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`, `Peter`, `Arnold`
 - Each person has a unique favorite drink: `milk`, `water`, `tea`
 - Each person prefers a unique type of vacation: `mountain`, `city`, `beach`
 - Each person lives in a unique style of house: `colonial`, `victorian`, `ranch`
 - The people keep unique animals: `cat`, `bird`, `horse`
 - Each person has a unique birthday month: `jan`, `sept`, `april`

## Clues:
1. The person living in a colonial-style house is somewhere to the left of the person who likes milk.
2. The person who prefers city breaks is directly left of the person residing in a Victorian house.
3. The person whose birthday is in January is directly left of the cat lover.
4. The one who only drinks water is the person who enjoys mountain retreats.
5. The person who keeps horses is Peter.
6. The person residing in a Victorian house is somewhere to the right of the person who loves beach vacations.
7. Peter is the person who prefers city breaks.
8. The person who enjoys mountain retreats is the person whose birthday is in April.
9. Eric is the one who only drinks water.
 What is the value of attribute Drink for the person whose attribute HouseStyle is ranch? Please reason step by step, and put your final answer within \boxed{}

Answer: milk


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

## Clues:
1. The person in a Craftsman-style house is Arnold.
2. The person who loves the rose bouquet is somewhere to the right of Peter.
3. The photography enthusiast is the person who owns a dog.
4. The person who loves a bouquet of daffodils is not in the fourth house.
5. The person who loves the rose bouquet is the person whose favorite color is red.
6. The person in a Craftsman-style house is in the second house.
7. Eric is the person residing in a Victorian house.
8. The person with an aquarium of fish is the person who loves white.
9. The person who loves cooking is somewhere to the right of the person whose favorite color is red.
10. The person who loves white is the person who loves a carnations arrangement.
11. The person who loves white is somewhere to the right of the person who enjoys gardening.
12. The person who loves a bouquet of daffodils is the person who loves yellow.
13. The person living in a colonial-style house is the person whose favorite color is red.
14. The person who has a cat is Eric.
 What is the value of attribute Color for the person whose attribute Pet is fish? Please reason step by step, and put your final answer within \boxed{}

Answer: white


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`, `Alice`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `roses`, `lilies`
 - People have unique heights: `very short`, `short`, `tall`, `average`
 - The mothers' names in different houses are unique: `Janelle`, `Kailyn`, `Holly`, `Aniya`
 - Each person has an occupation: `engineer`, `doctor`, `teacher`, `artist`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `soccer`

## Clues:
1. The person who loves swimming is the person who loves the rose bouquet.
2. The person who loves the rose bouquet is Eric.
3. Arnold is the person who is tall.
4. The person who loves a bouquet of daffodils is somewhere to the right of the person who is an engineer.
5. The person who loves soccer is the person who is short.
6. The person who is a teacher is in the first house.
7. The person whose mother's name is Janelle is the person who loves a carnations arrangement.
8. The person who loves basketball is the person who has an average height.
9. Arnold is not in the third house.
10. The person whose mother's name is Holly is somewhere to the right of the person who has an average height.
11. Peter is the person who is a doctor.
12. The person whose mother's name is Aniya is Alice.
13. Arnold is the person who loves the boquet of lilies.
 What is the value of attribute Name for the person whose attribute Flower is carnations? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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`, `Eric`, `Peter`, `Alice`, `Arnold`
 - Each person lives in a unique style of house: `victorian`, `ranch`, `modern`, `craftsman`, `mediterranean`, `colonial`
 - Everyone has a unique favorite cigar: `yellow monster`, `blue master`, `blends`, `pall mall`, `dunhill`, `prince`

## Clues:
1. The person living in a colonial-style house is not in the second house.
2. The person partial to Pall Mall is somewhere to the left of the person in a ranch-style home.
3. The person in a ranch-style home is not in the fourth house.
4. The Prince smoker is not in the fourth house.
5. The person in a Mediterranean-style villa is not in the fourth house.
6. The person who smokes many unique blends is in the third house.
7. The person in a modern-style house is directly left of Carol.
8. The person residing in a Victorian house is somewhere to the left of the person in a ranch-style home.
9. Eric is not in the sixth house.
10. Alice and the person in a Craftsman-style house are next to each other.
11. There are two houses between the person partial to Pall Mall and Bob.
12. The person in a ranch-style home is not in the third house.
13. The Prince smoker is somewhere to the right of the person in a Craftsman-style house.
14. The person who smokes Yellow Monster is not in the second house.
15. There are two houses between Arnold and Bob.
16. The person living in a colonial-style house is directly left of the person in a Mediterranean-style villa.
17. The person in a Mediterranean-style villa is not in the second house.
18. The person who smokes Blue Master is somewhere to the right of the person living in a colonial-style house.
19. The person living in a colonial-style house is directly left of Peter.
 What is the value of attribute House for the person whose attribute Cigar is prince? 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`, `Alice`, `Eric`, `Peter`, `Carol`
 - People have unique favorite sports: `volleyball`, `soccer`, `basketball`, `baseball`, `swimming`, `tennis`
 - Each person has a unique hobby: `gardening`, `cooking`, `painting`, `woodworking`, `knitting`, `photography`
 - Each mother is accompanied by their child: `Bella`, `Meredith`, `Alice`, `Samantha`, `Fred`, `Timothy`
 - Each person has a unique birthday month: `feb`, `jan`, `mar`, `may`, `sept`, `april`
 - Everyone has something unique for lunch: `stir fry`, `grilled cheese`, `pizza`, `spaghetti`, `stew`, `soup`

## Clues:
1. The person who loves soccer is somewhere to the right of Eric.
2. The person who paints as a hobby is somewhere to the left of the woodworking hobbyist.
3. The person whose birthday is in February is not in the third house.
4. The person who is the mother of Timothy is directly left of Arnold.
5. The person who loves cooking is somewhere to the left of the person's child is named Fred.
6. Alice is the person who paints as a hobby.
7. The person's child is named Alice is the person who loves cooking.
8. The person whose birthday is in March is the person's child is named Samantha.
9. The person who loves the soup is the person whose birthday is in May.
10. Carol is somewhere to the right of the person whose birthday is in May.
11. The person whose birthday is in March and the person who is a pizza lover are next to each other.
12. The person whose birthday is in February is not in the fourth house.
13. There are two houses between the person who loves baseball and the person who loves the stew.
14. The person whose birthday is in May is somewhere to the left of the person who loves volleyball.
15. The person whose birthday is in January is somewhere to the right of Arnold.
16. The person who loves cooking is not in the third house.
17. The person who enjoys knitting is the person whose birthday is in September.
18. The person who enjoys gardening is the person's child is named Meredith.
19. The person who is a pizza lover is the person who loves swimming.
20. There is one house between the person who loves basketball and the person whose birthday is in May.
21. The person who loves the soup is directly left of the person who loves the spaghetti eater.
22. The person who loves the stew is the person's child is named Bella.
23. The person who enjoys knitting is somewhere to the left of the person who loves the soup.
24. Bob is somewhere to the left of the person who loves the stew.
25. Arnold is the person who loves stir fry.
26. The woodworking hobbyist is directly left of the person who loves the soup.
 What is the value of attribute Food for the person whose attribute Children is Meredith? Please reason step by step, and put your final answer within \boxed{}

Answer: stir fry


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Carol`, `Bob`, `Alice`, `Peter`
 - Each person prefers a unique type of vacation: `mountain`, `camping`, `beach`, `cultural`, `city`, `cruise`
 - Everyone has something unique for lunch: `spaghetti`, `soup`, `stir fry`, `stew`, `grilled cheese`, `pizza`
 - Each person has a unique favorite drink: `boba tea`, `root beer`, `water`, `milk`, `coffee`, `tea`
 - People have unique heights: `tall`, `average`, `very tall`, `short`, `very short`, `super tall`

## Clues:
1. The person who enjoys mountain retreats and the person who prefers city breaks are next to each other.
2. Alice is in the fourth house.
3. The coffee drinker is the person who is a pizza lover.
4. The person who is very tall is directly left of the person who enjoys mountain retreats.
5. Peter is not in the first house.
6. The boba tea drinker is in the fourth house.
7. The person who loves stir fry is somewhere to the left of Eric.
8. The person who likes milk is Bob.
9. The person who loves the soup is not in the second house.
10. The person who loves the spaghetti eater is not in the second house.
11. The person who loves the stew is Eric.
12. The root beer lover is the person who is short.
13. The person who loves beach vacations is in the first house.
14. The person who loves stir fry is the person who enjoys mountain retreats.
15. The person who loves the soup is somewhere to the left of the person who loves the spaghetti eater.
16. The boba tea drinker is somewhere to the left of the person who is a pizza lover.
17. Arnold is the person who likes going on cruises.
18. The person who is very short is somewhere to the right of the person who has an average height.
19. The person who is tall is not in the fourth house.
20. The person who prefers city breaks is the coffee drinker.
21. Bob and the one who only drinks water are next to each other.
22. The person who loves the stew is the person who enjoys camping trips.
23. Arnold is the person who is very short.
24. Carol is in the fifth house.
 What is the value of attribute Drink for the person whose attribute House is 6? Please reason step by step, and put your final answer within \boxed{}

Answer: root beer


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Bob`, `Eric`, `Carol`, `Alice`
 - The people keep unique animals: `horse`, `rabbit`, `fish`, `cat`, `bird`, `dog`
 - Each person has an occupation: `engineer`, `nurse`, `lawyer`, `teacher`, `artist`, `doctor`
 - People have unique favorite sports: `basketball`, `volleyball`, `soccer`, `tennis`, `baseball`, `swimming`
 - People have unique heights: `average`, `tall`, `short`, `very short`, `very tall`, `super tall`

## Clues:
1. The person who is an engineer is the dog owner.
2. The person who has an average height is somewhere to the left of the person who is short.
3. The person who has an average height is directly left of the rabbit owner.
4. The person who is tall is somewhere to the left of the person who is very short.
5. Arnold is the cat lover.
6. The person who keeps horses is the person who is a teacher.
7. Carol is the person who loves soccer.
8. The person who is tall is the person who loves volleyball.
9. The person who is a lawyer is in the fifth house.
10. The person who loves tennis is the person who is a teacher.
11. The person who has an average height is the person who loves swimming.
12. The person who loves baseball is directly left of the person who is an engineer.
13. Peter is the person who is a nurse.
14. Bob is somewhere to the right of the person who is an artist.
15. The person who is a teacher is directly left of the person who loves soccer.
16. The rabbit owner is Alice.
17. The fish enthusiast is Carol.
18. The person who loves baseball is in the first house.
19. The cat lover is somewhere to the right of the person who is very short.
20. The person who is super tall is in the fifth house.
 What is the value of attribute Height for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: very tall


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Alice`, `Carol`, `Arnold`, `Eric`
 - Each person has a unique favorite drink: `milk`, `coffee`, `tea`, `root beer`, `boba tea`, `water`
 - They all have a unique favorite flower: `iris`, `roses`, `carnations`, `daffodils`, `lilies`, `tulips`
 - Each person has a unique birthday month: `sept`, `jan`, `feb`, `april`, `may`, `mar`
 - Each person has a unique hobby: `photography`, `cooking`, `woodworking`, `gardening`, `knitting`, `painting`

## Clues:
1. The root beer lover is in the fifth house.
2. The person whose birthday is in February is Peter.
3. The boba tea drinker is not in the fourth house.
4. The person who loves the boquet of iris is somewhere to the right of the person whose birthday is in April.
5. The coffee drinker is Arnold.
6. Eric is the photography enthusiast.
7. The person who paints as a hobby is the person who loves the vase of tulips.
8. The person whose birthday is in April is in the third house.
9. The tea drinker is the person who loves a bouquet of daffodils.
10. There are two houses between the person who loves the boquet of lilies and the person who likes milk.
11. The person whose birthday is in March is somewhere to the left of the person who enjoys gardening.
12. There are two houses between the person who loves a bouquet of daffodils and the person who paints as a hobby.
13. The person whose birthday is in February is in the fourth house.
14. The person who loves cooking is not in the fourth house.
15. The person who enjoys gardening is somewhere to the left of the person who loves a carnations arrangement.
16. The person whose birthday is in January is Bob.
17. Carol is the person whose birthday is in April.
18. The woodworking hobbyist is in the second house.
19. The person who likes milk is in the sixth house.
20. Peter is the person who loves a carnations arrangement.
21. The person whose birthday is in May is the woodworking hobbyist.
 What is the value of attribute House for the person whose attribute Flower is carnations? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`
 - Each person has a unique birthday month: `jan`, `april`, `sept`
 - Each person has a unique type of pet: `cat`, `dog`, `fish`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `city`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`

## Clues:
1. The person whose birthday is in January is somewhere to the left of the person who is a pizza lover.
2. The person who owns a dog is in the second house.
3. The person whose birthday is in September is the person who loves the spaghetti eater.
4. The person whose birthday is in January is the person who enjoys mountain retreats.
5. The Watermelon smoothie lover is the person with an aquarium of fish.
6. The person whose birthday is in September is not in the third house.
7. Eric is somewhere to the left of the person who prefers city breaks.
8. The person who has a cat is Peter.
9. The person who loves beach vacations is in the third house.
10. The person who likes Cherry smoothies is the person whose birthday is in April.
 What is the value of attribute Food for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: grilled cheese


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 House for the person whose attribute Vacation is camping? 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`, `Arnold`, `Peter`, `Alice`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `pizza`, `stew`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `oneplus 9`
 - Each person has a unique hobby: `painting`, `cooking`, `photography`, `gardening`

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

Answer: spaghetti


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

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

Answer: tall


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

## Clues:
1. The person's child is named Samantha is somewhere to the left of the person who loves biography books.
2. Arnold is the person who loves biography books.
3. The person who is the mother of Timothy is the person who loves science fiction books.
4. The person's child is named Bella is Peter.
5. There is one house between the person who loves biography books and Bob.
6. Alice is somewhere to the right of Eric.
7. The person who loves mystery books is in the fourth house.
8. The person who loves biography books is directly left of the person who is the mother of Timothy.
9. The person who loves fantasy books is directly left of the person's child is named Meredith.
 What is the value of attribute BookGenre for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: fantasy


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`, `Eric`, `Arnold`, `Bob`, `Carol`
 - The people are of nationalities: `dane`, `brit`, `norwegian`, `swede`, `german`, `chinese`
 - People have unique favorite book genres: `historical fiction`, `mystery`, `romance`, `fantasy`, `science fiction`, `biography`
 - The mothers' names in different houses are unique: `Holly`, `Aniya`, `Penny`, `Kailyn`, `Sarah`, `Janelle`

## Clues:
1. The person whose mother's name is Janelle is the Swedish person.
2. Alice is the person who loves science fiction books.
3. The person who loves romance books is not in the fifth house.
4. Carol and Peter are next to each other.
5. The person whose mother's name is Sarah is the Norwegian.
6. There is one house between Arnold and the British person.
7. The German and the person who loves historical fiction books are next to each other.
8. There is one house between the German and Carol.
9. The Swedish person is the person who loves fantasy books.
10. Arnold is in the second house.
11. The person whose mother's name is Sarah is in the third house.
12. The person who loves historical fiction books is somewhere to the right of The person whose mother's name is Penny.
13. The Chinese is not in the sixth house.
14. There are two houses between Alice and the person who loves mystery books.
15. Carol is somewhere to the right of Alice.
16. Arnold is the Dane.
17. The person whose mother's name is Aniya is in the fourth house.
18. The person whose mother's name is Penny is somewhere to the left of the Swedish person.
19. The person whose mother's name is Holly and Eric are next to each other.
 What is the value of attribute Mother for the person whose attribute Nationality is chinese? Please reason step by step, and put your final answer within \boxed{}

Answer: Holly


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

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

Answer: xiaomi mi 11


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
 - Each person has a unique hobby: `cooking`, `gardening`, `painting`, `photography`, `knitting`
 - People have unique favorite sports: `swimming`, `tennis`, `soccer`, `baseball`, `basketball`
 - Each person lives in a unique style of house: `ranch`, `craftsman`, `victorian`, `modern`, `colonial`
 - Each mother is accompanied by their child: `Timothy`, `Samantha`, `Bella`, `Meredith`, `Fred`
 - People have unique heights: `average`, `very tall`, `very short`, `short`, `tall`

## Clues:
1. The person who has an average height is the person's child is named Meredith.
2. The person who is tall is in the second house.
3. Peter is directly left of the person residing in a Victorian house.
4. Alice is the person who is tall.
5. The person who loves baseball is the person who is very tall.
6. The person's child is named Meredith and the person who is the mother of Timothy are next to each other.
7. Bob is the person who paints as a hobby.
8. The person who enjoys gardening is in the second house.
9. The person who is very short is somewhere to the right of Eric.
10. The person who loves tennis is the person's child is named Samantha.
11. The person who loves soccer is not in the first house.
12. The person's child is named Samantha is the person in a modern-style house.
13. The person in a Craftsman-style house is the person who has an average height.
14. The person's child is named Fred is the person residing in a Victorian house.
15. The person who is short is the person who loves basketball.
16. Peter is the person who is very tall.
17. The person in a ranch-style home is somewhere to the left of the person who loves cooking.
18. The person who enjoys knitting and the person who enjoys gardening are next to each other.
19. The person in a modern-style house is the person who loves cooking.
20. The person residing in a Victorian house is in the fifth house.
 What is the value of attribute Hobby for the person whose attribute HouseStyle is craftsman? Please reason step by step, and put your final answer within \boxed{}

Answer: knitting


Given $n=2020$, sort the 6 values: $$n^{n^2},\, 2^{2^{2^n}},\, n^{2^n},\, 2^{2^{n^2}},\, 2^{n^n},\, \text{and}\, 2^{n^{2^2}}$$ from least to greatest. Provide your answer as a 6-digit permutation of the string "123456", where the number $i$ corresponds to the $i$-th expression in the list, from left to right.

Answer: 163542


A $7 \times 7$ board has a chessboard coloring. In one move, you can select any $m \times n$ rectangle of cells and change all its cells to the opposite color (so that black cells become white, and white cells become black). What is the minimum number of moves required to make the entire board a single color?

Answer: 6


After obtaining his magical staff, Sun Wukong borrowed weapons for the other monkeys. It is known that Sun Wukong borrowed a total of 600 jin (a unit of weight), with each weapon weighing no more than 30 jin. Each monkey can carry a maximum of 50 jin. To ensure all borrowed weapons are transported back, how many monkeys are needed at minimum? (Sun Wukong doesn't carry any weapons)

Answer: 23


In the diagram, each street segment is 500 meters long. Lele starts from point $A$ and chooses the shortest path to point $B$. Dongdong starts from point $B$ and chooses the shortest path to point $A$. Lele walks 18 meters per minute, and Dongdong walks 24 meters per minute. The probability that they meet on their way is ________.

Answer: \frac{97}{245}


The faces of a regular octahedron are painted in white and black. Any two faces that share an edge are painted in different colors. Prove that for any point inside the octahedron, the sum of the distances to the planes of the white faces is equal to the sum of the distances to the planes of the black faces.

Answer: 0


There are mittens in a bag: right and left. A total of 12 pairs: 10 red and 2 blue. How many mittens do you need to pull out to surely get a pair of mittens of the same color?

Answer: 13


Buratino, Karabas-Barabas, and Duremar are running along a path around a circular pond. They start simultaneously at the same point, with Buratino running in one direction and Karabas-Barabas and Duremar running in the opposite direction. Buratino runs three times faster than Duremar and four times faster than Karabas-Barabas. After Buratino meets Duremar, he meets Karabas-Barabas 150 meters later. What is the length of the path around the pond?

Answer: 3000


An equilateral triangle with side length 3 is divided into nine triangular cells, as shown in the figure. Initially, zeros are written in these cells. In one move, you can choose two numbers located in adjacent cells and either add one to both or subtract one from both. Petya wants to make several moves so that after these moves, the cells contain, in some order, consecutive natural numbers $n, n+1, \ldots, n+8$. For which $n$ can he achieve this?

Answer: 2


A book contains 30 stories. Each story has a different number of pages under 31. The first story starts on page 1 and each story starts on a new page. What is the largest possible number of stories that can begin on odd page numbers?

Answer: 23


Alvin, Bingyi, and Cheska play a two-player game that never ends in a tie. In a recent tournament between the three players, a total of 60 games were played and each pair of players played the same number of games.

- When Alvin and Bingyi played, Alvin won 20% of the games.
- When Bingyi and Cheska played, Bingyi won 60% of the games.
- When Cheska and Alvin played, Cheska won 40% of the games.

How many games did Bingyi win?
(A) 12
(B) 24
(C) 28
(D) 30
(E) 36

Answer: 28


There is 1 day of travel between neighboring camps. An expedition needs to carry 1 can of preserves to a camp located 5 days from the base and return back. The conditions are:

- Each member of the expedition can carry no more than 3 cans of preserves.
- Each member consumes 1 can of preserves per day.
- Preserves can only be left at camps.

What is the minimum number of cans of preserves that need to be taken from the base camp for this purpose?

Answer: 243


At a certain point on the bank of a wide and turbulent river, 100 meters away from a bridge, a siren is installed that emits sound signals at regular intervals. Another identical siren was taken by Glafira, who got on a bike and positioned herself at the beginning of the bridge on the same bank. Gavrila got into a motorboat, located on the bank halfway between the first siren and the beginning of the bridge. The experimenters start simultaneously, and the speeds of the bicycle and the motorboat relative to the water are both 20 km/h and directed perpendicular to the bank. It turned out that the sound signals from both sirens reach Gavrila simultaneously. Determine the distance from the starting point where Gavrila will be when he is 40 meters away from the bank. Round your answer to the nearest whole number of meters. The riverbank is straight, and the current speed at each point is directed along the bank.

Answer: 41


10.2. To transport 80 tons of cargo, a truck required 5 more trips than to transport 40 tons of the same cargo. On all trips, except possibly the last one in each transportation, the truck was fully loaded. What values can the truck's carrying capacity take?

Answer: (\dfrac{80}{11}, \dfrac{80}{9})


B3 In the figure, two circles with radius 6 are drawn. The centers of the circles are also 6 units apart.

Determine the area of the shaded region, which is bounded by two circular arcs and a common tangent.

![](https://cdn.mathpix.com/cropped/2024_04_17_2582a2b087f6953c1edag-2.jpg?height=363&width=537&top_left_y=721&top_left_x=1316)

Answer: 36 - 9\sqrt{3} - 6\pi


Three. (25 points) Given a positive integer $M$ has $k$ positive divisors, among which, only two divisors are prime, and the sum of the reciprocals of these $k$ positive divisors is $\frac{342}{287}$. Find all values of $M$.

---

The text has been translated while preserving the original formatting and line breaks.

Answer: 2009


Problem 5. In triangle $A B C$, the angle at vertex $B$ is $120^{\circ}$, point $M$ is the midpoint of side $A C$. Points $E$ and $F$ are chosen on sides $A B$ and $B C$ respectively such that $A E=E F=F C$. Find $\angle E M F$.

Answer: 90^\circ


# Task № 4.1

## Condition:

On an island of knights, who always tell the truth, and liars, who always lie, a five-story building was built. One resident lived on each floor. One day, each of them made the same statement:

"There are more liars above me than knights below me!"

How many liars can live in this building?

Answer: 3


1. Solve the equation $2 \cdot 4^{2 x}-17 \cdot 4^{x}+8=0$.

Answer: \frac{3}{2}


52. Given a triangle $T$.

a) Place a centrally symmetric polygon $m$ of the largest possible area inside $T$.

What is the area of $m$ if the area of $T$ is 1?

b) Enclose $T$ in a convex centrally symmetric polygon $M$ of the smallest possible area.

What is the area of $M$ if the area of $T$ is 1?

Answer: \dfrac{1}{2}


Given the parabola equation as $y=a x^{2}+b x+c(a \neq 0)$. Inside it, construct a circle $\odot C_{1}$ with radius $\frac{1}{|a|}$ that is internally tangent to the parabola. Construct $\odot C_{2}$ to be externally tangent to $\odot C_{1}$ and internally tangent to the parabola. Construct $\odot C_{3}$ to be externally tangent to $\odot C_{2}$ and internally tangent to the parabola, and so on, resulting in a sequence of circles $C_{1}, C_{2}, C_{3}, \cdots$, $C_{n}, \cdots$. Find the sum of the areas of the first $n$ circles.

Answer: \dfrac{\pi n(n+1)(2n+1)}{6 a^{2}}


[ Two pairs of similar triangles]

In triangle $ABC$, point $D$ bisects side $AB$, and point $E$ lies on side $BC$, such that segment $BE$ is one-third the length of side $BC$. Segments $AE$ and $CD$ intersect at point $O$. Find $AB$, given that $AE=5$, $OC=4$, and $\angle AOC=120^{\circ}$.

Answer: 2\sqrt{7}


 1
If \(0. \dot{1} + 0.0 \dot{2} + 0.00 \dot{3} + \ldots + 0.00000000 \dot{9} = a\), find the value of \(a\) (Give your answer in decimal).

 2
The circle has center \(O\) and radius \(1\). \(A\) and \(B\) are points on the circle. Given that \(\frac{\text{Area of shaded part}}{\text{Area of unshaded part}} = \frac{\pi - 2}{3\pi + 2}\) and \(\angle AOB = b^\circ\), find the value of \(b\).

 3
A sequence of figures \(S_0, S_1, S_2, \ldots\) are constructed as follows. \(S_0\) is obtained by removing the middle third of the \([0,1]\) interval; \(S_1\) by removing the middle third of each of the two intervals in \(S_0\); \(S_2\) by removing the middle third of each of the four intervals in \(S_1\); \(S_3, S_4, \ldots\) are obtained similarly. Find the total length \(c\) of the intervals removed in the construction of \(S_5\) (Give your answer in fraction).

 4
All integers are coded as shown in the following table. If the sum of all integers coded from 101 to 200 is \(d\), find the value of \(d\).

\[
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Integer & \ldots & \ldots & -3 & -2 & -1 & 0 & 1 & 2 & 3 & \ldots & \ldots \\
\hline
Code & \ldots & \ldots & 7 & 5 & 3 & 1 & 2 & 4 & 6 & \ldots & \ldots \\
\hline
\end{tabular}
\]

Answer: 0.13717421


10. Person A and Person B start from points $A$ and $B$ respectively (Person A starts from $A$), walking towards each other and continuously moving back and forth between the two points. Person A's speed is 4 times that of Person B. It is known that the distance between $A$ and $B$ is $S$ kilometers, where $S$ is a positive integer and has 8 factors. The first time they meet at point $C$ (note: meeting here can refer to facing each other or catching up from behind), the length of $AC$ is an integer; the second time they meet at point $D$, the length of $AD$ is also an integer; after the second meeting, Person B feels that his speed is too slow, so he borrows a motorcycle from a villager near $D$. By the time he borrows the motorcycle and returns to $D$, Person A has already reached point $E$ (Person A has not yet reached $A$), and the length of $AE$ is also an integer; finally, Person B rides the motorcycle to catch up with Person A, with the motorcycle's speed being 14 times that of Person A, and they both reach point $A$ at the same time. Therefore, the distance between $A$ and $B$ is $\qquad$ kilometers.

Answer: 105


8. In a right trapezoid, the upper base is shorter than the lower base. If the volume of the solid of revolution obtained by rotating it around its lower base is $80 \pi$,

the volume of the solid of revolution obtained by rotating it around its upper base is $112 \pi$, and the volume of the solid of revolution obtained by rotating it around its right leg is $156 \pi$, then the perimeter of the right trapezoid is

Answer: 16 + 2\sqrt{13}


## Task A-2.3.

Let $A B C$ be a triangle such that $3|B C|=|A B|+|C A|$. Let $T$ be a point on the side $\overline{A C}$ such that $4|A T|=|A C|$, and let $K$ and $L$ be points on the sides $\overline{A B}$ and $\overline{C A}$, respectively, such that $K L \| B C$ and the line $K L$ is tangent to the incircle of triangle $A B C$.

In what ratio does the segment $\overline{B T}$ divide the segment $\overline{K L}$?

Answer: \dfrac{1}{2}


Problem 4. Each of the two workers was assigned to process the same number of parts. The first completed the work in 8 hours. The second spent more than 2 hours on setting up the equipment and with its help finished the work 3 hours earlier than the first. It is known that the second worker processed as many parts in 1 hour after the equipment started as the first did by that time. By what factor does the equipment increase labor productivity?

Answer: 4


2. There is a cube fixed on legs, and six different paints. In how many ways can all the faces of the cube be painted (each in one color, not all paints have to be used) so that adjacent faces (having a common edge) are of different colors? (16 points)

Answer: 4080


8. As shown in the figure, in the right triangle $\triangle A B C$, $A B=3, A C=4$, points $D, E, F, G, H, I$ are all on the rectangle $K L M J$, and $A B E D, A C H I, B C G F$ are all squares. Then the area of $K L M J$ is $\qquad$.

Answer: 110


11.1. Find all solutions in natural numbers for the equation: $x!+9=y^{3}$.

Answer: x = 6


4A. Two squares with side 1 are given, such that one vertex of one square is the center of the other square. What is the smallest possible value of the area of the intersecting part?

Answer: \dfrac{1}{4}


A robot is standing on the bottom left vertex $(0,0)$ of a $5\times5$ grid, and wants to go to $(5,5)$, only moving to the right $(a,b)\mapsto(a+1,b)$ or upward $(a,b)\mapsto(a,b+1)$. However this robot is not programmed perfectly, and sometimes takes the upper-left diagonal path $(a,b)\mapsto(a-1,b+1)$. As the grid is surrounded by walls, the robot cannot go outside the region $0\leq a,b\leq5$. Supposing that the robot takes the diagonal path exactly once, compute the number of different routes the robot can take.

Answer: 1650


9,10,11 |

Authors: Bverlov S.L., $\underline{\text { Yakubov A. }}$.

Given a parallelogram $A B C D$, where $A B<A C<B C$. Points $E$ and $F$ are chosen on the circumcircle $\omega$ of triangle $A B C$ such that the tangents to $\omega$ at these points pass through point $D$; moreover, segments $A D$ and $C E$ intersect. It turns out that $\angle A B F=\angle D C E$. Find the angle $A B C$.

#

Answer: 60^\circ


4. (3 points) A right trapezoid, its top base is $60 \%$ of the bottom base. If the top base increases by 24 meters, it can become a square. The original area of the right trapezoid is $\qquad$ square meters.

Answer: 2880


Triangle $ABC$ is inscribed in a unit circle $\omega$. Let $H$ be its orthocenter and $D$ be the foot of the perpendicular from $A$ to $BC$. Let $\triangle XY Z$ be the triangle formed by drawing the tangents to $\omega$ at $A, B, C$. If $\overline{AH} = \overline{HD}$ and the side lengths of $\triangle XY Z$ form an arithmetic sequence, the area of $\triangle ABC$ can be expressed in the form $\tfrac{p}{q}$ for relatively prime positive integers $p, q$. What is $p + q$?

Answer: 11


10. On a plane, a circle divides the plane into 2 parts, a circle and a line can divide the plane into at most 4 parts, a circle and 2 lines can divide the plane into at most 8 parts. A circle and 5 lines can divide the plane into $\qquad$ parts.

Answer: 26


6. On the front and back of four cards, 0 and 1, 0 and 2, 3 and 4, 5 and 6 are written respectively. By placing any three of them side by side to form a three-digit number, a total of $\qquad$ different three-digit numbers can be obtained.

Answer: 124


Nonnegative reals $x_1$, $x_2$, $\dots$, $x_n$ satisfies $x_1+x_2+\dots+x_n=n$. Let $||x||$ be the distance from $x$ to the nearest integer of $x$ (e.g. $||3.8||=0.2$, $||4.3||=0.3$). Let $y_i = x_i ||x_i||$. Find the maximum value of $\sum_{i=1}^n y_i^2$.

Answer: \dfrac{2n^2 - 2n + 1}{8}


Problem 9.1. In a notebook, a triangular grid is drawn (see figure). Tanya placed integers at the nodes of the grid. We will call two numbers close if they are in adjacent nodes of the grid. It is known that

- the sum of all ten numbers is 43;
- the sum of any three numbers such that any two of them are close is 11.

What is the central number?

![](https://cdn.mathpix.com/cropped/2024_05_06_ce887724b4f1896529bag-1.jpg?height=458&width=519&top_left_y=671&top_left_x=467)

Answer: 10


In a given trapezoid, a triangle is drawn with a base that coincides with one of the non-parallel sides $A C$ of the trapezoid, and its vertex is located at the midpoint of the side opposite to $A C$. Show that the area of this triangle is half of the area of the trapezoid.

Answer: \frac{1}{2}


From vertex \( A \) of the square \( ABCD \), two rays are drawn inside the square. Perpendiculars \( BK \), \( BL \), \( DM \), and \( DN \) are dropped from vertices \( B \) and \( D \) onto these rays. Prove that segments \( KL \) and \( MN \) are equal and perpendicular.

Answer: KL = MN \text{ and } KL \perp MN


68. Four foreign friends, each of them can speak 2 out of 4 languages: English, French, Japanese, and Chinese. One language is spoken by three of them, but there is no language that all of them can speak. It is also known that:
(1) A can speak Japanese, but D cannot, yet they communicate in the same language;
(2) B does not speak English, and when A and C communicate, he acts as a translator;
(3) B, C, and D cannot find a common language to communicate;
(4) None of the four can speak both Japanese and French.
The number of people who can speak Chinese is $\qquad$.

Answer: 3


Example 3 In a cube with edge length 1, let the volume of the right prism with the midpoints of the edges of the top and bottom faces as vertices be $V_{1}$, and the volume of the right prism with the midpoints of the edges of the left and right faces as vertices be $V_{2}$. Try to find the volume $V$ of the common part of $V_{1}$ and $V_{2}$.

Answer: \dfrac{1}{3}


Nathan has discovered a new way to construct chocolate bars, but it's expensive! He starts with a single $1 \times 1$ square of chocolate and then adds more rows and columns from there. If his current bar has dimensions $w \times h$ (where $w$ is the number of columns and $h$ is the number of rows), then it costs $w^2$ dollars to add another row and $h^2$ dollars to add another column. What is the minimum cost to get his chocolate bar to size $20 \times 20$?

Answer: 5339


The cross-section of a prism with an index of refraction $1.5$ is an equilateral triangle, as shown in the diagram. A ray of light enters horizontally from the air into the prism and exits at an angle $\theta$ with respect to the surface of the triangle. Find $\theta$ in degrees and round to the nearest whole number.

Answer: 13


For a complex number constant $c$, and real number constants $p$ and $q$, there exist three distinct complex values of $x$ that satisfy the equation:
\[
x^3 + cx + p(1 + qi) = 0
\]
Suppose $c$, $p$, and $q$ were chosen such that all three complex roots $x$ satisfy the inequality:
\[
\frac{5}{6} \leq \frac{\mathrm{Im}(x)}{\mathrm{Re}(x)} \leq \frac{6}{5}
\]
where $\mathrm{Im}(x)$ and $\mathrm{Re}(x)$ are the imaginary and real parts of $x$, respectively. The largest possible value of $|q|$ can be expressed as a common fraction \( \frac{m}{n} \), where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Answer: 11


Let $\Omega$ be a circle with radius $18$ and let $\mathcal{S}$ be the region inside $\Omega$ that the centroid of $\triangle XYZ$ sweeps through as $X$ varies along all possible points lying outside of $\Omega$, $Y$ varies along all possible points lying on $\Omega$ and $XZ$ is tangent to the circle. Compute the greatest integer less than or equal to the area of $\mathcal{S}$.

Answer: 904


Pedro must choose two irreducible fractions, each with a positive numerator and denominator, such that:

1. The sum of the fractions is equal to $2$.
2. The sum of the numerators of the fractions is equal to $1000$.

In how many ways can Pedro do this?

Answer: 200


Let $x_1, x_2, \ldots, x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p + x_q + x_r$ (with $1 \leq p < q < r \leq 5$) are equal to $0$, then $x_1 = x_2 = \cdots = x_5 = 0$.

Answer: 7


Sixty points are marked on a circle, divided as follows: thirty are red, twenty are blue, and ten are green. These points divide the circle into sixty arcs. Each arc is assigned a number based on the colors of its endpoints:

- An arc between a red and a green point is assigned the number $1$.
- An arc between a red and a blue point is assigned the number $2$.
- An arc between a blue and a green point is assigned the number $3$.
- An arc between two points of the same color is assigned the number $0$.

What is the greatest possible sum of all the numbers assigned to the arcs?

Answer: 100


On the evening, more than $\frac{1}{3}$ of the students of a school are going to the cinema. On the same evening, more than $\frac{3}{10}$ are going to the theatre, and more than $\frac{4}{11}$ are going to the concert. At least how many students are there in this school?

Answer: 173


Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$. Pick points $E, F, G, H$ such that the angles $\angle AEB$, $\angle BFC$, $\angle CGD$, $\angle AHD$ are all right. What is the largest possible area of quadrilateral $EFGH$?

Answer: 98


Consider a random permutation of the set $\{1, 2, \ldots, 2015\}$. In other words, for each $1 \le i \le 2015$, $i$ is sent to the element $a_i$ where $a_i \in \{1, 2, \ldots, 2015\}$ and if $i \neq j$, then $a_i \neq a_j$. What is the expected number of ordered pairs $(a_i, a_j)$ with $i - j > 155$ and $a_i - a_j > 266$?

Answer: 651222


Starting with a $5 \times 5$ grid, choose a $4 \times 4$ square in it. Then, choose a $3 \times 3$ square in the $4 \times 4$ square, and a $2 \times 2$ square in the $3 \times 3$ square, and a $1 \times 1$ square in the $2 \times 2$ square. Assuming all squares chosen are made of unit squares inside the grid, in how many ways can the squares be chosen so that the final $1 \times 1$ square is the center of the original $5 \times 5$ grid?

Answer: 36


A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut. What is the smallest possible volume of the largest of the three cuboids?

Answer: 350


Around a circle are written $2009$ integers, not necessarily distinct, so that if two numbers are neighbors, their difference is $1$ or $2$. We will say that a number is *huge* if it is greater than its two neighbors, and that it is *tiny* if it is less than its two neighbors. The sum of all the huge numbers is equal to the sum of all the tiny numbers plus $1810$. Determine how many odd numbers there can be around the circumference.

Answer: 1005


Let $S$ be the set of all ordered pairs $(x, y)$ of integer solutions to the equation:
\[ 6x^2 + y^2 + 6x = 3xy + 6y + x^2y. \]
$S$ contains a unique ordered pair $(a, b)$ with a maximal value of $b$. Compute $a + b$.

Answer: 34


On a straight line lie $100$ points and another point outside the line. What is the maximum number of isosceles triangles that can be formed using these $101$ points as vertices?

Answer: 50


The Venusian prophet Zabruberson sent his pupils a $10000$-letter word, with each letter being either $A$ or $E$. This is known as the Zabrubic word. The pupils consider that for $1 \leq k \leq 10000$, each word comprised of $k$ consecutive letters of the Zabrubic word is a prophetic word of length $k$. It is known that there are at most $7$ prophetic words of length $3$. Find the maximum number of prophetic words of length $10$.

Answer: 504


Ash writes the positive integers from $1$ to $2016$ inclusive as a single positive integer $n = 1234567891011\ldots2016$. What is the result obtained by successively adding and subtracting the digits of $n$? In other words, compute $1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 1 + 0 - 1 + \ldots$.

Answer: -1008


There are $n$ circular paper pieces, and every two pieces have a common point. It is possible to hook all $n$ pieces with 7 hooks.

Answer: 7


Along the path between Neznaika's and Syneglazka's houses, there were 15 peonies and 15 tulips growing in a row, mixed together.

On her way to visit Neznaika, Syneglazka watered all the flowers in a row. After watering the 10th tulip, she ran out of water, and 10 flowers remained unwatered.

The next day, on his way to visit Syneglazka, Neznaika picked all the flowers in a row. After picking the 6th tulip, he decided he had enough for a bouquet.

How many flowers remained growing along the path?

Answer: 19


On the Cartesian grid, Johnny wants to travel from \((0,0)\) to \((5,1)\), and he wants to pass through all twelve points in the set \(S=\{(i, j) \mid 0 \leq i \leq 1,0 \leq j \leq 5, i, j \in \mathbb{Z}\}\). Each step, Johnny may go from one point in \(S\) to another point in \(S\) by a line segment connecting the two points. How many ways are there for Johnny to start at \((0,0)\) and end at \((5,1)\) so that he never crosses his own path?

Answer: 252


There are 20 cards, each inscribed with a natural number greater than 0, and the sum of the natural numbers on any 9 cards is not greater than 63. If the cards with numbers greater than 7 are called "Dragon Cards," how many "Dragon Cards" can there be at most? What is the maximum sum of the natural numbers on all "Dragon Cards"?

Answer: 61


In a certain meeting, there are 30 participants. Each person knows at most 5 others among the remaining participants, and in any group of 5 people, there are always two who do not know each other. Find the largest positive integer \( k \) such that there exists a subset of \( k \) people from these 30 people who do not know each other.

Answer: 6


Two girls knit at constant, but different speeds. The first girl takes a tea break every 5 minutes, and the second girl every 7 minutes. Each tea break lasts exactly 1 minute. When the girls went for a tea break together, it turned out that they had knitted the same amount. By what percentage is the first girl's productivity higher if they started knitting at the same time?

Answer: 5


On the plane, there are two triangles, each with side lengths 18, 24, and 30. If the two triangles do not completely overlap but share the same circumcircle as well as the same inscribed circle, find the area of the region common to both triangles.

Answer: 132


Joe B. is frustrated with chess. He breaks the board, leaving a \(4 \times 4\) board, and throws 3 black knights and 3 white kings at the board. Miraculously, they all land in distinct squares! What is the expected number of checks in the resulting position? (Note that a knight can administer multiple checks and a king can be checked by multiple knights.)

Answer: \frac{9}{5}


Given that $A$ and $B$ are the left and right vertices of the ellipse $C: \frac{x^{2}}{25}+\frac{y^{2}}{9}=1$, the line $l$ intersects the ellipse $C$ at points $M$ and $N$. Let the slope of line segment $AM$ be $k_{1}$ and the slope of line segment $BN$ be $k_{2}$, with $k_{1}: k_{2} = 1: 9$.

(1) Prove that the line $l$ passes through a fixed point;
(2) Let the area of triangle $AMN$ be $S_{1}$ and the area of triangle $BMN$ be $S_{2}$. Find the maximum value of $S_{1} - $S_{2}$.

Answer: 15


In $\triangle ABC$, $\angle ABC = 90^{\circ}$, and $BC = 8$. Points $D$ and $E$ lie on $AB$ and $AC$ respectively, such that $CD = DE$ and $\angle DCB = \angle EDA$. If the area of $\triangle EDC$ is 50, what is the length of $AB$?

Answer: 56


In a regular decagon, all diagonals were drawn. How many pairwise non-similar triangles are there in this figure?

Answer: 8


A person left a will dictating that a certain amount should be separated from their estate and invested at $4 \frac{1}{2} \%$ compound interest. From this, the municipality of their hometown should receive an annuity of 5000 florins at the end of each year for 26 years. However, the relatives contested the will and it took 3 years before it could be executed. According to the court's decision, the municipality had the option either to take the accrued three payments at once or extend the period of the annuity payments beyond the 26 years. The municipality chose the latter. For how many years will the municipality receive the annuity?

Answer: 34


As shown in the figure, the area of square \(ABCD\) is 12, \(AE = ED\), and \(EF = 2FC\). What is the area of \(\triangle ABF\)?

Answer: 5


Around a circle are written the natural numbers from 1 to \( N \) with \( N > 2 \), such that each number is written only once and each pair of neighboring numbers has at least one digit in common. Find the smallest \( N > 2 \) for which this is possible.

Answer: 29


For any real number \( x \), let \([x]\) represent the greatest integer less than or equal to \( x \). When \( 0 \leq x \leq 100 \), determine the number of different integers in the range of \( f(x) = [3x] + [4x] + [5x] + [6x] \).

Answer: 1201


2.76 Determine for which values of $m$ one of the roots of the equation $z^{3}-\left(m^{2}-m+7\right) z-\left(3 m^{2}-3 m-6\right)=0$ is equal to -1. Find the two remaining roots of the equation for these values of $m$.

Answer: -2


11.6. The sheriff believes that if the number of bandits he catches on a certain day is a prime number, then he is lucky. On Monday and Tuesday, the sheriff was lucky, and starting from Wednesday, the number of bandits he caught was equal to the sum of the number from the day before yesterday and twice the number from yesterday. What is the maximum number of consecutive days the sheriff could have been lucky this week? Justify your answer.

Answer: 5


4. The last two digits of the integer $\left[\frac{10^{93}}{10^{31}+3}\right]$ are $\qquad$ (write the tens digit first, followed by the units digit; where $[x]$ denotes the greatest integer not exceeding $x$).

Answer: 08


Exercise 8. A coloring of the integers $\{1,2, \cdots, 2020\}$ in blue and red is said to be pleasant if there do not exist two distinct integers in $\{1,2, \cdots, 2020\}$ of the same color whose sum is a power of 2. How many such colorings exist?

Answer: 2048


The Garfield Super Winners play $100$ games of foosball, in which teams score a non-negative integer number of points and the team with more points after ten minutes wins (if both teams have the same number of points, it is a draw). Suppose that the Garfield Super Winners score an average of $7$ points per game but allow an average of $8$ points per game. Given that the Garfield Super Winners never won or lost by more than $10$, what is the largest possible number of games that they could win?

[i]2019 CCA Math Bonanza Lightning Round #4.1[/i]

Answer: 81


7. There is a conical container with its vertex at the bottom and its base horizontal, and its axial section is an equilateral triangle with a side length of 6. The container is filled with water. Now, a square prism with a base side length of $a(a<6)$ is vertically immersed in the container. To make the water overflow from the container as much as possible, the value of $a$ should be $\qquad$ .

Answer: 2\sqrt{2}


8. Given that the three sides of $\triangle A B C$ are $13,14,15$. There are 4 circles $O, O_{1}, O_{2}, O_{3}$ with the same radius $r$ placed inside $\triangle A B C$, and circle $O_{1}$ is tangent to sides $A B$ and $A C$, circle $O_{2}$ is tangent to sides $B A$ and $B C$, circle $O_{3}$ is tangent to sides $C B$ and $C A$, and circle $O$ is tangent to circles $O_{1}, O_{2}, O_{3}$. Then $r=$ $\qquad$ .

Answer: \dfrac{260}{129}


2. For the arithmetic sequence $\left.\mid a_{n}\right\}$, the first term $a_{1}=8$, and there exists a unique $k$ such that the point $\left(k, a_{k}\right)$ lies on the circle $x^{2}+y^{2}=10^{2}$. Then the number of such arithmetic sequences is $\qquad$.

Answer: 17


Knot must travel from point \( K \) to point \( T \) on a figure where \( OK = 2 \) and \( OT = 4 \). Before reaching \( T \), Knot must first reach both solid lines depicted in the figure in order to collect the two pieces of the Lineforce. What is the minimal distance Knot must travel to accomplish this?

Answer: 2\sqrt{5}


Let $\mathcal{P}$ be a set of monic polynomials with integer coefficients of the least degree, with root $k \cdot \cos\left(\frac{4\pi}{7}\right)$, as $k$ spans over the positive integers. Let $P(x) \in \mathcal{P}$ be the polynomial so that $|P(1)|$ is minimized. Find the remainder when $P(2017)$ is divided by $1000$.

[i]Proposed by [b] eisirrational [/b][/i]

Answer: 167


9.4. Under the Christmas tree, there are 2012 cones. Winnie-the-Pooh and donkey Eeyore are playing a game: they take cones for themselves in turns. On his turn, Winnie-the-Pooh takes 1 or 4 cones, and Eeyore takes 1 or 3. Pooh goes first. The player who cannot make a move loses. Who among the players can guarantee a win, no matter how the opponent plays?

Answer: Winnie-the-Pooh


14. (15 points) From the sequence of consecutive natural numbers $1,2,3, \cdots, 2014$, select $n$ numbers such that no two of these $n$ numbers satisfy the condition where one is 7 times the other. Determine the maximum value of $n$ and provide a justification.

Answer: 1763


A positive integer that is the same when read forwards or backwards is called a palindrome. A table is created, as shown, in which the first column contains all palindromes between 1000 and 10000 , listed in

| Palindrome | Difference |
| :---: | :---: |
| 1001 | 110 |
| 1111 | 110 |
| 1221 | 110 |
| 1331 | 110 |
| 1441 | $\vdots$ |
| $\vdots$ |  |

poritive diference between each pair of consecutive palindromes. For example, 1001 and 1111 are consecutive palindromes and their positive difference is $1111-1001=110$.

(a) What are the eighth and ninth palindromes in the first column of the table?

(b) The second column of the table contains only two different numbers: 110 and $x$. Determine the value of $x$.

(c) When the table is complete, there are $N$ palindromes in the first column. What is the value of $N$ ?

(d) Determine the average of all of the $N-1$ numbers in the second column of the table.

Answer: \dfrac{8998}{89}


12. Let $0<\theta<\pi$, the complex numbers $z_{1}=1-\cos \theta+i \sin \theta, z_{2}=a^{2}+a i, u \in \mathbf{R}, z_{1} z_{2}$ is a pure imaginary number, and $\bar{a}=z_{1}^{2}+z_{2}^{2}-2 z_{1} z_{2}$. Then $\theta=$ $\qquad$ when it is a negative real number.

Answer: \dfrac{\pi}{2}


1. Given that the set $M$ is a subset of $\{1,2, \cdots, 2011\}$, and the sum of any four elements in $M$ cannot be divisible by 3. Then $|M|_{\text {max }}=$ $\qquad$

Answer: 1342


Tokorevev. S.

Among 2000 indistinguishable balls, half are aluminum with a mass of 10 g, and the rest are duralumin with a mass of 9.9 g. It is required to separate the balls into two piles such that the masses of the piles are different, but the number of balls in them is the same. What is the smallest number of weighings on a balance scale without weights that can achieve this?

Answer: 1


2. Using the six digits $0,1,2,3,4,5$ to form a four-digit even number without repeating digits, and arrange these even numbers in ascending order, then the 99th number is $\qquad$ .

Answer: 4052


7. A pony and a donkey together carry 120 catties of rice. The owner first takes some rice from the pony and puts it on the donkey, making the donkey carry twice as much. Then, he takes some rice from the donkey and puts it on the pony, making the pony carry twice as much. At this point, the pony and the donkey carry the same amount of rice. The pony initially carried catties of rice. $\qquad$

Answer: 75


3. Let $\Sigma(n)$ denote the sum of the digits of the number $n$. Find the smallest three-digit $n$ such that $\Sigma(n)=\Sigma(2 n)=\Sigma(3 n)=\ldots=\Sigma\left(n^{2}\right)$

Answer: 999


8.3. What is the maximum number of rooks that can be placed on an 8x8 chessboard so that each rook attacks no more than one other? A rook attacks all squares on the same row and column it occupies.

Answer: 10


Consider a $ n \times n$ checkerboard with $ n > 1, n \in \mathbb{N}.$ How many possibilities are there to put $ 2n \minus{} 2$ identical pebbles on the checkerboard (each on a different field/place) such that no two pebbles are on the same checkerboard diagonal. Two pebbles are on the same checkerboard diagonal if the connection segment of the midpoints of the respective fields are parallel to one of the diagonals of the $ n \times n$ square.

Answer: 2^n


## Task A-1.4.

In container $A$, there are four kilograms of chickpeas, of which $45 \%$ are chickpeas. In container $B$, there are five kilograms of chickpeas, of which $48 \%$ are chickpeas. In container $C$, there is one kilogram of chickpeas. A certain portion is transferred from this container to container $A$, and the remainder to container $B$, in such a way that the proportion of chickpeas in both parts is equal and amounts to $k \%$. After this, the proportion of chickpeas in both container $A$ and container $B$ is exactly $50 \%$. Determine $k$.

Answer: 80


20. (3 points) As shown in the figure, the electronic clock can display times from 00:00:00 to 23:59:59. In one day (24 hours), the number of times the clock displays a time exactly composed of the digits $1,2,3,4,5,6$ is $\qquad$ kinds.

Answer: 96


4. (10 points) Xiao Ming bought 4 apples, 6 pears, and 8 peaches, while Xiao Hong bought 5 apples, 7 pears, and 6 peaches. In the following 18 days, each of them eats one fruit per day, with three days both eating apples; two days both eating pears, and three days one eating an apple and the other eating a pear. Therefore, there are $\qquad$ days both eating peaches.

Answer: 4


5. For several identical pictures, 104 rubles were paid. How much does one picture cost if more than 10 but less than 60 were bought and the price of one picture is a natural number.

Answer: 2,4,8


5. What is the maximum number of cells in an $8 \times 8$ square that can be colored so that the centers of any four colored cells do not form the vertices of a rectangle with sides parallel to the edges of the square?

Answer: 24


5. In the cash register of the commercial director Vasily, money appeared for the first time in a long time - 2019 coins. Unfortunately, it is known that one of them is counterfeit, differing in weight from the genuine ones. Enraged workers employed by Vasily are demanding immediate payment of their salaries, and only with genuine coins. The director has balance scales without weights. As soon as it becomes clear which coins are genuine, they are paid out to the next workers and, naturally, do not participate in further weighings. The curious Vasily wants to determine whether the counterfeit coin is lighter or heavier than the genuine ones. Can he definitely do this?

Answer: no


In the diagram, $A B$ is parallel to $C D$. Points $E$ and $F$ are on $A B$ and $C D$, respectively, so that $\angle F A B=30^{\circ}$ and $\angle A F E=\angle E F B=\angle B F D=x^{\circ}$. What is the value of $x$ ?

![](https://cdn.mathpix.com/cropped/2024_04_17_5042e00a68bea6f5e8e9g-1.jpg?height=290&width=545&top_left_y=1107&top_left_x=1359)

Answer: 50


Let $S$ be the set of all polynomials of the form $z^3 + az^2 + bz + c$ , where $a$ , $b$ , and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $|z| = 20$ or $|z| = 13$ .
Please give the answer directly without any intermediate steps.

Answer: 540


In the figure, the number filled in "$O$" is equal to the product of the numbers in the three connecting "$\triangle$s", and the number in the "$\square$" is equal to the sum of the numbers in the three connecting "$\bigcirc$s". Given the numbers $5, 6, 7, 8, 9$ are filled into the five "$\triangle$s", what is the maximum number that can be in the "$\square$"?

Answer: 1251


2. Given the functions $y=2 \cos \pi x(0 \leqslant x \leqslant 2)$ and $y=2(x \in$ $R$ ) whose graphs enclose a closed plane figure. Then the area of this figure is $\qquad$ .

Answer: 4


Consider all points on the surface of the globe whose geographical latitude is equal to their longitude. Find the geometric locus of the projections of these points onto the plane of the equator.

Answer: \left(x - \frac{R}{2}\right)^2 + y^2 = \left(\frac{R}{2}\right)^2


6. Given four points $A, B, C, D$ on a sphere with radius 3. If $AB=3, CD=4$, then the maximum volume of tetrahedron $ABCD$ is $\qquad$ .

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


The diagram shows a cube with 6 unit edges, and the points $K$ and $L$ are the trisection points of the edge $A E$. The cube is divided into parts by the planes $L H G$ and $K F G$. What is the volume of the part containing vertex $B$?

![](https://cdn.mathpix.com/cropped/2024_05_02_5cf585cbd165991a3e19g-1.jpg?height=339&width=396&top_left_y=264&top_left_x=862)

Answer: 138


Suppose that $n\ge 8$ persons $P_1,P_2,\dots,P_n$ meet at a party. Assume that $P_k$ knows $k+3$ persons for $k=1,2,\dots,n-6$. Further assume that each of $P_{n-5},P_{n-4},P_{n-3}$ knows $n-2$ persons, and each of $P_{n-2},P_{n-1},P_n$ knows $n-1$ persons. Find all integers $n\ge 8$ for which this is possible.

(It is understood that "to know" is a symmetric nonreflexive relation: if $P_i$ knows $P_j$ then $P_j$ knows $P_i$; to say that $P_i$ knows $p$ persons means: knows $p$ persons other than herself/himself.)

Answer: 8


B1. Above the campsite where Huub is camping, there is a cable car with $n$ numbered gondolas.

The gondolas hang in a loop at equal distances from each other. After gondola 1 comes gondola 2, after gondola 2 comes gondola 3, and so on, and after gondola $n$ comes gondola 1 again. At a certain moment, Huub looks straight up and sees gondolas 42 and 94 passing right next to each other, with 42 moving to the left and 94 moving to the right. A while later, Huub looks up from the same spot again and sees gondolas 185 and 35 passing next to each other, with 185 moving to the left and 35 moving to the right.

What is $n$?

Answer: 202


Two straight lines intersect at a right angle, and the centers of two circles move toward the intersection point along each line. The radius of the first circle is $981 \mathrm{~m}$, and the speed of its center is $7 \mathrm{~m}$ per second. The distance of the center from the intersection point at the start is $2442 \mathrm{~m}$. The radius of the second circle is $980 \mathrm{~m}$; the speed of its center is $5 \mathrm{~m}$ per second; the center is $1591 \mathrm{~m}$ away from the intersection point at the moment of simultaneous start. How many seconds after the start will the circumferences of the moving circles first touch externally, first before the intersection point, and then beyond the intersection point, assuming that they do not interfere with each other's movement?

Answer: 111


2. A kitten bites off a quarter of a sausage from one end, after which a puppy bites off a third of the remaining piece from the opposite end, then the kitten again - a quarter from its end, and the puppy - a third from its end, and so on. It is required to tie a thread around the sausage in advance so that no one eats the thread. In what ratio should it divide the sausage?

Answer: \dfrac{1}{1}


Problem 4. Joseph solved 6 problems correctly, Darko solved 5 problems incorrectly, and Petre had the same number of correct and incorrect solutions. How many problems did all three solve correctly together, if Joseph solved twice as many problems correctly as Darko?

Answer: 13


A square is divided into five rectangles such that the four corners of the square are the corners of four of the rectangles, and the areas of these four rectangles are equal. The fifth rectangle does not share any points with the sides of the square. Prove that this fifth rectangle is a square.

Answer: \text{The fifth rectangle is a square.}


Consider a rectangle $R$ with sides $x$ and $y$ where $x < y$. Suppose we remove a square with side $x$ from $R$, resulting in a new rectangle $R'$. It is known that if $R'$ is similar to $R$, then $R$ is a "golden rectangle" and $\frac{y}{x} = \frac{1+\sqrt{5}}{2}$. However, suppose $R'$ is not similar to $R$. Perform the same operation on $R'$, removing a square with side equal to the smaller side of $R'$. The result is a rectangle $R''$. Under what conditions is $R''$ similar to $R$?

Answer: \frac{y}{x} = \sqrt{2}


Prove that the three lines passing through the vertices of a triangle and bisecting its perimeter intersect at a single point $N$ (Nagel point). Let $M$ be the centroid of the triangle, $I$ be the incenter, and $S$ be the center of the incircle of the triangle formed by the midpoints of the sides of the given triangle. Prove that the points $N, M, I$ and $S$ lie on a single line, and that $|MN|=2|IM|$ and $|IS|=|SN|$.

Answer: N, M, I, S \text{ lie on a single line with } |MN| = 2|IM| \text{ and } |IS| = |SN|


11. How many four-digit numbers greater than 5000 can be formed from the digits $0,1,2$. $3,4,5,6,7,8,9$ if only the digit 4 may be repeated?

Answer: 2645


7. A die (a uniform cube with faces numbered $1,2,3,4$, 5,6) is rolled three times. What is the probability that the three numbers on the top faces can form the side lengths of a triangle whose perimeter is divisible by 3? $\qquad$

Answer: \dfrac{11}{72}


Alex is an entrepreneur who measures success by the financial growth of their business and the assets they accumulate. Last year, Alex's business made a profit of $120,000. This year, Alex set a goal to increase the profit by 25%. Additionally, Alex plans to invest $50,000 in a new project, expecting a 10% return on this investment by the end of the year. What is the total expected financial gain for Alex this year from both the increased profit and the investment return?

Answer: 35000


Professor Smith is an art history professor specializing in the miniature-painting genre in England. She has a collection of 48 miniature paintings. Each painting measures 4 inches by 6 inches. She wants to arrange these paintings in a special exhibit where each painting is placed in a frame with an additional border of 2 inches on all sides.

If she arranges the framed paintings in a single row along a gallery wall, how long will the row be in inches?

Answer: 480


Fabián Basabe and his old school friend, who always had a knack for getting into trouble, decided to sneak into a carnival without paying. The entrance fee to the carnival is $8. Fabián’s friend managed to get in and convinced Fabián to join him. Once inside, they bought 3 cotton candies for $2 each, 2 popcorns for $3 each, and a toy that Fabián’s friend wanted, which cost $5. If they originally had $30 between them, how much money did they have left after their carnival adventure?

Answer: 13


A business traveler named Alex is planning a 5-day trip to attend a conference in a new city. Alex wants a comfortable and hassle-free accommodation experience, so they have chosen a hotel that charges $150 per night. The hotel offers a special package that includes breakfast and dinner at an additional $40 per day. To make the travel even more convenient, Alex decides to use the hotel’s airport shuttle service, which costs $25 for a round trip.

What is the total cost of Alex’s hotel stay, including the meal package and shuttle service, for the entire 5-day trip?

Answer: 975


Jack "Thunderbolt" Thompson is a hot-headed motorcycle racer known for his incredible speed and daring stunts. In a recent race, he decided to perform a breathtaking series of stunts to wow the crowd. He started by accelerating his bike to a speed of 120 miles per hour. After maintaining this speed for 10 minutes, he performed a daring jump across 15 flaming hoops, each jump taking him 30 seconds. After completing the stunts, he continued racing at a reduced speed of 90 miles per hour for another 5 minutes to the finish line. How many miles did Jack travel in total during the race?

Answer: 42.5


Alex is a teenager who is trying to balance his schoolwork with the responsibilities at home. Every day, he spends 3 hours on his homework and 2 hours helping with chores around the house. On Saturdays, he spends an additional 4 hours helping his younger siblings with their school projects. If Alex continues this routine for 5 days during the school week and includes Saturday chores and sibling help, how many total hours does he spend on homework, chores, and helping his siblings in one week?

Answer: 31


Jamie is a K-pop superfan who likes to buy albums to support her favorite groups. She has a collection of 50 albums from various groups, but only 5 are from Loona despite appreciating their music. If Jamie decides to buy 3 more albums from her favorite groups other than Loona, what will be the new ratio of Loona albums to non-Loona albums in her collection?

Answer: 5:48


A performance artist is planning to incorporate their illustrations into a live art show. They have created 5 sets of illustrations, each set containing 12 unique drawings. The artist wants to display 3 sets of these illustrations during each of the 4 shows they plan to perform. How many unique illustrations will the artist use in total across all 4 shows, if each set of illustrations can only be used once during the entire series of shows?

Answer: 60


Sarah is a direct descendent of General Joseph Spencer. She is planning a historical exhibit about her ancestor's contributions during the American Revolution. She wants to create a timeline display, and she has gathered the following information: General Spencer was born in 1714 and joined the Continental Army in 1775. He retired from military service in 1782.

Sarah decides to create a timeline on a long strip of paper, where each year is represented by 2 centimeters. She begins by marking the year of General Spencer's birth and ends with the year of his retirement. How long, in centimeters, will the complete timeline be?

Answer: 138


Ali is a Turkish marine engineer who works on a project exploring oil reserves beneath the sea. His team uses a special type of drill that can penetrate 150 meters into the seabed each day. They have identified two potential drilling sites. The first site is estimated to have oil reserves 600 meters beneath the seabed, while the second site is 750 meters deep.

Ali's team decides to start with the first site. After drilling for 2 days, another team member suggests that they should move to the second site, as it might have a higher yield. How many total days will Ali's team need to drill to reach the oil reserves at the second site if they switch after the first 2 days of drilling at the first site?

Answer: 7


Mr. Harmon is a teacher known for his calm and patient nature. Every day, he takes a peaceful 25-minute walk in the morning to start his day. His partner, Alex, on the other hand, loves to run and completes the same distance in just 15 minutes. Over the course of a week, Mr. Harmon walks every morning and Alex runs every other day. How many minutes do they spend exercising together in one week?

Answer: 60


Maria, a biologist and a politician from Angola, is studying the growth of a rare plant species found in the rainforests of Angola. She is also planning a community event to discuss environmental policies. Maria observes that each plant grows 3 new leaves every week. She starts with 5 plants, each having 10 leaves. If Maria adds 2 more plants to her study every 4 weeks, how many leaves will all the plants have in total after 8 weeks?

Answer: 234


8.4. In the country, there are 15 cities, some of which are connected by roads. Each city is assigned a number equal to the number of roads leading out of it. It turned out that there are no roads between cities with the same number. What is the maximum number of roads that can be in the country?

Answer: 85


1. Today's date is written as: 22.11.2015. Name the last past date that is written with the same set of digits.

Answer: 15.12.2012


Three thin metal rods of lengths $8, 15, 17 \mathrm{~cm}$ were welded to form a triangle, and a solid sphere with a radius of $5 \mathrm{~cm}$ was placed on the horizontally held triangle frame. In what ratio does the plane of the triangle divide the volume of the sphere?

Answer: \dfrac{7}{243}


## Task 4 - 280914

Three workpieces $W_{1}, W_{2}, W_{3}$ go through an assembly line with four processing machines $M_{1}, M_{2}, M_{3}, M_{4}$. Each workpiece must pass through the machines in the order $M_{1}, M_{2}, M_{3}, M_{4}$, and the order of the three workpieces must be the same at each machine.

The processing times of the workpieces on the individual machines are (in hours) given in the following table:

|  | $M_{1}$ | $M_{2}$ | $M_{3}$ | $M_{4}$ |
| :--- | :---: | :---: | :---: | :---: |
| $W_{1}$ | 4 | 1 | 2 | 1.5 |
| $W_{2}$ | 2 | 2.5 | 1 | 0.5 |
| $W_{3}$ | 2 | 3.5 | 1 | 1 |

Two workpieces can never be processed simultaneously on the same machine. The times for changing the workpieces at the machines are so small that they can be neglected.

Give an order of the three workpieces for passing through the assembly line such that the total time (the time from the entry of the first workpiece into machine $M_{1}$ to the exit of the last workpiece from machine $M_{4}$) is as small as possible! Show that the order you give with its total time is better than any other order!

Answer: W_3, W_1, W_2


Example 5: A and B take turns picking distinct numbers from $0,1, \cdots, 81$, with A starting first, and each person picks one number from the remaining numbers each time. After all 82 numbers are picked, let $A$ and $B$ be the sums of all numbers chosen by A and B, respectively. During the process of picking numbers, A hopes that the greatest common divisor (GCD) of $A$ and $B$ is as large as possible, while B hopes that the GCD of $A$ and $B$ is as small as possible. Under the best strategies for both A and B, find the GCD of $A$ and $B$ after all numbers are picked. ${ }^{[4]}$

Answer: 41


41st Putnam 1980 Problem A6 Let R be the reals and C the set of all functions f : [0, 1] → R with a continuous derivative and satisfying f(0) = 0, f(1) = 1. Find inf C ∫ 0 1 | f '(x) - f(x) | dx.

Answer: \dfrac{1}{e}


Answer: T_{n+1} = 4^n + T_n


Given is the function $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}$ that satisfies the properties:
(i) $f(p)=1$ for all prime numbers $p$,
(ii) $f(x y)=y f(x)+x f(y)$ for all $x, y \in \mathbb{Z}_{>0}$.

Determine the smallest $n \geq 2016$ with $f(n)=n$.

Answer: 3125


3. This figure consists of 6 identical squares. Its perimeter is 84 cm. Find the area of this figure.

![](https://cdn.mathpix.com/cropped/2024_05_21_e53d4ff4f2278a1c0d12g-073.jpg?height=308&width=414&top_left_y=1265&top_left_x=821)

Answer: 216


4. [3] Three brothers Abel, Banach, and Gauss each have portable music players that can share music with each other. Initially, Abel has 9 songs, Banach has 6 songs, and Gauss has 3 songs, and none of these songs are the same. One day, Abel flips a coin to randomly choose one of his brothers and he adds all of that brother's songs to his collection. The next day, Banach flips a coin to randomly choose one of his brothers and he adds all of that brother's collection of songs to his collection. Finally, each brother randomly plays a song from his collection with each song in his collection being equally likely to be chosen. What is the probability that they all play the same song?

Answer: \dfrac{1}{288}


2. In a fairy-tale country, some cities are connected by roads in such a way that from any city you can reach any other city, possibly through some other cities. No two cities are directly connected by more than one road.

We will call a road amazing if, upon its closure, it becomes impossible to travel from some city to some other city. It is known that there are a total of 30 roads in the country, 25 of which are amazing. How many cities can there be in the country? If there are several correct answers, list them separated by commas or semicolons.

Answer: 29


10. Given the sequence $\left\{a_{n}\right\}$ defined by $a_{1}=\frac{2}{3}, a_{n+1}=a_{n}^{2}+a_{n-1}^{2}+\cdots+a_{1}^{2}\left(n \in \mathbf{N}^{*}\right)$. If for any $n$ $\in \mathbf{N}^{*}, \frac{1}{a_{1}+1}+\frac{1}{a_{2}+1}+\cdots+\frac{1}{a_{n}+1}<M$ always holds, find the minimum value of $M$.

Answer: \dfrac{57}{20}


## Task B-4.4.

From point $A$ outside a circle, tangents are drawn to circle $k$ with points of tangency at points $B$ and $C$. A line parallel to $A C$ passes through point $B$ and intersects circle $k$ again at point $D$. If $|A B|=49$ and $|C D|=28$, determine the length of $|A D|$.

Answer: 63


## 5. Imagined Number

Grga thought of a three-digit number, and his friends are trying to guess it.

These are their attempts:
Boris: 218
Robert: 571
Marko: 732
Darko: 853

Grga told them: "One of you has guessed all the digits, and the others have only guessed one each, but none of the guessed digits are in the correct position." To this, his friends replied: "Based on this information, we cannot determine which number you thought of because there are multiple such numbers." Determine the sum of all such possible numbers.

## Result: $\quad 712$

Answer: 712


7. As shown in Figure 2, in $\triangle A B C$, $D$ is a point on side $A C$, and $\angle A B D = \angle C$. Point $E$ is on side $A B$, and $B E = D E$. $M$ is the midpoint of side $C D$, and $A H \perp D E$ at point $H$. Given that $A H = 2 - \sqrt{3}$ and $A B = 1$. Find the degree measure of $\angle A M E$.

Answer: 15


The expression $\eta(x)-\eta(x-7\pi) = \left\{\begin{array}{l}1, \quad x \in [0,7\pi) \\ 0, \quad x \in (-\infty, 0) \cup [7\pi, +\infty) \end{array}\right.$

Case 1. \(x \in [0, 7\pi)\). The equation takes the form

\[ \sin x = 1 + \cos x \]

Applying the method of introduction of an auxiliary argument, we get

\[ \sin \left(x - \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

Solving this equation, we get

\[ \left\{\begin{array}{l} x = \frac{\pi}{2} + 2\pi k, \quad k \in \mathbb{Z} \\ x = \pi + 2\pi m, \quad m \in \mathbb{Z} \end{array}\right. \]

Considering \(x \in [0, 7\pi)\), we have

\[ \left\{\begin{array}{l} x = \frac{\pi}{2} + 2\pi k, \quad k = 0, 1, 2, 3 \\ x = \pi + 2\pi m, \quad m = 0, 1, 2 \end{array}\right. \]

Case 2. \(x \in (-\infty, 0) \cup [7\pi, +\infty)\). The equation takes the form

\[ 1 + \cos x = 0 \]

Solving this equation, we get

\[ x = \pi + 2\pi m, \quad m \in \mathbb{Z} \]

Considering \(x \in (-\infty, 0) \cup [7\pi, +\infty)\), we have

\[ x = \pi + 2\pi m, \quad m = -1, -2, -3, \ldots, 3, 4, \ldots \]

Combining the solutions obtained in the above cases, we find \( x = \frac{\pi}{2} + 2\pi k, \quad k = 0, 1, 2, 3;\quad x = \pi(2m + 1), \quad m \in \mathbb{Z} \).

Answer: x = \frac{\pi}{2} + 2\pi k


61. As shown in the figure, in the Cartesian coordinate system $x O y$, the vertices of the polygon $O A B C D E$ have coordinates $O(0, 0)$, $A(0,6)$, $B(4,6)$, $C(4,4)$, $D(6,4)$, $E(6,0)$. If the line $l$ passes through the point $M(2,3)$ and divides the polygon $O A B C D E$ into two parts of equal area, then the function expression of the line $l$ is . $\qquad$

Answer: y = -\dfrac{1}{3}x + \dfrac{11}{3}


## Task A-4.7.

Determine all natural numbers $n$ that have exactly 12 positive divisors

$$
1=d_{1}<d_{2}<\cdots<d_{12}=n
$$

such that $d_{4}=5$ and $d_{5}^{2}+1=d_{7}$.

Answer: 2020


2. Using a suitable substitution, determine the number of roots of the equation

$$
8 x\left(1-2 x^{2}\right)\left(8 x^{4}-8 x^{2}+1\right)=1
$$

that lie within the interval $[0,1]$.

Answer: 4


24. The numbers $a, b, c$ satisfy the equations

$$
a b+a+b=c, \quad b c+b+c=a, \quad c a+c+a=b
$$

Find all possible values of $a$.

Answer: -2


[The product of the lengths of the segments of chords and the lengths of the segments of secants] [Properties and characteristics of an isosceles triangle.] [Thales' theorem and the theorem of proportional segments]

On the extension of side $A D$ of rhombus $A B C D$ beyond point $D$, point $K$ is taken. Lines $A C$ and $B K$ intersect at point $Q$. It is known that $A K=14$ and that points $A, B$ and $Q$ lie on a circle with radius 6, the center of which lies on segment $A A$. Find $B K$.

Answer: 7


2.1. Trapezoid $A B C D$ with base $A D=6$ is inscribed in a circle. The tangent to the circle at point $A$ intersects lines $B D$ and $C D$ at points $M$ and $N$ respectively. Find $A N$, if $A B \perp M D$ and $A M=3$.

Answer: 12


Let $a_{0}, a_{1}, a_{2}, \ldots$ be the sequence defined by: $a_{0}=2, \quad a_{k+1}=2 a_{k}^{2}-1 \quad$ for $k \geq 0$. Show that if an odd prime number $p$ divides $a_{n}$, then $2^{n+3}$ divides $p^{2}-1$.

Answer: 2^{n+3} \text{ divides } p^{2} - 1


37. In Rt $\triangle ABC$, $\angle A=90^{\circ}$, the angle bisectors of $\angle ABC$ and $\angle ACB$ intersect at point $I$, and intersect the opposite sides at points $D$ and $E$. If the area of $\triangle BIC$ is 8, then the area of quadrilateral $BCDE$ is

Answer: 16


4. Sharik and Matroskin are skiing on a circular track, half of which is an uphill climb and the other half is a downhill descent. Their speeds on the climb are the same and four times slower than their speeds on the descent. The minimum distance by which Sharik lags behind Matroskin is 4 km, and the maximum distance is 13 km. Find the length of the track.

Answer: 24


In the following figure, we have a $2 \times 2 \times 2$ cube made with pieces of wire. The edge of each cube is a piece of $1 \mathrm{~cm}$ of wire and a total of 54 such pieces were used. To make a $10 \times 10 \times 10$ cube, how many pieces of wire will be used?

![](https://cdn.mathpix.com/cropped/2024_05_01_e67c0440f1fa8bc62cdcg-02.jpg?height=352&width=365&top_left_y=638&top_left_x=719)

#

Answer: 3630


## Task 4 - 280824

Let $k_{1}$ and $k_{2}$ be two concentric circles with center $M$, whose radii are in the ratio $3:1$. Two diameters $A_{1} B_{1}$ and $C_{1} D_{1}$ of $k_{1}$ intersect $k_{2}$ at points $A_{2}, B_{2}$ and $C_{2}, D_{2}$, respectively, as shown in the figure.

a) Determine the ratio of the areas of the sector $A_{2} M D_{2}$ and the annular sector $D_{2} B_{2} B_{1} D_{1}$, assuming that $\angle A_{1} M D_{1}$ is a right angle!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0785.jpg?height=440&width=377&top_left_y=634&top_left_x=1359)

b) How should the size of the angle $\angle A_{1} M D_{1}$ be chosen so that the area of the sector $A_{2} M D_{2}$ is equal to the area of the annular sector $D_{2} B_{2} B_{1} D_{1}$?

Answer: \dfrac{1}{8}


3. On the hundredth year of his reign, the Immortal Treasurer decided to start issuing new coins. In this year, he put into circulation an unlimited supply of coins with a value of $2^{100}-1$, the following year - with a value of $2^{101}-1$, and so on. As soon as the value of the next new coin can be exactly matched using the previously issued new coins, the Treasurer will be removed from office. In which year of his reign will this happen? (I. Bogdanov)

Answer: 200


In a triangle $ABC, L$ and $K$ are the points of intersections of the angle bisectors of $\angle ABC$ and $\angle BAC$ with the segments $AC$ and $BC$, respectively. The segment $KL$ is angle bisector of $\angle AKC$, determine $\angle BAC$.

Answer: 120^\circ


1. Given the sequence $\left\{a_{n}\right\}$ satisfies the first term $a_{1}=\frac{1}{2}, a_{n+1}$ $=3 a_{n}+7$. Find the general formula for the sequence $\left\{a_{n}\right\}$.

Answer: 4 \cdot 3^{n-1} - \dfrac{7}{2}


8. Given a non-empty set $M$ satisfying
$$
M \subseteq\{0,1, \cdots, n\}\left(n \geqslant 2, n \in \mathbf{Z}_{+}\right) \text {. }
$$

If there exists a non-negative integer $k(k \leqslant n)$, such that for any $a \in M$, we have $2 k-a \in M$, then the set $M$ is said to have property $P$. Let the number of sets $M$ with property $P$ be $f(n)$. Then the value of $f(9)-f(8)$ is $\qquad$

Answer: 31


Four. (20 points) Given a positive integer $n(n \geqslant 5)$, let the side lengths of rectangle $ABCD$ be $AB=2a, BC=2b (a>b)$. Points $P_{1}, P_{2}, \cdots, P_{n}=C$, $Q_{1}, Q_{2}, \cdots, Q_{n}=D$, $R_{0}=D, R_{1}, R_{2}, \cdots, R_{n}=C$ satisfy
$$
\begin{array}{c}
BP_{1}=P_{1}P_{2}=\cdots=P_{n-1}P_{n}=\frac{2b}{n}, \\
AQ_{1}=Q_{1}Q_{2}=\cdots=Q_{n-1}Q_{n}=\frac{2b}{n}, \\
R_{0}R_{1}=R_{1}R_{2}=\cdots=R_{n-1}R_{n}=\frac{2a}{n}.
\end{array}
$$
(1) For $i=1,2, \cdots, n$, find the equation of the conic section $\Gamma_{1}$ on which the intersection points of $AP_{i}$ and $BR_{n-i}$, $AR_{i}$ and $BQ_{i}$ lie;
(2) If points $S_{1}, S_{2}, \cdots, S_{n}$ on the extension of $DC$ satisfy $CS_{1}=S_{1}S_{2}=\cdots=S_{n-1}S_{n}=\frac{2a}{n}$, for $i=1,2, \cdots, n$, find the equation of the conic section $\Gamma_{2}$ on which the intersection points of $AP_{i}$ and $BS_{i}$ lie;
(3) Let $M$ be the point on the conic section $\Gamma_{1}$ that is farthest from $AB$. Find the minimum distance between $M$ and a point on the conic section $\Gamma_{2}$.

Answer: \dfrac{(x - a)^2}{a^2} + \dfrac{y^2}{b^2} = 1


2. Acute $\triangle A B C$ is inscribed in $\odot O$ with radius $R$, $H$ is the orthocenter of $\triangle A B C$, and the extension of $A O$ intersects $B C$ at point $M$. If $O H \perp A O, B C=10, O A=6$, then $O M=$ $\qquad$ -

Answer: \dfrac{11}{3}


11. Solve the equation $(\sin x)^{\arctan (\sin x+\cos x)}=(\csc x)^{\arctan (\sin 2 x)+\frac{\pi}{4}}$, where $2 n \pi<x<$ $(2 n+1) \pi, n$ is any integer.

Answer: \frac{\pi}{2} + 2n\pi


9. An electronic watch at 9:15:12 AM shows the time as $09: 15: 12$, and at 1:11:29 PM, it shows the time as 13:11:29. Then, within 24 hours a day, the number of times the six digits on the electronic watch form a symmetrical time (i.e., reading from left to right is the same as reading from right to left, such as 01:33:10) is $\qquad$ .

Answer: 96


In a right-angled triangle $A B C$, from vertices $B$ and $C$, one point each starts moving simultaneously towards point $A$ along the hypotenuse $B A=c=85$ m and the leg $C A=b=75$ m, respectively. The points move at speeds of $8.5 \mathrm{~m} / \mathrm{sec}$ and $5 \mathrm{~m} / \mathrm{sec}$. At what time will the distance between the two points be $26 \mathrm{~m}$. (Without using trigonometry!)

Answer: 4


Koschei the Deathless thinks of three two-digit numbers: $a, b, c$. Ivan Tsarevich must name three numbers: $X, Y$, $Z$, after which Koschei tells him the sum $aX + bY + cZ$. Tsarevich must guess the numbers thought of, or else he will lose his head. What numbers should he guess to stay alive?

Answer: 10000


\left[\begin{array}{l}\text { Integral and Area } \\ \text { [Quadratic Equations. Vieta's Theorem] }\end{array}\right]

Author: Sergiev I.N.

The numbers $p$ and $q$ are such that the parabolas $y=-2 x^{2}$ and $y=x^{2}+p x+q$ intersect at two points, enclosing a certain figure.

Find the equation of the vertical line that divides the area of this figure in half.

Answer: x = -\dfrac{p}{6}


Question 15 Five monkeys divide a pile of peanuts. The first monkey divides the peanuts into five piles, eats one left over, and takes away one pile. The second monkey again divides the remaining peanuts into five piles, with exactly one left over, eats it, and takes away one pile. This continues until the fifth monkey comes to divide the remaining peanuts, and can also divide them into five piles with exactly one left over. How many peanuts were there in the pile at least?

Answer: 3121