There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Bob`, `Eric`, `Arnold`
 - Everyone has a favorite smoothie: `lime`, `dragonfruit`, `desert`, `watermelon`, `cherry`
 - The people keep unique animals: `horse`, `dog`, `bird`, `fish`, `cat`
 - The people are of nationalities: `german`, `swede`, `norwegian`, `brit`, `dane`

## Clues:
1. The Swedish person is directly left of the dog owner.
2. There are two houses between the dog owner and the British person.
3. The Dane is the person who keeps horses.
4. The bird keeper is somewhere to the right of the cat lover.
5. The dog owner is directly left of the person who drinks Lime smoothies.
6. Eric is the cat lover.
7. Bob is the bird keeper.
8. The person who likes Cherry smoothies is directly left of Peter.
9. The bird keeper is the Watermelon smoothie lover.
10. The Desert smoothie lover is the dog owner.
11. The person who keeps horses is in the third house.
12. The Norwegian is Alice.
 What is the value of attribute Animal for the person whose attribute Smoothie is lime? 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: `Peter`, `Alice`, `Bob`, `Eric`, `Arnold`
 - The people are of nationalities: `norwegian`, `german`, `dane`, `brit`, `swede`
 - People have unique favorite book genres: `fantasy`, `biography`, `romance`, `mystery`, `science fiction`
 - Everyone has something unique for lunch: `stir fry`, `grilled cheese`, `pizza`, `spaghetti`, `stew`
 - Each person has a favorite color: `red`, `green`, `blue`, `yellow`, `white`
 - The people keep unique animals: `bird`, `dog`, `cat`, `horse`, `fish`

## Clues:
1. The person who loves fantasy books is the Norwegian.
2. The cat lover and the person who loves biography books are next to each other.
3. The German is Bob.
4. The person who loves yellow is Bob.
5. The person whose favorite color is green is Peter.
6. There is one house between the Dane and the person who is a pizza lover.
7. The person who loves blue is somewhere to the left of the Dane.
8. The person who loves eating grilled cheese is somewhere to the left of the Norwegian.
9. The person who loves the spaghetti eater is Peter.
10. The person who keeps horses is Alice.
11. The fish enthusiast is directly left of the person who loves science fiction books.
12. There is one house between the Norwegian and Arnold.
13. The person who loves romance books is the British person.
14. There are two houses between the Norwegian and Alice.
15. The bird keeper is the person whose favorite color is red.
16. The dog owner is directly left of the fish enthusiast.
17. The person who loves the stew is the Norwegian.
 What is the value of attribute Nationality for the person whose attribute BookGenre is biography? Please reason step by step, and put your final answer within \boxed{}

Answer: swede


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 FavoriteSport for the person whose attribute PhoneModel is iphone 13? Please reason step by step, and put your final answer within \boxed{}

Answer: soccer


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Carol`, `Eric`, `Peter`, `Bob`, `Arnold`
 - People have unique favorite music genres: `classical`, `hip hop`, `jazz`, `pop`, `rock`, `country`
 - The mothers' names in different houses are unique: `Sarah`, `Penny`, `Aniya`, `Janelle`, `Kailyn`, `Holly`
 - Each mother is accompanied by their child: `Alice`, `Fred`, `Timothy`, `Bella`, `Samantha`, `Meredith`
 - People have unique heights: `very short`, `tall`, `short`, `very tall`, `super tall`, `average`
 - The people keep unique animals: `bird`, `dog`, `horse`, `rabbit`, `cat`, `fish`

## Clues:
1. The person who loves pop music is the cat lover.
2. The rabbit owner is directly left of The person whose mother's name is Aniya.
3. The person whose mother's name is Holly is directly left of Carol.
4. The person whose mother's name is Holly is the person's child is named Alice.
5. The person whose mother's name is Holly is the person who loves classical music.
6. The person who loves jazz music is The person whose mother's name is Sarah.
7. The person's child is named Meredith is somewhere to the right of The person whose mother's name is Aniya.
8. The person who is super tall is The person whose mother's name is Holly.
9. The person who is the mother of Timothy is Bob.
10. The person who is very short is somewhere to the left of The person whose mother's name is Aniya.
11. Eric is the fish enthusiast.
12. The person's child is named Samantha is somewhere to the right of the person who is very tall.
13. The person who loves rock music is The person whose mother's name is Janelle.
14. There is one house between the person who keeps horses and the person's child is named Meredith.
15. The person's child is named Bella is somewhere to the right of Peter.
16. The fish enthusiast is somewhere to the left of the bird keeper.
17. The fish enthusiast is somewhere to the right of the person's child is named Alice.
18. There is one house between the person's child is named Bella and the person who loves rock music.
19. The person who is short is the cat lover.
20. Alice is directly left of the person who loves classical music.
21. The person's child is named Bella is The person whose mother's name is Aniya.
22. There are two houses between The person whose mother's name is Penny and the person who is short.
23. The person who loves hip-hop music is in the first house.
24. Carol is the person who is tall.
 What is the value of attribute MusicGenre for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: rock


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

## Clues:
1. The person with a pet hamster is somewhere to the right of the person whose birthday is in March.
2. The person whose birthday is in January is somewhere to the left of the person whose birthday is in September.
3. The person whose birthday is in May is in the second house.
4. The person living in a colonial-style house is in the second house.
5. Carol is in the third house.
6. The person in a Mediterranean-style villa is not in the sixth house.
7. The person with an aquarium of fish is somewhere to the right of Bob.
8. Eric is in the sixth house.
9. There is one house between the person who has a cat and the person residing in a Victorian house.
10. There are two houses between the person residing in a Victorian house and the person with a pet hamster.
11. The person in a Craftsman-style house is Arnold.
12. The person living in a colonial-style house is somewhere to the left of the person in a modern-style house.
13. The person with an aquarium of fish is not in the second house.
14. Peter is the person living in a colonial-style house.
15. The person whose birthday is in January is directly left of the person whose birthday is in April.
16. There is one house between the person who keeps a pet bird and the person in a modern-style house.
17. Carol is the person whose birthday is in March.
18. The person in a Craftsman-style house is in the fourth house.
19. The person who owns a dog is in the fourth house.
 What is the value of attribute House for the person whose attribute Birthday is mar? 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: `Eric`, `Carol`, `Peter`, `Bob`, `Alice`, `Arnold`
 - People have unique favorite sports: `swimming`, `basketball`, `volleyball`, `tennis`, `soccer`, `baseball`
 - Each person has a unique birthday month: `mar`, `feb`, `may`, `sept`, `april`, `jan`
 - Everyone has a favorite smoothie: `cherry`, `lime`, `desert`, `watermelon`, `blueberry`, `dragonfruit`
 - People have unique favorite music genres: `country`, `hip hop`, `jazz`, `rock`, `pop`, `classical`

## Clues:
1. The person whose birthday is in March is not in the fifth house.
2. Peter is not in the sixth house.
3. The person whose birthday is in September is not in the sixth house.
4. The person whose birthday is in March is somewhere to the left of the Desert smoothie lover.
5. The person who loves jazz music is Eric.
6. The person who loves swimming is somewhere to the right of the person who loves basketball.
7. The person who loves swimming is the person who loves jazz music.
8. There is one house between the person whose birthday is in September and the Desert smoothie lover.
9. The Watermelon smoothie lover is the person who loves jazz music.
10. There are two houses between the person whose birthday is in January and the Desert smoothie lover.
11. The person whose birthday is in February and the person who drinks Lime smoothies are next to each other.
12. There is one house between Bob and the person who loves jazz music.
13. The Watermelon smoothie lover is somewhere to the right of the person whose birthday is in April.
14. The person who loves rock music is Carol.
15. The person who loves volleyball is directly left of the Dragonfruit smoothie lover.
16. The person who loves rock music is directly left of the person whose birthday is in March.
17. The Desert smoothie lover is the person who loves soccer.
18. The person whose birthday is in January is the person who loves pop music.
19. There are two houses between the person who drinks Blueberry smoothies and the Desert smoothie lover.
20. The person whose birthday is in January is somewhere to the left of the person whose birthday is in March.
21. The Dragonfruit smoothie lover is directly left of the person who loves baseball.
22. Alice is the person who loves hip-hop music.
23. The person who loves classical music is in the sixth house.
 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


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

## Clues:
1. The person who loves a bouquet of daffodils is not in the first house.
2. The person's child is named Fred is Bob.
3. The person's child is named Fred is not in the second house.
4. The person's child is named Fred is somewhere to the left of the person who loves a carnations arrangement.
5. Peter is not in the first house.
6. The person's child is named Fred and the person who loves the boquet of lilies are next to each other.
7. Arnold is the person who is the mother of Timothy.
8. The person's child is named Bella is somewhere to the right of Eric.
9. The person who is the mother of Timothy is not in the first house.
10. Eric is the person who loves the boquet of lilies.
11. There is one house between the person's child is named Meredith and the person's child is named Fred.
12. The person who loves the vase of tulips is in the second house.
 What is the value of attribute House for the person whose attribute Flower is tulips? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Carol`, `Bob`, `Eric`
 - Each person has an occupation: `artist`, `nurse`, `engineer`, `lawyer`, `teacher`, `doctor`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Aniya`, `Janelle`, `Sarah`, `Kailyn`
 - People have unique favorite music genres: `pop`, `rock`, `classical`, `jazz`, `hip hop`, `country`

## Clues:
1. The person who is an engineer is directly left of the person who is an artist.
2. The person whose mother's name is Janelle is the person who is an artist.
3. The person who loves classical music is directly left of Peter.
4. The person whose mother's name is Janelle and the person who loves pop music are next to each other.
5. The person whose mother's name is Sarah is somewhere to the left of Arnold.
6. The person who loves country music is The person whose mother's name is Aniya.
7. The person whose mother's name is Penny is not in the second house.
8. Bob is the person who loves jazz music.
9. Carol is not in the sixth house.
10. The person who is an engineer is the person who loves classical music.
11. The person who is a nurse is somewhere to the right of the person who loves rock music.
12. There are two houses between Bob and The person whose mother's name is Penny.
13. The person who is a lawyer is directly left of the person who is a doctor.
14. The person whose mother's name is Penny is not in the third house.
15. The person who is an engineer is not in the fourth house.
16. The person who is a doctor is The person whose mother's name is Kailyn.
17. Carol is somewhere to the right of the person who loves country music.
18. The person who loves classical music is Alice.
19. The person who is a nurse is The person whose mother's name is Holly.
 What is the value of attribute MusicGenre for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: jazz


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
 - People own unique car models: `toyota camry`, `bmw 3 series`, `ford f150`, `honda civic`, `tesla model 3`
 - Each person has a unique favorite drink: `tea`, `milk`, `water`, `root beer`, `coffee`
 - The people are of nationalities: `dane`, `brit`, `german`, `norwegian`, `swede`
 - People use unique phone models: `oneplus 9`, `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `huawei p50`

## Clues:
1. The person who uses a Google Pixel 6 is somewhere to the right of the Dane.
2. The person who uses a Google Pixel 6 is somewhere to the right of the root beer lover.
3. The person who owns a Tesla Model 3 is the British person.
4. The person who owns a Ford F-150 is the Dane.
5. Eric is the tea drinker.
6. The British person is directly left of the person who owns a Honda Civic.
7. Peter is somewhere to the right of the person who likes milk.
8. Bob is directly left of the Swedish person.
9. The person who uses an iPhone 13 is somewhere to the right of the person who uses a Huawei P50.
10. Arnold is the person who uses a Google Pixel 6.
11. Eric is not in the third house.
12. The person who owns a BMW 3 Series is somewhere to the right of the person who uses a Samsung Galaxy S21.
13. The person who owns a Tesla Model 3 is not in the first house.
14. Alice is somewhere to the left of the person who owns a Ford F-150.
15. The person who uses a OnePlus 9 is somewhere to the right of the person who uses a Samsung Galaxy S21.
16. The German is somewhere to the right of the person who owns a Tesla Model 3.
17. Eric is not in the fourth house.
18. Peter is directly left of the person who uses a Samsung Galaxy S21.
19. The coffee drinker is Bob.
 What is the value of attribute Name for the person whose attribute CarModel is ford f150? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Eric`, `Peter`, `Arnold`
 - Each person has a unique type of pet: `hamster`, `bird`, `fish`, `dog`, `cat`
 - Each person has a unique favorite drink: `coffee`, `milk`, `root beer`, `water`, `tea`
 - People have unique favorite sports: `soccer`, `swimming`, `tennis`, `baseball`, `basketball`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `iphone 13`, `huawei p50`, `samsung galaxy s21`

## Clues:
1. The root beer lover is the person who owns a dog.
2. The person who uses an iPhone 13 is the root beer lover.
3. There is one house between the person who owns a dog and the person who uses a Google Pixel 6.
4. Bob is somewhere to the right of the person who uses a OnePlus 9.
5. Alice is not in the fourth house.
6. Eric is in the fifth house.
7. The person who likes milk is the person who uses a Samsung Galaxy S21.
8. The person who loves baseball is the person who uses a Google Pixel 6.
9. The coffee drinker is in the third house.
10. Peter is the tea drinker.
11. The person who loves basketball is the person who owns a dog.
12. The person with a pet hamster is the person who loves tennis.
13. There are two houses between Alice and the tea drinker.
14. The person who loves soccer is the tea drinker.
15. The person who keeps a pet bird is directly left of the person with an aquarium of fish.
16. The person with an aquarium of fish is the person who likes milk.
 What is the value of attribute Drink for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: tea


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Peter`, `Carol`, `Alice`, `Eric`
 - Everyone has something unique for lunch: `stir fry`, `stew`, `soup`, `pizza`, `grilled cheese`, `spaghetti`
 - Each person has an occupation: `doctor`, `artist`, `engineer`, `teacher`, `lawyer`, `nurse`
 - Each person has a unique type of pet: `dog`, `bird`, `hamster`, `cat`, `fish`, `rabbit`
 - People have unique favorite music genres: `pop`, `country`, `jazz`, `classical`, `rock`, `hip hop`
 - Everyone has a favorite smoothie: `blueberry`, `desert`, `watermelon`, `cherry`, `lime`, `dragonfruit`

## Clues:
1. The person who loves the spaghetti eater is the Dragonfruit smoothie lover.
2. The person who drinks Lime smoothies is in the second house.
3. Carol is somewhere to the left of the Desert smoothie lover.
4. The person who is an engineer is not in the second house.
5. The person who loves the soup is in the second house.
6. The person who owns a dog is not in the second house.
7. The person who is a pizza lover is Alice.
8. Arnold is somewhere to the left of Eric.
9. The person who loves classical music is the Dragonfruit smoothie lover.
10. The person who owns a rabbit is somewhere to the left of the Desert smoothie lover.
11. The person who is a nurse is not in the sixth house.
12. There are two houses between the person who is a nurse and the person who loves the spaghetti eater.
13. The person who likes Cherry smoothies is the person with an aquarium of fish.
14. The person who has a cat is Alice.
15. The person who likes Cherry smoothies is somewhere to the right of the Desert smoothie lover.
16. There is one house between the person who drinks Blueberry smoothies and the person who loves eating grilled cheese.
17. The person who loves classical music is somewhere to the left of the person who loves the stew.
18. The person who is a lawyer is the person who loves classical music.
19. The person who loves jazz music is Bob.
20. Bob is the person who drinks Lime smoothies.
21. The person who loves pop music is the person who is a teacher.
22. The person who is an engineer is directly left of the person who loves rock music.
23. The person who loves hip-hop music is not in the first house.
24. The person with an aquarium of fish is Peter.
25. The person who drinks Blueberry smoothies is the person who keeps a pet bird.
26. The person who has a cat is in the first house.
27. The person who is a doctor is not in the sixth house.
28. The person who owns a rabbit is not in the second house.
29. Carol is not in the fourth house.
 What is the value of attribute Food for the person whose attribute Pet is cat? Please reason step by step, and put your final answer within \boxed{}

Answer: pizza


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`, `Alice`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `pizza`, `stew`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `oneplus 9`
 - Each person has a unique hobby: `painting`, `cooking`, `photography`, `gardening`

## Clues:
1. Peter is the person who is a pizza lover.
2. The person who is a pizza lover and the person who loves the spaghetti eater are next to each other.
3. The person who loves eating grilled cheese is directly left of Eric.
4. Eric is the person who loves cooking.
5. The person who uses a Google Pixel 6 is in the fourth house.
6. The photography enthusiast is in the second house.
7. The person who is a pizza lover is the person who uses a OnePlus 9.
8. The person who uses a Samsung Galaxy S21 is in the third house.
9. Arnold is directly left of the person who enjoys gardening.
 What is the value of attribute Food for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: 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`, `Bob`, `Eric`, `Arnold`
 - The people keep unique animals: `fish`, `dog`, `horse`, `bird`, `cat`
 - People own unique car models: `honda civic`, `bmw 3 series`, `ford f150`, `toyota camry`, `tesla model 3`
 - Each person has a unique favorite drink: `coffee`, `tea`, `root beer`, `milk`, `water`

## Clues:
1. Arnold is not in the fifth house.
2. Alice and Bob are next to each other.
3. The person who owns a Tesla Model 3 is the person who keeps horses.
4. There are two houses between Arnold and the root beer lover.
5. Eric is in the second house.
6. The person who likes milk is the dog owner.
7. Arnold is the person who owns a Tesla Model 3.
8. The bird keeper is the person who owns a Honda Civic.
9. The one who only drinks water and the person who owns a Toyota Camry are next to each other.
10. The one who only drinks water is directly left of the cat lover.
11. There is one house between the person who owns a Ford F-150 and the fish enthusiast.
12. The dog owner is not in the fifth house.
13. The tea drinker is Bob.
 What is the value of attribute Name for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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 Height for the person whose attribute Cigar is blue master? Please reason step by step, and put your final answer within \boxed{}

Answer: very short


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Carol`, `Peter`, `Bob`
 - Each person has a unique level of education: `high school`, `trade school`, `bachelor`, `doctorate`, `master`, `associate`
 - Each person has a favorite color: `purple`, `blue`, `white`, `red`, `green`, `yellow`
 - Each person prefers a unique type of vacation: `cultural`, `cruise`, `beach`, `mountain`, `camping`, `city`
 - People have unique favorite book genres: `science fiction`, `biography`, `historical fiction`, `fantasy`, `romance`, `mystery`

## Clues:
1. Alice is the person with a master's degree.
2. There is one house between the person with an associate's degree and Peter.
3. The person who loves historical fiction books is somewhere to the left of the person who loves biography books.
4. The person who loves historical fiction books is the person with a high school diploma.
5. Bob is the person with a bachelor's degree.
6. Carol is somewhere to the right of Eric.
7. The person with a master's degree is somewhere to the right of the person who enjoys camping trips.
8. The person who loves science fiction books is somewhere to the left of the person who loves yellow.
9. The person who loves historical fiction books is somewhere to the left of the person whose favorite color is green.
10. The person with a high school diploma is not in the second house.
11. The person who loves beach vacations is the person with a doctorate.
12. The person who loves mystery books is somewhere to the left of the person with a bachelor's degree.
13. The person with a doctorate is in the first house.
14. The person with a high school diploma is directly left of the person who enjoys camping trips.
15. The person who loves romance books is not in the third house.
16. There is one house between the person who loves purple and the person with a bachelor's degree.
17. The person who loves biography books is the person whose favorite color is red.
18. There is one house between the person who loves fantasy books and the person who loves blue.
19. The person who goes on cultural tours is the person who loves blue.
20. Carol is not in the fifth house.
21. The person who loves purple is the person with a doctorate.
22. The person who enjoys mountain retreats is somewhere to the right of the person who loves science fiction books.
23. The person who prefers city breaks is the person whose favorite color is green.
 What is the value of attribute House for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Bob`, `Alice`, `Eric`
 - People own unique car models: `toyota camry`, `tesla model 3`, `bmw 3 series`, `honda civic`, `ford f150`
 - Each person has a favorite color: `blue`, `red`, `green`, `white`, `yellow`
 - The mothers' names in different houses are unique: `Janelle`, `Penny`, `Holly`, `Kailyn`, `Aniya`

## Clues:
1. The person who owns a Honda Civic is the person who loves yellow.
2. The person whose favorite color is red is the person who owns a Tesla Model 3.
3. The person who owns a BMW 3 Series is not in the fourth house.
4. The person whose mother's name is Aniya is the person who loves blue.
5. The person whose favorite color is green is Eric.
6. The person whose favorite color is red is somewhere to the left of the person who owns a Ford F-150.
7. Alice is directly left of Eric.
8. The person whose mother's name is Holly is in the first house.
9. Arnold is the person who loves white.
10. The person whose mother's name is Janelle is the person who loves white.
11. The person whose mother's name is Kailyn is Alice.
12. Arnold is somewhere to the left of Peter.
13. The person who owns a BMW 3 Series is Eric.
14. Bob and the person who owns a Ford F-150 are next to each other.
 What is the value of attribute CarModel for the person whose attribute Mother is Kailyn? Please reason step by step, and put your final answer within \boxed{}

Answer: honda civic


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

## Clues:
1. The person with a master's degree is the person partial to Pall Mall.
2. The person with a high school diploma is in the fourth house.
3. The person's child is named Samantha is in the first house.
4. The Norwegian and the person with a master's degree are next to each other.
5. The German is the Dunhill smoker.
6. The person who is the mother of Timothy is directly left of the person with a master's degree.
7. Arnold and the Dane are next to each other.
8. The Swedish person is the person who loves mystery books.
9. The Swedish person is Alice.
10. Arnold is the person with an associate's degree.
11. The person with a doctorate is in the second house.
12. Peter is somewhere to the right of the person who loves biography books.
13. The person who smokes many unique blends is Alice.
14. The person who loves fantasy books is in the first house.
15. Peter is the Norwegian.
16. The person with a doctorate is the person's child is named Meredith.
17. The Prince smoker is somewhere to the left of the person who smokes many unique blends.
18. The person partial to Pall Mall is the person's child is named Bella.
19. The person who loves romance books is Bob.
 What is the value of attribute Nationality for the person whose attribute Children is Samantha? 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: `Eric`, `Arnold`, `Bob`, `Peter`, `Alice`, `Carol`
 - People have unique heights: `very tall`, `tall`, `average`, `short`, `super tall`, `very short`
 - Each person has a unique birthday month: `sept`, `may`, `jan`, `april`, `feb`, `mar`
 - People have unique hair colors: `blonde`, `red`, `black`, `auburn`, `gray`, `brown`
 - People have unique favorite book genres: `science fiction`, `romance`, `biography`, `mystery`, `fantasy`, `historical fiction`
 - Each person has a unique type of pet: `bird`, `hamster`, `fish`, `dog`, `cat`, `rabbit`

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

Answer: rabbit


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`
 - People have unique favorite book genres: `romance`, `mystery`, `fantasy`, `science fiction`
 - People have unique heights: `tall`, `very short`, `short`, `average`
 - Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `desert`, `watermelon`
 - Each person has a favorite color: `red`, `green`, `yellow`, `white`

## Clues:
1. Arnold is not in the fourth house.
2. Peter is the person who loves mystery books.
3. The person who likes Cherry smoothies is somewhere to the right of the Desert smoothie lover.
4. The person who loves science fiction books is directly left of Eric.
5. The Desert smoothie lover is the person who is very short.
6. The person who loves white is the person who loves mystery books.
7. The person who loves fantasy books is the person who loves yellow.
8. The Desert smoothie lover is somewhere to the right of the person who is short.
9. The Dragonfruit smoothie lover and the person whose favorite color is green are next to each other.
10. The Dragonfruit smoothie lover is the person who loves mystery books.
11. The person who loves yellow is in the second house.
12. The person who likes Cherry smoothies is not in the third house.
13. The person who has an average height is the person who likes Cherry smoothies.
 What is the value of attribute Smoothie for the person whose attribute Height is short? Please reason step by step, and put your final answer within \boxed{}

Answer: watermelon


8. A thin beam of light falls normally on a plane-parallel glass plate. Behind the plate, at some distance from it, stands an ideal mirror (its reflection coefficient is equal to one). The plane of the mirror is parallel to the plate. It is known that the intensity of the beam that has passed through this system is 16 times less than the intensity of the incident beam. The reflection coefficient at the glass-air boundary is considered constant regardless of the direction of the beam. Neglect absorption and scattering of light in air and glass. Find the reflection coefficient at the glass-air boundary under these conditions. (10 points)

Answer: \dfrac{1}{2}


## Task 21/72

Given the equation $x^{4}+x^{3}+x^{2}+(62-k) x+k=0$ with the real solutions $x_{i}(i=1 ; 2 ; 3 ; 4)$. What real values can $k$ take if

$$
\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\frac{1}{x_{4}}>5
$$

is to be satisfied?

Answer: (-\frac{31}{2}, 0)


Example 3: On a plane, there are $n$ lines, no two of which are parallel, and no three of which intersect at the same point. How many intersection points do these $n$ lines have in total?
Example 4: Following the previous example, now we want to know how many non-overlapping regions will be formed by such $n$ lines dividing the plane?

Answer: \dfrac{n(n + 1)}{2} + 1


6. (10 points) A convoy of trucks is delivering supplies to a disaster victim resettlement point. Each truck has a carrying capacity of 10 tons. If each tent is allocated 1.5 tons of supplies, there will be less than one truck's worth of supplies left over. If each tent is allocated 1.6 tons of supplies, there will be a shortage of more than 2 trucks' worth of supplies. How many tents are there at the resettlement point at a minimum?

Answer: 213


Folkpor

In a convex quadrilateral $A B C D: A C \perp B D, \angle B C A=10^{\circ}, \angle B D A=20^{\circ}, \angle B A C=40^{\circ}$. Find $\angle B D C$.

Answer: 60^\circ


8. Let the rational number $r=\frac{p}{q} \in(0,1)$, where $p, q$ are coprime positive integers, and $pq$ divides 3600. The number of such rational numbers $r$ is $\qquad$ .

Answer: 112


12.313. The height of the cone is $H$, and the angle between the generatrix and the base plane is $\alpha$. The total surface area of this cone is divided in half by a plane perpendicular to its height. Find the distance from this plane to the base of the cone.

Answer: H \left(1 - \cos \frac{\alpha}{2}\right)


15. The teachers and students of a township primary school went to the county town for a visit. It was stipulated that the bus would depart from the county town and arrive at the school at 7:00 AM to pick up the visiting teachers and students and immediately head to the county town. However, the bus broke down on its way to the school and had to stop for repairs. The teachers and students at the school waited until 7:10 AM but still did not see the bus, so they started walking towards the county town. On their way, they met the repaired bus, immediately got on, and headed to the county town, arriving 30 minutes later than the originally scheduled time. If the speed of the bus is 6 times the walking speed, how long did the bus spend on the road for repairs?

Answer: 38


7. Given that the modulus of the complex number $z$ is 1, if when $z=z_{1}$ and $z=z_{2}$, $|z+1+\mathrm{i}|$ attains its maximum and minimum values respectively, then $z_{1}-z_{2}=$ $\qquad$ .

Answer: \sqrt{2}(1+i)


Example 3 In the tetrahedron $A-B C D$, $\angle D A B+$ $\angle B A C+\angle D A C=90^{\circ}$, and $\angle A D B=\angle B D C$ $=\angle A D C=90^{\circ}$. If $D B=a, D C=b$, try to calculate the volume of the tetrahedron $A-B C D$.

Answer: \dfrac{ab(a + b)}{6}


Problem 6. Calculate the maximum number of distinct real roots that a polynomial $P$ can have, which satisfies the following property: the product of two distinct roots of $P$ is still a root of $P$.

Answer: 4


# 1.1. Condition:

Twelve figures are made of matches - 3 triangles, 4 squares, and 5 pentagons. The figures have no common sides. Petya and Vasya take turns removing one match at a time. Vasya wants to leave as few untouched figures as possible, while Petya wants to leave as many untouched figures as possible. How many figures will remain after 10 moves? Each of the boys makes 5 moves, with Petya starting first.

Answer: 6


Answer: \dfrac{9 + \sqrt{3}}{9}


(Infinite Solitaire)(^) You all know the rules of solitaire: there are marbles on a board, and you eliminate marbles by jumping over them with another marble, etc. Now consider the plane and place marbles on all integer points of the negative half-plane. The goal of the game is to place a marble as high as possible in a finite number of moves. The drawing shows how to place a marble at height 1. What is the maximum height that can be reached?

![](https://cdn.mathpix.com/cropped/2024_05_10_59677aa68548da326d08g-019.jpg?height=369&width=1459&top_left_y=1873&top_left_x=214)

- Solutions to the exercises -

Answer: 4


9. 2 (IMO 18) In a planar convex quadrilateral with an area of $32 \mathrm{~cm}^{2}$, the sum of one diagonal and two opposite sides is $16 \mathrm{~cm}$. Determine all possible lengths of the other diagonal.

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

---

Note: The provided text is already translated. If you meant to request the translation again, here it is:

9. 2 (IMO 18) In a planar convex quadrilateral with an area of $32 \mathrm{~cm}^{2}$, the sum of one diagonal and two opposite sides is $16 \mathrm{~cm}$. Determine all possible lengths of the other diagonal.

Answer: 8\sqrt{2}


11. Let the plane point sets be
$$
\begin{array}{l}
A=\left\{(x, y) \left\lvert\,(y-x)\left(y-\frac{18}{25 x}\right) \geqslant 0\right.\right\}, \\
B=\left\{(x, y) \mid(x-1)^{2}+(y-1)^{2} \leqslant 1\right\} .
\end{array}
$$

If $(x, y) \in A \cap B$, find the minimum value of $2 x-y$.

Answer: -1


4. A circle is inscribed with 1001 numbers. For any two adjacent numbers $x$ and $y$, the inequalities $|x-y| \geqslant 4, x+y \geqslant 6$ are satisfied. Find the smallest possible sum of the recorded numbers.

Answer: 3009


Let $G$ be any convex region tightly sandwiched between parallel lines $l_{1}$ and $l_{2}$ (i.e., the line segment connecting any two points on its boundary is contained within its region), and its boundary $C$ intersects with $l_{1}$ and $l_{2}$.
Parallel to $l_{1}$, the line $l$ divides $G$ into two parts $A$ and $B$ as shown in the figure, and the distances between $l$ and $l_{1}$ and $l_{2}$ are $a$ and $b$, respectively.
(1) What shape should $G$ be for the ratio of the areas of $A$ and $B$, $\frac{S_{\mathrm{A}}}{S_{\mathrm{B}}}$, to reach its maximum? Explain your reasoning;
(2) Try to find the maximum value of $\frac{S_{\mathrm{A}}}{S_{\mathrm{B}}}$.

Answer: \left( \frac{a}{b} \right)^2 + 2 \left( \frac{a}{b} \right)


## Task 3 - 310613

Elke, Regina, Gerd, and Joachim are comparing their stamp collections. They notice:

(1) Joachim has more stamps than Gerd.

(2) Elke and Regina together have exactly as many stamps as Joachim and Gerd together.

(3) Elke and Joachim together have fewer stamps than Regina and Gerd together.

Determine whether these statements can only be satisfied by one specific order of the number of stamps Elke, Regina, Gerd, and Joachim have!

If this is the case, determine this order, sorted in descending numbers!

Answer: R > J > G > E


## Task 3 - 040623

A rectangular school garden is to be fenced. On each of the shorter sides, which are each 40 m long, there are 21 concrete posts, and on the longer sides, there are 15 more each. The distance between any two adjacent posts is the same. A gate will be installed between two of these posts.

What are the costs if 1 m of fence costs 9.50 MDN, 1 post costs 11.00 MDN, and the gate costs 64.00 MDN?

The thickness of the posts is not taken into account.

Answer: 3345


52. In the expression $x_{1}: x_{2}: \ldots: x_{n}$ for indicating the order of operations, parentheses are placed and the result is written as a fraction:

$$
\frac{x_{i_{1}} x_{i_{2}} \ldots x_{i_{k}}}{x_{i_{1}} x_{l_{2}} \ldots x_{i_{n-k}}}
$$

(every letter $x_{1}, x_{2}, \ldots, x_{n}$ stands either in the numerator or in the denominator of the fraction). How many different expressions can be obtained in this way for all possible ways of placing parentheses?

Answer: 2^{n-2}


63. Two circles touch each other internally at point $A$. From the center of the larger circle, a radius $O B$ is drawn, touching the smaller circle at point $C$. Find $\widehat{B A C}$.

Answer: 45^\circ


$162$ pluses and $144$ minuses are placed in a $30\times 30$ table in such a way that each row and each column contains at most $17$ signs. (No cell contains more than one sign.) For every plus we count the number of minuses in its row and for every minus we count the number of pluses in its column. Find the maximum of the sum of these numbers.

Answer: 2592


Pedrinho is playing with three triangular pieces with sides $(5,8,10),(5,10,12)$ and $(5,8,12)$ as shown in the drawing below. He can join two pieces by exactly gluing the sides of the same length. For example, he can join the side 10 of the first piece with the side 10 of the second, but he cannot join the side 10 of the first piece with the side 8 of the third, as they do not have the same length. What is the largest perimeter that Pedrinho can obtain by joining the three pieces?

![](https://cdn.mathpix.com/cropped/2024_05_01_80f7f1185898b39e7876g-19.jpg?height=157&width=345&top_left_y=1760&top_left_x=427)

10

![](https://cdn.mathpix.com/cropped/2024_05_01_80f7f1185898b39e7876g-19.jpg?height=157&width=411&top_left_y=1758&top_left_x=891)

12

![](https://cdn.mathpix.com/cropped/2024_05_01_80f7f1185898b39e7876g-19.jpg?height=114&width=412&top_left_y=1802&top_left_x=1416)

12

#

Answer: 49


14. A 31-digit integer, if each pair of adjacent digits of this integer is considered as a two-digit number, any such two-digit number can be divisible by 17 or 23. Additionally, this 31-digit integer contains only one 7. Then the sum of all the digits of this 31-digit number is $\qquad$

Answer: 151


6. In two baskets, there are apples. If we take out $\frac{5}{6}$ from the first basket and $\frac{3}{4}$ of the number of apples in the second basket, 20 apples will remain in each basket. What is the total number of apples, and how many will remain in both baskets together if we take out $\frac{2}{3}$ of the apples from the first basket and leave $\frac{3}{5}$ of the apples in the second basket?

Answer: 200


8. Petya saw a bank advertisement: "Deposit 'Super-Income' - up to $10 \%$ annually!". He became interested in this offer and found it on the bank's website. The conditions turned out to be slightly different, as reflected in the table.

| Interest period | from 1 to 3 months | from 4 to 6 months | from 7 to 9 months | from 10 to 12 months |
| :---: | :---: | :---: | :---: | :---: |
| Rate | $10 \%$ | $8 \%$ | $8 \%$ | $8 \%$ |

*     - interest is credited at the end of the period.

What is the actual effective annual interest rate of this deposit? Round your answer to the tenths.

Answer: 8.8


## Task 1 - 100611

A LPG had planted potatoes on two fields. A total of 810 t was harvested from the first field, and a total of 640 t from the second field. The average yield on the first field was 180 dt per ha, and on the second field, it was 200 dt per ha.

Which of the two fields has the larger area? By how many ares do the two areas differ?

Answer: 1300


[b]p1.[/b] Consider a parallelogram $ABCD$ with sides of length $a$ and $b$, where $a \ne b$. The four points of intersection of the bisectors of the interior angles of the parallelogram form a rectangle $EFGH$. A possible configuration is given below.
Show that $$\frac{Area(ABCD)}{Area(EFGH)}=\frac{2ab}{(a - b)^2}$$
[img]https://cdn.artofproblemsolving.com/attachments/e/a/afaf345f2ef7c8ecf4388918756f0b56ff20ef.png[/img]

[b]p2.[/b] A metal wire of length $4\ell$ inches (where $\ell$ is a positive integer) is used as edges to make a cardboard rectangular box with surface area $32$ square inches and volume $8$ cubic inches. Suppose that the whole wire is used.
(i) Find the dimension of the box if $\ell= 9$, i.e., find the length, the width, and the height of the box without distinguishing the different orders of the numbers. Justify your answer.
(ii) Show that it is impossible to construct such a box if $\ell = 10$.

[b]p3.[/b] A Pythagorean n-tuple is an ordered collection of counting numbers $(x_1, x_2,..., x_{n-1}, x_n)$ satisfying the equation $$x^2_1+ x^2_2+ ...+ x^2_{n-1} = x^2_{n}.$$
For example, $(3, 4, 5)$ is an ordinary Pythagorean $3$-tuple (triple) and $(1, 2, 2, 3)$ is a Pythagorean $4$-tuple.
(a) Given a Pythagorean triple $(a, b, c)$ show that the $4$-tuple $(a^2, ab, bc, c^2)$ is Pythagorean.
(b) Extending part (a) or using any other method, come up with a procedure that generates Pythagorean $5$-tuples from Pythagorean $3$- and/or $4$-tuples. Few numerical examples will not suffice. You have to find a method that will generate infinitely many such $5$-tuples.
(c) Find a procedure to generate Pythagorean $6$-tuples from Pythagorean $3$- and/or $4$- and/or $5$-tuples.

Note. You can assume without proof that there are infinitely many Pythagorean triples.

[b]p4.[/b] Consider the recursive sequence defined by $x_1 = a$, $x_2 = b$ and $$x_{n+2} =\frac{x_{n+1} + x_n - 1}{x_n - 1}, n \ge 1 .$$
We call the pair $(a, b)$ the seed for this sequence. If both $a$ and $b$ are integers, we will call it an integer seed.
(a) Start with the integer seed $(2, 2019)$ and find $x_7$.
(b) Show that there are infinitely many integer seeds for which $x_{2020} = 2020$.
(c) Show that there are no integer seeds for which $x_{2019} = 2019$.

[b]p5.[/b] Suppose there are eight people at a party. Each person has a certain amount of money. The eight people decide to play a game. Let $A_i$, for $i = 1$ to $8$, be the amount of money person $i$ has in his/her pocket at the beginning of the game. A computer picks a person at random. The chosen person is eliminated from the game and their money is put into a pot. Also magically the amount of money in the pockets of the remaining players goes up by the dollar amount in the chosen person's pocket. We continue this process and at the end of the seventh stage emerges a single person and a pot containing $M$ dollars. What is the expected value of $M$? The remaining player gets the pot and the money in his/her pocket. What is the expected value of what he/she takes home?

PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Answer: \dfrac{2ab}{(a - b)^2}


5. For which negative integer values of $n$ is the function $f$, defined by the equation

$$
f(x)=\cos 7 x \cdot \sin \frac{25 x}{n^{2}}
$$

a periodic function with period $T=7 \pi$?

Answer: -5


Given that \( p \) and \( r \) are constants with \( p > r > 0 \), and the sequence \( \{a_n\} \) is defined by \( a_1 = r \) and \( a_{n+1} = p a_n \). Additionally, it is given that \( r^{n+1} \) for \( n \in \mathbf{N} \). Find the general term \( a_n \).

Answer: \dfrac{r(p^n - r^n)}{p - r}


1. Find all pairs of integers $a, b$ such that the sum $a+b$ is a root of the equation $x^{2}+a x+b=0$.

Answer: (0, 0)


Problem 2. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that

$$
f\left(x^{2}+y+f(y)\right)=2 y+(f(x))^{2}
$$

for any $x, y \in \mathbb{R}$.

Answer: f(x) = x


Let $k>1$ be a positive integer and $n>2018$ an odd positive integer. The non-zero rational numbers $x_1,x_2,\ldots,x_n$ are not all equal and:
$$x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1}$$
Find the minimum value of $k$, such that the above relations hold.

Answer: 4


19. For what values of $k$ does the equation
$$
|x-2007|+|x+2007|=k
$$
have $(-\infty,-2007) \cup(2007,+\infty)$ as its solution set?

Answer: k > 4014


9. (16 points) Given complex numbers $z_{1}, z_{2}$ satisfy $\left|z_{1}\right|<1$, $\left|z_{2}\right|<1, z_{1}+z_{2}$ is a real number, and $z_{1}+z_{2}+z_{1} z_{2}=0$. Find the range of $z_{1} z_{2}$.

Answer: \left( -\frac{1}{2}, 1 \right)


In Figure 1, \( AEFD \) is a unit square. The ratio of the length of the rectangle \( ABCD \) to its width is equal to the ratio of the length of the rectangle \( BCF E \) to its width. If the length of \( AB \) is \( W \) units, find the value of \( W \).

Answer: \frac{1+\sqrt{5}}{2}


Each diagonal of a convex pentagon ABCDE cuts off a triangle with an area of one unit. Calculate the area of the pentagon ABCDE.

Answer: \frac{5+\sqrt{5}}{2}


Let \( S = \{1, 2, \cdots, 50\} \). Find the smallest natural number \( k \) such that in any \( k \)-element subset of \( S \), there exist two distinct numbers \( a \) and \( b \) satisfying \( (a+b) \mid ab \).

Answer: 39


Let \( f(x) = x^2 - r_2 x + r_3 \) for all real numbers \( x \), where \( r_2 \) and \( r_3 \) are some real numbers. Define a sequence \(\{ g_n \}\) for all nonnegative integers \( n \) by \( g_0 = 0 \) and \( g_{n+1} = f(g_n) \). Assume that \(\{ g_n \}\) satisfies the following three conditions:
(i) \( g_{2i} < g_{2i+1} \) and \( g_{2i+1} > g_{2i+2} \) for all \( 0 \leq i \leq 2011 \);
(ii) there exists a positive integer \( j \) such that \( g_{i+1} > g_i \) for all \( i > j \);
(iii) \(\{ g_n \}\) is unbounded. If \( A \) is the greatest number such that \( A \leq |r_2| \) for any function \( f \) satisfying these properties, find \( A \).

Answer: 2


In parallelogram \(ABCD\), the side \(AD\) is divided into equal parts by points \(A_1, A_2, \ldots, A_{2022}\). Point \(E_1\) is the intersection of lines \(BA_1\) and \(AC\). Determine what fraction of the diagonal \(AC\) is the segment \(AE_1\).

Answer: \frac{1}{2024}


3. Given triangle $\mathrm{ABC}: \mathrm{BK}, \mathrm{CL}$ - angle bisectors, M - the point of their intersection. It turns out that triangle $\mathrm{AMC}$ is isosceles, one of whose angles is 150 degrees. Find what the perimeter of triangle $\mathrm{ABC}$ can be, if it is known that $\mathrm{BK}=4-2 \sqrt{3}$.

Answer: 4


10. (15 points) A turtle and a rabbit are in a 1000-meter race. The rabbit's speed is 5 times that of the turtle. When they start from the starting line at the same time, the turtle runs continuously, while the rabbit runs to a certain point and then starts sleeping. When the rabbit wakes up, the turtle has already taken the lead. The rabbit then tries to catch up, but when the turtle reaches the finish line, the rabbit is still 10 meters behind. How many meters did the turtle run while the rabbit was sleeping?

Answer: 802


6. Given a positive integer $k$. If a sequence of subsets of the set of integers $\mathbf{Z}$, $I_{1} \supseteq I_{2} \supseteq \cdots \supseteq I_{k}$, satisfies that for $i=1,2, \cdots, k, 168$ $\in I_{i}$, and for any $x, y \in I_{i}$, we have $x-y \in I_{i}$ $\left(x, y\right.$ can be equal), then $\left(I_{1}, I_{2}, \cdots, I_{k}\right)$ is called a $k$-chain on $\mathbf{Z}$. How many $k$-chains are there on $\mathbf{Z}$? Please provide a reason.

Answer: \dfrac{(k + 1)^3 (k + 2)(k + 3)}{6}


5. Let the set $I=\{1,2, \cdots, 2020\}$. We define
$$
\begin{array}{l}
W=\{w(a, b)=(a+b)+a b \mid a, b \in I\} \cap I, \\
Y=\{y(a, b)=(a+b) \cdot a b \mid a, b \in I\} \cap I, \\
X=W \cap Y,
\end{array}
$$

as the "Wu" set, "Yue" set, and "Xizi" set, respectively. Find the sum of the largest and smallest numbers in the Xizi set.

Answer: 2020


Problem 2. In a small town, there is only one tram line. It is circular, and trams run along it in both directions. There are stops on the loop: Circus, Park, and Zoo. The journey from Park to Zoo via Circus is three times longer than not via Circus. The journey from Circus to Zoo via Park is twice as short as not via Park. Which route from Park to Circus is shorter - via Zoo or not via Zoo - and by how many times?

(A.V. Shapovalov)

Answer: 11


4. A sphere is inscribed in a regular triangular prism, touching all three sides and both bases of the prism. Find the ratio of the surface areas of the sphere and the prism.

Answer: \dfrac{2\pi\sqrt{3}}{27}


27.26. Provide an example of positive irrational numbers $a$ and $b$ such that the number $a^{b}$ is an integer.

## 27.6. Some Notable Limits

Answer: 3


10. As shown in Figure 2, the volume of tetrahedron $D A B C$ is $\frac{1}{6}$, and it satisfies $\angle A C B=45^{\circ}$, $A D+B C+\frac{A C}{\sqrt{2}}=3$.
Then $C D=$ . $\qquad$

Answer: \sqrt{3}


Let $ P(x)$ be a nonzero polynomial such that, for all real numbers $ x$, $ P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x)$. Determine the maximum possible number of real roots of $ P(x)$.

Answer: 4


# 2. Option 1

Masha wrote the number 547654765476 on a piece of paper. She erased several digits so that the resulting number is the largest possible multiple of 9. What is this number?

Answer: 5476547646


Example 4 In the complex plane, point $A$ corresponds to the complex number -2, and moving point $B$ is on a circle centered at the origin with a radius of 1. A regular triangle $ABC$ (with $A, B, C$ in counterclockwise order) is constructed with $AB$ as one side. Find the trajectory of moving point $C$.

Answer: |z + 1 - \sqrt{3}i| = 1


There are $281 n(n>3)$ football teams that have played $k$ matches, with each pair of teams playing at most one match, and there are no ties. It turns out that exactly one team remains undefeated. Additionally, it is found that for any two teams $A$ and $B$, there exists another team $C$ such that either $A$ beats $C$ and $B$ beats $C$; or $C$ beats $A$ and $C$ beats $B$. Find the minimum value of $k$, denoted as $f(n)$.

Answer: 2n - 2


Among the right-angled triangles with equal perimeter, determine the one into which the largest incircle can be drawn. What is the radius of the circle if the perimeter of the triangle is $k$?

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


6. In the Cartesian coordinate system $x O y$, it is known that points $A(6,4)、B(4,0)、C(0,3)$, and the line $l: y=k x$ bisects the area of $\triangle A B C$, $k=\frac{-b+\sqrt{c}}{a}(a、b、c$ are all positive integers, and $c$ has no square factors). Then $a+b+c=$ $\qquad$ .

Answer: 584


On the bases $A D$ and $B C$ of trapezoid $A B C D$, squares $A D E F$ and $B C G H$ are constructed, located outside the trapezoid. The diagonals of the trapezoid intersect at point $O$. Find the length of segment $A D$, if $B C=2, G O=7$, and $G F=18$.

Answer: \dfrac{22}{7}


6.2. How many minutes before $16:00$ was the angle between the hour and minute hands exactly the same in the previous instance?

Answer: \dfrac{240}{11}


3. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{7}{6}$, and for any positive integer $n$, the quadratic equation $a_{n} x^{2}-$ $a_{n+1} x+1=0$ has two roots $\alpha_{n}, \beta_{n}$ that satisfy $6 \alpha_{n}-2 \alpha_{n} \beta_{n}+6 \beta_{n}=3$. Then the general term formula for $\left\{a_{n}\right\}$ is $a_{n}=$ $\qquad$

Answer: \dfrac{2^{n+1} + 3}{3 \cdot 2^n}


4. A rectangle of 19 cm $\times$ 65 cm is divided by lines parallel to its sides into squares with a side of 1 cm. Into how many parts will this rectangle be divided if we also draw its diagonal?

Answer: 1318


Find the largest number $n$ having the following properties:

(a) No two digits of $n$ are equal.

(b) The number formed by reversing the digits of $n$ is divisible by 8 .

Remark. $n$ cannot start with 0 , but it can end with 0 .

Answer: 8697543210


Five, Given that line $l$ is tangent to the unit circle $\odot O$ at point $P$, point $A$ is on the same side of line $l$ as $\odot O$, and the distance from $A$ to line $l$ is $h(h>2)$, two tangents are drawn from point $A$ to $\odot O$, intersecting line $l$ at points $B$ and $C$. Find the product of the lengths of segments $P B$ and $P C$.

Answer: \dfrac{h}{h - 2}


32. Given an equilateral triangle with side 1. What is the smallest $d$ such that a segment of length $d$ can, by sliding its ends along the sides of the triangle, sweep over the entire triangle?

Answer: \dfrac{2}{3}


4. There are two fleas at the two endpoints of the line segment $[0, 1]$. Some points are marked within the line segment. Each flea can jump over the marked points such that the positions before and after the jump are symmetric about the marked point, and they must not jump out of the range of the segment $[0, 1]$. Each flea can independently jump once or stay in place, which counts as one step. How many steps are needed at minimum to ensure that the two fleas can always jump to the same small segment divided by the marked points on $[0, 1]$?

Answer: 2


5. Given the inequality $\sqrt{2}(2 a+3) \cos \left(\theta-\frac{\pi}{4}\right)+\frac{6}{\sin \theta+\cos \theta}-2 \sin 2 \theta<3 a+6$, for $\theta \in\left[0, \frac{\pi}{2}\right]$ to always hold. Find the range of $\theta$. (1st China Southeast Mathematical Olympiad)

Answer: (3, +\infty)


One, (20 points) Given the equation in terms of $x$
$$
x^{2}+2(a+2 b+3) x+\left(a^{2}+4 b^{2}+99\right)=0
$$

has no distinct real roots. How many ordered pairs of positive integers $(a, b)$ satisfy this condition?

Answer: 16


# Task 3. Maximum 20 points

At the conference "Economics of the Present," an intellectual tournament was held, in which more than 198 but fewer than 230 scientists, including doctors and candidates of sciences, participated. Within one match, participants had to ask each other questions and record correct answers within a certain time. Each participant played against each other exactly once. The winner of the match received one point, the loser received no points; in the event of a draw, both participants received half a point. At the end of the tournament, it turned out that in matches against doctors of sciences, each participant scored half of all their points. How many candidates of sciences participated in the tournament? Provide the smallest of all possible answers.

Answer: 105


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


86. What happened to the Goat?

- I have seen many of the most interesting trials in my life, - continued the White Knight. - I remember one particularly curious trial! I even remember all the defendants!

"At least some variety!" - thought Alice.

- It was a very nice trial! On the bench of the accused were three: a Goat, a Beetle, and a Mosquito.
- I remember them well! - exclaimed Alice, recalling her adventures with the mirror insects.
- The heaviest suspicions fell on the Goat, continued the White Knight. At first, the Goat accused one of the insects, but I don't remember whether it was the Beetle or the Mosquito. Then the Beetle accused either the Mosquito or the Goat, I don't remember exactly who. Then the Mosquito accused either the Beetle or the Goat. Unfortunately, I have forgotten who.

"What a task! - thought Alice. It seems I've gotten into another fine mess again!"

- I remember well, - continued the White Knight, that after hearing other testimonies, the court established that either the Goat was lying, or both insects were telling the truth. What exactly the court established, I don't remember, it could have been both.
- Was the Goat found guilty or not? - asked Alice.

![](https://cdn.mathpix.com/cropped/2024_05_21_5afa4e43618f5800e92dg-112.jpg?height=677&width=828&top_left_y=1829&top_left_x=637)

- I don't remember what happened, - answered the White Knight. - I only remember that the court either found the Goat guilty, or released him from custody, or did neither.
- But one of these three things must have happened! - Alice got angry. - Such things don't need to be remembered. It's just plain logic.
- You are absolutely right! - said the White Knight. And yet I remember it as if it were today!
- What else do you remember? - asked Alice.

A few days ago, I met the Man in White Paper, with whom you traveled on the train. He didn't manage to attend the trial, but he was very interested in everything that happened there, as he was personally acquainted with all the defendants. I told him everything you already know. In addition, at that time, I still remembered whether the Goat was lying or the insects were telling the truth. When I told all this to the Man in White Paper, he was able to determine through reasoning whether the Goat was found guilty, released from custody, or the court did not come to any definite decision.

So, was the Goat found guilty, released from custody, or did the court fail to reach a definite decision?

Answer: \text{The Goat was found guilty}


18. The diagram shows a semicircle with diameter $P Q$ inscribed in a rhombus $A B C D$. The rhombus is tangent to the arc of the semicircle in two places. Points $P$ and $Q$ lie on sides $B C$ and $C D$ of the rhombus respectively. The line of symmetry of the semicircle is coincident with the diagonal $A C$ of the rhombus. It is given that $\angle C B A=60^{\circ}$. The semicircle has radius 10 . The area of the rhombus can be written in the form $a \sqrt{b}$ where $a$ and $b$ are integers and $b$ is prime. What is the value of $a b+a+b ?$

Answer: 603


Дубанов И.С.

In five pots standing in a row, Rabbit poured three kilograms of honey (not necessarily into each and not necessarily equally). Winnie-the-Pooh can take any two adjacent pots. What is the maximum amount of honey that Winnie-the-Pooh can guarantee to eat?

#

Answer: 1


4. [4] Circle $\omega$ has radius 5 and is centered at $O$. Point $A$ lies outside $\omega$ such that $O A=13$. The two tangents to $\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $B C$ is tangent to $\omega$ and $\omega$ lies outside triangle $A B C$. Compute $A B+A C$ given that $B C=7$.

Answer: 17


7th CanMO 1975 Problem 2 Define the real sequence a 1 , a 2 , a 3 , ... by a 1 = 1/2, n 2 a n = a 1 + a 2 + ... + a n . Evaluate a n .

Answer: \dfrac{1}{n(n+1)}


(Hungary 2004). A palace has the shape of a square divided into $2003 \times 2003$ rooms, like the squares of a large chessboard. There is a door between two rooms if and only if they share a wall. The main door allows, coming from outside the palace, to enter the palace through the room located at the northwest corner. A person enters the palace, visits some of the rooms, then leaves the palace, through the entrance door, when they return to the northwest corner room for the first time. It turns out that they visited each of the other rooms exactly 100 times, except for the room located at the southeast corner. How many times did the visitor go into the southeast corner room?

Answer: 99


$$
\begin{array}{l}
\text { Three. (20 points) (1) The quadratic function } \\
f(x)=a x^{2}+b x+c(a, b, c \in \mathbf{R}, a \neq 0)
\end{array}
$$

satisfies
(i) For $x \in \mathbf{R}$, there is $4 x \leqslant f(x) \leqslant \frac{1}{2}(x+2)^{2}$

always holds;
(ii) $f(-4+2 \sqrt{3})=0$.
Find $f(x)$.
(2) Let $f_{1}(x)=\frac{3}{2+x}, f_{n+1}(x)=f_{1}\left(f_{n}(x)\right)$, find $f_{2009}(0)$.

Answer: \dfrac{1}{3}x^2 + \dfrac{8}{3}x + \dfrac{4}{3}


A building has \( n \) floors and two staircases that go from the first floor to the last floor. On each staircase, between every two floors, there is a door at an intermediate landing that separates the floors (it is possible to access the floors from the staircase even if the door is locked). The superintendent decided that too many open doors is not good and therefore locked exactly half of the doors, choosing them randomly. What is the probability that it is possible to go from the first floor to the last floor using only open doors?

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


For all integers $n\geq 1$ we define $x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2$, where $x_1$ is a positive integer. Find the least $x_1$ such that 2006 divides $x_{2006}$.

Answer: 531


[ [tangents drawn from one point] [ Midline of a triangle $\quad$]

A circle of radius 1 is inscribed in triangle $ABC$, where $\cos \angle B=0.8$. This circle touches the midline of triangle $ABC$, parallel to side $AC$. Find the side $AC$.

Answer: 3


II Let $a, d \geqslant 0, b, c>0$ and $b+c \geqslant a+d$. Find the minimum value of $\frac{b}{c+d}+\frac{c}{a+b}$.

Answer: \dfrac{2\sqrt{2} -1}{2}


The side of the base of a regular triangular prism $A B C A_1 B_1 C_1$ is 4, and the lateral edge is 3. On the edge $B B_1$, a point $F$ is taken, and on the edge $C C_1$, a point $G$ is taken such that $B_1 F=1, C G=\frac{2}{3}$. Points $E$ and $D$ are the midpoints of edges $A C$ and $B_1 C_1$ respectively. Find the minimum possible value of the sum $E P+P Q$, where point $P$ belongs to the segment $A_1 D$, and point $Q$ belongs to the segment $F G$.

Answer: \dfrac{\sqrt{102}}{2}


10.138. Three equal circles of radius $r$ touch each other pairwise. Calculate the area of the figure located outside the circles and bounded by their arcs between the points of tangency.

Answer: \left( \sqrt{3} - \frac{\pi}{2} \right) r^2


For any positive integer, if the number of $2$'s in its digits is greater than the number of $3$'s in its digits, we call that is a [b]good[/b] number. And if the number of $3$'s in its digits is more than the number of $2$'s in its digits, we call that is a [b]bad[/b] number. For example, there are two $2$'s and one $3$ in the number $2023$, so $2023$ is a good number. But in the number $123$, the number of $2$ and $3$ are both one, so $123$ is neither a good number nor a bad number.

Find the difference of numbers of good numbers and bad numbers among the positive integer not greater than $2023$.

Answer: 22


10.023. Given a square, two of its vertices lie on a circle of radius $R$, and the other two lie on a tangent to this circle. Find the length of the diagonal of the square.

Answer: \dfrac{8\sqrt{2}}{5} R


9. (10 points) The Fifth Plenary Session of the 18th Central Committee of the Communist Party of China, which concluded on October 29, 2015, determined the policy allowing universal second children. Xiao Xiao's father, after seeing the news on that day, told Xiao Xiao: The total age of our family this year is 7 times your age. If we add a younger brother or sister next year, the total age of our family in 2020 will be 6 times your age at that time. How old is Xiao Xiao this year? $\qquad$ years old.

Answer: 11


7. Given that the line $y=x$ intersects the cosine curve $y=\cos x$ at point $A$. Then the distance from the origin $O$ to point $A$, $|O A|=$ (accurate to $10^{-4}$ ).

Answer: 1.0452


One, (40 points) Integers $a, b, c, d$ satisfy $ad - bc = 1$. Find the minimum value of $a^2 + b^2 + c^2 + d^2 + ab + cd - ac - bd - bc$, and determine all quadruples $(a, b, c, d)$ that achieve this minimum value.

Answer: 2


10.5. Uncle Chernomor assigns 9 or 10 of his thirty-three bogatyrs (knights) to duty each evening. What is the smallest number of days after which it can happen that all bogatyrs have been on duty the same number of times?

Answer: 7


Each pair $(x, y)$ of nonnegative integers is assigned number $f(x, y)$ according the conditions:
$f(0, 0) = 0$;
$f(2x, 2y) = f(2x + 1, 2y + 1) = f(x, y)$,
$f(2x + 1, 2y) = f(2x, 2y + 1) = f(x ,y) + 1$ for $x, y \ge 0$.
Let $n$ be a fixed nonnegative integer and let $a$, $b$ be nonnegative integers such that $f(a, b) = n$. Decide how many numbers satisfy the equation $f(a, x) + f(b, x) = n$.

Answer: 2^n


67. As shown in the figure, in isosceles $\triangle A B C$, $\angle A C B=90^{\circ}, M, N$ are two points on the hypotenuse $A B$, and $\angle M C N=45^{\circ}$. Given $A M=3, B N=5$, then $M N=$ $\qquad$ -

Answer: \sqrt{34}


Answer: 1742


10.6. A square $100 \times 100$ is divided into squares $2 \times 2$. Then it is divided into dominoes (rectangles $1 \times 2$ and $2 \times 1$). What is the smallest number of dominoes that could end up inside the squares of the division?

(C. Berlov)

#

Answer: 100


# Problem 1. (2 points)

Let $x, y, z$ be pairwise coprime three-digit natural numbers. What is the greatest value that the GCD $(x+y+z, x y z)$ can take?

Answer: 2994


4*. Find all such positive integers $k$ for which the number

$$
\underbrace{1 \ldots 1 \overbrace{2 \ldots 2}^{k}}_{2000}-\underbrace{2 \ldots 2}_{1001}
$$

is a perfect square.

Answer: 2


Triangle  $ABC$  is the right angled triangle with the vertex  $C$  at the right angle. Let  $P$  be the point of reflection of  $C$  about  $AB$ . It is known that  $P$  and two midpoints of two sides of  $ABC$  lie on a line. Find the angles of the triangle.

Answer: 30


A multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get  $-1$  mark. He will get 0 mark for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.)

Answer: 501


The cells of a  $5\times5$  grid are each colored red, white, or blue. Sam starts at the bottom-left cell of the grid and walks to the top-right cell by taking steps one cell either up or to the right. Thus, he passes through  $9$  cells on his path, including the start and end cells. Compute the number of colorings for which Sam is guaranteed to pass through a total of exactly  $3$  red cells, exactly  $3$  white cells, and exactly  $3$  blue cells no matter which route he takes.

Answer: 1680


$22$  football players took part in the football training. They were divided into teams of equal size for each game ( $11:11$ ). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training.

Answer: 5


A *chaisri* figure is a triangle which the three vertices are vertices of a regular  $2019$ -gon. Two different chaisri figure may be formed by different regular  $2019$ -gon.
A *thubkaew* figure is a convex polygon which can be dissected into multiple chaisri figure where each vertex of a dissected chaisri figure does not necessarily lie on the border of the convex polygon.

Determine the maximum number of vertices that a thubkaew figure may have.

Answer: 4038


Suppose that  $ABCDE$  is a convex pentagon with  $\angle A = 90^o,\angle B = 105^o,\angle C = 90^o$  and  $AB = 2,BC = CD = DE =\sqrt2$ . If the length of  $AE$  is  $\sqrt{a }- b$  where  $a, b$  are integers, what is the value of  $a + b$ ?

Answer: 4


Let the sequence  $u_n$  be defined by  $u_0=0$  and  $u_{2n}=u_n$ ,  $u_{2n+1}=1-u_n$  for each  $n\in\mathbb N_0$ .
(a) Calculate  $u_{1990}$ .
(b) Find the number of indices  $n\le1990$  for which  $u_n=0$ .
(c) Let  $p$  be a natural number and  $N=(2^p-1)^2$ . Find  $u_N$ .

Answer: 0


Let  $MATH$  be a square with  $MA = 1$ . Point  $B$  lies on  $AT$  such that  $\angle MBT = 3.5 \angle BMT$ . What is the area of  $\vartriangle BMT$ ?

Answer: \frac{\sqrt{3}-1}{2}


It is now 3:25:15 PM, as shown on a 12-hour digital clock. After adding 137 hours, 59 minutes, and 59 seconds to the current time, what is the sum of all digits shown on the clock at the new time?

Answer: 21


The real function $f$ has the property that whenever $c,$ $d,$ $m$ are positive integers such that $c + d = 3^m,$ the equation
\[ f(c) + f(d) = m^2 \] holds. What is $f(243)$?

Answer: 25


A square and a regular pentagon are coplanar and share a common side \( \overline{AD} \). Determine the degree measure of the exterior angle \( BAC \).

Answer: 162^\circ


In Zuminglish, all words consist only of the letters $M, O, P$. $O$ is a vowel and $M$ and $P$ are consonants. A valid word in Zuminglish requires that between any two $O's$ there must be at least two consonants. Define $N$ as the number of 8-letter words in Zuminglish. Find the remainder when $N$ is divided by 1000.

Answer: 368


In triangle $PQR,$ the side lengths $p,$ $q,$ and $r$ satisfy $p + 3q + 3r = p^2$ and $p + 3q - 3r = -1.$ Find the largest angle of the triangle, in degrees.

Note: $p$ is the side length opposite $\angle P,$ etc.

Answer: 120^\circ


An equilateral triangle is divided into two 30-60-90 triangles by drawing a line from one vertex to the midpoint of the opposite side. If the length of the median to the longest side (hypotenuse in the 30-60-90 triangle) is $15$ units, what is the length of the shortest side of one of the 30-60-90 triangles, in units?

Answer: 15 \text{ units}


Three distinct vertices of a decagon are chosen at random. What is the probability that all three vertices form a side of the decagon?

Answer: \frac{1}{12}


The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $s$. Four vertical edges connect the base to a point directly above the center of the base at height $h$. Given that $s = 8$ and $h = 6$, what is the volume of the solid?

Answer: 128


Consider two shapes made up of unit squares. The first shape is a rectangle 4 units long and 3 units wide. Inside this rectangle, an additional vertical line divides one of its halves into two rectangles each 2 units long. The second shape is a simple 6x1 rectangle. What is the positive difference in their perimeters?

Answer: 0


According to the given data, what was the average daily high temperature in Brookdale from October 1st, 2023 through October 7th, 2023, inclusive? Daily high temperatures in degrees Fahrenheit during that period were as follows: 51, 67, 64, 61, 50, 65, 47.

Answer: 57.9


Alice chooses randomly two integers between 1 and 100, inclusive. What is the probability that at least one of the integers Alice chooses is a multiple of 4?

Answer: \frac{7}{16}


Consider four function graphs, labelled (2) through (5). The domain of function (3) is now $$\{-5, -4, -3, -2, -1, 0, 1, 2, 3\}.$$ Determine the product of the labels of the functions which are invertible. The function expressions and domains for the other functions are:
- Function (2): $f(x) = x^2 - 4x + 3$ over the domain $[-1, 4]$
- Function (4): $g(x) = -\tan(x)$ over the domain $(-\frac{\pi}{2}, \frac{\pi}{2})$
- Function (5): $h(x) = 5/x$ over the domain $(-\infty, -0.2) \cup (0.2, \infty)$

Answer: 60


In triangle $ABC$, $AB$ is congruent to $AC$, and the measure of angle $ABC$ is $60^{\circ}$. Segment $BD$ bisects angle $ABC$ with point $D$ on side $AC$. Point $E$ is on side $BC$ such that segment $DE$ is parallel to side $AB$, and point $F$ is on side $AC$ such that segment $EF$ is parallel to segment $BD$ and $EF$ bisects $\angle DEC$. How many isosceles triangles are present?

Answer: 7


A botanical garden map shows that one inch represents 1200 feet on the ground. A line segment on the map is 6.5 inches long. An error in the map indicates a part of the path that doesn’t exist, which is shown as a 0.75-inch line segment. How many feet on the ground does the correct path cover?

Answer: 6900 \text{ feet}


Suppose we have an equilateral triangle with vertices at \( (0, 0) \), \( (6, 0) \), and \( (3, 3\sqrt{3}) \), and we place 36 points equally spaced along the perimeter (including the vertices and dividing each side into 12 congruent sections). If \( P \), \( Q \), and \( R \) are any three non-collinear points among these, how many different possible positions are there for the centroid of \( \triangle PQR \)?

Answer: 361


It takes 18 men working steadily 3 days to build a wall. If 3 of the men work twice as efficiently as the others, how many days would it take 30 men to build the same wall, assuming 5 of them are twice as efficient as the others?

Answer: 1.8


A square and a regular hexagon are coplanar and share a common side $\overline{AD}$. Determine the degree measure of exterior angle $BAC$. Use a diagram for reference if needed.

Answer: 150^\circ


A region is bounded by four quarter-circle arcs constructed at each corner of a square whose sides measure $4/\pi$. Calculate the perimeter of this region.

Answer: 4


A frustum of a right circular cone was generated by slicing a smaller cone from the top of a larger cone. The larger cone originally had a height of 30 cm before the top was cut off to create the frustum. Now, the frustum that remains has a height of 18 cm and the radius of its upper base is 6 cm, while the radius of its lower base is 10 cm. Determine the altitude of the smaller cone that was cut off.

Answer: 12 \text{ cm}


1. Determine all pairs $(p, q)$ of real numbers such that the equation $x^{2}+p x+q=0$ has a solution $v$ in the set of real numbers, and if $t$ is a root of this equation, then $|2 t-15|$ is also a root.

Answer: (-30, 225)


9. Given $z \in \mathbf{C}$. If the equation in $x$
$$
4 x^{2}-8 z x+4 i+3=0
$$

has real roots. Then the minimum value of $|z|$ is $\qquad$

Answer: 1


A circle passes through vertices $A$ and $B$ of triangle $A B C$ and is tangent to line $A C$ at point $A$. Find the radius of the circle if $\angle B A C=\alpha, \angle A B C=\beta$ and the area of triangle $A B C$ is $S$.

Answer: \sqrt{ \dfrac{S \sin(\alpha + \beta)}{2 \sin^3 \alpha \sin \beta} }


8. Given that $A B C D$ and $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ are two rhombuses with side lengths of $\sqrt{3}+1$. If $A C \perp A^{\prime} C^{\prime}$, $\angle A B C=\angle A^{\prime} B^{\prime} C^{\prime}=120^{\circ}$, then the perimeter of the shaded part is $\qquad$

Answer: 8


1. Arrange the consecutive natural numbers from 1 to 99 in sequence to form a large number:
1234567891011…979899,
By extracting four consecutive digits, you can get a four-digit number, such as 5678, 1011, etc. Among the four-digit numbers obtained by this method, the largest is . $\qquad$

Answer: 9909


## Task 2 - 010932

Kurt is riding a tram along a long straight street. Suddenly, he sees his friend walking in the opposite direction on this street at the same level. After one minute, the tram stops. Kurt gets off and runs after his friend at twice the speed of his friend, but only at a quarter of the average speed of the tram.

How many minutes will it take for him to catch up? How did you arrive at your result?

Answer: 9


Example 10. In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, $B D=10 \mathrm{~cm}, \angle D_{1} B D=30^{\circ}, \angle A O D=60^{\circ}$. Find the distance between $A C$ and $B D_{1}$ (Figure 18).

Answer: \dfrac{5\sqrt{39}}{13}


The medians $A A^{\prime}$ and $B B^{\prime}$ of triangle $A B C$ intersect at point $M$, and $\angle A M B = 120^{\circ}$. Prove that the angles $A B^{\prime} M$ and $B A^{\prime} M$ cannot both be acute or both be obtuse.

Answer: \text{The angles } AB'M \text{ and } BA'M \text{ cannot both be acute or both be obtuse.}


Let $n$ be a positive integer. Ana and Banana are playing the following game:
First, Ana arranges $2n$ cups in a row on a table, each facing upside-down. She then places a ball under a cup
and makes a hole in the table under some other cup. Banana then gives a finite sequence of commands to Ana,
where each command consists of swapping two adjacent cups in the row.
Her goal is to achieve that the ball has fallen into the hole during the game. Assuming Banana has no information
about the position of the hole and the position of the ball at any point, what is the smallest number of commands
she has to give in order to achieve her goal?

Answer: n(2n - 1)


XXXV OM - I - Problem 9

Three events satisfy the conditions:
a) their probabilities are equal,
b) any two of them are independent,
c) they do not occur simultaneously.
Determine the maximum value of the probability of each of these events.

Answer: \dfrac{1}{2}


1. In some cells of a $1 \times 2021$ strip, one chip is placed in each. For each empty cell, the number equal to the absolute difference between the number of chips to the left and to the right of this cell is written. It is known that all the written numbers are distinct and non-zero. What is the minimum number of chips that can be placed in the cells?

Answer: 1347


Nine cells of a $10 \times 10$ diagram are infected. At each step, a cell becomes infected if it was already infected or if it had at least two infected neighbors (among the 4 adjacent cells).

(a) Can the infection spread everywhere?

(b) How many initially infected cells are needed to spread the infection everywhere?

Answer: 10


5. As shown in Figure 1, the plane containing
square $A B C D$ and the plane containing
square $A B E F$ form a $45^{\circ}$ dihedral angle.
Then the angle formed by the skew lines $A C$ and
$B F$ is $\qquad$

Answer: \arccos \frac{2 - \sqrt{2}}{4}


4. Among 25 coins, two are counterfeit. There is a device into which two coins can be placed, and it will show how many of them are counterfeit. Can both counterfeit coins be identified in 13 tests.

Answer: Yes


Problem 3. Inside the cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, consider the regular quadrilateral pyramid $S A B C D$ with base $A B C D$, such that $\angle\left(\left(S A^{\prime} B^{\prime}\right),\left(S C^{\prime} D^{\prime}\right)\right)=30^{\circ}$. Let point $M$ be on the edge $A^{\prime} D^{\prime}$ such that $\angle A^{\prime} B^{\prime} M=30^{\circ}$.

a) Find the angle between the apothem of the pyramid $S A B C D$ and the plane $(A B C)$.

b) Determine the tangent of the angle between the planes $(MAB')$ and $(S A B)$.

Answer: 60^\circ


209. "Fibonacci Tetrahedron". Find the volume of the tetrahedron whose vertices are located at the points with coordinates $\left(F_{n}, F_{n+1}, F_{n+2}\right), \quad\left(F_{n+3}, F_{n+4}, F_{n+5}\right), \quad\left(F_{n+6}, F_{n+7}, F_{n+8}\right)$ and $\left(F_{n+9}, F_{n+10}, F_{n+11}\right)$, where $F_{i}$ is the $i$-th term of the Fibonacci sequence: $1,1,2,3,5,8 \ldots$.

Answer: 0


II. Positive integers $x_{1}, x_{2}, \cdots, x_{n}\left(n \in \mathbf{N}_{+}\right)$ satisfy $x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=111$.
Find the maximum possible value of $S=\frac{x_{1}+x_{2}+\cdots+x_{n}}{n}$.

Answer: \dfrac{21}{4}


Given the side \( a \) and angle \( \alpha \) of the triangle \( ABC \). Additionally, it is known that the length of the angle bisector from vertex \( C \) is also \( a \). Can the triangle be constructed using these data with Euclidean construction methods?

Answer: \text{No}


1. Given $\min _{x \in R} \frac{a x^{2}+b}{\sqrt{x^{2}+1}}=3$.
(1) Find the range of $b$;
(2) For a given $b$, find $a$.

Answer: [3, \infty)


A regular quadrilateral pyramid has a circumscribed sphere whose center coincides with the center of the inscribed sphere. What is the angle between two adjacent side edges?

Answer: 90^\circ


Problem 8.1. (15 points) In a $5 \times 5$ grid, each cell is painted one of three colors: red, blue, or green. To the right of each row, the total number of blue and red cells in that row is written, and below each column, the total number of blue and green cells in that column is written.

To the right of the table, the numbers $1,2,3,4,5$ appear in some order. Could the numbers $1,2,3,4,5$ also appear below the table in some order?

Answer: Yes


4. In a water-filled and tightly sealed aquarium in the shape of a rectangular parallelepiped measuring 3 m $\times 4$ m $\times 2$ m, there are two small balls: an aluminum one and a wooden one. At the initial moment, the aquarium is at rest, and the distance between the balls is 2 m. What is the greatest distance between the balls that can be observed if the aquarium starts to move with constant acceleration? Provide an example of the motion in which the maximum distance is achieved.

Answer: \sqrt{29}


Three, (50 points) In an $8 \times 8$ grid, each cell contains a knight or a knave. Knights always tell the truth, and knaves always lie. It is known that each resident in the grid says: “The number of knaves in my column is (strictly) greater than the number of knaves in my row.” Determine the number of arrangements of residents that satisfy this condition.

Answer: 255


14.9. (SFRY, 74). For arbitrary $n \geqslant 3$ points $A_{1}$, $A_{2}, \ldots, A_{n}$ on a plane, no three of which lie on the same line, let $\alpha$ denote the smallest of the angles $\angle A_{i} A_{j} A_{k}$ formed by triples $A_{i}, A_{j}, A_{k}$ of distinct points. For each value of $n$, find the greatest value of $\alpha$. Determine for which arrangements of points this value is achieved.

Answer: \dfrac{180^\circ}{n}


49. In $\triangle A B C$, the sides opposite to $\angle A, \angle B, \angle C$ are $a, b, c$ respectively. There is an inequality $\left|\frac{\sin A}{\sin A+\sin B}+\frac{\sin B}{\sin B+\sin C}+\frac{\sin C}{\sin C+\sin A}-\frac{3}{2}\right|<m$, then $m=\frac{8 \sqrt{2}-5 \sqrt{5}}{6}$ is the smallest upper bound.

Answer: \dfrac{8\sqrt{2} - 5\sqrt{5}}{6}


11.1. Anya left the house, and after some time Vanya left from the same place and soon caught up with Anya. If Vanya had walked twice as fast, he would have caught up with Anya three times faster. How many times faster would Vanya have caught up with Anya (compared to the actual time) if, in addition, Anya walked twice as slow

Answer: 7


Right triangle $A B C$ is divided by the height $C D$, drawn to the hypotenuse, into two triangles: $B C D$ and $A C D$. The radii of the circles inscribed in these triangles are 4 and 3, respectively. Find the radius of the circle inscribed in triangle $A B C$.

Answer: 5


Let $ A \equal{} \{(a_1,\dots,a_8)|a_i\in\mathbb{N}$ , $ 1\leq a_i\leq i \plus{} 1$ for each $ i \equal{} 1,2\dots,8\}$.A subset $ X\subset A$ is called sparse if for each two distinct elements $ (a_1,\dots,a_8)$,$ (b_1,\dots,b_8)\in X$,there exist at least three indices $ i$,such that $ a_i\neq b_i$.
   Find the maximal possible number of elements in a sparse subset of set $ A$.

Answer: 5040


43. There are 100 consecutive natural numbers. Please arrange them in a certain order, then calculate the sum of every three adjacent numbers, and find the maximum number of sums that are even. $\qquad$

Answer: 97


11.6. A stack consists of 300 cards: 100 white, 100 black, and 100 red. For each white card, the number of black cards lying below it is counted; for each black card, the number of red cards lying below it is counted; and for each red card, the number of white cards lying below it is counted. Find the maximum possible value of the sum of the three hundred resulting numbers.

Answer: 20000


Triangle $ABC$ satisfies $AB=104$, $BC=112$, and $CA=120$.  Let $\omega$ and $\omega_A$ denote the incircle and $A$-excircle of $\triangle ABC$, respectively.  There exists a unique circle $\Omega$ passing through $A$ which is internally tangent to $\omega$ and externally tangent to $\omega_A$.  Compute the radius of $\Omega$.

Answer: 49


22. Try to write two sets of integer solutions for the indeterminate equation $x^{2}-2 y^{2}=1$ as $\qquad$ .

Answer: (3, 2)


7、The volunteer service team delivers newspapers to elderly people with mobility issues in the community. Xiao Ma is responsible for an elderly person living on the 7th floor. It takes 14 seconds to go up or down one floor, so it takes Xiao Ma a total of $\qquad$ seconds for a round trip up and down.

Answer: 168


Example 6. Find the one-sided limits of the function $f(x)=\frac{6}{x-3}$ as $x \rightarrow 3$ from the left and from the right.

Answer: -\infty


Shaovalov A.v.

For which $n>2$ can the integers from 1 to $n$ be arranged in a circle so that the sum of any two adjacent numbers is divisible by the next number in the clockwise direction?

Answer: 3


In an isosceles triangle, the legs are equal to \(a\) and the base is equal to \(b\). An inscribed circle in this triangle touches its sides at points \(M\), \(N\), and \(K\). Find the area of triangle \(MNK\).

Answer: \dfrac{b^2 (2a - b) \sqrt{4a^2 - b^2}}{16 a^2}


Yamenniko i.v.

The numbers 2, 3, 4, ..., 29, 30 are written on the board. For one ruble, you can mark any number. If a number is already marked, you can freely mark its divisors and numbers that are multiples of it. What is the minimum number of rubles needed to mark all the numbers on the board?

Answer: 5


8. It is known that 99 wise men are seated around a large round table, each wearing a hat of one of two different colors. Among them, 50 people's hats are of the same color, and the remaining 49 people's hats are of the other color. However, they do not know in advance which 50 people have the same color and which 49 people have the other color. They can only see the colors of the hats on others' heads, but not their own. Now they are required to simultaneously write down the color of their own hat on the paper in front of them. Question: Can they pre-arrange a strategy to ensure that at least 74 of them write the correct color?

Answer: 74


22. There are four cards, each with a number written on both sides. The first card has 0 and 1, the other three cards have 2 and 3, 4 and 5, 7 and 8 respectively. Now, any three of these cards are taken out and placed in a row, forming a total of $\qquad$ different three-digit numbers.

Answer: 168


2. On a very long road, a race was organized. 22 runners started at different times, each running at a constant speed. The race continued until each runner had overtaken all the slower ones. The speeds of the runners who started first and last were the same, and the speeds of the others were different from theirs and distinct from each other.

What could be the number of overtakes, if each involved exactly two people? In your answer, indicate the largest and smallest possible numbers in any order, separated by a semicolon.

Example of answer format:

$10 ; 20$

Answer: 20 ; 210


## Task 4 - 260924

For a right triangle $A B C$ with the right angle at $C$, it is required that this right angle be divided into four equal angles by the median of side $A B$, the angle bisector of $\angle A C B$, and the altitude perpendicular to side $A B$.

Investigate whether there exists a triangle $A B C$ that meets these requirements, and whether all such triangles are similar to each other!

Determine, if this is the case, the measures of the angles $\angle B A C$ and $\angle A B C$!

Answer: 22.5^\circ


5. For different real numbers $m$, the equation $y^{2}-6 m y-4 x+$ $9 m^{2}+4 m=0$ represents different parabolas. A line intersects all these parabolas, and the length of the chord intercepted by each parabola is $\frac{8 \sqrt{5}}{9}$. Then the equation of this line is $\qquad$ .

Answer: y = 3x - \dfrac{1}{3}


7. Dijana used sticks of length $5 \mathrm{~cm}$ to form a row of equilateral triangles (as shown in the figure). If Dijana used 99 sticks, what is the distance between the two farthest points in the row of triangles formed?

![](https://cdn.mathpix.com/cropped/2024_06_03_acca65fe0c81619e2622g-04.jpg?height=197&width=916&top_left_y=1244&top_left_x=570)

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

## SCHOOL/CITY COMPETITION IN MATHEMATICS

Answer: 125


The sequence \(\{a_n\}\) is defined as follows: \(a_1\) is any positive integer. For integers \(n \geq 1\), \(a_{n+1}\) is the smallest positive integer that is coprime with \(\sum_{i=1}^{n} a_i\) and is different from \(a_1, \cdots, a_n\). Prove that every positive integer appears in the sequence \(\{a_n\}\).

Answer: \text{Every positive integer appears in the sequence }\{a_n\}


Problem 4. A field in the shape of a rectangle, with a length $20 \mathrm{~m}$ longer than its width, is fenced with 3 strands of wire, using a total of $840 \mathrm{~m}$ of wire. What is the perimeter and what are the lengths of the sides of the field?

Answer: 280


A kindergarten group consisting of 5 girls and 7 boys is playing "house." They select from among themselves a bride and groom, a mother, two flower girls, a best man for the bride and one for the groom, a bridesmaid for the bride and one for the groom. We know that three of the girls each have a brother in the group and there are no other sibling pairs. How many ways can the selection be made if siblings cannot be the bride and groom and the mother cannot be a sibling of either the bride or the groom? (Flower girls can only be girls, best men can only be boys, and there are no gender restrictions for the witnesses.)

Answer: 626400


10.228. One end of the diameter of a semicircle coincides with the vertex of the angle at the base of an isosceles triangle, and the other lies on this base. Find the radius of the semicircle if it touches one lateral side and divides the other into segments of lengths 5 and 4 cm, measured from the base.

Answer: \dfrac{15\sqrt{11}}{11}