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

## Clues:
1. The person who prefers city breaks is Bob.
2. The person who has brown hair is the one who only drinks water.
3. Eric is directly left of the person who uses a Samsung Galaxy S21.
4. There is one house between the person who loves beach vacations and the person who enjoys camping trips.
5. The one who only drinks water is somewhere to the left of the person who uses a Xiaomi Mi 11.
6. The person who has brown hair is the Dunhill smoker.
7. The person who has red hair is Arnold.
8. The person who has auburn hair is the person who uses an iPhone 13.
9. The person who has gray hair is not in the sixth house.
10. The person who likes milk is the person who has auburn hair.
11. The person who smokes Blue Master is Alice.
12. The person who uses a OnePlus 9 is the person who likes going on cruises.
13. The coffee drinker is the Prince smoker.
14. The coffee drinker is the person who uses a Huawei P50.
15. Eric is the tea drinker.
16. The Prince smoker is in the third house.
17. The person who smokes Yellow Monster is somewhere to the right of the coffee drinker.
18. The person who enjoys mountain retreats is Peter.
19. The person who uses a Huawei P50 is the person who has blonde hair.
20. There are two houses between the person who loves beach vacations and the person who uses a Google Pixel 6.
21. The one who only drinks water is in the fourth house.
22. Eric is the person partial to Pall Mall.
23. Bob is the boba tea drinker.
 What is the value of attribute PhoneModel for the person whose attribute Vacation is cruise? 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`, `Carol`, `Peter`, `Alice`, `Eric`, `Arnold`
 - They all have a unique favorite flower: `daffodils`, `iris`, `lilies`, `roses`, `tulips`, `carnations`
 - People have unique hair colors: `gray`, `red`, `blonde`, `auburn`, `brown`, `black`
 - Each person has a favorite color: `white`, `blue`, `green`, `purple`, `red`, `yellow`
 - People own unique car models: `tesla model 3`, `ford f150`, `honda civic`, `toyota camry`, `chevrolet silverado`, `bmw 3 series`
 - The mothers' names in different houses are unique: `Janelle`, `Holly`, `Kailyn`, `Sarah`, `Penny`, `Aniya`

## Clues:
1. There is one house between Arnold and the person who loves blue.
2. The person whose mother's name is Janelle is directly left of the person who has auburn hair.
3. The person who owns a Chevrolet Silverado and the person who owns a BMW 3 Series are next to each other.
4. Carol is the person who loves a carnations arrangement.
5. The person who loves the boquet of iris is the person who has gray hair.
6. The person who loves the rose bouquet is The person whose mother's name is Kailyn.
7. Bob is the person who loves the boquet of lilies.
8. The person who loves a bouquet of daffodils is somewhere to the left of the person who owns a Chevrolet Silverado.
9. Bob is in the third house.
10. The person who owns a Honda Civic is in the sixth house.
11. The person who loves the vase of tulips is somewhere to the left of The person whose mother's name is Penny.
12. The person who has brown hair is in the fourth house.
13. The person who has brown hair is somewhere to the left of the person who has blonde hair.
14. The person whose mother's name is Holly is Bob.
15. The person who owns a Ford F-150 is in the third house.
16. Eric is directly left of the person who has auburn hair.
17. The person who owns a Tesla Model 3 is the person who loves yellow.
18. There are two houses between the person who loves purple and The person whose mother's name is Aniya.
19. The person who loves purple is in the first house.
20. The person whose favorite color is green is not in the fourth house.
21. The person who has gray hair is the person who owns a BMW 3 Series.
22. The person who has gray hair is somewhere to the left of Alice.
23. The person whose mother's name is Holly is somewhere to the left of the person who loves the vase of tulips.
24. The person who has red hair is the person who loves white.
 What is the value of attribute Flower for the person whose attribute CarModel is ford f150? Please reason step by step, and put your final answer within \boxed{}

Answer: lilies


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Bob`, `Eric`, `Arnold`
 - They all have a unique favorite flower: `tulips`, `daffodils`, `lilies`, `roses`, `carnations`
 - Each mother is accompanied by their child: `Fred`, `Bella`, `Timothy`, `Meredith`, `Samantha`
 - Each person has a unique level of education: `high school`, `bachelor`, `doctorate`, `master`, `associate`
 - The people keep unique animals: `cat`, `fish`, `dog`, `horse`, `bird`

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

Answer: lilies


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

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

Answer: 4


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

## Clues:
1. The person who loves biography books is Eric.
2. The person who loves biography books is directly left of the person who loves mystery books.
3. The person who loves science fiction books is directly left of the person who loves fantasy books.
4. There are two houses between Bob and the person who likes going on cruises.
5. The person who enjoys mountain retreats is in the first house.
6. The person who loves mystery books is somewhere to the right of the person who prefers city breaks.
7. The person who enjoys camping trips is Arnold.
8. Alice is not in the third house.
9. The person who loves mystery books is not in the fifth house.
10. The person who loves beach vacations is the person who loves mystery books.
 What is the value of attribute House for the person whose attribute Name is Alice? 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`, `Alice`, `Eric`, `Peter`
 - People have unique favorite music genres: `pop`, `classical`, `rock`, `jazz`
 - Each person has a unique type of pet: `fish`, `dog`, `bird`, `cat`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`, `honda civic`
 - Each person has a unique favorite drink: `water`, `milk`, `tea`, `coffee`

## Clues:
1. The one who only drinks water is the person who owns a dog.
2. The one who only drinks water is Eric.
3. The tea drinker is somewhere to the right of the person who loves rock music.
4. The person who likes milk is the person who owns a Tesla Model 3.
5. The person who loves rock music and the person who loves classical music are next to each other.
6. There is one house between the person who likes milk and the person who loves classical music.
7. The person who owns a Honda Civic is Arnold.
8. The person who has a cat and the coffee drinker are next to each other.
9. Alice is not in the second house.
10. The person who loves classical music is not in the second house.
11. The person with an aquarium of fish is the tea drinker.
12. The person who owns a Ford F-150 is somewhere to the right of the person with an aquarium of fish.
13. The person who owns a Toyota Camry is not in the third house.
14. The tea drinker and the person who loves jazz music are next to each other.
 What is the value of attribute House for the person whose attribute CarModel is ford f150? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Carol`, `Eric`, `Arnold`, `Bob`
 - Each person has a unique birthday month: `mar`, `feb`, `jan`, `sept`, `may`, `april`
 - Each person has a unique hobby: `knitting`, `gardening`, `woodworking`, `painting`, `cooking`, `photography`

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

Answer: photography


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

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

Answer: grilled cheese


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Eric`
 - Each person has a favorite color: `yellow`, `red`, `white`, `green`
 - Everyone has a unique favorite cigar: `dunhill`, `blue master`, `pall mall`, `prince`
 - Each person has a unique level of education: `high school`, `associate`, `master`, `bachelor`

## Clues:
1. The person whose favorite color is green is somewhere to the left of the person with a high school diploma.
2. Eric is the person whose favorite color is red.
3. There are two houses between Alice and the person who loves white.
4. The person who smokes Blue Master and the person partial to Pall Mall are next to each other.
5. The person who loves white is the person with a master's degree.
6. Arnold is in the fourth house.
7. The person who loves yellow is the person who smokes Blue Master.
8. The Dunhill smoker is Alice.
9. The person partial to Pall Mall is the person with a master's degree.
10. The person who loves yellow is the person with a bachelor's degree.
 What is the value of attribute Color for the person whose attribute Education is associate? Please reason step by step, and put your final answer within \boxed{}

Answer: green


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

## Clues:
1. The boba tea drinker is the person who uses a Xiaomi Mi 11.
2. Eric is the root beer lover.
3. The person who loves science fiction books is the person who likes milk.
4. The person who likes milk is somewhere to the right of Eric.
5. Arnold is the person who uses a Huawei P50.
6. The person who loves romance books is not in the fourth house.
7. Alice is the person who uses an iPhone 13.
8. The person who loves fantasy books is somewhere to the right of Bob.
9. There are two houses between the person who uses an iPhone 13 and the person who loves fantasy books.
10. The root beer lover is directly left of Peter.
11. The person who uses an iPhone 13 and the person who loves science fiction books are next to each other.
12. The person who uses a Samsung Galaxy S21 is the person who loves romance books.
13. The tea drinker is the person who uses a Huawei P50.
14. There is one house between the person who loves romance books and the person who uses a OnePlus 9.
15. There are two houses between the person who loves biography books and the boba tea drinker.
16. The person who loves historical fiction books is somewhere to the right of the root beer lover.
17. There is one house between the one who only drinks water and the person who uses a Huawei P50.
 What is the value of attribute House for the person whose attribute PhoneModel is iphone 13? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


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

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

Answer: Janelle


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Eric`
 - People use unique phone models: `iphone 13`, `oneplus 9`, `google pixel 6`, `samsung galaxy s21`
 - Each person prefers a unique type of vacation: `mountain`, `cruise`, `beach`, `city`
 - They all have a unique favorite flower: `carnations`, `daffodils`, `lilies`, `roses`

## Clues:
1. The person who uses an iPhone 13 is the person who enjoys mountain retreats.
2. Arnold is not in the second house.
3. The person who loves a bouquet of daffodils is directly left of the person who loves a carnations arrangement.
4. The person who uses a OnePlus 9 is not in the first house.
5. The person who uses an iPhone 13 is somewhere to the right of Eric.
6. The person who loves the rose bouquet is the person who uses a Samsung Galaxy S21.
7. The person who enjoys mountain retreats is in the third house.
8. The person who loves a carnations arrangement is directly left of the person who loves the boquet of lilies.
9. The person who prefers city breaks is not in the second house.
10. The person who uses a OnePlus 9 is Arnold.
11. The person who prefers city breaks is somewhere to the left of the person who loves the boquet of lilies.
12. The person who likes going on cruises is Peter.
 What is the value of attribute House for the person whose attribute Vacation is cruise? 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: `Peter`, `Arnold`, `Bob`, `Alice`, `Eric`, `Carol`
 - The mothers' names in different houses are unique: `Holly`, `Aniya`, `Kailyn`, `Janelle`, `Penny`, `Sarah`
 - Everyone has something unique for lunch: `stir fry`, `stew`, `spaghetti`, `soup`, `pizza`, `grilled cheese`
 - The people keep unique animals: `dog`, `horse`, `cat`, `fish`, `rabbit`, `bird`
 - Everyone has a favorite smoothie: `watermelon`, `desert`, `lime`, `cherry`, `blueberry`, `dragonfruit`
 - Each person lives in a unique style of house: `mediterranean`, `craftsman`, `victorian`, `colonial`, `modern`, `ranch`

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

Answer: stew


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`, `Arnold`, `Bob`
 - Everyone has a favorite smoothie: `lime`, `cherry`, `desert`, `dragonfruit`, `watermelon`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `pizza`, `stew`, `stir fry`
 - People have unique hair colors: `gray`, `black`, `red`, `brown`, `blonde`

## Clues:
1. The person who has blonde hair is the person who loves eating grilled cheese.
2. The Desert smoothie lover is somewhere to the left of the person who has red hair.
3. The person who loves the spaghetti eater is the person who has red hair.
4. Bob and the person who has blonde hair are next to each other.
5. The person who likes Cherry smoothies is in the fourth house.
6. Arnold is the person who has black hair.
7. The Desert smoothie lover is in the third house.
8. The Desert smoothie lover is the person who loves eating grilled cheese.
9. Eric is the person who has blonde hair.
10. The person who has gray hair is somewhere to the left of the person who has black hair.
11. The person who has black hair is the Dragonfruit smoothie lover.
12. Bob is the Watermelon smoothie lover.
13. The person who loves stir fry is the person who has black hair.
14. The person who is a pizza lover is the person who has brown hair.
15. Peter is the person who has gray hair.
 What is the value of attribute Name for the person whose attribute HairColor is gray? 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`, `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 PhoneModel for the person whose attribute Vacation is mountain? Please reason step by step, and put your final answer within \boxed{}

Answer: iphone 13


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Arnold`, `Eric`, `Alice`, `Carol`
 - People have unique hair colors: `red`, `gray`, `auburn`, `blonde`, `brown`, `black`
 - People have unique favorite book genres: `historical fiction`, `biography`, `romance`, `fantasy`, `mystery`, `science fiction`
 - People have unique heights: `very short`, `very tall`, `tall`, `short`, `super tall`, `average`
 - Each person has a unique level of education: `associate`, `doctorate`, `high school`, `trade school`, `bachelor`, `master`
 - Each person has a unique favorite drink: `boba tea`, `milk`, `water`, `root beer`, `coffee`, `tea`

## Clues:
1. The person with a doctorate and Bob are next to each other.
2. There is one house between Carol and the person who has black hair.
3. Peter is the person with a doctorate.
4. The person who loves science fiction books is Eric.
5. The one who only drinks water is the person who has an average height.
6. Alice is the person who loves mystery books.
7. The person who loves romance books is Bob.
8. The person who has brown hair is in the second house.
9. The person who has an average height is in the fourth house.
10. The boba tea drinker is the person who is very tall.
11. The person with a high school diploma is Bob.
12. The person who is short is somewhere to the left of the person who is super tall.
13. The person who loves historical fiction books and the person with a bachelor's degree are next to each other.
14. The person who attended trade school is the person who has red hair.
15. Carol is the person who has auburn hair.
16. The person who is very short is somewhere to the right of the person who has gray hair.
17. There are two houses between Eric and the person who loves fantasy books.
18. There are two houses between the tea drinker and the person who is very short.
19. Eric is somewhere to the left of the person with a bachelor's degree.
20. The root beer lover is Alice.
21. The person with an associate's degree is Arnold.
22. The person who is tall is not in the second house.
23. The person who loves science fiction books is the boba tea drinker.
24. The person who has black hair is not in the fourth house.
25. The person who is short is the coffee drinker.
26. The coffee drinker and the person with a high school diploma are next to each other.
27. The coffee drinker is not in the second house.
 What is the value of attribute Height for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: very short


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

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

Answer: jan


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`, `Eric`, `Arnold`, `Alice`
 - People have unique hair colors: `brown`, `black`, `gray`, `blonde`, `red`
 - Everyone has a unique favorite cigar: `pall mall`, `dunhill`, `prince`, `blue master`, `blends`
 - Each person has a unique favorite drink: `milk`, `coffee`, `water`, `tea`, `root beer`
 - People have unique favorite music genres: `rock`, `classical`, `pop`, `jazz`, `hip hop`

## Clues:
1. The person who has gray hair and Bob are next to each other.
2. Arnold is not in the first house.
3. The person who has blonde hair is not in the second house.
4. Peter is not in the first house.
5. The Dunhill smoker is the person who likes milk.
6. Arnold is the one who only drinks water.
7. The person who loves classical music is the person partial to Pall Mall.
8. The person who loves pop music is directly left of the person who loves jazz music.
9. The person who has brown hair is Bob.
10. The person who has red hair is the person who loves hip-hop music.
11. The root beer lover is Alice.
12. The person who has black hair is in the third house.
13. There are two houses between the coffee drinker and Arnold.
14. The person who smokes Blue Master is somewhere to the right of the person who loves hip-hop music.
15. The person who has blonde hair is the person who smokes many unique blends.
16. Alice is the person who loves classical music.
17. The Dunhill smoker is the person who loves jazz music.
 What is the value of attribute MusicGenre for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: pop


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

## Clues:
1. Alice is not in the fifth house.
2. The person in a Mediterranean-style villa is directly left of the person in a Craftsman-style house.
3. The person in a Craftsman-style house is in the fifth house.
4. Carol is directly left of Alice.
5. The person in a modern-style house is Eric.
6. The person residing in a Victorian house is not in the third house.
7. Bob is in the first house.
8. Bob and Eric are next to each other.
9. Peter is not in the third house.
10. The person in a ranch-style home is somewhere to the right of the person residing in a Victorian house.
11. The person living in a colonial-style house is somewhere to the left of Carol.
 What is the value of attribute House for the person whose attribute HouseStyle is mediterranean? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Peter`, `Bob`, `Arnold`
 - Each person has a favorite color: `red`, `white`, `green`, `blue`, `yellow`
 - Everyone has a favorite smoothie: `watermelon`, `dragonfruit`, `cherry`, `desert`, `lime`
 - Each person has a unique birthday month: `feb`, `sept`, `april`, `mar`, `jan`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`, `fantasy`, `biography`
 - People have unique heights: `very short`, `very tall`, `tall`, `short`, `average`

## Clues:
1. The person who drinks Lime smoothies is in the second house.
2. The person who loves fantasy books is directly left of the Dragonfruit smoothie lover.
3. The person whose birthday is in September is not in the third house.
4. There is one house between the person whose birthday is in March and Peter.
5. The person whose birthday is in February is the Desert smoothie lover.
6. The person who is tall is not in the fifth house.
7. The person whose birthday is in January is somewhere to the left of the person whose birthday is in February.
8. The person who is very short is the person who loves blue.
9. The person whose favorite color is red is Arnold.
10. The person who loves science fiction books is directly left of the person who loves biography books.
11. Bob is not in the fifth house.
12. The person whose birthday is in January is the person whose favorite color is green.
13. The person who has an average height is the person who likes Cherry smoothies.
14. The person who loves fantasy books is the person whose birthday is in February.
15. The person whose favorite color is red is directly left of the person who is very short.
16. The person who loves science fiction books is the person whose birthday is in March.
17. The person who is very tall is in the second house.
18. The person who drinks Lime smoothies is somewhere to the left of the person who loves mystery books.
19. Eric is in the third house.
20. The person who loves fantasy books is the person who loves white.
 What is the value of attribute Smoothie for the person whose attribute Birthday is april? Please reason step by step, and put your final answer within \boxed{}

Answer: watermelon


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Eric`, `Arnold`, `Peter`
 - The mothers' names in different houses are unique: `Aniya`, `Janelle`, `Penny`, `Kailyn`, `Holly`
 - People have unique favorite music genres: `pop`, `jazz`, `rock`, `classical`, `hip hop`
 - Everyone has something unique for lunch: `grilled cheese`, `stir fry`, `stew`, `spaghetti`, `pizza`

## Clues:
1. Eric is somewhere to the left of the person who loves classical music.
2. The person who loves eating grilled cheese is directly left of The person whose mother's name is Janelle.
3. The person whose mother's name is Penny is directly left of Alice.
4. The person who loves stir fry is Peter.
5. The person who loves the spaghetti eater is Alice.
6. Eric is somewhere to the left of the person who loves rock music.
7. The person whose mother's name is Kailyn is directly left of The person whose mother's name is Aniya.
8. The person who loves rock music is not in the third house.
9. The person who loves eating grilled cheese is directly left of the person who loves the stew.
10. The person who loves pop music is The person whose mother's name is Kailyn.
11. The person who loves the spaghetti eater is the person who loves jazz music.
12. The person who loves pop music is Arnold.
 What is the value of attribute House for the person whose attribute Food is spaghetti? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Peter`, `Alice`, `Carol`, `Eric`
 - Everyone has something unique for lunch: `stew`, `grilled cheese`, `stir fry`, `soup`, `pizza`, `spaghetti`
 - People have unique heights: `tall`, `average`, `super tall`, `very short`, `very tall`, `short`
 - Each person has a unique favorite drink: `root beer`, `boba tea`, `coffee`, `water`, `tea`, `milk`
 - Each person has a unique type of pet: `hamster`, `fish`, `cat`, `dog`, `bird`, `rabbit`
 - People use unique phone models: `samsung galaxy s21`, `xiaomi mi 11`, `google pixel 6`, `iphone 13`, `huawei p50`, `oneplus 9`

## Clues:
1. The person who uses an iPhone 13 is in the third house.
2. Bob is the person who is tall.
3. The person who loves the soup is in the second house.
4. The root beer lover is directly left of the person who uses a Xiaomi Mi 11.
5. The person who uses a Huawei P50 is directly left of the person who loves eating grilled cheese.
6. The person who loves stir fry is the person who likes milk.
7. The person who loves eating grilled cheese is the person who is tall.
8. The person who uses a Xiaomi Mi 11 is the coffee drinker.
9. The person who uses a OnePlus 9 is Arnold.
10. The person who owns a rabbit is not in the fifth house.
11. The person with a pet hamster is somewhere to the right of the person who uses a Google Pixel 6.
12. The person who is super tall is the person with an aquarium of fish.
13. The person with an aquarium of fish is Alice.
14. The tea drinker is directly left of the person who is a pizza lover.
15. The person who uses a Samsung Galaxy S21 is Carol.
16. The person who is a pizza lover is the person who is short.
17. Arnold is the person who is very tall.
18. The person who loves the spaghetti eater is the person who uses a Google Pixel 6.
19. The boba tea drinker is somewhere to the right of the person who loves the soup.
20. The person with a pet hamster is not in the fifth house.
21. The person who is very tall is not in the second house.
22. The person who is super tall is somewhere to the left of Peter.
23. The person who is very short is the person who loves the spaghetti eater.
24. The person who keeps a pet bird is somewhere to the left of the person who loves the spaghetti eater.
25. The person with an aquarium of fish is directly left of Eric.
26. The person who owns a dog is the person who likes milk.
 What is the value of attribute PhoneModel for the person whose attribute Pet is hamster? Please reason step by step, and put your final answer within \boxed{}

Answer: xiaomi mi 11


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`, `Carol`, `Arnold`, `Bob`, `Alice`
 - Each person has a favorite color: `blue`, `red`, `green`, `yellow`, `purple`, `white`
 - Each person prefers a unique type of vacation: `city`, `beach`, `cruise`, `cultural`, `mountain`, `camping`
 - People have unique hair colors: `blonde`, `black`, `gray`, `brown`, `red`, `auburn`
 - Each person has an occupation: `artist`, `lawyer`, `engineer`, `teacher`, `doctor`, `nurse`
 - They all have a unique favorite flower: `roses`, `iris`, `daffodils`, `carnations`, `tulips`, `lilies`

## Clues:
1. The person who loves a bouquet of daffodils is the person who is an artist.
2. The person who enjoys camping trips is the person whose favorite color is green.
3. The person who has brown hair is in the first house.
4. The person who loves blue is somewhere to the right of the person who enjoys mountain retreats.
5. The person who loves the vase of tulips is the person who enjoys mountain retreats.
6. Eric is in the second house.
7. The person who likes going on cruises and Arnold are next to each other.
8. The person who loves yellow and the person who enjoys camping trips are next to each other.
9. The person who loves yellow is somewhere to the left of Peter.
10. The person who is a lawyer and Arnold are next to each other.
11. The person who has red hair is Arnold.
12. Alice is the person who is a doctor.
13. The person who loves purple is directly left of the person who loves the vase of tulips.
14. The person who has brown hair is somewhere to the left of the person who is an engineer.
15. The person who has gray hair is somewhere to the left of the person whose favorite color is red.
16. The person who is an artist is somewhere to the right of Arnold.
17. Alice is the person who loves the boquet of lilies.
18. The person whose favorite color is red is directly left of the person who prefers city breaks.
19. The person who prefers city breaks is somewhere to the right of the person who has blonde hair.
20. There is one house between Alice and Bob.
21. The person who loves the rose bouquet is directly left of the person who is a nurse.
22. The person whose favorite color is green is in the fourth house.
23. The person who loves beach vacations is somewhere to the left of the person who has red hair.
24. The person who has black hair is Alice.
25. The person who loves blue is the person who loves a carnations arrangement.
 What is the value of attribute Color for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: purple


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Eric`, `Peter`, `Bob`, `Carol`
 - They all have a unique favorite flower: `daffodils`, `roses`, `tulips`, `iris`, `carnations`, `lilies`
 - People have unique favorite sports: `basketball`, `swimming`, `soccer`, `volleyball`, `tennis`, `baseball`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `lime`, `blueberry`, `dragonfruit`, `desert`
 - Each person has an occupation: `lawyer`, `doctor`, `teacher`, `artist`, `nurse`, `engineer`

## Clues:
1. The person who loves a bouquet of daffodils is the person who is an artist.
2. The person who loves tennis is not in the second house.
3. The person who is an artist is somewhere to the right of the person who likes Cherry smoothies.
4. The person who loves soccer is somewhere to the right of the person who is a teacher.
5. The person who loves basketball is directly left of the person who loves the vase of tulips.
6. The person who is a nurse is in the sixth house.
7. The person who loves a bouquet of daffodils and the Dragonfruit smoothie lover are next to each other.
8. The person who loves tennis is not in the third house.
9. The person who loves the boquet of iris is not in the second house.
10. The person who is a teacher is not in the second house.
11. The person who loves a carnations arrangement is the Watermelon smoothie lover.
12. There are two houses between the person who loves a bouquet of daffodils and the person who is a nurse.
13. The person who loves volleyball is the Dragonfruit smoothie lover.
14. Peter is the person who is a nurse.
15. There are two houses between the person who is a doctor and the person who is a lawyer.
16. The person who loves baseball is somewhere to the left of the person who drinks Blueberry smoothies.
17. The person who drinks Lime smoothies is somewhere to the left of Carol.
18. The person who loves the vase of tulips is the Desert smoothie lover.
19. The person who drinks Blueberry smoothies is Alice.
20. There is one house between the person who is an artist and Arnold.
21. Eric is directly left of Arnold.
22. There is one house between the person who is a doctor and Alice.
23. The person who loves the boquet of lilies is in the first house.
24. The person who is a doctor is not in the fifth house.
 What is the value of attribute Occupation for the person whose attribute House is 5? Please reason step by step, and put your final answer within \boxed{}

Answer: teacher


A circular grass plot with a diameter of 12 feet is intersected by a straight gravel path that is 3 feet wide. One edge of the path passes through the center of the plot. Calculate the number of square feet in the remaining grass area. The original answer is in the form $k\pi - m\sqrt{n}$, please provide the value of k + m + n.

Answer: 42


How many ten-digit positive integers with distinct digits are multiples of $11111$? Provide your answer as an integer.

Answer: 3456


A vertical mass-spring oscillator is displaced $2.0\, \text{cm}$ from equilibrium. The $100\, \text{g}$ mass passes through the equilibrium point with a speed of $0.75\, \text{m/s}$. What is the spring constant of the spring? If the spring constant is $k\, \text{N/m}$, what is the value of $k$?

Answer: 140


In the figure below, $ABCD$ is a rectangle and $E$ is the midpoint of $AD$. The segment $FG$ passes through the midpoint $M$ of $CE$. Determine the ratio between the lengths of $GM$ and $MF$.

![](https://cdn.mathpix.com/cropped/2024_05_01_360f8ce7ec440aed2c7ag-16.jpg?height=351&width=580&top_left_y=1115&top_left_x=652)

Answer: \dfrac{1}{3}


5. Natural numbers divisible by 3 were painted in two colors: red and blue, such that the sum of a blue and a red number is red, and the product of a blue and a red number is blue. In how many ways can the numbers be colored so that the number 546 is blue?

Answer: 7


A Pretti number is a seven-digit positive integer with the following properties:

- The integer formed by its leftmost three digits is a perfect square.
- The integer formed by its rightmost four digits is a perfect cube.
- Its ten thousands digit and ones (units) digit are equal.
- Its thousands digit is not zero.

How many Pretti numbers are there?

Answer: 30


Let $(a_{n})_{n\geq 1}$ be a sequence defined by $a_{n}=2^{n}+49$. Find all values of $n$ such that $a_{n}=pg, a_{n+1}=rs$, where $p,q,r,s$ are prime numbers with $p<q, r<s$ and $q-p=s-r$.

Answer: 7


## Task A-3.2.

Four cubes with edge lengths 1, 2, 3, and 4 are placed next to each other as shown in the figure. Determine the length of the segment $\overline{X Y}$ that lies within the cube with an edge length of 3.

![](https://cdn.mathpix.com/cropped/2024_05_30_a897c7e1109e2574c8d7g-16.jpg?height=349&width=711&top_left_y=1987&top_left_x=981)

Answer: \dfrac{3\sqrt{33}}{5}


In a regular quadrilateral pyramid \(S A B C D\), the side of the base \(A B C D\) is equal to \(a\), and the height is \(2 a \sqrt{2}\). A plane is drawn through vertex \(A\) parallel to the diagonal \(B D\) of the base such that the angle between the line \(A B\) and this plane is \(30^{\circ}\). Find the area of the cross-section.

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


One, (20 points) Real numbers $x, y, z, w$ satisfy $x \geqslant y \geqslant z \geqslant w \geqslant 0$, and $5 x+4 y+3 z+6 w=100$. Find the maximum and minimum values of $x+y+z+w$.

Answer: 25


$\underline{\text { Khachaturyan A.V. }}$.

On a grid paper, a rectangle 200 cells wide and 100 cells high is drawn. It is filled cell by cell, starting from the top left and moving in a spiral (when reaching the edge or an already filled part, turn right, see fig.). Which cell will be filled last? (Specify the number of its row and column. For example, the bottom right cell is in the $100-$th row and $200-$th column.)

![](https://cdn.mathpix.com/cropped/2024_05_06_f36000480089c1ccffcbg-16.jpg?height=269&width=544&top_left_y=823&top_left_x=757)

Answer: 51


2. Given real numbers $a, b$ satisfy $a^{3}+b^{3}+3 a b=1$. Then $a+b$ $=$ . $\qquad$

Answer: 1


For every positive integer $x$, obtain the number $z$ according to the following process:
(i) Move the last digit of $x$ to the first position to get $y$;
(ii) Then take the square root of $y$ to get its arithmetic square root $z$.
For example, if $x=9, y=9, z=\sqrt{9}=3$;

and if $x=5002, y=2500$,
$$
z=\sqrt{2500}=50 \text {. }
$$

Find all positive integers $x$ such that $z=\frac{1}{2} x$.

Answer: 4


Example 1. Calculate the mass of the tetrahedron bounded by the planes $x=0, y=0, z=0$ and $x / 10+y / 8+z / 3=1$, if the density distribution of mass at each point is given by the function $\rho=(1+x / 10+y / 8+z / 3)^{-6}$.

Answer: 2


In $\triangle ABC$, let $a$, $b$, and $c$ represent the lengths of the sides. Prove:
1. $2(a+b+c)\left(a^{2}+b^{2}+c^{2}\right) \geqslant 3\left(a^{3}+b^{3}+c^{3}+3abc\right)$;
2. $(a+b+c)^{3} \leqslant 5[bc(b+c)+ca(c+a)+ab(a+b)]-3abc$;
3. $abc < a^{2}(p-a) + b^{2}(p-b) + c^{2}(p-c) \leqslant \frac{3}{2}abc$, where $p=\frac{1}{2}(a+b+c)$;
4. $1 < \cos A + \cos B + \cos C \leqslant \frac{3}{2}$.

Answer: 2(a+b+c)\left(a^{2}+b^{2}+c^{2}\right) \geqslant 3\left(a^{3}+b^{3}+c^{3}+3abc\right)


Elisa has 24 science books and also books on math and literature. If Elisa had one more math book, then $\frac{1}{9}$ of her books would be math books and one-fourth would be literature books. If Elisa has fewer than 100 books, how many math books does she have?

Answer: 7


4. Quadrilateral $ABCD$ is inscribed in a circle, $BC=CD=4$, $AC$ and $BD$ intersect at $E$, $AE=6$, and the lengths of $BE$ and $DE$ are both integers. Then the length of $BD$ is $\qquad$

Answer: 7


10.9. In the vertices of a regular 100-gon, 100 chips numbered $1, 2, \ldots, 100$ were placed, in exactly that order clockwise. In one move, it is allowed to swap two adjacent chips if their numbers differ by no more than $k$. For what smallest $k$ can a series of such moves result in a configuration where each chip is shifted one position clockwise relative to its initial position?

(S. Berlov)

Answer: 50


15. On an $8 \times 8$ square grid of a chessboard, some squares are marked with stars, such that:
(1) No two starred squares share a common edge or vertex;
(2) Every unstarred square shares a common edge or vertex with at least one starred square.
What is the minimum number of squares that need to be starred? Explain your reasoning.
(1990 Chinese National Training Team Practice Problem)

Answer: 9


Example 2 On the blackboard, there are $n$ (where $n$ is an odd number) expressions: $* x^{2}+* x+*=0$. Two players, A and B, take turns to play the following game: each person is only allowed to replace one of the * symbols with a non-zero real number at a time. After a total of $3 n$ operations, $n$ quadratic equations are obtained. A tries to maximize the number of equations without real roots, while B tries to minimize the number of equations without real roots. Try to determine: Under the condition that A does not depend on how B operates, what is the maximum number of equations without real roots that A can achieve?
（19th All-Russian Mathematical Olympiad problem）

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


Given any point \( P \) on a hyperbola, draw a line \( l \) with an inclination angle \(\alpha\) (a constant value). The line \( l \) intersects the asymptotes of the hyperbola at points \( Q \) and \( R \). Prove that \( |PQ| \cdot |PR| \) is a constant value.

Answer: \frac{a^2 b^2}{|a^2 \sin^2 \alpha - b^2 \cos^2 \alpha|}


Answer: 23


Hen Hao randomly selects two distinct squares on a standard $8 \times 8$ chessboard. Given that the two squares touch (at either a vertex or a side), the probability that the two squares are the same color can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m + n$.

Answer: 715


We are given some three-element subsets of $\{1,2, \dots ,n\}$ for which any two of them have at most one common element. We call a subset of $\{1,2, \dots ,n\}$ nice if it doesn't include any of the given subsets. If, no matter how the three-element subsets are selected initially, we can add one more element to every 29-element nice subset while keeping it nice, find the minimum value of $n$.

Answer: 436


How many ways are there to arrange the numbers $1$ through $8$ into a $2 \times 4$ grid such that:
1. The sum of the numbers in each of the two rows is a multiple of $6$.
2. The sum of the numbers in each of the four columns is a multiple of $3$?

Answer: 288


Sitting at a desk, Alice writes a nonnegative integer $N$ on a piece of paper, with $N \le 10^{10}$. Interestingly, Celia, sitting opposite Alice at the desk, is able to properly read the number upside-down and gets the same number $N$, without any leading zeros. (Note that the digits 2, 3, 4, 5, and 7 will not be read properly when turned upside-down.) Find the number of possible values of $N$.

Answer: 4999


Let $p_1, p_2, p_3, p_4$ be four distinct prime numbers, and let $1 = d_1 < d_2 < \ldots < d_{16} = n$ be the divisors of $n = p_1 p_2 p_3 p_4$. Determine all $n < 2001$ such that $d_9 - d_8 = 22$.

Answer: 1995


Determine the maximal size of a set of positive integers with the following properties:

1. The integers consist of digits from the set $\{ 1,2,3,4,5,6\}$.

2. No digit occurs more than once in the same integer.

3. The digits in each integer are in increasing order.

4. Any two integers have at least one digit in common (possibly at different positions).

5. There is no digit which appears in all the integers.

Answer: 32


On a board, there are integers written from $1$ to $119$. Two players, $A$ and $B$, take turns making a move. A move consists of erasing $9$ numbers from the board. The player after whose move two numbers remain on the board wins, and their score is equal to the positive difference between the two remaining numbers. Player $A$ makes the first move. Find the highest integer $k$ such that player $A$ can ensure that their score is not smaller than $k$.

Answer: 64


We call a path Valid if:

1. It only comprises the following types of steps:
   - $(x, y) \rightarrow (x + 1, y + 1)$
   - $(x, y) \rightarrow (x + 1, y - 1)$

2. It never goes below the x-axis.

Let $M(n)$ be the set of all valid paths from $(0,0)$ to $(2n,0)$, where $n$ is a natural number.

Consider a Valid path $T \in M(n)$. Denote:

$$\phi(T) = \prod_{i=1}^{2n} \mu_i$$

where $\mu_i$ is defined as:

- $1$, if the $i^{th}$ step is $(x, y) \rightarrow (x + 1, y + 1)$
- $y$, if the $i^{th}$ step is $(x, y) \rightarrow (x + 1, y - 1)$

Now, let $f(n) = \sum_{T \in M(n)} \phi(T)$. Evaluate the number of zeroes at the end of the decimal expansion of $f(2021)$.

Answer: 0


David, Kevin, and Michael each choose an integer from the set $\{1, 2, \ldots, 100\}$ randomly, uniformly, and independently of each other. The probability that the positive difference between David's and Kevin's numbers is \emph{strictly} less than that of Kevin's and Michael's numbers is $\frac{m}{n}$, for coprime positive integers $m$ and $n$. Find $100m + n$.

Answer: 1005100


7. A chessboard $(8 \times 8)$ was cut into several equal parts in such a way that all white cells remained uncut, while each black cell was cut. How many parts could have been obtained?

Answer: 32


12.087. Find the angle between the slant height and the height of a cone, the lateral surface of which is the mean proportional between the area of the base and the total surface area.

Answer: \arcsin\left( \dfrac{\sqrt{5} - 1}{2} \right)


VIII-3. Given a right triangle ABC ( $\angle \mathrm{C}=90^{\circ}$ ), whose median to the hypotenuse is equal to $20 \mathrm{~cm}$. From the midpoint $D$ of the hypotenuse, a perpendicular to the hypotenuse is drawn, intersecting one of the legs at point $E$, and $\overline{D E}=15 \mathrm{~cm}$. Calculate the legs of the triangle.

Answer: 24


Let \( S \) be the sum of all positive integers less than \( 10^6 \) which can be expressed as \( m! + n! \), where \( m \) and \( n \) are nonnegative integers. Determine the last three digits of \( S \).

Answer: 130


Given image \begin{tabular}{|llll|} \hline 7 & 1 & 6 & 0 \\ 3 & 3 & 7 & 6 \\ 6 & 6 & 5 & 7 \\ \hline \end{tabular} , and the bit-depth of the image is 4. Suppose you want to use the thresholding technique to segment the image. What is the appropriate threshold value based on the histogram of the image? Follow the following rule when you do thresholding or grouping: pixel $(i, j) \in$ Group A pixels if $g(i, j) \leq$ current threshold $\mathrm{T}$; pixel $(i, j) \in$ Group B pixels otherwise, where $g(i, j)$ is the intensity value of pixel $(i, j)$.
Please give the answer directly without any intermediate steps.

Answer: 4


$\underline{\text { Folklore }}$

In Chicago, there are 36 criminal gangs, some of which are at war with each other. Each gangster belongs to several gangs, and every two gangsters belong to different sets of gangs. It is known that no gangster belongs to two gangs that are at war with each other. Additionally, it turned out that every gang in which a certain gangster does not belong is at war with some gang in which this gangster does belong. What is the maximum number of gangsters that can be in Chicago?

Answer: 3^{12}


2. Find the smallest natural $k$ such that for some natural number $a$, greater than 500,000, and some natural number $b$, the equality $\frac{1}{a}+\frac{1}{a+k}=\frac{1}{b}$ holds. (I. Bogdanov)

Answer: 1001


4. Mom asked Xiaoming to boil water and make tea for the guests. It takes 1 minute to clean the kettle, 15 minutes to boil water, 1 minute to clean the teapot, 1 minute to clean the teacup, and 2 minutes to get the tea leaves. Xiaoming estimated that it would take 20 minutes to complete these tasks. To make the guests drink tea as soon as possible, in what you think is the most reasonable arrangement, how many minutes can the tea be made?

Answer: 16


59. Form the equation of the geometric locus of points that are equally distant from the $O x$ axis and from the point $F(0 ; 2)$. Sketch the curve.

Answer: y = \frac{1}{4}x^2 + 1


As shown in Figure 1, points $D$ and $E$ are on the sides $AB$ and $AC$ of equilateral $\triangle ABC$, respectively, satisfying $BD = AE$. Connecting $CD$ and $BE$ intersect at point $O$. Given that $BO = 2$ and $CO = 5$, find the length of $AO$.

---

The above text translated into English, preserving the original text's line breaks and format, is as follows:

As shown in Figure 1, points $D$ and $E$ are on the sides $AB$ and $AC$ of equilateral $\triangle ABC$, respectively, satisfying $BD = AE$. Connecting $CD$ and $BE$ intersect at point $O$. Given that $BO = 2$ and $CO = 5$, find the length of $AO$.

Answer: \sqrt{19}


Exercise 2. We place the integers from 1 to 9 in each of the cells of a $3 \times 3$ grid. For $i=1$, 2 and 3, we denote $\ell_{i}$ as the largest integer present in the $i$-th row and $c_{i}$ as the smallest integer present in the $i^{\text{th}}$ column.

How many grids exist such that $\min \left\{\ell_{1}, \ell_{2}, \ell_{3}\right\}=\max \left\{\mathbf{c}_{1}, \mathrm{c}_{2}, \mathrm{c}_{3}\right\}=4$?

Answer: 25920


## Task B-3.6.

During the New Year's celebration, an irresponsible person fired two fireworks. One of the fireworks lightly injured a passerby. Forensic experts need to discover from which location the two fireworks were fired. They found a mark (black dot) in the wall of a nearby building that had ricocheted and hit the passerby $\frac{5 \sqrt{3}}{2}$ meters away from the wall. From the mark, they concluded that the path of the projectile formed a $60^{\circ}$ angle with the wall and that it ricocheted at the same angle. The second projectile remained in the wall 1 meter away from the mark of the first projectile, and its path formed a $45^{\circ}$ angle with the wall. The projectile was on the same side of the perpetrator as the passerby. How far was the passerby from the perpetrator? (The paths of both projectiles were in the same plane, parallel to the pavement, and the wall was perpendicular to the pavement.)

Answer: \sqrt{24 - 3\sqrt{3}}


[ Median to the hypotenuse ] [ Area of a triangle (using height and base).]

Outside the right triangle $A B C$ on its legs $A C$ and $B C$, squares $A C D E$ and $B C F G$ are constructed.

The extension of the median $C M$ of triangle $A B C$ intersects the line $D F$ at point $N$. Find the segment $C N$, if $A C=$ $4, B C=1$.

Answer: \dfrac{4\sqrt{17}}{17}


5. By definition, $n!=1 \cdot 2 \cdot 3 \cdot \ldots \cdot n$. Which factor should be removed from the product $1! \cdot 2! \cdot 3! \cdot \ldots \cdot 20!$ so that the remaining product becomes a square of some natural number?

Answer: 10!


6. (15 points) Looking down from the edge of the stream bank, Vovochka decided that the height of his rubber boots would be enough to cross the stream. However, after crossing, Vovochka got his legs wet up to his knees ($H=52$ cm). Estimate the height $h$ of Vovochka's boots. Assume the depth of the stream is constant, and the refractive index of water $n=1.33$.

Rector of SPbPU

![](https://cdn.mathpix.com/cropped/2024_05_06_080bff219c908526ae49g-1.jpg?height=379&width=776&top_left_y=2502&top_left_x=699)

A.I. Rudskoy S.A. Staroytov

## Instructions for Evaluating Final Round Solutions of the Polytechnic Physics Olympiad

The version given to the participant contains 6 problems of varying difficulty.

The evaluation of the work is based on the points received for each individual problem.

The score for solving each problem in the version is one of the following coefficients:

1.0 - the problem is solved correctly;

0.8 - the problem is solved correctly and the answer is given in a general form; there is an error in the unit of measurement of the obtained physical quantity or an arithmetic error;

0.6 - the problem is not fully solved; all necessary physical relationships for solving the problem are present; there is an error in algebraic transformations;

0.4 - the problem is not fully solved; some physical relationships necessary for solving the problem are missing;

0.2 - the problem is not solved; the work contains only some notes related to solving the problem or describing the phenomenon considered in the problem;

0.0 - the solution to the problem or any notes related to it are absent in the work.

The coefficient is entered in the first column "For the Teacher" of the "Answer Sheet". In the second column of the section, the score is entered, which is the product of the coefficient received for the solution and the maximum score for the given problem.

The obtained scores are summed and entered in the "Total" row.

Table for converting coefficients to points.

| \multirow{coefficient}{max points}{} | 10 | 15 | 20 | 25 |
| :---: | :---: | :---: | :---: | :---: |
| 1 | 10 | 15 | 20 | 25 |
| 0.8 | 8 | 12 | 16 | 20 |
| 0.6 | 6 | 9 | 12 | 15 |
| 0.4 | 4 | 6 | 8 | 10 |
| 0.2 | 2 | 3 | 4 | 5 |
| 0 | 0 | 0 | 0 | 0 |

Answer: 39


3. In a convex quadrilateral $A B C D$, the lengths of sides $A B$ and $B C$ are equal, $D B$ is the bisector of angle $A D C$, and $A D: D C=4: 3$. Find the cosine of angle $A K B$, where $K$ is the intersection point of diagonals $A C$ and $B D$, and $B K: K D=1: 3$. (16 points)

Answer: \dfrac{1}{4}


One-and-a-half-liter mineral water bottles are narrowed in the shape of an "hourglass" to make them easier to grip. The normal circumference of the bottle is $27.5 \mathrm{~cm}$, while at the waist - which is a $1 \mathrm{~cm}$ tall cylindrical section - it is only $21.6 \mathrm{~cm}$. The cylindrical sections with different circumferences are connected above and below the waist by truncated cone sections each $2 \mathrm{~cm}$ high. How much taller are such bottles compared to their regular counterparts with the same volume and normal circumference, but without the narrowing?

Answer: 1.18 \text{ cm}


Given $M$ as the set of all rational numbers within the open interval $(0,1)$. Does there exist a subset $A$ of $M$ such that every element in $M$ can be uniquely represented as the sum of a finite number of distinct elements from $A$?

Answer: \text{No}


As shown in the figure, point $D$ is an interior point of $\triangle ABC$. $BD$ intersects $AC$ at $E$, and $CD$ intersects $AB$ at $F$, with $AF = BF = CD$ and $CE = DE$. Find the measure of $\angle BFC$.

Answer: 120^\circ


On the fields $A$, $B$, and $C$ in the bottom left corner of a chessboard, there are white rooks. You can make moves according to the usual chess rules. However, after any move, each rook must be protected by another rook. Is it possible, in several moves, to rearrange the rooks so that each lands on the square marked with the same letter in the top right corner?

Answer: \text{No}


A motorboat departed from dock A downstream. Simultaneously, a boat departed from dock B upstream towards the motorboat. After some time, they met. At the moment of their meeting, a second boat departed from B and after some time met the motorboat. The distance between the points of the first and second meetings is $\frac{56}{225}$ of the distance AB. The flow velocities, the motorboat, and the boats' speeds are constant (the boats' speeds are the same), and the ratio of the motorboat's speed to the boat's speed is $2:3$. What fraction of the distance AB had the motorboat traveled by the time it met the second boat?

Answer: $\frac{161}{225}$ or $\frac{176}{225}$.

Answer: \frac{176}{225}


The MK-97 microcalculator can perform only three operations on numbers stored in its memory:

1. Check if two selected numbers are equal.
2. Add selected numbers.
3. Given selected numbers \(a\) and \(b\), find the roots of the equation \(x^2 + ax + b = 0\), and indicate if there are no roots.

All results of these actions are stored in memory. Initially, one number \(x\) is stored in memory. How can one use the MK-97 to determine if this number is equal to one?

Answer: x = 1


Divide an equilateral triangle into $n^{2}$ equally sized smaller equilateral triangles. Number some of these smaller triangles with the numbers $1, 2, 3, \cdots, m$ such that adjacent triangles have adjacent numbers. Show that $m \leq n^{2} - n + 1$.

Answer: m \leq n^2 - n + 1


The famous Fibonacci sequence is defined as follows:

$$
\begin{gathered}
F_{0}=0, F_{1}=1 \\
F_{n+2}=F_{n+1}+F_{n}
\end{gathered}
$$

1. Let $\varphi$ and $\varphi'$ be the two roots of the equation $x^2 - x - 1 = 0$.

$$
\varphi=\frac{1+\sqrt{5}}{2}, \varphi^{\prime}=\frac{1-\sqrt{5}}{2}
$$

Show that there exist two real numbers $\lambda$ and $\mu$ such that

$$
F_{n}=\lambda \cdot \varphi^{n}+\mu \cdot \varphi^{\prime n}
$$

Can you generalize the result to any sequences of the type $u_{n+2} = a u_{n+1} + b u_{n}$?

2. Show that

$$
\lim_{n \rightarrow \infty} \frac{F_{n+1}}{F_{n}} = \varphi
$$

3. Show the following formulas:

$$
\begin{gathered}
\sum_{i=1}^{n} F_{i}^{2}=F_{n} F_{n+1} \\
F_{2n-1}=F_{n}^{2} + F_{n+1}^{2} \\
m \mid n \Rightarrow F_{m} \mid F_{n}
\end{gathered}
$$

Answer: \varphi


A steamboat travels from point $A$ to point $B$ in $a$ hours and back in $b$ hours. How many hours will a raft take to drift from $A$ to $B$?

Answer: \frac{2ab}{b-a}


In an $n$-throw game of dice, the sum of the points from all throws must be greater than $2^n$ to pass the round.

1. How many rounds can a person pass at most?
2. What is the probability that the person passes the first three rounds in succession?

(Note: The dice is a fair six-sided cube with faces numbered 1 through 6. When the dice come to rest after being thrown, the number on the upper face is the point scored.)

Answer: \frac{100}{243}


As shown in the figure, an equilateral triangle $ABC$ with a side length of 4 is folded along a line $EF$ parallel to side $BC$ in such a way that plane $AEF$ is perpendicular to plane $BCFE$. Let $O$ be the midpoint of $EF$.
(1) Find the cosine value of the dihedral angle $F-AE-B$.
(2) If $BE$ is perpendicular to plane $AOC$, find the length of the fold $EF$.

Answer: \frac{8}{3}


In the figure below, triangle \(ABC\) and rectangle \(PQRS\) have the same area and the same height of 1. For each value of \(x\) between 0 and 1, trapezoid \(ABED\) of height \(x\) is drawn, and then rectangle \(PQNM\) of the same area as the trapezoid is drawn, as shown in the figure. Let \(f\) be the function that assigns to each \(x\) the height of rectangle \(PQNM\).

a) What is the ratio between \(AB\) and \(PQ\)?

b) What is the value of \(f\left(\frac{1}{2}\right)\)?

c) Find the expression of \(f(x)\) and draw the graph of \(f\).

Answer: 2x - x^2


Given distinct real numbers \(a_{1}, a_{2}, a_{3}\) and \(b\). It turns out that the equation \((x-a_{1})(x-a_{2})(x-a_{3}) = b\) has three distinct real roots \(c_{1}, c_{2}, c_{3}\). Find the roots of the equation \((x+c_{1})(x+c_{2})(x+c_{3}) = b\).

Answer: -a_1, -a_2, -a_3


The product of three consecutive even numbers equals $87_{* * * * *}8$. Find these numbers and fill in the blanks in this product.

Answer: 87526608


8. Distribute 16 identical books to 4 classes, with each class getting at least one book, and the number of books each class receives being different. The number of different distribution methods is $\qquad$ .(Answer with a number)

Answer: 216


let $a_{1}, a_{2}, \cdots, a_{9}$ be a permutation of $1,2, \cdots, 9$. If the permutation $C=\left(a_{1}, a_{2}, \cdots, a_{9}\right)$ can be obtained from $1,2, \cdots, 9$, by swapping two elements at most 4 times, the total number of permutations satisfying this condition is \_\_\_\_\_.

Answer: 27568


Two plane mirrors form the edges of a dihedral angle. A light ray perpendicular to the edge of the angle and parallel to the first mirror reflects off the second mirror, then off the first, then off the second again, then the first again, and finally reflects off the second mirror for the fifth time, returning along the same path. What is the measure of the dihedral angle between the mirrors?

(Remember that according to the laws of light reflection, the incident and reflected rays lie in the plane perpendicular to the mirror plane and form angles of equal magnitude with the mirror plane.)

Answer: 30^\circ


Given two complex coefficient functions \( f(x) = \sum_{i=0}^{n} a_{i} x^{n-i} \) and \( g(x) = \sum_{i=0}^{n} b_{i} x^{n-i} \) (where \( a_{0} = b_{0} = 1 \)), \(\sum_{i=1}^{\left\lfloor\frac{n}{2}\right\rfloor} b_{2 i} \) and \(\sum_{i=1}^{\left\lfloor\frac{n+1}{2}\right\rfloor} b_{2 i-1}\) are both real numbers, and the negations of the squares of all roots of \(g(x) = 0\) are exactly all the roots of \(f(x) = 0\). Prove that \(\sum_{i=1}^{n}(-1)^{i} a_{i} \) is a real number.

Answer: \sum_{i=1}^{n}(-1)^{i} a_{i} \text{ is a real number}


3. Find all integer pairs $(a, b)$ such that the system of equations
$$\left\{\begin{array}{l}
x^{2}+2 a x-3 a-1=0 \\
y^{2}-2 b y+x=0
\end{array}\right.$$

has exactly three distinct real solutions $(x, y)$.

Answer: (0, 1)


3. As shown in the figure, it is known that the area of $\triangle A B C$ is 1, and $\frac{B D}{D C}=\frac{C E}{E F}=\frac{A F}{F D}=\frac{1}{2}$. Therefore, the area of $\triangle D E F$ is

Answer: \dfrac{8}{27}


230. Given two equations

$$
x^{2}+a x+1=0 \text { and } x^{2}+x+a=0
$$

Find all values of $a$ for which these equations have at least one common root.

Answer: -2


In a competition there are $18$ teams and in each round $18$ teams are divided into $9$ pairs where the $9$ matches are played coincidentally. There are $17$ rounds, so that each pair of teams play each other exactly once. After $n$ rounds, there always exists $4$ teams such that there was exactly one match played between these teams in those $n$ rounds. Find the maximum value of $n$.

Answer: 7


9.3. Find the value of the expression $x-\sqrt{2022 x}+2023$,

if $x-\sqrt{\frac{2022}{x}}=2023$.

Answer: 2024


29.34. Calculate the length of the arc of the parabola $2 y=x^{2}$ from the point $(0,0)$ to the point $\left(x_{0}, \frac{x_{0}^{2}}{2}\right)$.

Answer: \dfrac{x_0 \sqrt{x_0^2 + 1} + \ln\left(x_0 + \sqrt{x_0^2 + 1}\right)}{2}


In a mathematics competition, there are 30 problems. Each correctly solved problem is worth 4 points, a wrong solution results in -1 point. If someone does not attempt a problem, they get 0 points for it. How many different total scores can a contestant achieve?

Answer: 145


13. B. If five pairwise coprime distinct integers $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ are randomly selected from $1,2, \cdots, n$, and one of these integers is always a prime number, find the maximum value of $n$.

Answer: 48


On Lineland there are 2018 bus stations numbered 1 through 2018 from left to right. A self-driving bus that can carry at most $N$ passengers starts from station 1 and drives all the way to station 2018, while making a stop at each bus station. Each passenger that gets on the bus at station $i$ will get off at station $j$ for some $j>i$ (the value of $j$ may vary over different passengers). Call any group of four distinct stations $i_1, i_2, j_1, j_2$ with $i_u< j_v$ for all $u,v\in \{1,2\}$ a [i]good[/i] group. Suppose that in any good group $i_1, i_2, j_1, j_2$, there is a passenger who boards at station $i_1$ and de-boards at station $j_1$, or there is a passenger who boards at station $i_2$ and de-boards at station $j_2$, or both scenarios occur. Compute the minimum possible value of $N$.

[i]Proposed by Yannick Yao[/i]

Answer: 1009


3. Inside triangle $ABC$, a point $P$ is chosen such that $AP=BP$ and $CP=AC$. Find $\angle CBP$, given that $\angle BAC = 2 \angle ABC$.

---

Here is the translation of the provided text, maintaining the original formatting and structure.

Answer: 30^\circ


3. In triangle $A B C$, the angle bisectors $A A_{1}, B B_{1}, C C_{1}$ are drawn, $L$ is the intersection point of segments $B_{1} C_{1}$ and $A A_{1}$, $K$ is the intersection point of segments $B_{1} A_{1}$ and $C C_{1}$, $M$ is the intersection of $B K$ and $A A_{1}$, $N$ is the intersection of $B L$ and $C C_{1}$. Find the ratio $M S: S N$, if $S$ is the intersection of the angle bisector $B B_{1}$ with segment $M N$, and $A B: B C: A C=2: 3: 4$. (16 points)

Answer: \dfrac{16}{15}


7. Given the elliptical region $D_{1}: \frac{x^{2}}{3}+\frac{y^{2}}{2} \leqslant 1$ and the circular region $D_{2}$ : $x^{2}+y^{2} \leqslant 2.5$. Then the area of the common region between $D_{1}$ and $D_{2}$ is $\qquad$ (accurate to 0.01).
Note: The following result can be used: In the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, $B$ is the upper vertex, and $P\left(x_{0}, y_{0}\right)$ is a point on the first quadrant part of the ellipse, then $S_{\text {sector } O B P}=\frac{a b}{2} \arcsin \frac{x_{0}}{a}$.

Answer: 7.27


18. There are three types of people, A, B, and C, totaling 25 people. A type people always tell the truth, B type people always lie, and C type people alternate between telling the truth and lying (for example, if a C type person tells the truth this time, the next thing they say will definitely be a lie, and the one after that will be the truth again).
The priest asked everyone, “Are you an A type person?” 17 people answered “yes”.
The priest then asked everyone, “Are you a C type person?” 12 people answered “yes”.
The priest then asked everyone, “Are you a B type person?” 8 people answered “yes”.

Among these 25 people, ( ) are C type people.

Answer: 16


9.4 In the thirtieth kingdom, there are three types of coins in circulation: bronze rubles, silver coins worth 9 rubles, and gold coins worth 81 rubles. From the treasury, which contains an unlimited supply of each type of coin, a certain amount was issued with 23 coins, which is less than 700 rubles. Find this amount, given that it is impossible to issue it with a smaller number of coins.

Answer: 647


39. The little trains on多多Island are busily working. Thomas and Percy travel from the shed to the harbor, while Edward travels from the harbor to the station. They all immediately return to their starting points after reaching their destinations, completing one round trip before finishing their work. The three trains start at the same time. When Edward arrives at the station, Thomas also happens to be passing the station, while Percy has only traveled $45 \mathrm{~km}$; when Edward returns to the harbor, Thomas is passing the station again. At this point, Thomas's steam engine malfunctions, and he continues at half speed, meeting Percy face-to-face at the midpoint of the line between the shed and the harbor. What is the length of the railway line between the shed and the harbor in $\mathrm{km}$?

Answer: 225


3. The seller has a balance scale. Help the seller come up with a set of 4 weights that can be used to weigh any whole number of kilograms from 1 to 12 on these scales. No more than two weights can be used for each weighing; weights can be placed on different pans of the scale.

Answer: 1,2,5,10


5. (5 points) As shown in the figure, in $\triangle A B C$, point $E$ is on $A B$, point $F$ is on $A C$, and $B F$ intersects $C E$ at point $P$. If $S$ quadrilateral $A E P F = S_{\triangle B E P} = S_{\triangle C F P} = 4$, then the area of $S_{\triangle B P C}$ is $\qquad$ .

Answer: 12


On four cards, there were four different digits, one of which was zero. Vojta formed the largest four-digit number from the cards, Martin then the smallest four-digit number. Adam wrote the difference between Vojta's and Martin's numbers on the board.

Then Vojta formed the largest three-digit number from the cards, and Martin the smallest three-digit number. Adam again wrote the difference between Vojta's and Martin's numbers on the board. Then Vojta and Martin similarly formed two-digit numbers, and Adam wrote their difference on the board. Finally, Vojta chose the largest one-digit number, and Martin the smallest non-zero one-digit number, and Adam wrote their difference on the board.

When Adam added all four differences on the board, he got 9090. Determine the four digits on the cards.

(L. Růžičková)

Hint. Express the result generally using the unknown digits on the cards.

Answer: 0,1,2,9


\section*{Problem 4 - 251034}

A natural number \(x\) is required to satisfy the following conditions (1) to (5):

(1) The number \(x\), when written in the binary system (base 2), has exactly ten digits.

(2) When \(x\) is written in the ternary system, the second digit is 1.

(3) When \(x\) is written in the quaternary system, the second digit is 0.

(4) The number \(x\), when written in the quinary system, has exactly four digits.

(5) When \(x\) is written in the decimal system, the last digit is 2.

Prove that there is exactly one natural number \(x\) that satisfies these conditions, and determine this number!

Answer: 572


3. There are 5 rooms $A, B, C, D, E$ arranged in a circular pattern, with the number of people living in them being $17, 9, 14, 16, 4$ respectively. Now, adjustments are to be made so that the number of people in each room is the same, and it is stipulated that people can only move to the adjacent left or right room. How many people should each room move to the left or right so that the total number of people moved is minimized?

Answer: 11


Example 2: In a certain year, the total coal production of a coal mine, apart from a certain amount of coal used for civilian, export, and other non-industrial purposes each year, the rest is reserved for industrial use. According to the standard of industrial coal consumption of a certain industrial city in that year, it can supply 3 such industrial cities for 6 years, or 4 such industrial cities for 5 years (of course, the fixed amount of coal for non-industrial use is deducted each year). How many years can it supply if it only supplies the industrial coal for this one city?
(1978, Chongqing Mathematical Competition)

Answer: 10


Poddlisky 0. .  Around a round table, 2015 people are sitting, each of them is either a knight or a liar. Knights always tell the truth, and liars always lie. Each of them was given a card with a number on it; all the numbers on the cards are different. After looking at their neighbors' cards, each person said: "My number is greater than the number of each of my two neighbors." After this, $k$ of those sitting said: "My number is less than the number of each of my two neighbors." For what maximum $k$ could this have happened?

Answer: 2013


6. (15 points) A pedestrian is moving towards a crosswalk along a straight path at a constant speed of 3.6 km/h. At the initial moment, the distance from the pedestrian to the crosswalk is 20 m. The length of the crosswalk is $5 \mathrm{~m}$. At what distance from the crosswalk will the pedestrian be after half a minute?

Answer: 5


## Task 1 - 080911

A FDJ meeting was so well attended that exactly 75 percent of the FDJ members found seats. Therefore, it was decided to hold a second meeting in another room. 150 of the young friends went there. The rest remained in the first room. As a result, exactly 5 seats were left free in this room.

Determine the total number of young friends who appeared at the originally scheduled event!

Answer: 580


4. Six schoolchildren decided to plant 5 trees in the schoolyard. It is known that each tree was planted by a different number of schoolchildren and each schoolchild participated in planting the same number of trees. Could this have happened?

Answer: Yes


13.398 Two friends decided to go hunting. One of them lives 46 km from the hunting base, the other, who has a car, lives 30 km from the base between the base and his friend's house. They set off at the same time, with the car owner driving towards his friend who was walking. Upon meeting, they drove together to the base and arrived there one hour after leaving their homes. If the pedestrian had left his house 2 hours and 40 minutes earlier than the car owner, they would have met 11 km from the pedestrian's house. What is the speed of the car?

Answer: 60


12. (10 points) In a math competition, each team can only score 0 points, 3 points, or 5 points per question. At the end of the competition, the total score of three teams is 32 points. If any team's total score can reach 32 points, how many different combinations of total scores are there for these three teams?

Answer: 255


$64$ teams with distinct skill levels participate in a knockout tournament. In each of the $6$ rounds, teams are paired into match-ups and compete; the winning team moves on to the next round and the losing team is eliminated. After the second-to-last round, winners compete for first and second and losers compete for third and fourth. Assume that the team with higher skill level always wins. What is the probability that the first, second, and third place teams have the highest, second highest, and third highest skill levels, respectively?

[i]2019 CCA Math Bonanza Lightning Round #3.3[/i]

Answer: \dfrac{512}{1953}


We will call two numbers almost equal if they are either equal to each other or differ from each other by no more than one unit. A grid rectangle with sides equal to the natural numbers \(a\) and \(b\) is such that it is impossible to cut out a rectangle along the grid lines that has an area almost equal to half of the original rectangle. What is the smallest value that the number \( |a-b| \) can take?

Answer: 4


A2. Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as
$$
(x+y+z) P(x, y, z)+(x y+y z+z x) Q(x, y, z)+x y z R(x, y, z)
$$
with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^{i} y^{j} z^{k} \in \mathcal{B}$ for all nonnegative integers $i, j, k$ satisfying $i+j+k \geqslant n$.

Answer: 4


5. On New Year's Day, January 1st, the Elderly Sage was reflecting on his life. He noticed that over the past 5 years, all days of the week had been equally represented, and 10 years ago, the New Year's celebration fell on a Friday. On which day of the week did the sage's somber reflections take place? It is known that the sage's reflections occurred in the $21-\mathrm{m}$ century. Don't forget to explain your answer.

Answer: Thursday


4. As shown in Figure 2, given that $\odot O$ is the incircle of rhombus $A B C D$ with side length 16, points $E$ and $F$ are on sides $A B$ and $B C$ respectively, and $E F$ is tangent to $\odot O$ at point $M$. If $B E=4, B F=13$, then the length of $E F$ is

Answer: \dfrac{21}{2}


Example 6 Divide the sides $AB$ and $CD$ of the convex quadrilateral $ABCD$ into $m$ equal parts, and connect the corresponding division points. Then divide the sides $AD$ and $BC$ into $n$ equal parts, and connect the corresponding division points, where $m$ and $n$ are both odd numbers greater than 1. Find the ratio of the area of the middle quadrilateral to the area of quadrilateral $ABCD$.

Answer: \dfrac{1}{mn}


}}[$
Inscribed and Circumscribed Circles
Two tangents drawn from one point
[The area of a figure is equal to the sum of the areas of the figures into which it is divided]

On the sides $B C, C A$, and $A B$ of a triangle, points $A_{1}, B_{1}, C_{1}$ are taken respectively, such that the radii of the circles inscribed in triangles $A_{1} B C_{1}, A B_{1} C_{1}$, and $A_{1} B_{1} C$ are equal to each other and equal to $r$. The radius of the circle inscribed in triangle $A_{1} B_{1} C_{1}$ is $r_{1}$. Find the radius of the circle inscribed in triangle $A B C$.

#

Answer: r + r_1


3. Let $x, y$, and $z$ be real numbers that satisfy the following system:
$$
\begin{aligned}
x+3 y+6 z & =1 \\
x y+2 x z+6 y z & =-8 \\
x y z & =2
\end{aligned}
$$

Find the smallest possible value of $x+y+z$.

Answer: -\dfrac{8}{3}


Two players are playing a game: they take turns naming prime numbers no greater than 100, such that the last digit of the number named by one player matches the first digit of the number named by the other player on their next turn (except for the very first prime number named in the game). Previously named numbers cannot be repeated. The player who cannot name a valid prime number according to these rules loses. Prove that one of the players can guarantee a win and find the minimum possible number of prime numbers that will be used by both players in such a game.

Answer: 3


How many of the first $1000$ positive integers can be expressed in the form $\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor$ , where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$ ?
Please give the answer directly without any intermediate steps.

Answer: 600


## Task 4 - 160814

Peter presents his friend Fritz with the following problem:

"Given a circle whose diameter is equal to the Earth's diameter, and a second concentric circle whose circumference is $1 \mathrm{~m}$ longer than the circumference of the first circle. Determine the distance between the two circle lines!"

After a brief moment of thought, Fritz names this distance and claims:

"If the first circle has only the diameter of a pinhead ( $1 \mathrm{~mm}$ ), and the circumference of the second concentric circle is again $1 \mathrm{~m}$ longer than that of the first circle, then the distance between these two circles is exactly the same as in your problem."

Is Fritz's claim correct?

How large is the distance between the concentric circle lines in both cases?

Answer: \dfrac{1}{2\pi}


444. Eight hockey teams play against each other in a round-robin tournament to determine the final four. What is the minimum number of points that guarantees a team's advancement to the final four?

Answer: 11


Let $n$ a positive integer. We call a pair $(\pi ,C)$ composed by a permutation $\pi$$:$ {$1,2,...n$}$\rightarrow${$1,2,...,n$} and a binary function $C:$ {$1,2,...,n$}$\rightarrow${$0,1$} "revengeful" if it satisfies the two following conditions:

$1)$For every $i$ $\in$ {$1,2,...,n$}, there exist $j$ $\in$ $S_{i}=${$i, \pi(i),\pi(\pi(i)),...$} such that $C(j)=1$.

$2)$ If $C(k)=1$, then $k$ is one of  the $v_{2}(|S_{k}|)+1$ highest elements of $S_{k}$, where $v_{2}(t)$ is the highest nonnegative integer such that $2^{v_{2}(t)}$ divides $t$, for every positive integer $t$.

Let $V$ the number of revengeful pairs and $P$ the number of partitions of $n$ with all parts powers of $2$.
Determine $\frac{V}{P}$.

Answer: n!


3、A and B are 36 kilometers apart. Supermen A and B start walking from $\mathbf{A}$ to $\mathbf{B}$ at the same time. Once they reach $\mathbf{B}$, they immediately walk back to $\mathbf{A}$, and upon reaching $\mathbf{A}$, they immediately walk back to $\mathbf{B}$...... They keep moving back and forth between $\mathbf{A}$ and $\mathbf{B}$. If Super A's speed is $2 k$ kilometers/hour, and Super B's speed is $k$ kilometers/hour. Suppose after $p$ hours, the distance between Super A and Super B reaches its maximum for the 2012th time; after $q$ hours, the distance between Super A and Super B reaches its maximum for the 2013th time. If $q-p$ is a positive integer, find: the maximum value of the positive integer $k$

Answer: 72


Let $ABCD$ be a parallelogram with diagonal $AC = 10$ such that the distance from $A$ to line $CD$ is $6$ and the distance from $A$ to line $BC$ is $7$. There are two non-congruent configurations of $ABCD$ that satisfy these conditions. The sum of the areas of these two parallelograms is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 4717


Sphere $S$ with radius $100$ has diameter $\overline{AB}$ and center $C$. Four small spheres all with radius $17$ have centers that lie in a plane perpendicular to $\overline{AB}$ such that each of the four spheres is internally tangent to $S$ and externally tangent to two of the other small spheres. Find the radius of the smallest sphere that is both externally tangent to two of the four spheres with radius $17$ and internally tangent to $S$ at a point in the plane perpendicular to $\overline{AB}$ at $C$.

Answer: 66


Antonio plays a game where he continually flips a fair coin to see the sequence of heads ($H$) and tails ($T$) that he flips. Antonio wins the game if he sees on four consecutive flips the sequence $TTHT$ before he sees the sequence $HTTH$. The probability that Antonio wins the game is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 17


Let $A_1, A_2, A_3, \ldots , A_{12}$ be the vertices of a regular $12-$gon (dodecagon). Find the number of points in the plane that are equidistant to at least $3$ distinct vertices of this $12-$gon.

Answer: 1


Find the number of sequences of 10 letters where all the letters are either $A$ or $B$, the first letter is $A$, the last letter is $B$, and the sequence contains no three consecutive letters reading $ABA$. For example, count $AAABBABBAB$ and $ABBBBBBBAB$ but not $AABBAABABB$ or $AAAABBBBBA$.

Answer: 86


In convex quadrilateral $ABCD$, $\angle BAD = \angle BCD = 90^o$, and $BC = CD$. Let $E$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$. Given that $\angle AED = 123^o$, find the degree measure of $\angle ABD$.

Answer: 78


For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0, t_1, \ldots, t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \geq t_0$ by the following properties:
(a) $f(t)$ is continuous for $t \geq t_0$, and is twice differentiable for all $t>t_0$ other than $t_1, \ldots, t_n$;
(b) $f\left(t_0\right)=1 / 2$;
(c) $\lim _{t \rightarrow t_k^{+}} f^{\prime}(t)=0$ for $0 \leq k \leq n$;
(d) For $0 \leq k \leq n-1$, we have $f^{\prime \prime}(t)=k+1$ when $t_k<t<t_{k+1}$, and $f^{\prime \prime}(t)=n+1$ when $t>t_n$.
Considering all choices of $n$ and $t_0, t_1, \ldots, t_n$ such that $t_k \geq t_{k-1}+1$ for $1 \leq k \leq n$, what is the least possible value of $T$ for which $f\left(t_0+T\right)=2023$?

Answer: 29


Mônica has six colors to paint the faces of a cube. In how many ways can she do this if:

a) All faces have the same color?

b) Five faces have the same color and one face has a different color from the rest?

c) All faces have different colors?

Note: remember, for example, that the two paintings below are the same, since if we rotate one of them appropriately, we will get the other!
![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-03.jpg?height=384&width=808&top_left_y=798&top_left_x=724)

Answer: 6


8. $n$ chess players participate in a chess tournament, with each pair of players competing in one match. The rules are: the winner gets 1 point, the loser gets 0 points, and in the case of a draw, both players get 0.5 points. If it is found after the tournament that among any $m$ players, there is one player who has won against the other $m-1$ players, and there is also one player who has lost to the other $m-1$ players, this situation is said to have property $P(m)$.

For a given $m(m \geqslant 4)$, find the minimum value of $n$, denoted as $f(m)$, such that for any tournament situation with property $P(m)$, all $n$ players have distinct scores.
(Wang Jianwei provided)

Answer: 2m - 3


In a coastal village, 7 people go fishing every day, 8 people go fishing every other day, 3 people go fishing every three days, and the rest do not fish at all. Yesterday, 12 people went fishing, today 10 people are fishing. How many people will go fishing tomorrow?

Answer: 15


Problem 4. The students of a class are going on a mountain hike, one behind the other. When Andrei arrived at the cabin, half the number of students still on the trail were already in the cabin. Bianca arrived tenth after Andrei, and after her, there were half as many students left as those who had arrived before her at the cabin. How many students are in that class?

Answer: 31


The bases of a trapezoid are \(a\) and \(b\). The trapezoid can be cut by a line parallel to the bases such that circles can be inscribed in the two resulting smaller trapezoids. What is the perimeter of the original trapezoid?

Answer: 2(a + b + \sqrt{ab})


8. In how many different ways can integers $a, b, c \in [1,100]$ be chosen so that the points with coordinates $A(-1, a), B(0, b)$, and $C(1, c)$ form a right triangle?

Answer: 974