There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Eric`, `Peter`, `Arnold`
 - People use unique phone models: `iphone 13`, `samsung galaxy s21`, `huawei p50`, `google pixel 6`, `oneplus 9`
 - People have unique favorite sports: `soccer`, `swimming`, `baseball`, `basketball`, `tennis`
 - Everyone has a favorite smoothie: `cherry`, `dragonfruit`, `lime`, `desert`, `watermelon`
 - Each person prefers a unique type of vacation: `city`, `cruise`, `beach`, `mountain`, `camping`
 - The people keep unique animals: `horse`, `fish`, `cat`, `dog`, `bird`

## Clues:
1. Peter is somewhere to the right of the person who uses an iPhone 13.
2. The person who loves tennis is directly left of the person who uses a Google Pixel 6.
3. There is one house between the Watermelon smoothie lover and the person who loves beach vacations.
4. The person who loves baseball is in the fourth house.
5. The dog owner is Peter.
6. There is one house between the person who prefers city breaks and the person who keeps horses.
7. The fish enthusiast is directly left of the cat lover.
8. Arnold is the Desert smoothie lover.
9. The person who likes Cherry smoothies is the person who enjoys mountain retreats.
10. The person who loves swimming is not in the first house.
11. Bob is the person who prefers city breaks.
12. The person who uses a Huawei P50 is the fish enthusiast.
13. The person who loves tennis is the person who enjoys mountain retreats.
14. The person who loves baseball is somewhere to the left of the person who loves soccer.
15. The person who uses a OnePlus 9 is Peter.
16. There is one house between the person who prefers city breaks and Alice.
17. The person who enjoys camping trips is the Dragonfruit smoothie lover.
18. The person who loves baseball is the person who enjoys camping trips.
19. The person who loves beach vacations is directly left of the person who enjoys camping trips.
20. Bob is the fish enthusiast.
 What is the value of attribute Animal for the person whose attribute Vacation is city? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - People own unique car models: `toyota camry`, `ford f150`, `tesla model 3`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `victorian`
 - Each person has a unique type of pet: `cat`, `dog`, `fish`
 - Each person has an occupation: `engineer`, `doctor`, `teacher`
 - Each person prefers a unique type of vacation: `city`, `mountain`, `beach`

## Clues:
1. The person with an aquarium of fish is in the first house.
2. The person who owns a Toyota Camry is in the second house.
3. The person who enjoys mountain retreats is not in the second house.
4. The person who prefers city breaks is not in the second house.
5. The person in a ranch-style home is somewhere to the left of Peter.
6. The person who owns a Toyota Camry is directly left of the person living in a colonial-style house.
7. Arnold is the person who has a cat.
8. Eric is somewhere to the left of the person who enjoys mountain retreats.
9. The person who is an engineer is not in the third house.
10. The person who owns a Tesla Model 3 is somewhere to the left of the person who is a teacher.
11. The person who owns a dog is the person who is an engineer.
 What is the value of attribute Name for the person whose attribute Pet is dog? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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

## Clues:
1. The person whose favorite color is red is the person who loves the spaghetti eater.
2. The person who loves the stew is the cat lover.
3. The person who loves yellow is not in the fourth house.
4. Alice is somewhere to the right of Arnold.
5. The person who loves the spaghetti eater is somewhere to the right of the person who loves soccer.
6. The person who loves white is the person who loves tennis.
7. The fish enthusiast is in the second house.
8. Arnold is the person who loves the stew.
9. The person who prefers city breaks is the person who is a pizza lover.
10. The person who prefers city breaks is directly left of the person who likes going on cruises.
11. The person who loves swimming is the person who loves beach vacations.
12. The person who keeps horses and the person whose favorite color is red are next to each other.
13. The person whose favorite color is green is directly left of Eric.
14. The person who prefers city breaks is somewhere to the left of the cat lover.
 What is the value of attribute House for the person whose attribute FavoriteSport is basketball? 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: `Eric`, `Bob`, `Carol`, `Arnold`, `Peter`, `Alice`
 - Everyone has a unique favorite cigar: `blends`, `prince`, `dunhill`, `yellow monster`, `blue master`, `pall mall`
 - People have unique hair colors: `auburn`, `gray`, `blonde`, `black`, `red`, `brown`
 - Each person has a unique favorite drink: `boba tea`, `root beer`, `water`, `tea`, `coffee`, `milk`
 - Each person has an occupation: `artist`, `nurse`, `teacher`, `doctor`, `engineer`, `lawyer`

## Clues:
1. Bob is the tea drinker.
2. The Dunhill smoker is not in the second house.
3. There is one house between Carol and Arnold.
4. The Prince smoker is the person who has black hair.
5. The person who has auburn hair is not in the third house.
6. Arnold is in the fourth house.
7. The Prince smoker is not in the fourth house.
8. The person who is an artist is the person partial to Pall Mall.
9. The person who is an engineer is directly left of the root beer lover.
10. The boba tea drinker is the person who has brown hair.
11. The person who is a lawyer is in the fifth house.
12. The person who is an engineer is the person who smokes Blue Master.
13. The person who likes milk is the person who has auburn hair.
14. Alice is somewhere to the left of the person who is a teacher.
15. The person who has blonde hair is the person who smokes Yellow Monster.
16. The person who has black hair is the person who is a doctor.
17. Arnold is somewhere to the right of the person who has gray hair.
18. The person who has auburn hair is not in the first house.
19. The one who only drinks water is the person who smokes many unique blends.
20. The person who smokes many unique blends is somewhere to the right of the person who has brown hair.
21. Arnold is the root beer lover.
22. The person who has brown hair is not in the third house.
23. The person who smokes many unique blends is somewhere to the left of Peter.
24. The person who likes milk is somewhere to the left of the person who has blonde hair.
 What is the value of attribute Cigar for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}

Answer: pall mall


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`, `Alice`
 - The people keep unique animals: `fish`, `cat`, `horse`, `bird`
 - Everyone has a unique favorite cigar: `pall mall`, `blue master`, `dunhill`, `prince`
 - They all have a unique favorite flower: `lilies`, `roses`, `daffodils`, `carnations`
 - People have unique favorite music genres: `classical`, `jazz`, `rock`, `pop`

## Clues:
1. The Dunhill smoker is Eric.
2. Eric is the cat lover.
3. The person who loves the boquet of lilies is the person who loves pop music.
4. The person partial to Pall Mall is the fish enthusiast.
5. The person who smokes Blue Master is the person who keeps horses.
6. Arnold is not in the third house.
7. The person who loves rock music is somewhere to the left of the person who loves a carnations arrangement.
8. The person who loves classical music is somewhere to the left of the fish enthusiast.
9. The person who loves rock music is Peter.
10. The person who loves jazz music is the person who smokes Blue Master.
11. The person who loves a bouquet of daffodils is in the fourth house.
 What is the value of attribute House for the person whose attribute Flower is carnations? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Bob`, `Peter`, `Alice`
 - Everyone has a unique favorite cigar: `dunhill`, `blue master`, `blends`, `pall mall`, `prince`
 - Each person has an occupation: `lawyer`, `artist`, `engineer`, `doctor`, `teacher`
 - People own unique car models: `ford f150`, `toyota camry`, `honda civic`, `bmw 3 series`, `tesla model 3`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `dragonfruit`, `desert`, `lime`
 - Everyone has something unique for lunch: `stew`, `spaghetti`, `grilled cheese`, `pizza`, `stir fry`

## Clues:
1. There are two houses between the person who owns a BMW 3 Series and the person who owns a Tesla Model 3.
2. The person who is an artist is Bob.
3. The person who owns a Ford F-150 is the person who smokes Blue Master.
4. The person who loves the spaghetti eater is in the second house.
5. The person who is an engineer is the Watermelon smoothie lover.
6. Alice is the person who drinks Lime smoothies.
7. The person who is an artist is the Dunhill smoker.
8. The person who likes Cherry smoothies is not in the fifth house.
9. Peter is not in the fourth house.
10. The Desert smoothie lover is the person who loves stir fry.
11. The person who likes Cherry smoothies is somewhere to the right of the person who smokes many unique blends.
12. The person who is a teacher is the person who loves eating grilled cheese.
13. Arnold is the person who is a lawyer.
14. The Dragonfruit smoothie lover is the person who owns a Honda Civic.
15. Peter is somewhere to the right of the Desert smoothie lover.
16. The Prince smoker is the Watermelon smoothie lover.
17. There is one house between the person who owns a Honda Civic and the person who loves the stew.
18. The person who owns a Tesla Model 3 is the person who likes Cherry smoothies.
19. The person who owns a Ford F-150 is directly left of the person who is a doctor.
20. The person who is an artist is the Desert smoothie lover.
 What is the value of attribute CarModel for the person whose attribute Smoothie is cherry? Please reason step by step, and put your final answer within \boxed{}

Answer: tesla model 3


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Peter`, `Bob`, `Carol`
 - Each person has an occupation: `engineer`, `artist`, `doctor`, `teacher`, `nurse`, `lawyer`
 - People own unique car models: `chevrolet silverado`, `ford f150`, `honda civic`, `toyota camry`, `bmw 3 series`, `tesla model 3`

## Clues:
1. The person who owns a Ford F-150 is in the fifth house.
2. The person who owns a Chevrolet Silverado is not in the second house.
3. The person who owns a Honda Civic and Peter are next to each other.
4. The person who is a lawyer is not in the fifth house.
5. The person who is a nurse is directly left of the person who is an artist.
6. Carol is somewhere to the right of Eric.
7. The person who is a doctor is Eric.
8. The person who is a teacher is somewhere to the left of the person who is a nurse.
9. Carol is not in the sixth house.
10. The person who is an engineer is Bob.
11. The person who owns a Toyota Camry is the person who is a nurse.
12. There is one house between Peter and the person who is a lawyer.
13. There is one house between the person who owns a Tesla Model 3 and Bob.
14. Arnold is the person who is an artist.
 What is the value of attribute Occupation for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: nurse


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Bob`, `Eric`, `Peter`
 - Each person has a unique hobby: `painting`, `photography`, `cooking`, `gardening`, `knitting`
 - Each person has a unique birthday month: `jan`, `sept`, `mar`, `feb`, `april`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `baseball`, `soccer`
 - Each person has an occupation: `engineer`, `lawyer`, `teacher`, `doctor`, `artist`
 - Each person has a unique level of education: `bachelor`, `master`, `high school`, `doctorate`, `associate`

## Clues:
1. The person with a high school diploma is directly left of the person whose birthday is in February.
2. The person who loves soccer is directly left of the photography enthusiast.
3. Arnold is the person whose birthday is in January.
4. The person who is a doctor is the person who loves baseball.
5. The person who is an artist is somewhere to the left of Bob.
6. The photography enthusiast is somewhere to the right of the person with an associate's degree.
7. The photography enthusiast is directly left of the person who paints as a hobby.
8. Eric is in the fifth house.
9. The person whose birthday is in February is directly left of the person who loves tennis.
10. The person who loves baseball is somewhere to the right of the person whose birthday is in January.
11. The person who is an engineer is not in the first house.
12. Alice is not in the second house.
13. The person who enjoys gardening is somewhere to the left of the person whose birthday is in September.
14. The person who loves swimming is not in the first house.
15. The person who is a teacher is Bob.
16. Arnold is the person who enjoys knitting.
17. The person who is an artist is the person whose birthday is in March.
18. Arnold is the person with a master's degree.
19. The person whose birthday is in March is directly left of the person who is a doctor.
20. The person whose birthday is in September is the person with a bachelor's degree.
 What is the value of attribute House for the person whose attribute Birthday is april? Please reason step by step, and put your final answer within \boxed{}

Answer: 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: `Carol`, `Bob`, `Alice`, `Eric`, `Arnold`, `Peter`
 - Each person has a unique type of pet: `hamster`, `cat`, `fish`, `rabbit`, `dog`, `bird`
 - Each person has a favorite color: `blue`, `purple`, `yellow`, `white`, `red`, `green`

## Clues:
1. The person who has a cat is not in the third house.
2. Arnold is not in the first house.
3. The person who owns a dog is the person whose favorite color is green.
4. The person who loves yellow is somewhere to the left of Bob.
5. The person who loves blue is the person who has a cat.
6. There are two houses between the person with an aquarium of fish and Eric.
7. The person who owns a dog is Carol.
8. The person who loves yellow is the person with a pet hamster.
9. The person who loves white and the person who keeps a pet bird are next to each other.
10. The person who owns a rabbit is somewhere to the left of the person who loves white.
11. The person who owns a dog is directly left of the person who loves blue.
12. The person who loves purple is directly left of Carol.
13. Bob is not in the third house.
14. Alice is the person who loves purple.
 What is the value of attribute Pet for the person whose attribute Color is yellow? Please reason step by step, and put your final answer within \boxed{}

Answer: hamster


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Alice`, `Bob`, `Arnold`, `Carol`
 - People have unique favorite book genres: `mystery`, `fantasy`, `romance`, `historical fiction`, `science fiction`, `biography`
 - Everyone has a favorite smoothie: `cherry`, `desert`, `lime`, `watermelon`, `blueberry`, `dragonfruit`
 - The people keep unique animals: `fish`, `rabbit`, `bird`, `cat`, `horse`, `dog`
 - People have unique favorite music genres: `classical`, `hip hop`, `country`, `jazz`, `rock`, `pop`
 - Everyone has a unique favorite cigar: `prince`, `dunhill`, `blends`, `pall mall`, `blue master`, `yellow monster`

## Clues:
1. The person who smokes many unique blends is the person who loves fantasy books.
2. The person who loves biography books is the Desert smoothie lover.
3. There are two houses between the person who loves pop music and the person who loves country music.
4. The rabbit owner is in the third house.
5. Arnold is the bird keeper.
6. The person who loves country music is in the fifth house.
7. The person who loves mystery books is not in the first house.
8. The person who drinks Lime smoothies is not in the third house.
9. The fish enthusiast is somewhere to the right of the person who loves rock music.
10. The Prince smoker is somewhere to the left of the person who likes Cherry smoothies.
11. Bob is the cat lover.
12. Eric is the person who likes Cherry smoothies.
13. The person who smokes Yellow Monster is in the sixth house.
14. The Prince smoker is the Dragonfruit smoothie lover.
15. Carol is directly left of Arnold.
16. The person who smokes Blue Master is directly left of the person who loves classical music.
17. The person who loves classical music is the person who loves historical fiction books.
18. The Dunhill smoker and the person who loves science fiction books are next to each other.
19. The person who keeps horses is directly left of Peter.
20. The Watermelon smoothie lover is the cat lover.
21. The person who smokes many unique blends is directly left of the person who loves jazz music.
22. The cat lover is in the first house.
23. The Desert smoothie lover is the person who loves jazz music.
24. The person who drinks Lime smoothies is directly left of the fish enthusiast.
25. The person who smokes Blue Master is not in the fifth house.
 What is the value of attribute Animal for the person whose attribute Smoothie is cherry? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


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

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

Answer: green


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

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

Answer: science fiction


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

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

Answer: Peter


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Alice`, `Eric`, `Arnold`
 - The mothers' names in different houses are unique: `Penny`, `Kailyn`, `Aniya`, `Janelle`, `Holly`
 - Everyone has a unique favorite cigar: `blue master`, `pall mall`, `prince`, `dunhill`, `blends`
 - The people keep unique animals: `bird`, `cat`, `horse`, `fish`, `dog`
 - Everyone has something unique for lunch: `grilled cheese`, `stir fry`, `pizza`, `spaghetti`, `stew`

## Clues:
1. The person who loves the stew is somewhere to the left of the fish enthusiast.
2. The person who is a pizza lover is somewhere to the right of the Dunhill smoker.
3. Peter is the cat lover.
4. The dog owner is directly left of the person who loves the spaghetti eater.
5. The person who loves stir fry is The person whose mother's name is Aniya.
6. There are two houses between the fish enthusiast and Eric.
7. The person who smokes many unique blends is The person whose mother's name is Kailyn.
8. The fish enthusiast is the person who loves the spaghetti eater.
9. The fish enthusiast is in the second house.
10. The person whose mother's name is Kailyn is the person who loves the stew.
11. The person whose mother's name is Janelle is not in the fifth house.
12. The person whose mother's name is Holly is the person who smokes Blue Master.
13. Arnold is The person whose mother's name is Aniya.
14. The person who loves the spaghetti eater and the person partial to Pall Mall are next to each other.
15. Peter is not in the third house.
16. Arnold is the person who keeps horses.
17. The person whose mother's name is Holly is Bob.
 What is the value of attribute Mother for the person whose attribute Cigar is dunhill? Please reason step by step, and put your final answer within \boxed{}

Answer: Janelle


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`
 - People have unique favorite sports: `soccer`, `tennis`, `basketball`
 - Each person has a unique birthday month: `april`, `sept`, `jan`
 - Each person prefers a unique type of vacation: `city`, `beach`, `mountain`
 - People have unique heights: `average`, `short`, `very short`
 - They all have a unique favorite flower: `lilies`, `carnations`, `daffodils`

## Clues:
1. The person who is very short is the person who prefers city breaks.
2. The person whose birthday is in January is Peter.
3. The person whose birthday is in January is directly left of the person who loves a carnations arrangement.
4. The person who loves beach vacations is somewhere to the right of the person who loves a bouquet of daffodils.
5. Eric is the person who loves basketball.
6. The person who is short is the person whose birthday is in April.
7. The person who enjoys mountain retreats is the person who loves a carnations arrangement.
8. Eric is the person who is short.
9. The person who loves tennis is the person who loves a carnations arrangement.
 What is the value of attribute Height for the person whose attribute Vacation is city? Please reason step by step, and put your final answer within \boxed{}

Answer: very short


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Bob`, `Alice`, `Arnold`
 - People have unique heights: `very short`, `average`, `tall`, `short`, `very tall`
 - People own unique car models: `toyota camry`, `bmw 3 series`, `honda civic`, `ford f150`, `tesla model 3`
 - People use unique phone models: `iphone 13`, `oneplus 9`, `samsung galaxy s21`, `google pixel 6`, `huawei p50`

## Clues:
1. Arnold is in the fifth house.
2. The person who is short is the person who owns a Tesla Model 3.
3. The person who has an average height is somewhere to the right of Alice.
4. The person who owns a Honda Civic is not in the third house.
5. The person who owns a BMW 3 Series is directly left of Arnold.
6. The person who uses a OnePlus 9 is directly left of the person who owns a Ford F-150.
7. Alice is directly left of the person who is very short.
8. The person who uses an iPhone 13 is not in the second house.
9. The person who uses a Google Pixel 6 is Arnold.
10. The person who uses a Google Pixel 6 is the person who is tall.
11. Peter is the person who uses a Huawei P50.
12. Peter is directly left of the person who uses a OnePlus 9.
13. Bob is the person who owns a Tesla Model 3.
 What is the value of attribute Name for the person whose attribute Height is very short? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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

## Clues:
1. The person who owns a Toyota Camry is in the first house.
2. Eric and the person who loves the spaghetti eater are next to each other.
3. The person who is an engineer is the person who loves eating grilled cheese.
4. Peter is the person who owns a Toyota Camry.
5. The person whose birthday is in April is directly left of the person whose birthday is in February.
6. The person who is a doctor is somewhere to the right of the person who is an artist.
7. The person who owns a Ford F-150 is the person who is a doctor.
8. The person whose birthday is in September is somewhere to the left of Alice.
9. Arnold is the person who is an artist.
10. The person who loves the spaghetti eater is the person who owns a Ford F-150.
11. The person who is an engineer is the person whose birthday is in April.
12. The person whose birthday is in February and the person who is a pizza lover are next to each other.
13. The person who owns a Honda Civic is Eric.
 What is the value of attribute Food for the person whose attribute CarModel is ford f150? Please reason step by step, and put your final answer within \boxed{}

Answer: spaghetti


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

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

Answer: oneplus 9


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Peter`
 - Each person has a unique favorite drink: `milk`, `water`, `coffee`, `tea`
 - They all have a unique favorite flower: `lilies`, `roses`, `daffodils`, `carnations`
 - People have unique favorite sports: `tennis`, `basketball`, `soccer`, `swimming`
 - Each person has an occupation: `doctor`, `teacher`, `artist`, `engineer`
 - Each mother is accompanied by their child: `Fred`, `Samantha`, `Bella`, `Meredith`

## Clues:
1. The person who loves basketball is somewhere to the left of the person who is an engineer.
2. Eric is the person who loves a carnations arrangement.
3. The person who loves the rose bouquet is somewhere to the left of Peter.
4. The person who is a doctor is the person who loves tennis.
5. The person's child is named Bella is directly left of the person's child is named Samantha.
6. The person who is an artist is not in the third house.
7. The person who loves the rose bouquet is in the second house.
8. The person who loves a bouquet of daffodils and the person who loves tennis are next to each other.
9. The person who loves a bouquet of daffodils is the person who is a teacher.
10. The person who loves the rose bouquet is the person who likes milk.
11. The person's child is named Samantha is Alice.
12. The person who loves tennis is directly left of the one who only drinks water.
13. The person who loves a carnations arrangement is in the fourth house.
14. Arnold is somewhere to the left of the person who is an artist.
15. The person's child is named Fred is not in the fourth house.
16. The coffee drinker is the person who loves swimming.
 What is the value of attribute House for the person whose attribute Children is Meredith? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`, `Alice`
 - Each person has a unique favorite drink: `coffee`, `water`, `tea`, `milk`
 - Each mother is accompanied by their child: `Samantha`, `Meredith`, `Bella`, `Fred`

## Clues:
1. The person's child is named Meredith is not in the fourth house.
2. The person who likes milk is the person's child is named Bella.
3. The person's child is named Samantha is somewhere to the right of Alice.
4. The tea drinker is the person's child is named Fred.
5. There is one house between the coffee drinker and Peter.
6. The person's child is named Samantha is somewhere to the left of the tea drinker.
7. Eric is somewhere to the right of the person's child is named Fred.
 What is the value of attribute Name for the person whose attribute Drink is tea? 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: `Peter`, `Arnold`, `Eric`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`
 - People have unique heights: `very short`, `average`, `short`
 - People have unique favorite sports: `basketball`, `tennis`, `soccer`

## Clues:
1. The person who loves tennis is somewhere to the left of The person whose mother's name is Aniya.
2. The person who is short is The person whose mother's name is Holly.
3. The person who loves tennis and the person who is very short are next to each other.
4. The person who loves soccer is somewhere to the right of Arnold.
5. Eric is not in the first house.
6. Arnold is in the second house.
7. The person whose mother's name is Holly is in the third house.
 What is the value of attribute Height for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: short


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

Answer: science fiction


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`
 - Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `desert`, `watermelon`
 - Everyone has a unique favorite cigar: `blue master`, `pall mall`, `dunhill`, `prince`
 - People have unique heights: `tall`, `average`, `short`, `very short`
 - People use unique phone models: `google pixel 6`, `samsung galaxy s21`, `iphone 13`, `oneplus 9`

## Clues:
1. The Dragonfruit smoothie lover is Eric.
2. The Dunhill smoker is the person who likes Cherry smoothies.
3. The person who uses a Samsung Galaxy S21 is directly left of the person who uses an iPhone 13.
4. The Dunhill smoker is somewhere to the right of the person who is very short.
5. The Watermelon smoothie lover is somewhere to the right of the Desert smoothie lover.
6. The Prince smoker is the person who uses a OnePlus 9.
7. The person who is tall is in the third house.
8. The person who is very short is the person who uses an iPhone 13.
9. The person who smokes Blue Master is not in the first house.
10. The Dunhill smoker is the person who is short.
11. Peter is not in the third house.
12. Arnold is the person who uses a Google Pixel 6.
13. The Dragonfruit smoothie lover is the person partial to Pall Mall.
 What is the value of attribute PhoneModel for the person whose attribute Smoothie is watermelon? Please reason step by step, and put your final answer within \boxed{}

Answer: oneplus 9


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`, `Alice`, `Eric`, `Carol`
 - The people are of nationalities: `swede`, `dane`, `german`, `norwegian`, `chinese`, `brit`
 - Each person has a favorite color: `red`, `green`, `purple`, `blue`, `white`, `yellow`

## Clues:
1. Alice is directly left of Bob.
2. The person whose favorite color is green is the British person.
3. Alice is somewhere to the left of the German.
4. The person who loves blue is the Dane.
5. Eric is the person who loves purple.
6. Carol is in the second house.
7. The Swedish person is not in the second house.
8. The Swedish person is not in the first house.
9. The person who loves white is directly left of the Norwegian.
10. There are two houses between Alice and the person who loves yellow.
11. The Dane is in the fifth house.
12. The person who loves blue is Peter.
 What is the value of attribute Color for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: white


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`
 - The mothers' names in different houses are unique: `Kailyn`, `Aniya`, `Holly`, `Janelle`
 - Each person lives in a unique style of house: `craftsman`, `victorian`, `ranch`, `colonial`
 - The people keep unique animals: `cat`, `fish`, `bird`, `horse`
 - Each person has a unique level of education: `bachelor`, `high school`, `master`, `associate`

## Clues:
1. The person with a bachelor's degree is directly left of The person whose mother's name is Aniya.
2. The person who keeps horses is not in the second house.
3. Eric is not in the second house.
4. The person with a high school diploma is the person in a Craftsman-style house.
5. Peter is somewhere to the left of the person with a master's degree.
6. Alice is in the third house.
7. The person in a Craftsman-style house is somewhere to the left of The person whose mother's name is Janelle.
8. The person living in a colonial-style house is Peter.
9. The fish enthusiast is the person with a master's degree.
10. The person with an associate's degree is in the third house.
11. The person whose mother's name is Kailyn is the bird keeper.
12. The person whose mother's name is Kailyn is the person residing in a Victorian house.
 What is the value of attribute Education for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: high school


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Bob`, `Arnold`, `Eric`, `Alice`
 - Each person has a favorite color: `red`, `white`, `green`, `blue`, `yellow`
 - Each person has a unique hobby: `gardening`, `cooking`, `painting`, `photography`, `knitting`
 - People have unique heights: `very tall`, `tall`, `average`, `short`, `very short`
 - People have unique hair colors: `black`, `blonde`, `brown`, `gray`, `red`

## Clues:
1. The person whose favorite color is green is somewhere to the right of the person who has black hair.
2. The person who has an average height is somewhere to the right of the person who loves blue.
3. The person who has red hair is Alice.
4. The person who loves cooking is not in the first house.
5. The person who loves yellow is the person who has gray hair.
6. The person who is short is somewhere to the left of the person who loves yellow.
7. The person who enjoys gardening is the person who loves white.
8. Arnold is somewhere to the left of the person who has black hair.
9. The person who is short is somewhere to the right of the person who has blonde hair.
10. Eric and the person who loves cooking are next to each other.
11. The person who has black hair is directly left of the person who loves blue.
12. The person who is very short is the person who has black hair.
13. Bob is the person who loves blue.
14. The person who is tall is directly left of the photography enthusiast.
15. The person who loves white is not in the first house.
16. The person who loves yellow is the person who enjoys knitting.
17. The person who loves blue is the person who has blonde hair.
 What is the value of attribute Height for the person whose attribute Color is blue? Please reason step by step, and put your final answer within \boxed{}

Answer: tall


12. Let the continuous monotonic function $f: \mathbf{R} \rightarrow \mathbf{R}, f(0)=1$, and satisfy the inequality
$$
f(x+y) \geqslant f(x) f(y)-f(x y)+1 .
$$

Find $f(x)$.

Answer: f(x) = x + 1


350. Among 18 coins, one is counterfeit. Genuine coins weigh the same, while the counterfeit coin differs in weight from the genuine ones. What is the minimum number of weighings on correct balance scales without weights needed to determine whether the counterfeit coin is lighter or heavier than the genuine ones? (There is no need to find the counterfeit coin.)

Answer: 2


10,11
[Application of trigonometric formulas (geometry) $]$
[ Identical transformations (trigonometry). ]

![](https://cdn.mathpix.com/cropped/2024_05_06_1560f9d9b3af2f34b251g-59.jpg?height=43&width=223&top_left_y=2307&top_left_x=0)
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


[ Diameter, main properties ]  [ Signs and properties of the tangent ]

A circle passes through the midpoints of the hypotenuse $A B$ and the leg $B C$ of a right triangle $A B C$ and is tangent to the leg $A C$. In what ratio does the point of tangency divide the leg $A C$.

Answer: \dfrac{1}{3}


4. Use a set of student's

triangular rulers to form quadrilateral $A B C D$ as shown in Figure 3, where $\angle C B D=\angle C D B=$ $45^{\circ}, \angle B A D=2 \angle B D A$ $=60^{\circ}$, and let the angle formed by diagonal $C A$ and side $C B$ be $\theta$. Then $\tan \theta=$ $\qquad$

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


Three lines parallel to the sides of a given triangle cut off three smaller triangles, leaving a regular hexagon.

Find the side length of the hexagon if the lengths of the sides of the triangle are $a$, $b$, and $c$.

Answer: \dfrac{abc}{ab + bc + ca}


7. Given a cube with edge length 6, there is a regular tetrahedron with edge length $x$ inside it, and the tetrahedron can rotate freely within the cube. Then the maximum value of $x$ is $\qquad$ .

Answer: 2\sqrt{6}


Let $ n$ be a given positive integer.  Find the smallest positive integer $ u_n$ such that for any positive integer $ d$, in any $ u_n$ consecutive odd positive integers, the number of them that can be divided by $ d$ is not smaller than the number of odd integers among $ 1, 3, 5, \ldots, 2n \minus{} 1$ that can be divided by $ d$.

Answer: 2n - 1


Exercise 8. Let $S$ be a set included in $\{1,2,3, \ldots, 199,200\}$. We say that $S$ is pretty if, for all elements $a$ and $b$ of $S$, the number $a-b$ is not a prime number. What is the maximum number of elements in a pretty set included in $\{1,2,3, \ldots, 199,200\}$?

Note: We say that a set $A$ is included in a set $B$ if every element of $A$ is also an element of $B$.

Answer: 100


13. As shown in the figure, a cube with an edge length of 12 cm is cut with one slice. The cut is made along $I J$ and exits at $L K$, such that $A I=D L=4$ cm, $J F=K G=3$ cm, and the cross-section $I J K L$ is a rectangle. The cube is cut into two parts, and the sum of the surface areas of these two parts is $\qquad$ square centimeters.

Answer: 1176


Answer: (2\sqrt{2}, 9)


5-147 Find all polynomials in two variables that satisfy the following conditions:
(1) $P$ is homogeneous of degree $n$. That is, for all real numbers $t, x, y$, we have $P(t x, t y)=t^{n} P(x, y)$;
(2) For all real numbers $a, b, c$,
$$
P(a+b, c)+P(b+c, a)+P(c+a, b)=0 ;
$$
(3) $P(1,0)=1$.

Answer: (x - 2y)(x + y)^{n-1}


Example 1. Plot the graph of the function $y=\frac{1}{2} x^{2}+2 x-1$.

Answer: y = \frac{1}{2}x^2 + 2x - 1


9. (3 points) Cars A and B start from locations $A$ and $B$ simultaneously and travel back and forth between $A$ and $B$ at a constant speed. If after the first meeting, Car A continues to drive for 4 hours to reach $B$, while Car B only drives for 1 hour to reach $A$, then when the two cars meet for the 15th time (meetings at $A$ and $B$ are not counted), they have driven $\qquad$ hours.

Answer: 86


4. Let's say a subset $\mathrm{P}$ of the set $\mathrm{M}=\{1,2,3, \ldots, 42\}$ is halfish if it contains 21 elements and each of the 42 numbers in the sets $\mathrm{P}$ and $\mathrm{Q}=\{7 x ; x \in \mathrm{P}\}$ gives a different remainder when divided by 43. Determine the number of halfish subsets of the set M.

(Josef Tkadlec)

Answer: 128


8、A container is already full of water. There are three lead balls: large, medium, and small. The first time, the small ball is submerged in the water; the second time, the small ball is taken out and the medium ball is submerged in the water; the third time, the medium ball is taken out and the large ball is submerged. It is known that the water overflowed the first time is 3 times that of the second time, and the water overflowed the third time is 3 times that of the first time. Find the volume ratio of the three balls $\qquad$

Answer: 3:4:13


11. (12 points) The distance between locations $A$ and $B$ is 291 kilometers. Two people, Jia and Yi, start from $A$ and travel at a constant speed to $B$ at the same time, while Bing starts from $B$ and travels at a constant speed to $A$. When Yi has walked $p$ kilometers and meets Bing, Jia has walked $q$ kilometers. After some time, when Jia and Bing meet, Yi has walked a total of $r$ kilometers. If $p$, $q$, and $r$ are all prime numbers, then the sum of $p$, $q$, and $r$ is . $\qquad$

Answer: 221


5. If the equation with respect to $x$
$$
4^{x}+(a+3) 2^{x}+5=0
$$

has at least one real root in the interval $[1,2]$, then the range of the real number $a$ is $\qquad$ .

Answer: [-\frac{33}{4}, -3 - 2\sqrt{5}]


4. Inside the rectangle $A B C D$, there are 9 equal rectangles as shown in the figure. All the gaps between the rectangles are equal - the gap between two smaller rectangles and the gap between a smaller rectangle and the side of rectangle $A B C D$. Calculate the area of the shaded part if the area of rectangle $A B C D$ is $697 \mathrm{~cm}^{2}$. The lengths of the sides of all rectangles and the length of the gaps are natural numbers in centimeters.

![](https://cdn.mathpix.com/cropped/2024_06_03_84104ab88f15e63b3af1g-1.jpg?height=339&width=737&top_left_y=2052&top_left_x=728)

Answer: 400


Majális, merriment .... Then with the twilight

The music fades, the song becomes quieter.

And when the full moon rises over the trees,

The troop returns home, the forest falls silent.

Only at the little kitchen hearth a song is heard,

As if they were filling a cask with wine:

It keeps flowing, gushing ... and as the breath passes,

The master speaks: "it will be exactly forty liters!"

"Twelve hours it takes ... to sip it away

As I please, in my own way."

Said one of the soldiers. "I'll bet my wages

$S$ on this, I'll drink it with ten!"

The other doubles down. "Eight is enough for me,

To see the bottom of the barrel clearly!"

Grumbled the third. "Well, just drink,

Said the master, you've served me well.

I'm tired, I'll sleep a bit,

Until the cart arrives. By then, you'll be drunk."

They started drinking ... drinking ... toasting,

Their dreams were torn by snoring, roaring.

They drank as much as each was used to,

But how much? ... They couldn't agree on that.

The last glass filled up just as

The carts arrived.

The master woke up, pondering for a long time,

How long had he slept ... they asked him too ...

"Well, if you drank as you bet:

I'll tell you bluntly, you don't need your watches."

He spoke, and in a moment the numbers were ready,

The soldiers stared in awe at the monstrous sight.

So who will be the master, who, calculating properly,

Will tell exactly how many hours have passed!?

Answer: \dfrac{120}{37}


57. As shown in the figure, points $D, E$ are on the sides $A C, A B$ of $\triangle A B C$ respectively, and the line $B D$ intersects $C E$ at point $F$. It is known that the areas of $\triangle C D F, \triangle B F E, \triangle B C F$ are 3, 4, 5 respectively. Then the area of quadrilateral $A E F D$ is $\qquad$ .

Answer: \dfrac{204}{13}


21 In a regular tetrahedron wooden block $ABCD$ with edge length 2, there is a point $P (AP<1)$ on the edge $AB$. A cross-section perpendicular to the edge $AB$ is to be sawed through point $P$. When the sawing stops at a certain position, it is measured that the saw cut $PM=1$ on the face $ABD$. The saw cut $PN=\frac{2}{3}$ on the face $ABC$, then $MN=$

Answer: 1


Along the road connecting Masha and Sasha's houses, there are 17 apple trees and 18 poplar trees. While Masha was walking to visit Sasha, she took pictures of all the trees. Right after the tenth apple tree, the memory on Masha's phone was filled, and she couldn't take pictures of the remaining 13 trees. The next day, when Sasha was walking to visit Masha, starting from the eighth apple tree, he plucked one leaf from each tree. How many leaves did Sasha pluck?

Answer: 22


2. There is one power outlet connected to the network, two extension cords with three outlets each, and one table lamp included. Nosy Nick randomly plugged all three plugs into 3 out of 7 outlets. What is the probability that the lamp will light up? (16 points)

Answer: \dfrac{13}{35}


8. Given $a_{1}, a_{2}, \cdots, a_{n}$ these $n$ people correspond to $A_{1}, A_{2}, \cdots, A_{n}$ these $n$ cards $\left(n \geqslant 2, a_{i}\right.$ corresponds to $\left.A_{i}\right)$. Now $a_{1}$ randomly picks a card from the deck, and then each person takes a card in sequence. When taking a card, if the deck contains their corresponding card, they take that card. Otherwise, they take any card. Then the probability that $a_{n}$ can get $A_{n}$ is $\qquad$ .

Answer: \dfrac{1}{2}


4. As shown in Figure 3, in the isosceles right triangle $\triangle ABC\left(\angle C=90^{\circ}\right)$, take a point $P$ inside, and $AP=AC=$ $a, BP=CP=b(a>b)$. Then $\frac{a^{2}+b^{2}}{a^{2}-b^{2}}=$ $\qquad$

Answer: \sqrt{3}


Each of the numbers in the set \( N = \{ 1, 2, 3, \ldots, n-1 \} \), where \( n \geq 3 \), is colored with one of two colors, say red or black, so that:
(i) \( i \) and \( n - i \) always receive the same color, and
(ii) for some \( j \in N \), relatively prime to \( n \), \( i \) and \( |j - i| \) receive the same color for all \( i \in N, i \neq j \).
Prove that all numbers in \( N \) must receive the same color.

Answer: \text{All numbers in } N \text{ must be the same color.}


Three, (16 points) Given that $\triangle X Y Z$ is an isosceles right triangle with legs of length 1 $\left(\angle Z=90^{\circ}\right)$, its three vertices lie on the three sides of isosceles right $\triangle A B C\left(\angle C=90^{\circ}\right)$. Find the maximum possible length of the legs of $\triangle A B C$.

Answer: \sqrt{5}


A week-old set's three-element subsets need to be colored such that if the intersection of two subsets is empty, then their colors differ. How many colors do we need at least?

Answer: 3


## Task B-1.3.

In an isosceles triangle, there are two circles. The first has a radius $R$ and touches all sides of the triangle, while the second has a radius $r$ and touches the legs of the triangle and the first circle. Determine the perimeter of the triangle.

Answer: \dfrac{4 R^2 \sqrt{\dfrac{R}{r}}}{R - r}


24. Brother Bear and Little Bear went on an adventure to find treasure and obtained a treasure chest. This treasure chest requires a ten-digit password consisting only of the digits 0 and 1 to open. Brother Bear first entered “1000000000”, and the treasure chest indicated that 1 digit was incorrect; Little Bear then entered “0110000000”, and the treasure chest indicated that 2 digits were incorrect. If the treasure chest no longer gives hints for subsequent password attempts, how many more times do the clever Brother Bear and Little Bear need to try to definitely open the treasure chest?

Answer: 3


A convex equilateral pentagon with side length $2$ has two right angles. The greatest possible area of the pentagon is $m+\sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m+n$.

[i]Proposed by Yannick Yao[/i]

Answer: 407


Let  $D$  be the midpoint of the arc  $B-A-C$  of the circumcircle of  $\triangle ABC (AB<AC)$ . Let  $E$  be the foot of perpendicular from  $D$  to  $AC$ . Prove that  $|CE|=\frac{|BA|+|AC|}{2}$ .

Answer: \frac{|BA| + |AC|}{2}


Determine the minimal number of operations needed to calculate $x^{2006}$ using only multiplication and division of any two given or already obtained powers of the number $x$. For example, $x \cdot x = x^{2}$, $x^{2} \cdot x^{2} = x^{4}$, $x^{4} : x = x^{3}$, etc.

Answer: 14


Let $f(x,y)$ be a function defined for all pairs of nonnegative integers $(x, y)$, such that $f(0,k) = f(k,0) = 2^k$ and
\[ f(a,b) + f(a+1,b+1) = f(a+1,b) + f(a,b+1) \]
for all nonnegative integers $a, b$. Determine the number of positive integers $n \leq 2016$ for which there exist two nonnegative integers $a, b$ such that $f(a,b) = n$.

Answer: 65


Let $p=2017$ be a prime. Find the remainder when \[ \left\lfloor \frac{1^p}{p} \right\rfloor + \left\lfloor \frac{2^p}{p} \right\rfloor + \left\lfloor \frac{3^p}{p} \right\rfloor + \cdots + \left\lfloor \frac{2015^p}{p} \right\rfloor \] is divided by $p$. Here $\lfloor \cdot \rfloor$ denotes the greatest integer function.

Answer: 1010


Leticia has a $9 \times 9$ board. She says that two squares are "friends" if they share a side, if they are at opposite ends of the same row, or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of three colors: green, blue, or red. In each square, a number will be written based on the following rules:

- If the square is green, write the number of red friends plus twice the number of blue friends.
- If the square is red, write the number of blue friends plus twice the number of green friends.
- If the square is blue, write the number of green friends plus twice the number of red friends.

Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares.

Answer: 486


Let $S$ be the set of all positive integers from 1 through 1000 that are not perfect squares. What is the length of the longest, non-constant, arithmetic sequence that consists of elements of $S$?

Answer: 333


The sequence of nonnegative integers $F_0, F_1, F_2, \dots$ is defined recursively as $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$ for all integers $n \geq 0$. Let $d$ be the largest positive integer such that, for all integers $n \geq 0$, $d$ divides $F_{n+2020} - F_n$. Compute the remainder when $d$ is divided by $1001$.

Answer: 638


Start by writing the integers $1, 2, 4, 6$ on the blackboard. At each step, write the smallest positive integer $n$ that satisfies both of the following properties on the board:
1. $n$ is larger than any integer currently on the board.
2. $n$ cannot be written as the sum of two distinct integers already on the board.

Find the $100$-th integer that you write on the board. Recall that at the beginning, there are already $4$ integers on the board.

Answer: 388


10. (20 points) The rules of a card game are as follows: Arrange nine cards labeled $1,2, \cdots, 9$ in a random row. If the number on the first card (from the left) is $k$, then reverse the order of the first $k$ cards, which is considered one operation. The game stops when no operation can be performed (i.e., the number on the first card is “1”). If a permutation cannot be operated on and is obtained from exactly one other permutation by one operation, it is called a “secondary terminal permutation”. Among all possible permutations, find the probability of a secondary terminal permutation occurring.

Answer: \dfrac{103}{2520}


Let $1 = d_1 < d_2 < ...< d_k = n$ be all natural divisors of the natural number $n$. Find all possible values of the number $k$ if  $n=d_2d_3 + d_2d_5+d_3d_5$.

Answer: 8


Take the x-axis as horizontal and the y-axis as vertical. A gun at the origin can fire at any angle into the first quadrant (x, y ≥ 0) with a fixed muzzle velocity \( v \). Assuming the only force on the pellet after firing is gravity (acceleration \( g \)), which points in the first quadrant can the gun hit?

Answer: y \leq \frac{v^2}{2g} - \frac{g x^2}{2v^2}


8. Given the dihedral angle $\alpha-l-\beta$ is $60^{\circ}$, a point $P$ inside this dihedral angle is 3 units away from plane $\alpha$ and 5 units away from plane $\beta$. If $A \in \alpha, B \in \beta$, then the minimum perimeter of $\triangle P A B$ is $\qquad$

Answer: 14


There are $N$ permutations $(a_1, a_2, \dots, a_{30})$ of $1, 2, \dots, 30$ such that for $m \in \{2,3,5\}$ , $m$ divides $a_{n+m} - a_n$ for all integers $n$ with $1 \le n < n+m \le 30$ . Find the remainder when $N$ is divided by 1000.
Please give the answer directly without any intermediate steps.

Answer: 440


2. Given are two circles $k_{1}$ and $k_{2}$ with centers $P$ and $R$ respectively, which touch each other externally at point $A$. Let $p$ be their common tangent, which does not pass through $A$, and touches $k_{1}$ and $k_{2}$ at points $B$ and $C$, respectively. The line $P R$ intersects the line $B C$ at point $E$, and the circle $k_{2}$ at points $A$ and $D$. If $\overline{A C}=\frac{\overline{A B}}{2}$, find the ratio $\frac{\overline{B C}}{\overline{D E}}$.

Answer: 6


20. N6 (POL) ${ }^{\mathrm{IMO} 06}$ Let $p$ be an odd prime. Find the number of $p$-element subsets $A$ of $\{1,2, \ldots, 2 p\}$ such that the sum of all elements of $A$ is divisible by $p$.

Answer: \dfrac{1}{p} \left( \dbinom{2p}{p} + 2(p - 1) \right)


2. Given that $P(x, y)$ is a point on the ellipse $\frac{x^{2}}{8}+\frac{y^{2}}{2}=1$. Then the minimum value of $3^{-x}+9^{y}$ is $\qquad$ .

Answer: \dfrac{2}{9}


Environmentalists have protested against the large volume of logging. The chairman of the forestry company calmed them down as follows: "99% of the trees in the forest are pines. Only pines will be cut, and after the logging, the percentage of pines will remain almost unchanged - pines will be $98\%$". What part of the forest is allocated for logging?

Answer: \dfrac{1}{2}


12.6. Consider the parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, where $m\left(\angle A_{1} A D\right)=m\left(\angle A_{1} A B\right)=m(\angle D A B)=60^{\circ}$, and $C_{1} A_{1}=\sqrt{7} \mathrm{~cm}, C_{1} B=\sqrt{13} \mathrm{~cm}, C_{1} D=\sqrt{19} \mathrm{~cm}$. Determine the distance from point $A$ to the plane $A_{1} B D$.

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


6. In the magical town, you can only buy things with gems. In the town, 1 blue gem can be exchanged for 2 yellow gems, 3 yellow gems can be exchanged for 6 green gems, and 2 green gems can buy 7 lollipops. Xiao Bu has 1 blue gem and 2 yellow gems, she can buy $\qquad$ lollipops.

Answer: 28


8-(22nd $I M O$ Problem) Let $P$ be any point inside triangle $A B C$, and let the distances from $P$ to the three sides $B C, C A, A B$ be $d_{1}, d_{2}, d_{3}$ respectively, with $B C=a, C A=b, A B=c$. Find the minimum value of $u=\frac{a}{d_{1}}+\frac{b}{d_{2}}+\frac{c}{d_{3}}$.

Answer: \dfrac{(a + b + c)^2}{2S}


Given points $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)$ and a non-negative number $\lambda$. Find the coordinates of point $M$ on the ray $A B$, for which $A M: A B=\lambda$.

Answer: \left( (1 - \lambda)x_1 + \lambda x_2, (1 - \lambda)y_1 + \lambda y_2 \right)


Alín has 8 keys on a keyring. The keys are indistinguishable by sight and both sides of each key are the same. To be able to tell the keys apart, Alín puts a colored cap on each one. How many colors does Alín need?

Answer: 2


Answer: \frac{1}{2\sqrt{x}} - 3 \sin 6x


6. Given that $A$ is a subset of $S=\{1,2,3,4,5,6\}$ with at least 2 elements, and $a, b$ are two distinct elements in $A$. When $A$ ranges over $S$ and $a, b$ range over $A$, the total sum of the product $ab$ is $M=$ $\qquad$

Answer: 2800


Let $a$, $b$, $c$, $a+b-c$, $a+c-b$, $b+c-a$, and $a+b+c$ be 7 distinct prime numbers, and suppose that the sum of two of $a$, $b$, and $c$ is 800. Let $d$ be the difference between the largest and smallest of these 7 prime numbers. Find the maximum possible value of $d$.
(Liang Darong, problem contributor)

Answer: 1594


An ant is crawling from the left end of a $4 \mathrm{~m}$ long rubber band towards the right end at a constant speed, covering exactly one meter per minute. After each minute, the horizontally placed rubber band, fixed at the left end, is uniformly stretched by one meter. In which minute does the ant reach the right end of the rubber band? The ant is considered point-like, the time taken to stretch the rubber band is negligible, and the rubber band can stretch indefinitely without breaking.

Answer: 7


Example 1. What day of the week was October 1, 1949?

Answer: Saturday


There are points $A$ and $B$ in the plane. Where should point $C$ be placed in the plane so that the distance $C A$ is 2.6 times the distance $C B$, and the angle $B A C$ is as large as possible?

Answer: \frac{5}{12} AB


## Task B-1.4.

Points $A, B, C, D$ and $E$ are connected by segments as shown in the figure. Segments $\overline{A B}$ and $\overline{B C}$ intersect segment $\overline{D E}$ at points $F$ and $G$, respectively. If $\varangle A B C=20^{\circ}$ and if $\varangle D F A=\varangle C G E$, determine $\varangle E A B+\varangle D E A$.

![](https://cdn.mathpix.com/cropped/2024_05_30_3d5a33db4db92485fe9eg-03.jpg?height=363&width=326&top_left_y=612&top_left_x=1276)

Answer: 80^\circ


In a field, there are $n$ sheep and a wolf disguised as a sheep. Some of the sheep are friends (friendship is mutual). The goal of the wolf is to eat all the sheep. Initially, the wolf chooses certain sheep to become friends with. Each subsequent day, the wolf eats one of its friends. Each time the wolf eats a sheep $A$: (a) If a friend of $A$ was initially a friend, they are no longer friends with the wolf. (b) If a friend of $A$ is not a friend of the wolf, they become friends with the wolf. The wolf repeats this procedure until it has eaten as many sheep as possible.

Find the largest integer $m$ in terms of $n$ satisfying the following condition: there exists a choice of friendships among the sheep such that there are exactly $m$ ways to choose the sheep initially friendly with the wolf, so that the wolf can eat all the sheep.

Answer: 2^{n-1}


Variant 132. $\frac{1}{6}$. Variant 133. $\frac{5}{6}$. Variant 134. $\frac{2}{9}$.

Grading criteria: 20 points - correct (not necessarily the same as above) solution and correct answer; $\mathbf{1 5}$ points - an arithmetic error was made in the final calculation, provided the solution is otherwise correct; 5 points - the expression for efficiency is written, the relationship between the specified powers is noted, but the answer is not obtained.

Problem 5 (variant 131).

The entire terrain is divided into squares, marked by two integer indices $M$ and $N$ such that, for example, the point with coordinates $x=12.25, y=20.9$ is in the square numbered

$[12 ; 20]$, and the point with coordinates $x=-12.34, y=0.1239$ is in the square numbered $[-13 ; 0]$, and so on. A mysterious object moves in the Oxy plane along the trajectory $y=\left(\left(\left(x^{5}-2013\right)^{5}-2013\right)^{5}-2013\right)^{5}$, while the radar beam on the terrain is directed along the line $y=x+2013$. Indicate the numbers of all squares in which the radar will register the appearance of the mysterious object.

Answer: [4; 2017]


10. $A$ is the center of a semicircle, with radius $A D$ lying on the base. $B$ lies on the base between $A$ and $D$, and $E$ is on the circular portion of the semicircle such that $E B A$ is a right angle. Extend $E A$ through $A$ to $C$, and put $F$ on line $C D$ such that $E B F$ is a line. Now $E A=1, A C=\sqrt{2}, B F=\frac{2-\sqrt{2}}{4}, C F=\frac{2 \sqrt{5}+\sqrt{10}}{4}$, and $D F=\frac{2 \sqrt{5}-\sqrt{10}}{4}$. Find $D E$.

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


61. In quadrilateral $A B C D$ with angles $\angle A=60^{\circ}, \angle B=90^{\circ}, \angle C=120^{\circ}$, the point of intersection of the diagonals $M$ is marked. It turns out that $B M=1$ and $M D=2$. Find the area of $A B C D$.

Answer: \dfrac{9}{2}


6. Determine all integers $n \geq 3$ such that

$$
n!\mid \prod_{\substack{p<q \leq n \\ p, q \text { prime }}}(p+q)
$$

Remark: The expression on the right-hand side denotes the product over all sums of two distinct primes less than or equal to $n$. For $n=6$, this is equal to $(2+3)(2+5)(3+5)$.

Answer: 7


## 162. Math Puzzle $11 / 78$

Assume that a fly lays 120 eggs at the beginning of summer, on June 21st, and after 20 days, fully developed insects emerge from these eggs, each of which then lays 120 eggs. How many "descendants" would this fly have in total by the beginning of autumn?

Answer: 209102520


A set $S$ consists of $k$ sequences of $0,1,2$ of length $n$. For any two sequences $(a_i),(b_i)\in S$ we can construct a new sequence $(c_i)$ such that $c_i=\left\lfloor\frac{a_i+b_i+1}2\right\rfloor$ and include it in $S$. Assume that after performing finitely many such operations we obtain all the $3n$ sequences of $0,1,2$ of length $n$. Find the least possible value of $k$.

Answer: n+1


On a clock face, the hour, minute, and second hands all start at 12 o'clock. The question is, after how many seconds will the second hand divide the angle formed by the hour and minute hands for the $n$-th time in the given ratio $\frac{H O S}{S O M}=k$?

Answer: \dfrac{43200n(k + 1)}{708k + 719}


4. On the side $AC$ of triangle $ABC$, a point $K$ is taken which divides $AC$ in the ratio $1:3$, and on the side $BC$ a point $L$ is taken which divides $BC$ in the ratio $1:4$. Let $M$ be the intersection point of segments $AL$ and $BK$. Determine the ratio of segments $KM$ and $MB$.

Answer: \dfrac{1}{1}


7. 5 people participate in 4 groups, each group has two people, and each person must participate in at least one group. Then, the number of different groupings is
$\qquad$ kinds.

Answer: 135


Let  $S$  be a set of  $n$  points in the plane such that no four points are collinear. Let  $\{d_1,d_2,\cdots ,d_k\}$  be the set of distances between pairs of distinct points in  $S$ , and let  $m_i$  be the multiplicity of  $d_i$ , i.e. the number of unordered pairs  $\{P,Q\}\subseteq S$  with  $|PQ|=d_i$ . Prove that  $\sum_{i=1}^k m_i^2\leq n^3-n^2$ .

Answer: n^3 - n^2


\[
\begin{aligned} & {\left[\begin{array}{l}\text { Angle between the tangent and the chord }\end{array}\right]} \\ & \text { [Inscribed angle is half of the central angle]}\end{aligned}
\]

Secant $A B C$ intercepts arc $B C$, which contains $112^{\circ}$; tangent $A D$ at point of tangency $D$ divides this arc in the ratio $7: 9$. Find $\angle B A D$.

Answer: 7^\circ


60th Putnam 1999 Problem B3 Let R be the reals. Define f : [0, 1) x [0, 1) → R by f(x, y) = ∑ x m y n , where the sum is taken over all pairs of positive integers (m, n) satisfying m ≥ n/2, n ≥ m/2. Find lim (x, y)→(1, 1) (1 - xy 2 )(1 - x 2 y)f(x, y).

Answer: 3


Answer: \{(1, 2, 3, 4), (1, 2, 3, -4), (-1, -2, -3, 4), (-1, -2, -3, -4)\}


3. In the cottage village where Gavrila spends his summers, there is a water supply with cold water. The boy's parents installed a water heater, which has a fixed power as long as the temperature of the water in it is below $100^{\circ} \mathrm{C}$. After the water pipe enters the house, a tee was installed so that part of the water goes through the heater to the hot tap, and the rest goes directly to the cold tap. Before exiting, the hot and cold water mix. Gavrila fully opened the cold tap and found that the water temperature was $20^{\circ} \mathrm{C}$. When he closed the cold tap and opened the hot one, water with the same flow rate came out at a temperature of $40^{\circ} \mathrm{C}$. Then Gavrila opened both taps equally so that the flow rate remained the same. What is the water temperature in this case?

Answer: 40


Problem A3. Find all pairs $(x, y)$ of real numbers such that $|x|+|y|=1340$ and $x^{3}+y^{3}+2010 x y=670^{3}$.

Answer: (-670, -670)


 $n \geq 4$  players participated in a tennis tournament. Any two players have played exactly one game, and there was no tie game. We call a company of four players  $bad$  if one player was defeated by the other three players, and each of these three players won a game and lost another game among themselves. Suppose that there is no bad company in this tournament. Let  $w_i$  and  $l_i$  be respectively the number of wins and losses of the  $i$ -th player. Prove that \[\sum^n_{i=1} \left(w_i - l_i\right)^3 \geq 0.\]

*Proposed by Sung Yun Kim, South Korea*

Answer: \sum^n_{i=1} \left(w_i - l_i\right)^3 \geq 0


$n$ students take a test with $m$ questions, where $m,n\ge 2$ are integers. The score given to every question is as such: for a certain question, if $x$ students fails to answer it correctly, then those who answer it correctly scores $x$ points, while those who answer it wrongly scores $0$. The score of a student is the sum of his scores for the $m$ questions. Arrange the scores in descending order $p_1\ge p_2\ge \ldots \ge p_n$. Find the maximum value of $p_1+p_n$.

Answer: m(n -1)


Example 2 (to $2^{\circ}, 3^{\circ}$ ). Find $\int \cos ^{6} 2 x d x$.

Answer: \frac{5}{16}x + \frac{1}{8}\sin(4x) + \frac{3}{128}\sin(8x) - \frac{1}{96}\sin^3(4x) + C


Let $d(n)$ be the number of positive divisors of a positive integer $n$.  Let $\mathbb{N}$ be the set of all positive integers.  Say that a function $F$ from $\mathbb{N}$ to $\mathbb{N}$ is [i]divisor-respecting[/i] if $d(F(mn)) = d(F(m)) d(F(n))$ for all positive integers $m$ and $n$, and $d(F(n)) \le d(n)$ for all positive integers $n$.  Find all divisor-respecting functions.  Justify your answer.

Answer: 1


7. Let $A=\{2,4, \cdots, 2014\}, B$ be any non-empty subset of $A$, and $a_{i} 、 a_{j}$ be any two elements in set $B$. There is exactly one isosceles triangle with $a_{i} 、 a_{j}$ as side lengths. Then the maximum number of elements in set $B$ is $\qquad$

Answer: 10


A quadrilateral $ABCD$ is circumscribed around a circle with center $O$. Points $K, L, M, N$ are the points of tangency of sides $AB, BC, CD,$ and $AD$ respectively. $KP, LQ, MR,$ and $NS$ are the altitudes of triangles $OKB, OLC, OMD,$ and $ONA$. Given $OP=15, OA=32,$ and $OB=64$.

Find the length of segment $QR$.

Answer: 30


Given a triangle $ABC$ with $I$ as the incenter. Let $D, E,$ and $F$ be the points of tangency of the incircle with the segments $[BC], [CA],$ and $[AB]$ respectively. Let $X$ be the second point of intersection of the circumcircles of triangles $ABC$ and $AEF$. Show that \(\widehat{BXD} = \widehat{CXD}\).

Answer: \widehat{BXD} = \widehat{CXD}


6. The function $f(x)$ defined on $\mathbf{R}$ satisfies: when $x \in[0,1)$, $f(x)=2^{x}-x$, and for any real number $x$, $f(x)+f(x+1)=1$. Let $a=\log _{2} 3$, then the value of the expression $f(a)+f(2 a)+f(3 a)$ is $\qquad$ .

Answer: \dfrac{17}{16}


3. Given a regular hexagon $A B C D E F$, with side $10 \sqrt[4]{27}$. Find the area of the union of triangles ACE and BDF.

![](https://cdn.mathpix.com/cropped/2024_05_06_b7cc51e9ffdd04fa7b93g-2.jpg?height=526&width=560&top_left_y=191&top_left_x=931)

Answer: 900


10. (16 points) Figure 1 is a rhombus paper piece composed of 2 small equilateral triangles; Figure 2 is a fixed regular hexagonal board $A B C D E F$, which is made up of 24 equally sized small equilateral triangles. Now, 12 rhombus paper pieces are used to completely cover the hexagonal board, there are $\qquad$ different covering methods.

Answer: 20


8. Given that $n$ is a positive integer, $P$ is a prime, $p n$ has exactly 8 positive divisors, which are arranged in ascending order as $1=d_{1}<d_{2}<$ $\cdots<d_{8}=p n$, and $d_{17 p-d_{3}}=\left(d_{1}+d_{2}+d_{3}\right)\left(d_{3}+d_{4}+13 p\right)$, then $n=$ $\qquad$ .

Answer: 2021


M2. The diagram shows a circle that has been divided into six sectors of different sizes.
Two of the sectors are to be painted red, two of them are to be painted blue, and two of them are to be painted yellow. Any two sectors which share an edge are to be painted in different colours.
In how many ways can the circle be painted?

Answer: 24


C1. Sara has 10 blocks numbered $1 \mathrm{t} / \mathrm{m}$ 10. She wants to stack all the blocks into a tower. A block can only be placed on top of a block with a higher number, or on a block with a number that is exactly one lower. An example is, from top to bottom: 2, 1, 5, 4, 3, 6, 7, 9, 8, 10. How many different towers are possible?

Answer: 512


(solved by Anca Arnautu and Adrien Laroche). Let $\mathscr{P}$ be the parabola in the plane with equation $y=x^{2}$. Let $\Gamma_{1}$ be the circle of diameter 1 tangent internally to $\mathscr{P}$ at the origin. By recurrence, we define $\Gamma_{n+1}$ as the circle tangent to $\Gamma_{n}$ and twice to $\mathscr{P}$. Calculate the diameter of $\Gamma_{2007}$.

Answer: 4013


6 unit radius spheres are arranged so that each one touches exactly four others. What is the volume of the convex body defined by the points of contact?

Answer: \dfrac{5\sqrt{2}}{3}


[ Numerical inequalities. Comparisons of numbers. ]  [ Examples and counterexamples. Constructions ]

Write five numbers in a row so that the sum of any two adjacent numbers is negative, while the sum of all numbers is positive.

Answer: 3


2. The distance between points A and B is 90 km. At 9:00, a bus left point A for point B at a speed of 60 km/h. Starting from 9:00, every 15 minutes, buses leave point B towards point A at a speed of 80 km/h. The bus that left point A, after traveling 45 km, reduces its speed to 20 km/h due to a breakdown and continues at this speed. A passenger in this bus is considering whether to continue the slow journey to point B or return to point A in the first passing bus. How much time does the passenger have to make a decision?

Answer: 3


C2 $\underline{\text { ROU (Romania) }}$
For any integer $n \geq 2$, let $N(n)$ be the maximal number of triples $\left(a_{i}, b_{i}, c_{i}\right), i=1, \ldots, N(n)$, consisting of nonnegative integers $a_{i}, b_{i}$ and $c_{i}$ such that the following two conditions are satisfied:
(1) $a_{i}+b_{i}+c_{i}=n$ for all $i=1, \ldots, N(n)$,
(2) If $i \neq j$, then $a_{i} \neq a_{j}, b_{i} \neq b_{j}$ and $c_{i} \neq c_{j}$.

Determine $N(n)$ for all $n \geq 2$.

Comment. The original problem was formulated for $m$-tuples instead for triples. The numbers $N(m, n)$ are then defined similarly to $N(n)$ in the case $m=3$. The numbers $N(3, n)$ and $N(n, n)$ should be determined. The case $m=3$ is the same as in the present problem. The upper bound for $N(n, n)$ can be proved by a simple generalization. The construction of a set of triples attaining the bound can be easily done by induction from $n$ to $n+2$.

Answer: \left\lfloor \dfrac{2n}{3} \right\rfloor + 1


A \( 2012 \times 2012 \) table is to be filled with integers in such a way that each of the 4026 rows, columns, and main diagonals has a different sum. What is the smallest number of distinct values that must be used in the table?

Answer: 3


A castle has infinitely many rooms labeled \(1, 2, 3, \ldots\), which are divided into several halls. Suppose room \(n\) is in the same hall as rooms \(3n+1\) and \(n+10\) for every \(n\). Determine the maximum possible number of different halls in the castle.

Answer: 3


8. Let $A B C$ be an equilateral triangle with side length 8 . Let $X$ be on side $A B$ so that $A X=5$ and $Y$ be on side $A C$ so that $A Y=3$. Let $Z$ be on side $B C$ so that $A Z, B Y, C X$ are concurrent. Let $Z X, Z Y$ intersect the circumcircle of $A X Y$ again at $P, Q$ respectively. Let $X Q$ and $Y P$ intersect at $K$. Compute $K X \cdot K Q$.

Answer: 304


Problem 3. Let $\mathbb{R}^{*}$ be the set of non-zero real numbers. Find all functions $f: \mathbb{R}^{*} \rightarrow \mathbb{R}^{*}$ such that

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

for all $x, y \in \mathbb{R}^{*}, y \neq-x^{2}$.

Alexander Ivanov

Answer: f(x) = x


## Problem 2

The feet of the angle bisectors of the triangle $\mathrm{ABC}$ form a right-angled triangle. If the right-angle is at $\mathrm{X}$, where $\mathrm{AX}$ is the bisector of angle $\mathrm{A}$, find all possible values for angle $\mathrm{A}$.

Answer: 120^\circ


A quarter circle with a radius of $12 \mathrm{~cm}$ has a semicircle cut out over one of its bounding radii, as if the radius were a diameter. What is the radius of the largest circle that can be inscribed in the resulting shape?

Answer: 3


8. The sum of the ages of three people, A, B, and C, represented by $x, y, z$ is 120, and $x, y, z \in (20,60)$. The number of ordered triples $(x, y, z)$ is $\qquad$ .

Answer: 1141


Problem 4. For an infinite sequence of numbers $x_{1}, x_{2}, x_{3}, \ldots$, for all natural $n \geq 4$, the relation $x_{n}=x_{n-1} \cdot x_{n-3}$ holds. It is known that $x_{1}=1, x_{2}=1, x_{3}=-1$. Find $x_{2022}$.

Answer: 1


8. Let one edge of a tetrahedron be 6, and the remaining edges

all be 5. Then the radius of the circumscribed sphere of this tetrahedron is $\qquad$ .

Answer: \dfrac{20\sqrt{39}}{39}


3. In tetrahedron $ABCD$,
\[
\begin{array}{l}
\angle ABC=\angle BAD=60^{\circ}, \\
BC=AD=3, AB=5, AD \perp BC,
\end{array}
\]
$M, N$ are the midpoints of $BD, AC$ respectively. Then the size of the acute angle formed by lines $AM$ and $BN$ is \qquad (express in radians or inverse trigonometric functions).

Answer: \arccos\left(\dfrac{40}{49}\right)


2. As shown in Figure 1, PB is a secant of circle $\odot O$ with radius 5, and the lengths of $PA$ and $PB$ are the two roots of the equation $x^{2}-10 x+16=0$ $(PA<PB)$, $PC$ is a tangent to $\odot O$, and $C$ is the point of tangency. Then the area of quadrilateral $PAOC$ is $\qquad$

Answer: 14


2. (1993 National High School Competition Question) There are 900 three-digit numbers, $100, 101, \cdots, 999$. When printing these three-digit numbers on cards, for example, 198 looks like 861 when flipped, while some cards do not, such as 531 which looks like IE9 when flipped. Therefore, some cards can be used for two numbers, thus the maximum number of cards that can be saved is $\qquad$.

Answer: 34


A pyramid with a parallelogram base has base edges of lengths \(a\) and \(2a\), and lateral edges of length \(a \sqrt{3}\). The pyramid is cut by a plane that intersects the base at one of its diagonals and is parallel to a lateral edge. What is the area of the cross-section of the plane?

Answer: \dfrac{\sqrt{51}}{8} a^2


5-4. Andrei, Boris, Vladimir, and Dmitry each made two statements. For each boy, one of his statements turned out to be true, and the other false.

Andrei: "Boris is not the tallest among us four." "Vladimir is the shortest among us four."

Boris: "Andrei is the oldest in the room." "Andrei is the shortest in the room."

Vladimir: "Dmitry is taller than me." "Dmitry is older than me."

Dmitry: "Both of Vladimir's statements are true." "I am the oldest person in the room."

It is known that there are no two boys of the same height or age. What is the name of the tallest boy?

Answer: Vladimir


7th Irish 1994 Problem B2 p, q, r are distinct reals such that q = p(4-p), r = q(4-q), p = r(4-r). Find all possible values of p+q+r.

Answer: 6


In a regular quadrilateral pyramid $S A B C D$, where each edge is equal to 2, a section is constructed by a plane parallel to the diagonal of the base $A C$ and the lateral edge $S B$ of the pyramid, intersecting the edge $A B$. Find the perimeter of the polygon obtained in this section, if the lower base of the section is equal to $\sqrt{2}$.

Answer: 2 + \sqrt{2} + \sqrt{3}


## Task 6A - 291246A

In two urns $A$ and $B$, there are exactly $m$ red and exactly $n$ blue balls in total. The total number of balls is greater than 2; at least one of the balls is red.

At the beginning, $A$ contains all the red and $B$ all the blue balls.

By alternately taking one randomly selected ball from $A$ and $B$ and placing it in the other urn, the balls are to be mixed.

The process begins with the removal from urn $A$.

Determine all pairs $(m ; n)$ of the numbers $m$ and $n$ for which the fourth ball moved has a probability of $\frac{1}{2} 2$ of being red.

Hint:

If an urn contains exactly $Z$ balls, then by random selection of a ball, it is understood that the probability of selecting any of the $Z$ balls is $\frac{1}{Z}$.

More generally, if from $M$ possible events $G$ are considered favorable and $M-G$ are considered unfavorable, and all $M$ events are equally likely, then the probability of a favorable event occurring is $\frac{G}{M}$.

Answer: (4;2)


9. (20 points) Inside an acute triangle $A B C$, a point $M$ is marked. The lines $A M, B M$, $C M$ intersect the sides of the triangle at points $A_{1}, B_{1}$ and $C_{1}$ respectively. It is known that $M A_{1}=M B_{1}=M C_{1}=3$ and $A M+B M+C M=43$. Find $A M \cdot B M \cdot C M$.

Answer: 441


(IMO SL 2019 C4)

On a flat plain in Camelot, King Arthur builds a labyrinth $\mathfrak{L}$ consisting of $n$ lines, no two of which are parallel and no three of which are concurrent. For each wall, Merlin colors one side red and the other side blue.

At the intersection of two walls, there are 4 corners: two corners where a red side and a blue side intersect, one corner where two red sides intersect, and one corner where two blue sides intersect. At this intersection, there is a door connecting the two opposite corners where sides of different colors intersect.

After Merlin has colored the walls, Morgana places knights in the labyrinth. The knights can walk through the doors but not through the walls.

Let $k(\mathfrak{L})$ be the largest integer $k$ such that: no matter how Merlin colors the labyrinth $\mathfrak{L}$, Morgana can always place $k$ knights such that no two of them can ever meet. For a fixed $n$, what can the values of $k(\mathfrak{L})$ be for labyrinths $\mathfrak{L}$ consisting of $n$ lines?

Answer: n+1


4. There are circles $\omega_{1}$ and $\omega_{2}$. They intersect in two points, one of which is the point $A$. $B$ lies on $\omega_{1}$ such that $A B$ is tangent to $\omega_{2}$. The tangent to $\omega_{1}$ at $B$ intersects $\omega_{2}$ at $C$ and $D$, where $D$ is the closer to $B$. $A D$ intersects $\omega_{1}$ again at $E$. If $B D=3$ and $C D=13$, find $E B / E D$.

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


The positive integers are written into rows so that Row \( n \) includes every integer \( m \) with the following properties:
(i) \( m \) is a multiple of \( n \),
(ii) \( m \leq n^{2} \), and
(iii) \( m \) is not in an earlier row.

The table below shows the first six rows.
(a) Determine the smallest integer in Row 10.
(b) Show that, for all positive integers \( n \geq 3 \), Row \( n \) includes each of \( n^{2}-n \) and \( n^{2}-2 n \).
(c) Determine the largest positive integer \( n \) with the property that Row \( n \) does not include \( n^{2}-10 n \).

Answer: 50


 The deputies in a parliament were split into $10$ fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation.
Find the smallest possible number of independent deputies after the vacation.

Answer: 50


6. Given that the tangents at two points $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$ on the parabola $y=4 x$ are perpendicular to each other. Then the equation of the locus of the intersection point of the tangents is $\qquad$ .

Answer: x = -1


2. Can 2010 be written as the sum of squares of $k$ distinct prime numbers? If so, find the maximum value of $k$; if not, please briefly explain the reason.

Answer: 7


Example 5 In a competition consisting of $n$ true/false questions, 8 contestants participate. It is known that for any pair of true/false questions $(A, B)$ (referred to as an ordered pair), exactly two people's answers are (true, true); exactly two people's answers are (true, false); exactly two people's answers are (false, true); exactly two people's answers are (false, false). Find the maximum value of $n$, and explain the reasoning.

Answer: 7


An arithmetic sequence with common difference $r$ is a sequence of numbers $s_{0}, s_{1}, s_{2}, \ldots$ that satisfies for all indices $k, s_{k+1}=s_{k}+r$. Determine an expression for $s_{n}$ in terms of $s_{0}, r$, and $n$.

Answer: s_0 + r n


Example 2.2.3 In the sequence $1,9,81, \cdots, 9^{2005}$, delete the terms whose highest digit is 9. How many terms are left in the remaining sequence?

Answer: 1914


Any finite sequence composed of 0s and 1s is called a word. Repeating a word \( A \) three times to form \( A A A \) is called the tripling of \( A \). For example, if \( A = 101 \), then the tripling of \( A \) is 101101101. Two operations can be performed on words:
1. Add the tripling of any word at any position (including the beginning or the end) of the word.
2. Remove the tripling of any word from any position of the word.

For example, from the word 0001, it is possible to obtain 0111001 or 1, among others.

Question: Is it possible, through a series of the above operations, to transform the word 10 into the word 01?

Answer: \text{No}


207. Young Bandits. Three young "highwaymen," returning from the cinema, met a vendor with apples. Tom grabbed half of all the apples, but threw 10 back into the basket. Ben took a third of the remaining apples, but returned 2 apples he didn't like. Jim took half of the remaining apples, but threw back one wormy one. The vendor was left with only 12 apples in the basket.

How many apples did the vendor have before the raid?

Answer: 40


Illustrate a semicircle with diameter $A G$. The arc of the semicircle is divided into six equal parts by points $B$, $C$, $D$, $E$, and $F$. $D F$ and $C G$ are both straight line segments. Given that the area of the semicircle is $60 \mathrm{~cm}^{2}$, what is the area of the shaded part in $\mathrm{cm}^{2}$?

Answer: 20


A rise of $600$ feet is required to get a railroad line over a mountain. The grade can be kept down by lengthening the track and curving it around the mountain peak. Find the additional length of track required to reduce the grade from $3\%$ to $2\%$. If the answer is $x$ ft., what is the value of $x$?

Answer: 10000


Chord $[AD]$ is perpendicular to the diameter $[BC]$ of a circle. Let $E$ and $F$ be the midpoints of the arcs $AC$ and $CD$, respectively. If $AD \cap BE = \{G\}$, $AF \cap BC = \{H\}$ and $m(AC) = \alpha$, find the measure of angle $BHC$ in terms of $\alpha$. The answer is in the form $x^\circ$, where $x = k - \frac{m\alpha}{2}$. Please find the value of $k + m$.

Answer: 91


Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. Find the least positive integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \dots, T_n$ returns the point $(1,0)$ back to itself.

Answer: 359


Let $D$ be a point on side $[BC]$ of triangle $ABC$ where $|BC|=11$ and $|BD|=8$. The circle passing through the points $C$ and $D$ touches $AB$ at $E$. Let $P$ be a point on the line which is passing through $B$ and is perpendicular to $DE$. If $|PE|=7$, then what is $|DP|$?

Answer: 5


If $ \log 2 = 0.3010$ and $ \log 3 = 0.4771$, find the value of $ x$ when $ 3^{x+3} = 135$. If the answer is $y$, what is the value of $100y$?

Answer: 147


What is the sum \( \frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{2 \cdot 3 \cdot 4} + \cdots + \frac{1}{1996 \cdot 1997 \cdot 1998} \)? Provide your answer in the form of \(\frac{k}{m} - \frac{n}{p \cdot q \cdot r}\). Please find the value of \(k + m + n + p + q + r\).

Answer: 4003


For how many pairs of consecutive integers in $\{1000,1001,1002^{}_{},\ldots,2000\}$ is no carrying required when the two integers are added?
Please give the answer directly without any intermediate steps.

Answer: 156


How many subsets of {1,2,3,4,5,6,7,8,9,10,11,12} have the property that no two of its elements diﬀer by more than 5? For example, count the sets {3}, {2,5,7}, and {5,6,7,8,9} but not the set {1,3,5,7}.

Answer: 256


## Zadatak A-2.5.

Dan je trapez $A B C D$ s osnovicama $\overline{A B}$ i $\overline{C D}$, takav da je trokut $A B C$ šiljastokutan. Neka je $O$ središte kružnice opisane trokutu $A B C$, a točka $E$ sjecište pravaca $O B$ i $C D$. Ako je $\varangle D B C=\varangle C E B+10^{\circ}$, odredi veličinu kuta između dijagonala trapeza $A B C D$.

Answer: 80^\circ


32. (1989 American Mathematical Invitational) When a fair coin is tossed 5 times, the probability of getting exactly one head is not 0, and it is the same as the probability of getting exactly two heads.
Let the irreducible fraction $\frac{i}{j}$ be the probability of getting exactly 3 heads in 5 tosses of the coin, find the value of $i+j$.

Answer: 283


5*. In rhombus $A B C D$ the measure of angle $B$ is $40^{\circ}, E-$ is the midpoint of $B C, F$ - is the foot of the perpendicular dropped from $A$ to $D E$. Find the measure of angle $D F C$.

Answer: 110^\circ


8. The function $J(x)$ is defined by:
$$
J(x)=\left\{\begin{array}{ll}
4+x & \text { for } x \leq-2, \\
-x & \text { for }-20 .
\end{array}\right.
$$

How many distinct real solutions has the equation $J(J(J(x)))=0$ ?

Answer: 4


10.120. Two circles of radius $R$ with centers at points $O_{1}$ and $O_{2}$ touch each other. They are intersected by a line at points $A, B, C$ and $D$ such that $A B=B C=C D$. Find the area of the quadrilateral $O_{1} A D O_{2}$.

Answer: \dfrac{5\sqrt{3}}{4} R^2


3. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$. What is the smaller part of the volume separated from it by a plane passing through the midpoint of edge $A D$ and points $X$ and $Y$ on edges $A A_{1}$ and $C C_{1}$ such that $A X: A_{1} X=C Y: C_{1} Y=1: 7 ?$

Answer: \dfrac{25}{192}


31. Around a round table, 300 people are sitting: some of them are knights, and the rest are liars. Anton asked each of them: "How many liars are among your neighbors?" and added up the numbers he received. Then Anya did the same. When answering the question, knights always tell the truth, while liars always lie, but they only say the numbers 0, 1, or 2. It turned out that the sum of the numbers Anton received is 400 more than the sum Anya received. How many liars are there at the table? Find all possible answers and prove that there are no others.

(A. Chukhnov)

Answer: 200