bachelor`, `high school`, `master`, `associate`, `doctorate`
 - Each person lives in a unique style of house: `craftsman`, `victorian`, `colonial`, `modern`, `ranch`

## Clues:
1. The person in a modern-style house is somewhere to the left of the person in a ranch-style home.
2. Eric is the person residing in a Victorian house.
3. The person in a ranch-style home is not in the fourth house.
4. Peter is in the first house.
5. The person with a doctorate is in the first house.
6. The person in a modern-style house is in the second house.
7. The person with a high school diploma is the person in a Craftsman-style house.
8. The person residing in a Victorian house is directly left of Bob.
9. The person residing in a Victorian house is the person with a bachelor's degree.
10. The person with an associate's degree is somewhere to the left of Arnold.
11. The person with a high school diploma is in the fifth house.
 What is the value of attribute House for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


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

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

Answer: 1


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Peter`
 - The mothers' names in different houses are unique: `Holly`, `Kailyn`, `Aniya`, `Janelle`
 - Each person has a unique birthday month: `april`, `feb`, `sept`, `jan`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `victorian`, `craftsman`

## Clues:
1. Peter is the person whose birthday is in February.
2. The person whose birthday is in February is not in the fourth house.
3. The person whose birthday is in January is somewhere to the right of the person in a ranch-style home.
4. The person whose birthday is in January is directly left of Eric.
5. The person in a ranch-style home is the person whose birthday is in April.
6. The person whose mother's name is Janelle is the person whose birthday is in September.
7. The person in a ranch-style home and The person whose mother's name is Kailyn are next to each other.
8. Arnold is The person whose mother's name is Holly.
9. The person whose mother's name is Kailyn is directly left of the person residing in a Victorian house.
10. The person whose birthday is in January is the person in a Craftsman-style house.
 What is the value of attribute Mother for the person whose attribute HouseStyle is victorian? Please reason step by step, and put your final answer within \boxed{}

Answer: Janelle


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Carol`, `Bob`, `Arnold`, `Peter`
 - People have unique favorite music genres: `hip hop`, `jazz`, `country`, `classical`, `rock`, `pop`
 - People use unique phone models: `iphone 13`, `oneplus 9`, `xiaomi mi 11`, `huawei p50`, `samsung galaxy s21`, `google pixel 6`
 - People have unique favorite book genres: `biography`, `historical fiction`, `fantasy`, `romance`, `science fiction`, `mystery`

## Clues:
1. Arnold is not in the sixth house.
2. There are two houses between the person who loves country music and the person who loves hip-hop music.
3. The person who loves romance books is Carol.
4. The person who loves biography books is somewhere to the left of the person who loves rock music.
5. The person who loves mystery books is somewhere to the right of Peter.
6. The person who loves romance books is the person who loves classical music.
7. The person who loves mystery books is directly left of the person who loves fantasy books.
8. Bob is the person who loves pop music.
9. The person who loves jazz music is somewhere to the left of the person who loves pop music.
10. Arnold is somewhere to the right of the person who loves hip-hop music.
11. The person who loves hip-hop music and the person who uses a OnePlus 9 are next to each other.
12. The person who loves science fiction books is the person who uses a Xiaomi Mi 11.
13. The person who loves classical music is in the second house.
14. The person who uses a Xiaomi Mi 11 is directly left of Eric.
15. The person who loves historical fiction books is somewhere to the left of the person who loves classical music.
16. The person who uses a Huawei P50 is Peter.
17. The person who uses a Samsung Galaxy S21 is in the fourth house.
18. The person who uses a Samsung Galaxy S21 is somewhere to the right of the person who uses an iPhone 13.
 What is the value of attribute MusicGenre for the person whose attribute BookGenre is fantasy? Please reason step by step, and put your final answer within \boxed{}

Answer: pop


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Arnold`, `Eric`
 - Each person lives in a unique style of house: `victorian`, `ranch`, `craftsman`, `colonial`
 - People have unique favorite music genres: `pop`, `classical`, `rock`, `jazz`
 - They all have a unique favorite flower: `roses`, `lilies`, `carnations`, `daffodils`
 - Each person prefers a unique type of vacation: `city`, `beach`, `mountain`, `cruise`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`

## Clues:
1. The person who loves classical music is Peter.
2. The person in a ranch-style home is not in the fourth house.
3. The person who loves a bouquet of daffodils is the person living in a colonial-style house.
4. The person living in a colonial-style house is the person who loves white.
5. The person who prefers city breaks and the person whose favorite color is red are next to each other.
6. The person who loves the rose bouquet is not in the fourth house.
7. The person who likes going on cruises is in the second house.
8. The person who prefers city breaks is the person who loves rock music.
9. The person who loves pop music is Alice.
10. The person who loves the rose bouquet is the person who prefers city breaks.
11. Eric is the person who enjoys mountain retreats.
12. The person who enjoys mountain retreats is directly left of the person who loves a bouquet of daffodils.
13. The person who loves yellow is somewhere to the left of the person who likes going on cruises.
14. The person who enjoys mountain retreats is the person residing in a Victorian house.
15. The person who loves classical music is the person who likes going on cruises.
16. Peter is somewhere to the right of the person who loves the boquet of lilies.
 What is the value of attribute Color for the person whose attribute Flower is daffodils? Please reason step by step, and put your final answer within \boxed{}

Answer: white


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - Each mother is accompanied by their child: `Bella`, `Meredith`, `Fred`
 - People use unique phone models: `samsung galaxy s21`, `iphone 13`, `google pixel 6`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`
 - Everyone has a unique favorite cigar: `blue master`, `prince`, `pall mall`
 - People have unique heights: `short`, `very short`, `average`

## Clues:
1. Eric is directly left of the Prince smoker.
2. The person who has an average height is somewhere to the right of Peter.
3. The person who uses a Samsung Galaxy S21 is in the second house.
4. The person who smokes Blue Master is in the third house.
5. The person who uses a Google Pixel 6 is Eric.
6. Arnold is somewhere to the right of the person who loves science fiction books.
7. The person who uses an iPhone 13 is the person's child is named Bella.
8. The person who is short is the Prince smoker.
9. The person who loves mystery books is the person who uses a Google Pixel 6.
10. The person's child is named Meredith and the person who is short are next to each other.
 What is the value of attribute BookGenre for the person whose attribute Children is Bella? Please reason step by step, and put your final answer within \boxed{}

Answer: romance


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`, `Alice`
 - The mothers' names in different houses are unique: `Holly`, `Kailyn`, `Aniya`, `Janelle`
 - People have unique favorite book genres: `science fiction`, `mystery`, `fantasy`, `romance`
 - Each person has a favorite color: `green`, `red`, `white`, `yellow`
 - Each person prefers a unique type of vacation: `mountain`, `cruise`, `beach`, `city`

## Clues:
1. The person who loves romance books and the person who likes going on cruises are next to each other.
2. Arnold is The person whose mother's name is Janelle.
3. The person who prefers city breaks is the person who loves yellow.
4. The person who loves science fiction books is directly left of Peter.
5. The person who loves science fiction books is somewhere to the left of The person whose mother's name is Kailyn.
6. The person who loves beach vacations is not in the first house.
7. There is one house between Alice and the person who likes going on cruises.
8. The person who loves mystery books is the person who loves white.
9. Peter is the person who prefers city breaks.
10. The person whose favorite color is green is in the third house.
11. The person whose mother's name is Aniya is Peter.
12. The person whose favorite color is green is The person whose mother's name is Janelle.
 What is the value of attribute Mother for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: Janelle


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

## Clues:
1. The coffee drinker is the person who likes Cherry smoothies.
2. The Watermelon smoothie lover is the person who loves rock music.
3. The person who loves yellow is the person who loves hip-hop music.
4. Eric is somewhere to the right of the person who loves classical music.
5. The one who only drinks water is the Prince smoker.
6. The person who smokes Blue Master is Eric.
7. The Dunhill smoker is Alice.
8. The root beer lover is the person whose favorite color is red.
9. The Dunhill smoker is somewhere to the left of the Prince smoker.
10. The Dragonfruit smoothie lover is directly left of the person who likes Cherry smoothies.
11. Arnold is the person whose favorite color is green.
12. Eric is the tea drinker.
13. The Dunhill smoker is the person who loves white.
14. The person who loves classical music and the person who drinks Lime smoothies are next to each other.
15. The person who drinks Lime smoothies is Eric.
16. The person who loves rock music is directly left of the root beer lover.
17. The person who loves pop music is the root beer lover.
18. The person whose favorite color is green is not in the third house.
19. The one who only drinks water is the person who loves blue.
20. The person who loves rock music is the person who loves white.
21. Peter is not in the third house.
22. The person who likes Cherry smoothies is somewhere to the right of the person partial to Pall Mall.
 What is the value of attribute House for the person whose attribute Smoothie is cherry? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`
 - Each person prefers a unique type of vacation: `mountain`, `beach`, `city`
 - Each person has a unique hobby: `gardening`, `cooking`, `photography`
 - People have unique favorite sports: `tennis`, `soccer`, `basketball`
 - They all have a unique favorite flower: `carnations`, `daffodils`, `lilies`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`

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

Answer: 1


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Eric`, `Peter`
 - Each person has a unique hobby: `cooking`, `painting`, `photography`, `gardening`
 - Each person has a unique birthday month: `april`, `jan`, `sept`, `feb`
 - Each person has a unique level of education: `master`, `bachelor`, `associate`, `high school`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `desert`, `dragonfruit`

## Clues:
1. The Desert smoothie lover is the person whose birthday is in January.
2. Eric is the person with a bachelor's degree.
3. The person whose birthday is in January is the person with a bachelor's degree.
4. The person with a high school diploma is in the third house.
5. The Watermelon smoothie lover is not in the third house.
6. The person with an associate's degree is Arnold.
7. The person with a master's degree is the person who paints as a hobby.
8. There is one house between the Dragonfruit smoothie lover and the person whose birthday is in September.
9. The person with a high school diploma is the person whose birthday is in September.
10. The person who loves cooking is Alice.
11. The person whose birthday is in April and the person who enjoys gardening are next to each other.
12. The person who paints as a hobby is the person whose birthday is in February.
 What is the value of attribute Smoothie for the person whose attribute Birthday is feb? Please reason step by step, and put your final answer within \boxed{}

Answer: watermelon


In $\triangle{ABC}$, $\angle{C} = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. The original answer for the perimeter of the triangle is in the form $k + m\sqrt{2}$. Please provide the value of $k + m$.

Answer: 24


Tetrahedron \(ABCD\) has \(AB=5\), \(AC=3\), \(BC=4\), \(BD=4\), \(AD=3\), and \(CD=\tfrac{12}{5}\sqrt{2}\). Find the volume of the tetrahedron. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

Answer: 29


Let $n$ be a positive integer greater than $4$ such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. Find the sum of the digits of $s$.

Answer: 8


A rectangular box measures $a \times b \times c$, where $a$, $b$, and $c$ are integers and $1\leq a \leq b \leq c$. The volume and the surface area of the box are numerically equal. Find the number of ordered triples $(a,b,c)$ that are possible.

Answer: 10


A collection of circles in the upper half-plane, all tangent to the \(x\)-axis, is constructed in layers as follows. Layer \(L_0\) consists of two circles of radii \(70^2\) and \(73^2\) that are externally tangent. For \(k \ge 1\), the circles in \(\bigcup\limits_{j=0}^{k-1}L_j\) are ordered according to their points of tangency with the \(x\)-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer \(L_k\) consists of the \(2^{k-1}\) circles constructed in this way. Let \(S=\bigcup\limits_{j=0}^{6}L_j\), and for every circle \(C\) denote by \(r(C)\) its radius. Find \(\sum\limits_{C\in S} \frac{1}{\sqrt{r(C)}}\). The original answer is in the form of \(\frac{k}{m}\), where the fraction is in its simplest form. Please give the value of k + m.

Answer: 157


A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$ she is exactly $a \sqrt{b} + c \sqrt{d}$ inches away from $P_0$, where $a$, $b$, $c$, and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. Find the value of $a+b+c+d$.

Answer: 2024


Square $ABCD$ and rectangle $BEFG$ are placed as shown in the figure. $AG = CE = 2$ cm. What is the difference in the area between square $ABCD$ and rectangle $BEFG$ in square centimeters?

Answer: 4


9. Given the ellipse $\Gamma: \frac{x^{2}}{9}+\frac{y^{2}}{5}=1$, a line passing through the left focus $F(-2,0)$ of the ellipse $\Gamma$ with a slope of $k_{1}\left(k_{1} \notin\{0\right.$, $\infty\})$ intersects the ellipse $\Gamma$ at points $A$ and $B$. Let point $R(1,0)$, and extend $A R$ and $B R$ to intersect the ellipse $\Gamma$ at points $C$ and $D$ respectively. The slope of line $C D$ is $k_{2}$. Write $\frac{k_{1}^{2}}{k_{2}^{2}}$ as a reduced fraction $\frac{a}{b}$ (where $a$ and $b$ are coprime positive integers). Then $a^{2}+b=$

Answer: 305


12. $A B C$ is a triangle with $A C=B C$ and $A \widehat{C} B<60^{\circ}$. Let $A^{\prime}$ and $B^{\prime}$ be points on sides $B C$ and $A C$ respectively such that $A A^{\prime}=B B^{\prime}=A B$. Let $C^{\prime}$ be the intersection of $A A^{\prime}$ and $B B^{\prime}$. Given that $A C^{\prime}=A B^{\prime}$ and $B C^{\prime}=B A^{\prime}$, what is the measure in degrees of the angle $A \widehat{C} B$?

Answer: 36


Three, as shown, in the triangular prism $A B C A_{1} B_{1} C_{1}$, all nine edges are equal to 1, and $\angle A_{1} A B$ $=\angle A_{1} A C$ $=\angle B A C$. Point $P$ is on the diagonal $A_{1} B$ of the side face $A_{1} A B B_{1}$, with $A_{1} P=\frac{\sqrt{3}}{3}$. Connect $P C_{1}$. Find the degree measure of the angle formed by the skew lines $P C_{1}$ and $A C$.

Answer: 30


In an acute triangle \( \triangle ABC \), \( AE \) and \( AF \) are the trisectors of \( \angle A \), and intersect the circumcircle of \( \triangle ABC \) at points \( M \) and \( N \), respectively. Points \( P \) and \( R \) are selected on sides \( AB \) and \( AC \) such that \( \angle PEA = \angle B \) and \( \angle AER = \angle C \). Let \( PR \) intersect \( AE \) at point \( L \), and \( LN \) intersect \( BC \) at point \( D \). Prove:
$$
\frac{1}{MN} + \frac{1}{EF} = \frac{1}{ED}.
$$

Answer: \frac{1}{MN} + \frac{1}{EF} = \frac{1}{ED}


51. Put 2017 balls into 10 boxes, requiring the number of balls in each box to contain the digit 7. Does such an arrangement exist?

Answer: Yes


60. Given $\triangle A B C$. On the ray $B A$, take a point $D$ such that $|B D|=|B A|+|A C|$. Let $K$ and $M$ be two points on the rays $B A$ and $B C$ respectively such that the area of $\triangle B D M$ is equal to the area of $\triangle B C K$. Find $\widehat{B K M}$, if $\widehat{B A C}=\alpha$.

Answer: \dfrac{\alpha}{2}


PROBLEM 3. Consider the sequence with the general term

$$
I_{n}=\int_{0}^{\frac{\pi}{4}} e^{n x}\left(\tan^{n-1} x+\tan^{n} x+\tan^{n+1} x\right) d x, n \geq 1
$$

Calculate $\lim _{n \rightarrow \infty} n\left(\sqrt[n^{2}]{n I_{n}}-1\right)$.

Answer: \dfrac{\pi}{4}


Let $A$, $B$, $C$, $D$ be four points on a circle in that order. Also, $AB=3$, $BC=5$, $CD=6$, and $DA=4$. Let diagonals $AC$ and $BD$ intersect at $P$. Compute $\frac{AP}{CP}$.

Answer: \dfrac{2}{5}


20.2.3 $\star \star$ Let $A=\{1,2, \cdots, 2002\}, M=\{1001,2003,3005\}$. For any non-empty subset $B$ of $A$, if the sum of any two numbers in $B$ does not belong to $M$, then $B$ is called an $M$-free set. If $A=$ $A_{1} \cup A_{2}, A_{1} \cap A_{2}=\varnothing$, and both $A_{1}$ and $A_{2}$ are $M$-free sets, then the ordered pair $\left(A_{1}, A_{2}\right)$ is called an $M$-partition of $A$. Find the number of all $M$-partitions of $A$.

Answer: 2^{501}


In triangle $P Q R$, the midpoint of side $P Q$ is $A$, the midpoint of $Q R$ is $B$, and the midpoint of $R P$ is $C$. We enlarge triangle $A B C$ from its centroid $S$ by a factor of $k$, where $1<k<4$. The sides of the enlarged triangle intersect the sides of triangle $P Q R$ at points $D_{1} D_{2}, E_{1} E_{2}$, and $F_{1} F_{2}$, respectively. For what value of $k$ will the area of the hexagon $D_{1} D_{2} \ldots F_{1} F_{2}$ be twice the area of triangle $A B C$?

Answer: 4 - \sqrt{6}


A positive integer $n$ has the property that there are three positive integers $x, y, z$ such that $\text{lcm}(x, y) = 180$, $\text{lcm}(x, z) = 900$, and $\text{lcm}(y, z) = n$, where $\text{lcm}$ denotes the lowest common multiple. Determine the number of positive integers $n$ with this property.

Answer: 9


5. The segment $AC = a$ with its internal point $B$ is divided in the ratio $3:2$. Squares $ABDE$ and $CBFG$ are constructed on segments $AB$ and $BC$, respectively, on different sides relative to segment $AC$. Let $O$ and $O'$ be the intersections of the diagonals of these squares. Determine the ratio of the area of quadrilateral $OO'C D$ to the area of the square with side length $AC$.

Answer: \dfrac{1}{4}


2. How many 100-digit natural numbers are there in whose decimal representation only odd digits appear, such that the difference between any two adjacent digits is equal to $2$?

Answer: 8 \cdot 3^{49}


There are 2 houses, numbered 1 to 2 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`
 - The people keep unique animals: `cat`, `horse`
 - People use unique phone models: `samsung galaxy s21`, `iphone 13`
 - Each person has a unique type of pet: `cat`, `dog`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`

## Clues:
1. Arnold and the person who uses a Samsung Galaxy S21 are next to each other.
2. The person partial to Pall Mall and Eric are next to each other.
3. The person who owns a dog is directly left of Eric.
4. The cat lover and the person who uses a Samsung Galaxy S21 are next to each other.
 What is the value of attribute PhoneModel for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: samsung galaxy s21


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`
 - Everyone has something unique for lunch: `spaghetti`, `pizza`, `stew`, `grilled cheese`
 - Each person has a unique favorite drink: `coffee`, `milk`, `tea`, `water`
 - Everyone has a favorite smoothie: `cherry`, `desert`, `dragonfruit`, `watermelon`

## Clues:
1. The Dragonfruit smoothie lover is the person who likes milk.
2. The Watermelon smoothie lover is not in the fourth house.
3. The person who loves eating grilled cheese is directly left of Peter.
4. The person who loves the stew is Alice.
5. The person who is a pizza lover is the Dragonfruit smoothie lover.
6. The tea drinker is directly left of the coffee drinker.
7. The person who loves the stew is the Watermelon smoothie lover.
8. The person who likes Cherry smoothies is Eric.
9. Eric is in the first house.
10. Alice is somewhere to the right of the person who is a pizza lover.
 What is the value of attribute Food for the person whose attribute Drink is milk? Please reason step by step, and put your final answer within \boxed{}

Answer: pizza


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`, `Eric`, `Bob`
 - The mothers' names in different houses are unique: `Aniya`, `Penny`, `Holly`, `Kailyn`, `Janelle`
 - The people are of nationalities: `swede`, `brit`, `dane`, `norwegian`, `german`
 - Each person has a unique birthday month: `mar`, `april`, `feb`, `sept`, `jan`

## Clues:
1. The Dane is the person whose birthday is in January.
2. Alice is the British person.
3. Eric is not in the second house.
4. The person whose birthday is in January is somewhere to the left of Arnold.
5. The British person is somewhere to the left of The person whose mother's name is Janelle.
6. Peter is the German.
7. The person whose mother's name is Aniya is Peter.
8. The person whose birthday is in March is The person whose mother's name is Janelle.
9. The person whose mother's name is Kailyn is the Dane.
10. The British person is in the third house.
11. The person whose birthday is in January is somewhere to the left of The person whose mother's name is Penny.
12. The British person is directly left of the person whose birthday is in February.
13. The Dane is somewhere to the left of the Norwegian.
14. Eric is the person whose birthday is in April.
 What is the value of attribute Nationality for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: german


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`, `Peter`, `Alice`, `Bob`, `Eric`
 - Each person has a unique level of education: `associate`, `bachelor`, `high school`, `trade school`, `doctorate`, `master`

## Clues:
1. The person with a doctorate is Bob.
2. The person who attended trade school is not in the third house.
3. Eric is somewhere to the right of the person who attended trade school.
4. Alice is not in the first house.
5. Alice is somewhere to the left of the person with an associate's degree.
6. The person with a master's degree is Carol.
7. The person with a doctorate is not in the first house.
8. Bob is somewhere to the left of the person who attended trade school.
9. Arnold is in the fifth house.
10. There is one house between Bob and the person with a bachelor's degree.
11. The person with a bachelor's degree is Eric.
 What is the value of attribute House for the person whose attribute Education is bachelor? Please reason step by step, and put your final answer within \boxed{}

Answer: 6


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Bob`, `Peter`, `Alice`
 - Everyone has something unique for lunch: `spaghetti`, `stew`, `grilled cheese`, `stir fry`, `pizza`
 - Each person has a favorite color: `red`, `green`, `white`, `blue`, `yellow`

## Clues:
1. The person who loves blue is somewhere to the left of the person who is a pizza lover.
2. Bob is somewhere to the left of the person who loves white.
3. Alice is directly left of the person who loves the stew.
4. The person who loves stir fry is in the fourth house.
5. The person who loves white is Peter.
6. The person who loves the spaghetti eater is Bob.
7. The person whose favorite color is green is Alice.
8. The person who loves blue is in the third house.
9. The person who loves yellow is somewhere to the right of the person who loves stir fry.
10. Alice is in the first house.
11. The person who loves yellow is Arnold.
 What is the value of attribute Food for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: stir fry


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Arnold`, `Peter`, `Alice`
 - People have unique favorite book genres: `romance`, `science fiction`, `biography`, `fantasy`, `mystery`
 - Each person prefers a unique type of vacation: `camping`, `cruise`, `beach`, `mountain`, `city`

## Clues:
1. The person who loves biography books is Eric.
2. The person who loves biography books is directly left of the person who loves mystery books.
3. The person who loves science fiction books is directly left of the person who loves fantasy books.
4. There are two houses between Bob and the person who likes going on cruises.
5. The person who enjoys mountain retreats is in the first house.
6. The person who loves mystery books is somewhere to the right of the person who prefers city breaks.
7. The person who enjoys camping trips is Arnold.
8. Alice is not in the third house.
9. The person who loves mystery books is not in the fifth house.
10. The person who loves beach vacations is the person who loves mystery books.
 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: 1


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Peter`, `Alice`, `Eric`
 - Each person has a unique favorite drink: `milk`, `root beer`, `coffee`, `tea`, `water`
 - Each person has a favorite color: `blue`, `green`, `white`, `yellow`, `red`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `tulips`, `carnations`
 - Each person has a unique hobby: `painting`, `cooking`, `photography`, `gardening`, `knitting`

## Clues:
1. Alice is not in the fourth house.
2. The root beer lover is the person who enjoys gardening.
3. The person whose favorite color is green is the coffee drinker.
4. The person whose favorite color is green is the person who loves the boquet of lilies.
5. The person who loves blue is somewhere to the right of the person who loves a bouquet of daffodils.
6. The person who loves cooking is the person who loves blue.
7. Eric is directly left of the tea drinker.
8. The one who only drinks water is Peter.
9. Arnold is the photography enthusiast.
10. The person who loves white is the person who loves the rose bouquet.
11. There is one house between the person who loves a carnations arrangement and the person whose favorite color is red.
12. The person who loves cooking is somewhere to the left of the person who paints as a hobby.
13. The one who only drinks water is in the third house.
14. The person who loves a carnations arrangement is the root beer lover.
15. The person who loves white is in the second house.
 What is the value of attribute House for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Peter`, `Eric`
 - Everyone has a unique favorite cigar: `pall mall`, `dunhill`, `blue master`, `prince`
 - Each person has a unique birthday month: `sept`, `april`, `feb`, `jan`
 - Each person has a unique hobby: `gardening`, `cooking`, `photography`, `painting`
 - People have unique favorite music genres: `pop`, `classical`, `jazz`, `rock`

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

Answer: painting


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`
 - Each mother is accompanied by their child: `Fred`, `Meredith`, `Bella`
 - The people keep unique animals: `cat`, `bird`, `horse`
 - Each person has a unique favorite drink: `tea`, `water`, `milk`

## Clues:
1. Peter is somewhere to the left of the person's child is named Meredith.
2. The person's child is named Fred is directly left of the person who keeps horses.
3. The person's child is named Fred is in the second house.
4. The person who likes milk is the person who keeps horses.
5. The person whose mother's name is Holly is Eric.
6. The tea drinker is the cat lover.
7. The person's child is named Bella is Eric.
8. The tea drinker and Arnold are next to each other.
9. The person whose mother's name is Janelle is in the second house.
 What is the value of attribute Children for the person whose attribute Mother is Janelle? 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: `Eric`, `Arnold`, `Peter`
 - Each mother is accompanied by their child: `Meredith`, `Fred`, `Bella`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `desert`
 - Each person prefers a unique type of vacation: `mountain`, `city`, `beach`
 - Each person has a favorite color: `white`, `red`, `yellow`

## Clues:
1. There is one house between the person's child is named Bella and the person's child is named Meredith.
2. The person who likes Cherry smoothies is the person who loves beach vacations.
3. The person who prefers city breaks is Arnold.
4. The person whose favorite color is red is the person's child is named Fred.
5. The person's child is named Fred is the Watermelon smoothie lover.
6. Eric is not in the first house.
7. The person's child is named Meredith is the person who loves white.
8. The person who loves white is the Desert smoothie lover.
9. The Desert smoothie lover is Eric.
 What is the value of attribute House for the person whose attribute Smoothie is desert? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Eric`, `Bob`, `Carol`, `Arnold`
 - People have unique favorite sports: `baseball`, `basketball`, `swimming`, `tennis`, `soccer`, `volleyball`
 - Each person has a favorite color: `purple`, `green`, `white`, `red`, `yellow`, `blue`
 - People have unique hair colors: `red`, `auburn`, `blonde`, `black`, `gray`, `brown`
 - Everyone has something unique for lunch: `stew`, `spaghetti`, `stir fry`, `grilled cheese`, `soup`, `pizza`

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

Answer: 5


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`
 - Each person has a unique hobby: `gardening`, `photography`, `cooking`
 - Each person prefers a unique type of vacation: `mountain`, `city`, `beach`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`, `blue master`
 - The mothers' names in different houses are unique: `Janelle`, `Holly`, `Aniya`
 - People have unique heights: `short`, `average`, `very short`

## Clues:
1. The person who loves beach vacations is somewhere to the right of the person who is very short.
2. The person whose mother's name is Aniya is somewhere to the right of the person who loves cooking.
3. Eric is not in the second house.
4. The person who is short is in the third house.
5. The person who prefers city breaks is the photography enthusiast.
6. The person who enjoys gardening is the person who enjoys mountain retreats.
7. The person partial to Pall Mall is the photography enthusiast.
8. The person who enjoys mountain retreats is The person whose mother's name is Holly.
9. The person who enjoys gardening is the person who smokes Blue Master.
10. Peter is The person whose mother's name is Aniya.
 What is the value of attribute Cigar for the person whose attribute Mother is Aniya? Please reason step by step, and put your final answer within \boxed{}

Answer: pall mall


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Alice`, `Eric`
 - They all have a unique favorite flower: `roses`, `daffodils`, `carnations`, `lilies`
 - Each person has a unique hobby: `photography`, `painting`, `cooking`, `gardening`
 - Each person has a unique type of pet: `dog`, `fish`, `bird`, `cat`
 - Each person has a favorite color: `red`, `yellow`, `green`, `white`
 - Each person lives in a unique style of house: `craftsman`, `colonial`, `ranch`, `victorian`

## Clues:
1. The person in a Craftsman-style house is Arnold.
2. The person who loves the rose bouquet is somewhere to the right of Peter.
3. The photography enthusiast is the person who owns a dog.
4. The person who loves a bouquet of daffodils is not in the fourth house.
5. The person who loves the rose bouquet is the person whose favorite color is red.
6. The person in a Craftsman-style house is in the second house.
7. Eric is the person residing in a Victorian house.
8. The person with an aquarium of fish is the person who loves white.
9. The person who loves cooking is somewhere to the right of the person whose favorite color is red.
10. The person who loves white is the person who loves a carnations arrangement.
11. The person who loves white is somewhere to the right of the person who enjoys gardening.
12. The person who loves a bouquet of daffodils is the person who loves yellow.
13. The person living in a colonial-style house is the person whose favorite color is red.
14. The person who has a cat is Eric.
 What is the value of attribute House for the person whose attribute Color is red? 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: `Peter`, `Bob`, `Arnold`, `Alice`, `Eric`
 - Each person has an occupation: `lawyer`, `teacher`, `artist`, `engineer`, `doctor`
 - Each person lives in a unique style of house: `victorian`, `craftsman`, `modern`, `ranch`, `colonial`
 - People have unique hair colors: `blonde`, `gray`, `red`, `brown`, `black`
 - People have unique favorite book genres: `romance`, `biography`, `science fiction`, `mystery`, `fantasy`

## Clues:
1. Alice is not in the fifth house.
2. Peter is not in the second house.
3. Peter is directly left of Eric.
4. The person living in a colonial-style house is somewhere to the left of the person who loves fantasy books.
5. Arnold is the person who is a doctor.
6. The person who has gray hair is the person who loves fantasy books.
7. The person who loves science fiction books is not in the fifth house.
8. Peter is the person who has red hair.
9. The person who loves romance books is the person who has black hair.
10. Eric is the person who has black hair.
11. The person who is a doctor is somewhere to the right of Peter.
12. The person who has blonde hair is the person who is a teacher.
13. Peter is the person who loves mystery books.
14. The person who is a lawyer is Eric.
15. The person living in a colonial-style house is the person who is an engineer.
16. The person residing in a Victorian house is the person who loves mystery books.
17. The person who is a teacher is not in the fourth house.
18. The person who is an artist is somewhere to the left of the person in a modern-style house.
19. The person who is a teacher is not in the third house.
20. The person in a ranch-style home is directly left of the person in a Craftsman-style house.
 What is the value of attribute Name for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


1. In the interior angles of an $n(n \geqslant 3)$-sided polygon, the maximum number of acute angles is $\qquad$ .

Answer: 3


8. Car A and Car B start from locations $A$ and $B$ respectively at the same time, heading towards each other. They meet after 3 hours, then Car A turns around and heads back to $\mathrm{A}$, while Car B continues on. After Car A reaches $A$ and turns around to head towards $B$, it meets Car B again half an hour later. Therefore, the total time for Car B to travel from $A$ to $B$ is $\qquad$ hours.

Answer: \dfrac{36}{5}


5. Let $\angle A, \angle B, \angle C$ be the three interior angles of $\triangle ABC$. If $\sin A=a, \cos B=b$, where $a>0, b>0$, and $a^{2}+b^{2} \leqslant 1$, then $\tan C=$ $\qquad$

Answer: \dfrac{a \sqrt{1 - a^2} + b \sqrt{1 - b^2}}{a^2 - b^2}


1. The sum of a set of numbers is the sum of all its elements. Let $S$ be a set of positive integers not exceeding 15, such that the sums of any two disjoint subsets of $S$ are not equal, and among all sets with this property, the sum of $S$ is the largest. Find the sum of the set $S$.
(4th American Invitational Mathematics Examination)

Answer: 61


5. Given a positive integer $n$ greater than 2004, fill the numbers $1,2,3, \cdots, n^{2}$ into the squares of an $n \times n$ chessboard (consisting of $n$ rows and $n$ columns) such that each square contains exactly one number. If a square contains a number that is greater than the numbers in at least 2004 squares in its row and at least 2004 squares in its column, then this square is called a “good square”. Find the maximum number of “good squares” on the chessboard.

Answer: n(n - 2004)


$\left[\begin{array}{l}\text { [Rectangles and squares. Properties and characteristics } \\ \text { Word problems (miscellaneous). }\end{array}\right]$

A frame for three square photographs has the same width everywhere (see figure). The perimeter of one opening is 60 cm, and the perimeter of the entire frame is 180 cm. What is the width of the frame?

![](https://cdn.mathpix.com/cropped/2024_05_06_53798a1e86d520071242g-12.jpg?height=557&width=238&top_left_y=2311&top_left_x=910)

Answer: 5


3. Given real numbers $a, b, c$ satisfy
$$
f(x)=a \cos x+b \cos 2 x+c \cos 3 x \geqslant-1
$$

for any real number $x$. Then the maximum value of $a+b+c$ is . $\qquad$

Answer: 3


7,8,9 |

In a deck of 16 cards, numbered from top to bottom. It is allowed to take a portion of the deck from the top, after which the removed and remaining parts of the deck, without flipping, are "interleaved" with each other. Can it happen that after several such operations, the cards end up numbered from bottom to top? If yes, in what minimum number of operations can this occur?

Answer: 4


As shown in Figure 2, in the acute triangle $\triangle ABC$, points $D$ and $E$ are on side $BC$. The circumcenters of $\triangle ABC$, $\triangle ABD$, and $\triangle ADC$ are $O$, $P$, and $Q$, respectively.
Prove:
1. $\triangle APQ \sim \triangle ABC$.
2. If $EO \perp PQ$, then $QO \perp PE$.

Answer: \triangle APQ \sim \triangle ABC


30 numbers are written in a circle. Each of these numbers is equal to the absolute value of the difference between two numbers that follow it in a clockwise direction. The sum of all the numbers is equal to 1. Find these numbers.

Answer: \frac{1}{20}


5. In a convex quadrilateral $A B C D$, the areas of triangles $A B D$ and $B C D$ are equal, and the area of $A C D$ is half the area of $A B D$. Find the length of the segment $C M$, where $M$ is the midpoint of side $A B$, if it is known that $A D=12$.

Answer: 18


7.4. Can several rectangles, the sum of the perimeters of which is 100, be combined to form a square with a side length of 16?

Answer: Yes


35. (5 points)
In the Longevity Village, there is an elderly person who was 101 years old in 2021! If we take the elder's age as the numerator and the year number as the denominator, we can write a fraction, such as in the year 2001 when the elder was 81 years old, the fraction $\frac{81}{2001}$ can be written. From the age of 1 to 100, the elder can write 100 fractions, among which the simplest fractions are $\qquad$ in number.

Answer: 26


5. How many five-digit natural numbers are there in which there are no zeros and the sum of all digits is greater than their product?

The use of a pocket calculator or any manuals is not allowed.

## Ministry of Science and Education of the Republic of Croatia Agency for Education and Education Croatian Mathematical Society

## County Competition in Mathematics February 26, 2024. 7th grade - elementary school

Each task is worth 10 points. In addition to the final result, the procedure is also graded. To earn all points, it is necessary to find all solutions and confirm that there are no others, write down the procedure, and justify your conclusions.

Answer: 91


[Perpendicular bisector of a segment_([MT)]

[ Inscribed angle subtended by a diameter ]

A circle is circumscribed around triangle $A M B$, and its center is 10 units away from side $A M$.

The extension of side $A M$ beyond vertex $M$ intercepts a segment $C B$ from the tangent to the circle at vertex $B$, which is 29 units long. Find the area of triangle $C M B$, given that the angle $A C B$ is $\operatorname{arctg} \frac{20}{21}$.

Answer: 210


10.017. Given a point $P$, which is 7 cm away from the center of a circle with a radius of $11 \mathrm{~cm}$. A chord of length 18 cm is drawn through this point. What are the lengths of the segments into which the chord is divided by point $P$?

Answer: 6


1. In a $3 \times 3$ table, 9 numbers are arranged such that the six products of these numbers in the rows and columns of the table are all different. What is the maximum number of these numbers that can equal one?

Answer: 5


A perpendicular dropped from the vertex $C$ of parallelogram $A B C D$ to the line $C D$ intersects at point $F$ a perpendicular dropped from vertex $A$ to the diagonal $B D$. A perpendicular dropped from point $B$ to the line $A B$ intersects at point $E$ the perpendicular bisector of segment $A C$. In what ratio does segment $E F$ divide side $B C$?

Answer: 1:2


Let \( a_{1}, a_{2}, \cdots, a_{n}, b_{1}, b_{2}, \cdots, b_{n} \in [1,2] \), and \( \sum_{i=1}^{n} a_{i}^{2} = \sum_{i=1}^{n} b_{i}^{2} \). Prove that \( \sum_{i=1}^{n} \frac{a_{i}^{3}}{b_{i}} \leq \frac{17}{10} \sum_{i=1}^{n} a_{i}^{2} \), and determine the necessary and sufficient conditions for equality to hold.

Answer: \frac{17}{10} \sum_{i=1}^{n} a_{i}^{2}


[ [ correct prism $]$

The side of the base $A B C$ of a regular prism $A B C A 1 B 1 C 1$ is 1, and each of the lateral edges has a length of $\frac{1}{\sqrt{3}}$. A right circular cylinder is positioned such that point $A 1$ and the midpoint $M$ of edge $C C 1$ lie on its lateral surface, and the axis of the cylinder is parallel to line $A B 1$ and is at a distance of $\frac{1}{4}$ from it. Find the radius of the cylinder.

#

Answer: \dfrac{\sqrt{7}}{4}


Let $ABC$ be a triangle, $P$ a point on $[BC]$, $Q$ a point on $[CA]$, and $R$ a point on $[AB]$. The circumcircles of $AQR$ and $BRP$ intersect at a second point $X$. Show that $X$ is also on the circumcircle of $CQP$.

Answer: X \text{ lies on the circumcircle of } \triangle CQP


3. In a square $A B C D$ with side length 2, $E$ is the midpoint of $A B$. Now, fold the figure along segments $E C, E D$ so that segments $E A, E B$ coincide. This results in a tetrahedron $C D E A$ (point $B$ has coincided with point $A$). The volume of this tetrahedron is

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


10.5. 10.5 A - a four-digit number composed of non-zero digits, B the number written with the same digits in reverse order. It is known that the sum A+B is divisible by 109. What can the sum of the digits of A be?

Answer: 14


5.2. Given three points $A(2 ; 1), B(3 ;-1), C(-4 ; 0)$, which are vertices of an isosceles trapezoid $A B D C$. Find the coordinates of point $D$, if $\overline{A B}=k \overline{C D}$.

Answer: \left( -\dfrac{7}{5}, -\dfrac{26}{5} \right)


2. What is the largest whole number that is equal to the product of its digits?

Answer: 9


\section*{Problem 2 - 121232}

In space, four points \(P_{1}, P_{2}, P_{3}\) and \(P_{4}\) are given, which do not lie in the same plane. Determine the number of all those planes that are equidistant from these four points.

Answer: 7


B2. In a circle with radius $r$, we inscribe a deltoid, one side of which is twice as long as the other. Calculate the ratio of the areas of the inscribed deltoid and the circle.

Answer: \dfrac{8}{5\pi}


6. As shown in Figure 1, given that the volume of the regular quadrilateral pyramid $P$ $A B C D$ is 1, $E$, $F$, $G$, $H$ are the midpoints of line segments $A B$, $C D$, $P B$, $P C$ respectively. Then the volume of the polyhedron $B E G-C F H$ is $\qquad$

Answer: \dfrac{5}{16}


A4 Square $A B C D$ has side length $10 \mathrm{~cm}$. Points $W, X, Y$, and $Z$ are the midpoints of segments $A B, B C, C D$ and $W X$, respectively. Determine the area of quadrilateral $X C Y Z$ (in $\mathrm{cm}^{2}$ ).

Answer: 25


19. (ITA 5) We are given a fixed point on the circle of radius 1, and going from this point along the circumference in the positive direction on curved distances $0,1,2, \ldots$ from it we obtain points with abscisas $n=0,1,2, \ldots$ respectively. How many points among them should we take to ensure that some two of them are less than the distance $1 / 5$ apart?

Answer: 20


3. A company was manufacturing toys. They received an order that 20 workers could complete in 90 days. Due to summer vacations and leaves, only 10 workers worked for the first 40 days. At that point, they realized they were falling behind schedule, so they immediately hired 40 more workers. How many days did it take them to fulfill the order? By how many days did they fall behind or exceed the 90-day deadline?

Answer: 68


Three unit radius circles each pass through the point $P$, and their other intersection points are $A, B$, and $C$. What is the radius of the circumcircle of triangle $ABC$?

Answer: 1


8. As shown in the figure, it is a rectangular prism with length, width, and height of $17 \mathrm{~cm}, 8 \mathrm{~cm}, 5 \mathrm{~cm}$ respectively. An ant starts from one vertex, crawling along the edges, with each edge being traversed at most once. The longest route the ant can crawl is $\qquad$ $\mathrm{cm}$.

Answer: 105


10.4. In triangle $A B C$, points $D$ and $F$ are marked on sides $A B$ and $A C$ respectively such that lines $D C$ and $B F$ are perpendicular to each other and intersect at point $E$ inside triangle $A B C$. It turns out that $A D=D C$ and $D E \cdot E C=F E \cdot E B$. What degree measure can angle $B A C$ have? (7 points)

Answer: 30


Example 1 A deck of cards consists of 52 cards, with 13 cards of each of the four suits: "Diamonds," "Clubs," "Hearts," and "Spades." The cards are numbered $2,3, \cdots, 10, \mathrm{~J}, \mathrm{Q}, \mathrm{K}, \mathrm{A}$. Two cards of the same suit and consecutive numbers are called "straight flush" cards, and $\mathrm{A}$ can be considered as 1, making it a straight flush with 2. Determine the number of ways to draw 13 cards from this deck such that each number appears and no straight flush cards are included.
(3rd China Southeast Mathematical Olympiad)

Answer: 1594320


21.33. If we consider digits in different positions as different, then in a $d$-ary numeral system, $n d$ digits allow us to write $d^{n}$ numbers (from 0 to $d^{n}-1$). Which numeral system is the most economical in this respect, i.e., allows recording the largest number of numbers using a given number of digits? (When comparing numeral systems with bases $d_{1}$ and $d_{2}$, we consider only sets of $m$ digits, where $m$ is divisible by $d_{1}$ and $d_{2}$.)

See also problems $20.15,20.17$.

### 21.10. Other representations of numbers

Answer: 3


11.5. On each of ten cards, a real number is written. For each non-empty set of these cards, the sum of all numbers written on the cards in this set was found. It is known that not all of the obtained sums are integers. What is the maximum possible number of integer sums that could have resulted?

Answer: 511


6. We have an $8 \times 8$ board. An inner edge is an edge between two $1 \times 1$ fields. We cut the board into $1 \times 2$ dominoes. For an inner edge $k$, $N(k)$ denotes the number of ways to cut the board such that the cut goes along the edge $k$. Calculate the last digit of the sum we get when we add all $N(k)$, where $k$ is an inner edge.

Answer: 0


## Task B-1.7.

In triangle $A B C$, point $D$ is on side $\overline{B C}$, and point $E$ is on side $\overline{A C}$. Segments $\overline{B E}$ and $\overline{A D}$ intersect at point $F$, and segment $\overline{B F}$ divides the area of triangle $A B D$ into two equal parts. If the area of triangle $A F E$ is 7, and the area of triangle $A F B$ is 13, calculate the areas of triangles $C E F$ and $C F D$.

Answer: \dfrac{70}{3}


Problem 8.3. How many lines exist that pass through the point $(0,2019)$ and intersect the parabola $y=x^{2}$ at two points with integer coordinates on the $y$-axis?

Answer: 9.

Answer: 9


11. Let the left focus of the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ be $F$, and let the line $l$ passing through $(1,1)$ intersect the ellipse at points $A$ and $B$. When the perimeter of $\triangle FAB$ is maximized, the area of $\triangle FAB$ is $\qquad$

Answer: 3


1. For which values of the parameters $p$ and $q$ does the equation

$$
\sqrt{x^{2} + p x + q} + x = 2017
$$

have more than 2017 different real solutions?

Answer: -4034


30th IMO 1989 shortlist Problem 23 155 birds sit on a circle center O. Birds at A and B are mutually visible iff ∠AOB ≤ 10 o . More than one bird may sit at the same point. What is the smallest possible number of mutually visible pairs? Solution

Answer: 270


Test $B-5$. Let $T$ be the inscribed trapezoid $ABCD$ (counterclockwise) in the unit circle $O$, $ABCDI, AB=s_{1}$, $CD=s_{2}, OE=d, E$ is the intersection point of the heights. When $d \neq 0$, determine the minimum upper bound of $\frac{s_{1}-s_{2}}{d}$. If the minimum upper bound can be achieved, determine all such cases.

Answer: 2


On a board, there are $n$ equations in the form $*x^2+*x+*$. Two people play a game where they take turns. During a turn, you are aloud to change a star into a number not equal to zero. After $3n$ moves, there will be $n$ quadratic equations. The first player is trying to make more of the equations not have real roots, while the second player is trying to do the opposite. What is the maximum number of equations that the first player can create without real roots no matter how the second player acts?

Answer: 0


6. Three cones with vertex $A$ and slant height 6 touch each other externally. For two of the cones, the angle between the slant height and the axis of symmetry is $\frac{\pi}{8}$, while for the third cone, it is $\frac{\pi}{4}$. Find the volume of the pyramid $O_{1} O_{2} O_{3} A$, where $O_{1}, O_{2}, O_{3}$ are the centers of the bases of the cones.

Answer: 9 \sqrt{1 + \sqrt{2}}


Problem 3. Let $\alpha$ and $\beta$ be two adjacent angles such that their bisectors are mutually perpendicular. If it is known that $\alpha$ is $30^{\circ}$ greater than $\beta$, calculate how many degrees each of them measures.

Answer: 105^\circ


13.25 Given 4 coins, one of which may be counterfeit, each genuine coin weighs 10 grams, and the counterfeit coin weighs 9 grams. Now there is a balance scale with one pan that can measure the total weight of the objects on the pan. To identify whether each coin is genuine or counterfeit, what is the minimum number of weighings required?

Answer: 3


2. Find all real numbers $p$ such that the cubic equation $5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p$ has three roots that are all natural numbers.

Answer: 76


A circle \( C \) has two parallel tangents \( L' \) and \( L'' \). A circle \( C' \) touches \( L' \) at \( A \) and \( C \) at \( X \). A circle \( C'' \) touches \( L'' \) at \( B \), \( C \) at \( Y \), and \( C' \) at \( Z \). The lines \( AY \) and \( BX \) meet at \( Q \). Show that \( Q \) is the circumcenter of \( XYZ \).

Answer: Q \text{ is the circumcenter of } \triangle XYZ


2 Let the medians of $\triangle A B C$ corresponding to sides $a, b, c$ be $m_{a}, m_{b}, m_{c}$, and the angle bisectors be $w_{a}, w_{b}, w_{c}$. Suppose $w_{a} \cap m_{b}=P, w_{b} \cap m_{c}=Q, w_{c} \cap m_{a}=R$. Let the area of $\triangle P Q R$ be $\delta$, and the area of $\triangle A B C$ be $F$. Find the smallest positive constant $\lambda$ such that the inequality $\frac{\delta}{F}<\lambda$ holds.

Answer: \dfrac{1}{6}


14 Let $x, y$ be real numbers greater than 1, and let $a=\sqrt{x-1}+\sqrt{y-1}, b=\sqrt{x+1}+$ $\sqrt{y+1}$, where $a, b$ are two non-consecutive positive integers. Find the values of $x, y$.

Answer: \dfrac{5}{4}


There is a pile with $15$ coins on a table. At each step, Pedro choses one of the piles in the table with $a>1$ coins and divides it in two piles with $b\geq1$ and $c\geq1$ coins and writes in the board the product $abc$. He continues until there are $15$ piles with $1$ coin each. Determine all possible values that the final sum of the numbers in the board can have.

Answer: 1120


There is number $1$ on the blackboard initially. The first step is to erase $1$ and write two nonnegative reals whose sum is $1$. Call the smaller number of the two $L_2$.
For integer $k \ge 2$, the ${k}$ the step is to erase a number on the blackboard arbitrarily and write two nonnegative reals whose sum is the number erased just now. Call the smallest number of the $k+1$ on the blackboard $L_{k+1}$.
Find the maximum of $L_2+L_3+\cdots+L_{2024}$.

Answer: \dfrac{1405}{256}


7. The Brazilian IMO leader chooses two natural numbers $n$ and $k$ with $n>k$, and then tells these to his deputy and a participant. The leader then whispers a binary sequence of length $n$ into the deputy's ear. The deputy writes down all binary sequences of length $n$ that differ from the leader's sequence at exactly $k$ positions. (Example for $n=3$ and $k=1$: If the leader chooses 101, the deputy writes down 001, 100, 111.) The participant looks at the sequences written down by the deputy. Now the participant tries to guess the original sequence from the leader.

How many times does he need to guess at minimum (depending on $n$ and $k$) to ensure he guesses correctly at least once?

Answer: 1


14. In a convex pentagon $A B C D E$, a point $M$ is taken on side $A E$, and a point $N$ is taken on side $D E$. Segments $C M$ and $B N$ intersect at point $P$. Find the area of pentagon $A B C D E$ if it is known that quadrilaterals $A B P M$ and $D C P N$ are parallelograms with areas param1 and param2, respectively, and the area of triangle $B C P$ is param3.

| param1 | param2 | param3 | Answer |
| :---: | :---: | :---: | :---: |
| 5 | 8 | 10 | 25 |
| 6 | 10 | 3 | 29 |
| 7 | 9 | 7 | 27.5 |
| 8 | 11 | 20 | 41.2 |
| 9 | 12 | 4 | 38.5 |

## Condition

In a convex pentagon $A B C D E$, a point $M$ is taken on side $A E$, and a point $N$ is taken on side $D E$. Segments $C M$ and $B N$ intersect at point $P$. Find the area of pentagon $A B C D E$ if it is known that quadrilaterals $A B P M$ and $D C P N$ are parallelograms with areas 5 and 8, respectively, and the area of triangle $B C P$ is 10.

## Answer.

25

Answer: 25


G1.4 Using numbers: $1,2,3,4,5,6$ to form a six-digit number: $A B C D E F$ such that $A$ is divisible by $1, A B$ is divisible by $2, A B C$ is divisible by $3, A B C D$ is divisible by $4, A B C D E$ is divisible by $5, A B C D E F$ is divisible by 6 . Find the greatest value of $A$.

Answer: 3


6. Fill two $a$s and two $b$s into a $4 \times 4$ grid, with at most one letter per cell, such that the same letters are neither in the same row nor in the same column. The number of ways to do this is $\qquad$

Answer: 3960


The circle $k_{1}$ with center $O_{1}$ and radius 2 units, and the circle $k_{2}$ with center $O_{2}$ and radius $r$, touch each other externally. The Thales circle of the segment $O_{1} O_{2}$ intersects the circle $k_{1}$ at points $A$ and $B$, and the circle $k_{2}$ at points $C$ and $D$. The reflection of $O_{1}$ over the segment $A B$ is $O_{1}^{\prime}$, and the reflection of $O_{2}$ over the segment $C D$ is $O_{2}^{\prime}$. The distance $O_{1}^{\prime} O_{2}^{\prime}$ is 1 unit.

What is the value of $r$?

Answer: \dfrac{5 + \sqrt{17}}{2}


Mat was digging a hole. Pat asked him how deep the hole would be. Mat answered with a riddle: "I am $90 \mathrm{~cm}$ tall and I have just dug half of the hole. When I finish digging the hole, the top of my head will be as deep below the ground as it is now above the ground." How deep was the hole Mat was digging?

Answer: 120


Acute triangle $ABC$ satisfies $AB=2AC$ and $AB^4+BC^4+CA^4 = 2006 \cdot 10^{10}$. A tetrahedron $DEFP$ is formed by choosing points $D$, $E$, and $F$ on the segments $BC$, $CA$, and $AB$ (respectively) and folding $A$, $B$, $C$ over $EF$, $FD$, and $DE$ (respectively) to the common point $P$. Let $R$ denote the circumradius of $DEFP$. Compute the smallest positive integer $N$ for which we can be certain that $N \geq R$. It may be helpful to use $\sqrt[4]{1239} = 5.9329109\ldots$.

Answer: 6


In a sequence of numbers, a term is called "golden" if it is divisible by the term immediately before it. What is the maximum possible number of golden terms in a permutation of $1, 2, 3, \ldots, 2021$?

Answer: 1010


Alexis and Joshua are walking along the beach when they decide to draw symbols in the sand. Alex draws only stars and only draws them in pairs, while Joshua draws only squares in trios. "Let's see how many rows of $15$ adjacent symbols we can make this way," suggests Josh. Alexis is game for the task and the two get busy drawing. Some of their rows look like:

\[
\begin{array}{ccccccccccccccc}
\vspace{10pt}*&*&*&*&*&*&*&*&*&*&*&*&\blacksquare&\blacksquare&\blacksquare\\
\vspace{10pt}\blacksquare&\blacksquare&\blacksquare&*&*&*&*&*&*&*&*&*&*&*&*\\
\vspace{10pt}\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&*&*&*&*&*&*&\blacksquare&\blacksquare&\blacksquare \\
\vspace{10pt}\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare\\
\vspace{10pt} *&*&*&*&*&*&\blacksquare&\blacksquare&\blacksquare&*&*&*&*&*&*
\end{array}
\]

The twins decide to count each of the first two rows above as distinct rows, even though one is the mirror image of the other. But the last of the rows above is its own mirror image, so they count it only once. Around an hour later, the twins realized that they had drawn every possible row exactly once using their rules of stars in pairs and squares in trios. How many rows did they draw in the sand?

Answer: 28


For real numbers $x$ and $y$, let $M$ be the maximum value of the expression $x^4y + x^3y + x^2y + xy + xy^2 + xy^3 + xy^4$, subject to $x + y = 3$. Find $\lfloor M \rfloor$.

Answer: 36


On a $5 \times 5$ board, pieces made up of $4$ squares are placed, as seen in the figure, each covering exactly $4$ squares of the board. The pieces can be rotated or turned over. They can also overlap, but they cannot protrude from the board. Suppose that each square on the board is covered by at most two pieces. Find the maximum number of squares on the board that can be covered (by one or two pieces).

Answer: 24


In a simple graph, there are two disjoint sets of vertices $A$ and $B$ where $A$ has $k$ vertices and $B$ has $2016$ vertices. Four numbers are written on each vertex using the colors red, green, blue, and black. Initially, there are no edges.

For each vertex in $A$, a color is chosen, and edges are drawn from this vertex to all vertices in $B$ that have a larger number with the chosen color. It is known that for each vertex in $B$, the set of vertices in $A$ connected to it is unique.

Find the minimal possible value of $k$.

Answer: 6


For positive real numbers $p$ and $q$, define the remainder when dividing $p$ by $q$ as the smallest nonnegative real $r$ such that $\frac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_1$ and $r_2$ be the remainders when $a\sqrt{2} + b\sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ respectively. Find the number of pairs $(a, b)$ such that $a, b \leq 20$ and $r_1 + r_2 = \sqrt{2}$.

Answer: 16


Let $AXYZB$ be a convex pentagon inscribed in a semicircle with diameter $AB$. Suppose that $AZ - AX = 6$, $BX - BZ = 9$, $AY = 12$, and $BY = 5$. Find the greatest integer not exceeding the perimeter of quadrilateral $OXYZ$, where $O$ is the midpoint of $AB$.

Answer: 23


Let $ABCDEF$ be a regular hexagon with side length $6$, and construct squares $ABGH$, $BCIJ$, $CDKL$, $DEMN$, $EFOP$, and $FAQR$ outside the hexagon. Find the perimeter of the dodecagon $HGJILKNMPORQ$.

Answer: 72


A large collection of congruent right triangles is given, each with side lengths $3$, $4$, and $5$. Find the maximal number of such triangles you can place inside a $20 \times 20$ square, with no two triangles intersecting in their interiors.

Answer: 66


Consider the sum $$S =\sum^{2021}_{j=1} \left|\sin \frac{2\pi j}{2021}\right|.$$
The value of $S$ can be written as $\tan \left( \frac{c\pi}{d} \right)$ for some relatively prime positive integers $c, d$, satisfying $2c < d$. Find the value of $c + d$.

Answer: 3031


Given an integer $a_0$, we define a sequence of real numbers $a_0, a_1, . . .$ using the relation $$a^2_i = 1 + ia^2_{i-1},$$
for $i \ge 1$. An index $j$ is called [i]good [/i] if $a_j$ can be an integer for some $a_0$. Determine the sum of the indices $j$ which lie in the interval $[0, 99]$ and which are not good.

Answer: 4946


Jack plays a game in which he first rolls a fair six-sided die and gets some number $n$, then, he flips a coin until he flips $n$ heads in a row and wins, or he flips $n$ tails in a row in which case he rerolls the die and tries again. What is the expected number of times Jack must flip the coin before he wins the game.

Answer: 40


Alice, Bob, and Carol are playing a game. Each turn, one of them says one of the $3$ players' names, chosen from {Alice, Bob, Carol} uniformly at random. Alice goes first, Bob goes second, Carol goes third, and they repeat in that order. Let $E$ be the expected number of names that are have been said when, for the first time, all $3$ names have been said twice. If $E = \tfrac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$. (Include the last name to be said twice in your count.)

Answer: 383


Have $b, c \in R$ satisfy $b \in (0, 1)$ and $c > 0$, then let $A,B$ denote the points of intersection of the line $y = bx+c$ with $y = |x|$, and let $O$ denote the origin of $R^2$. Let $f(b, c)$ denote the area of triangle $\vartriangle OAB$. Let $k_0 = \frac{1}{2022}$ , and for $n \ge 1$ let $k_n = k^2_{n-1}$. If the sum  $\sum^{\infty}_{n=1}f(k_n, k_{n-1})$ can be written as $\frac{p}{q}$ for relatively prime positive integers $p, q$, find the remainder when $p+q$ is divided by 1000.

Answer: 484


Triangle $\vartriangle ABC$ has sidelengths $AB = 10$, $AC = 14$, and, $BC = 16$. Circle $\omega_1$ is tangent to rays $\overrightarrow{AB}$, $\overrightarrow{AC}$ and passes through $B$. Circle $\omega_2$ is tangent to rays $\overrightarrow{AB}$, $\overrightarrow{AC}$ and passes through $C$. Let $\omega_1$, $\omega_2$ intersect at points $X, Y$ . The square of the perimeter of triangle $\vartriangle AXY$ is equal to $\frac{a+b\sqrt{c}}{d}$ , where $a, b, c$, and, $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$.

Answer: 6272


Pick $x, y, z$ to be real numbers satisfying $(-x+y+z)^2-\frac13 = 4(y-z)^2$, $(x-y+z)^2-\frac14 = 4(z-x)2$, and $(x+y-z)^2 -\frac15 = 4(x-y)^2$. If the value of $xy+yz +zx$ can be written as $\frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.

Answer: 1727


Let $\vartriangle ABC$ be a triangle with $AB = 5$, $BC = 8$, and, $CA = 7$. Let the center of the $A$-excircle be $O$, and let the $A$-excircle touch lines $BC$, $CA$, and,$ AB$ at points $X, Y$ , and, $Z$, respectively. Let $h_1$, $h_2$, and, $h_3$ denote the distances from $O$ to lines $XY$ , $Y Z$, and, ZX, respectively. If $h^2_1+ h^2_2+ h^2_3$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m + n$.

Answer: 2189


Let $\alpha, \beta, \gamma \in C$ be the roots of the polynomial $x^3 - 3x2 + 3x + 7$. For any complex number $z$, let $f(z)$ be defined as follows:
$$f(z) = |z -\alpha | + |z - \beta|+ |z-\gamma | - 2 \underbrace{\max}_{w \in \{\alpha, \beta, \gamma}\} |z - w|.$$
Let $A$ be the area of the region bounded by the locus of all $z \in C$ at which $f(z)$ attains its global minimum. Find $\lfloor A \rfloor$.

Answer: 12


In the complex plane, let $z_1, z_2, z_3$ be the roots of the polynomial $p(x) = x^3- ax^2 + bx - ab$. Find the number of integers $n$ between $1$ and $500$ inclusive that are expressible as $z^4_1 +z^4_2 +z^4_3$ for some choice of positive integers $a, b$.

Answer: 51