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`, `Eric`, `Arnold`, `Carol`, `Alice`, `Peter`
 - Each person has a unique hobby: `cooking`, `painting`, `knitting`, `photography`, `woodworking`, `gardening`
 - People have unique hair colors: `brown`, `auburn`, `black`, `red`, `gray`, `blonde`
 - People have unique favorite sports: `swimming`, `volleyball`, `basketball`, `baseball`, `tennis`, `soccer`
 - The mothers' names in different houses are unique: `Sarah`, `Holly`, `Penny`, `Janelle`, `Kailyn`, `Aniya`
 - Everyone has a unique favorite cigar: `dunhill`, `blends`, `pall mall`, `prince`, `yellow monster`, `blue master`

## Clues:
1. The person who has gray hair is directly left of The person whose mother's name is Janelle.
2. The person who loves soccer is in the sixth house.
3. Arnold is the person who has blonde hair.
4. The photography enthusiast is the person who smokes Yellow Monster.
5. There is one house between the Dunhill smoker and the person who loves cooking.
6. The person who paints as a hobby is the person who has blonde hair.
7. The Dunhill smoker is directly left of Alice.
8. The person who paints as a hobby is directly left of the person who loves volleyball.
9. The person who loves soccer is The person whose mother's name is Aniya.
10. The person partial to Pall Mall is somewhere to the left of Carol.
11. The person who enjoys knitting and the person who loves basketball are next to each other.
12. The person partial to Pall Mall is directly left of The person whose mother's name is Holly.
13. The person who loves soccer is the Prince smoker.
14. The person who loves swimming is The person whose mother's name is Penny.
15. The person whose mother's name is Kailyn is the person who loves volleyball.
16. The person who loves basketball is The person whose mother's name is Janelle.
17. The person who enjoys knitting is the person who loves volleyball.
18. The person who enjoys knitting is not in the third house.
19. The person who has auburn hair is somewhere to the left of the photography enthusiast.
20. The person whose mother's name is Sarah is not in the fifth house.
21. Bob is directly left of the person who has red hair.
22. The Dunhill smoker is the person who enjoys gardening.
23. The person who loves baseball is the person who has auburn hair.
24. Eric is the person who smokes many unique blends.
25. The person who has brown hair is somewhere to the right of The person whose mother's name is Penny.
 What is the value of attribute House for the person whose attribute FavoriteSport is baseball? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Alice`, `Eric`, `Arnold`
 - Each mother is accompanied by their child: `Meredith`, `Bella`, `Samantha`, `Fred`, `Timothy`
 - People have unique favorite music genres: `classical`, `hip hop`, `pop`, `jazz`, `rock`
 - Everyone has a favorite smoothie: `watermelon`, `lime`, `dragonfruit`, `cherry`, `desert`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `city`, `cruise`, `camping`

## Clues:
1. The person who loves rock music is somewhere to the left of the person who loves classical music.
2. The person who loves beach vacations and the person who loves rock music are next to each other.
3. Eric is the person who is the mother of Timothy.
4. The person who drinks Lime smoothies is not in the second house.
5. The Watermelon smoothie lover is somewhere to the left of the person who loves rock music.
6. Alice is the person who loves beach vacations.
7. The Dragonfruit smoothie lover is somewhere to the right of Peter.
8. The person who loves jazz music is the person's child is named Samantha.
9. Bob is the person's child is named Samantha.
10. Alice is somewhere to the left of Eric.
11. The person who loves pop music is the person who likes going on cruises.
12. The person's child is named Fred is in the fifth house.
13. The person who prefers city breaks is the person who loves jazz music.
14. The Dragonfruit smoothie lover is the person's child is named Samantha.
15. The person's child is named Meredith is the person who likes going on cruises.
16. The Desert smoothie lover is somewhere to the left of the person who likes Cherry smoothies.
17. The person who enjoys mountain retreats is not in the fifth house.
18. The person who drinks Lime smoothies is directly left of the Desert smoothie lover.
 What is the value of attribute House for the person whose attribute Smoothie is watermelon? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


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

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

Answer: sept


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Arnold`, `Carol`, `Eric`, `Peter`
 - People use unique phone models: `xiaomi mi 11`, `google pixel 6`, `samsung galaxy s21`, `oneplus 9`, `huawei p50`, `iphone 13`
 - People have unique favorite book genres: `romance`, `historical fiction`, `fantasy`, `mystery`, `science fiction`, `biography`
 - They all have a unique favorite flower: `tulips`, `roses`, `carnations`, `daffodils`, `iris`, `lilies`
 - The people keep unique animals: `dog`, `cat`, `bird`, `fish`, `rabbit`, `horse`
 - Each person has a unique level of education: `trade school`, `high school`, `master`, `associate`, `bachelor`, `doctorate`

## Clues:
1. The bird keeper is the person who uses a Google Pixel 6.
2. The person who loves a bouquet of daffodils is the person with an associate's degree.
3. The person who uses a OnePlus 9 is not in the fourth house.
4. There are two houses between the person with a master's degree and the rabbit owner.
5. The person who loves a carnations arrangement is Arnold.
6. The person who loves the boquet of lilies is not in the sixth house.
7. The person who loves the vase of tulips is Carol.
8. The person with a high school diploma is Bob.
9. The person who attended trade school is somewhere to the left of the person who keeps horses.
10. The person who loves mystery books is Eric.
11. The fish enthusiast is directly left of Carol.
12. Bob is the fish enthusiast.
13. Eric is the dog owner.
14. The person who loves a carnations arrangement is the person who uses a Google Pixel 6.
15. The person who loves fantasy books is in the second house.
16. The person who uses a Samsung Galaxy S21 is the person who loves the boquet of iris.
17. Alice is the person with an associate's degree.
18. The person who uses a Google Pixel 6 is not in the second house.
19. The dog owner is the person with a doctorate.
20. The fish enthusiast is the person who uses a Samsung Galaxy S21.
21. The person who uses an iPhone 13 is somewhere to the right of the person who loves fantasy books.
22. The person who loves mystery books is somewhere to the right of the person who loves the boquet of iris.
23. Alice is the person who uses an iPhone 13.
24. The person who attended trade school is not in the fourth house.
25. The person who loves biography books is directly left of Eric.
26. There is one house between the rabbit owner and the person who loves historical fiction books.
27. The person who loves the boquet of iris is somewhere to the right of the person who uses a Xiaomi Mi 11.
28. The person who loves science fiction books is the person who uses a Samsung Galaxy S21.
29. The person who loves the boquet of lilies is not in the second house.
 What is the value of attribute Education for the person whose attribute BookGenre is science fiction? Please reason step by step, and put your final answer within \boxed{}

Answer: high school


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`
 - They all have a unique favorite flower: `daffodils`, `lilies`, `carnations`
 - The people are of nationalities: `brit`, `dane`, `swede`
 - The mothers' names in different houses are unique: `Aniya`, `Holly`, `Janelle`
 - People have unique favorite sports: `basketball`, `tennis`, `soccer`

## Clues:
1. The Swedish person is not in the third house.
2. The person who loves basketball is directly left of the person who loves tennis.
3. Arnold is the person who loves the boquet of lilies.
4. The person whose mother's name is Janelle is the Dane.
5. The person whose mother's name is Holly is the person who loves a carnations arrangement.
6. The person who loves the boquet of lilies is somewhere to the right of Peter.
7. The Dane is directly left of the person who loves a carnations arrangement.
8. Eric is not in the first house.
9. The person whose mother's name is Janelle is directly left of the person who loves basketball.
 What is the value of attribute FavoriteSport for the person whose attribute Nationality is dane? Please reason step by step, and put your final answer within \boxed{}

Answer: soccer


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`
 - People have unique favorite sports: `basketball`, `tennis`, `soccer`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`
 - Each person has a unique birthday month: `sept`, `april`, `jan`
 - They all have a unique favorite flower: `lilies`, `daffodils`, `carnations`

## Clues:
1. Arnold is somewhere to the left of the person who loves romance books.
2. The person who loves soccer is the person who loves a carnations arrangement.
3. The person who loves mystery books is the person whose birthday is in September.
4. The person who loves tennis is in the second house.
5. Peter is the person who loves a bouquet of daffodils.
6. Eric is somewhere to the left of the person whose birthday is in April.
7. The person who loves science fiction books is somewhere to the right of the person who loves tennis.
 What is the value of attribute Birthday for the person whose attribute Flower is carnations? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Alice`
 - Each person lives in a unique style of house: `victorian`, `craftsman`, `colonial`, `ranch`
 - People have unique hair colors: `red`, `blonde`, `black`, `brown`

## Clues:
1. The person in a ranch-style home is somewhere to the right of the person who has black hair.
2. There are two houses between Alice and the person living in a colonial-style house.
3. The person who has red hair is directly left of Eric.
4. The person living in a colonial-style house is somewhere to the right of the person in a Craftsman-style house.
5. There is one house between Peter and the person who has blonde hair.
6. The person who has red hair and the person in a ranch-style home are next to each other.
7. The person residing in a Victorian house is in the second house.
 What is the value of attribute HouseStyle for the person whose attribute HairColor is brown? Please reason step by step, and put your final answer within \boxed{}

Answer: ranch


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

## Clues:
1. The boba tea drinker is the person who uses a Xiaomi Mi 11.
2. Eric is the root beer lover.
3. The person who loves science fiction books is the person who likes milk.
4. The person who likes milk is somewhere to the right of Eric.
5. Arnold is the person who uses a Huawei P50.
6. The person who loves romance books is not in the fourth house.
7. Alice is the person who uses an iPhone 13.
8. The person who loves fantasy books is somewhere to the right of Bob.
9. There are two houses between the person who uses an iPhone 13 and the person who loves fantasy books.
10. The root beer lover is directly left of Peter.
11. The person who uses an iPhone 13 and the person who loves science fiction books are next to each other.
12. The person who uses a Samsung Galaxy S21 is the person who loves romance books.
13. The tea drinker is the person who uses a Huawei P50.
14. There is one house between the person who loves romance books and the person who uses a OnePlus 9.
15. There are two houses between the person who loves biography books and the boba tea drinker.
16. The person who loves historical fiction books is somewhere to the right of the root beer lover.
17. There is one house between the one who only drinks water and the person who uses a Huawei P50.
 What is the value of attribute Drink for the person whose attribute Name is Bob? 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: `Alice`, `Arnold`, `Eric`, `Peter`
 - Each mother is accompanied by their child: `Samantha`, `Meredith`, `Fred`, `Bella`
 - People have unique hair colors: `red`, `black`, `brown`, `blonde`

## Clues:
1. The person who has red hair is directly left of Eric.
2. The person's child is named Meredith is directly left of the person who has brown hair.
3. The person who has blonde hair is the person's child is named Fred.
4. Peter is in the fourth house.
5. The person who has black hair is directly left of Arnold.
6. The person's child is named Bella is Arnold.
 What is the value of attribute Children for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: Fred


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

## Clues:
1. Bob is the person who loves fantasy books.
2. The person whose mother's name is Janelle and Eric are next to each other.
3. The person who enjoys camping trips is the person who loves biography books.
4. Peter is The person whose mother's name is Kailyn.
5. Peter and The person whose mother's name is Janelle are next to each other.
6. The person who loves beach vacations is Bob.
7. Peter is somewhere to the right of the person who loves romance books.
8. The person whose mother's name is Penny is the person who loves biography books.
9. Bob is directly left of the person who prefers city breaks.
10. The person who loves science fiction books is in the fourth house.
11. Arnold and Bob are next to each other.
12. The person whose mother's name is Aniya is the person who enjoys mountain retreats.
 What is the value of attribute House for the person whose attribute Mother is Kailyn? 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: `Arnold`, `Peter`, `Eric`, `Alice`
 - The mothers' names in different houses are unique: `Holly`, `Kailyn`, `Aniya`, `Janelle`
 - People have unique favorite book genres: `science fiction`, `mystery`, `fantasy`, `romance`
 - Each person has a favorite color: `green`, `red`, `white`, `yellow`
 - Each person prefers a unique type of vacation: `mountain`, `cruise`, `beach`, `city`

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

Answer: white


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Peter`, `Eric`, `Alice`, `Carol`
 - People have unique favorite music genres: `jazz`, `rock`, `pop`, `hip hop`, `classical`, `country`
 - Each person has a unique birthday month: `april`, `feb`, `jan`, `mar`, `sept`, `may`
 - Each person has a unique favorite drink: `root beer`, `coffee`, `tea`, `boba tea`, `water`, `milk`
 - Each person has an occupation: `nurse`, `doctor`, `teacher`, `lawyer`, `artist`, `engineer`

## Clues:
1. The person who loves pop music is not in the fourth house.
2. The boba tea drinker is the person whose birthday is in April.
3. The person whose birthday is in January is somewhere to the right of the person whose birthday is in May.
4. Arnold is not in the third house.
5. Eric is in the first house.
6. Carol is the person who loves jazz music.
7. The person who loves classical music is directly left of the root beer lover.
8. The person who loves country music is somewhere to the left of the person who is a teacher.
9. The person who is a teacher is somewhere to the left of the person who is a nurse.
10. The person who is a doctor is somewhere to the left of the person who is a teacher.
11. The root beer lover is the person who loves hip-hop music.
12. Arnold is the person who is a lawyer.
13. Carol is not in the fourth house.
14. The person who is an artist is somewhere to the left of the person who loves pop music.
15. The person whose birthday is in April is Bob.
16. The person who likes milk is not in the second house.
17. The one who only drinks water is somewhere to the right of the person whose birthday is in March.
18. The tea drinker is the person whose birthday is in September.
19. The person whose birthday is in January is the person who loves pop music.
20. The person who loves jazz music is directly left of the person who loves rock music.
21. Alice is the person who is a nurse.
22. The tea drinker is not in the sixth house.
23. The person who is a doctor is the root beer lover.
 What is the value of attribute Birthday for the person whose attribute MusicGenre is rock? Please reason step by step, and put your final answer within \boxed{}

Answer: feb


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Alice`, `Bob`, `Arnold`, `Carol`
 - Everyone has a unique favorite cigar: `yellow monster`, `blends`, `prince`, `dunhill`, `pall mall`, `blue master`
 - People use unique phone models: `xiaomi mi 11`, `iphone 13`, `google pixel 6`, `huawei p50`, `samsung galaxy s21`, `oneplus 9`
 - People own unique car models: `bmw 3 series`, `honda civic`, `tesla model 3`, `ford f150`, `chevrolet silverado`, `toyota camry`
 - The people are of nationalities: `dane`, `brit`, `chinese`, `swede`, `norwegian`, `german`

## Clues:
1. The person who uses a Xiaomi Mi 11 is somewhere to the left of the person who uses a Google Pixel 6.
2. Carol is somewhere to the left of the person who owns a Ford F-150.
3. Peter is not in the second house.
4. There is one house between the Dane and the person who uses a Google Pixel 6.
5. The Chinese is the person who uses a Google Pixel 6.
6. The person who smokes many unique blends is somewhere to the right of the person who owns a Honda Civic.
7. The person who owns a Tesla Model 3 is in the fourth house.
8. The person who uses a Samsung Galaxy S21 is Alice.
9. There is one house between the person who uses a Xiaomi Mi 11 and the person who smokes Blue Master.
10. The person partial to Pall Mall is directly left of the person who uses a Samsung Galaxy S21.
11. The person who smokes many unique blends is the German.
12. The person who owns a Chevrolet Silverado is the Prince smoker.
13. The person who owns a BMW 3 Series is Eric.
14. The person who owns a Ford F-150 is directly left of Bob.
15. The person who smokes Yellow Monster is not in the second house.
16. The Swedish person is somewhere to the right of the person who smokes Yellow Monster.
17. The person who owns a Chevrolet Silverado is directly left of the person who uses a Huawei P50.
18. The person who smokes Blue Master is in the fifth house.
19. The person who smokes Blue Master is directly left of the person who uses a OnePlus 9.
20. The Norwegian is the person who owns a Toyota Camry.
 What is the value of attribute CarModel for the person whose attribute Nationality is swede? Please reason step by step, and put your final answer within \boxed{}

Answer: honda civic


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`
 - People own unique car models: `ford f150`, `toyota camry`, `tesla model 3`
 - The people are of nationalities: `swede`, `dane`, `brit`

## Clues:
1. The person who owns a Ford F-150 is somewhere to the right of the Swedish person.
2. Peter is not in the third house.
3. The person who owns a Toyota Camry is in the third house.
4. Eric is the person who owns a Ford F-150.
5. The British person is not in the third house.
 What is the value of attribute CarModel for the person whose attribute Nationality is brit? Please reason step by step, and put your final answer within \boxed{}

Answer: ford f150


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Arnold`, `Eric`, `Peter`
 - The people keep unique animals: `fish`, `dog`, `cat`, `horse`, `bird`
 - People own unique car models: `tesla model 3`, `ford f150`, `honda civic`, `bmw 3 series`, `toyota camry`

## Clues:
1. Bob is the person who keeps horses.
2. The person who owns a Toyota Camry is the cat lover.
3. Arnold is in the fifth house.
4. Alice is directly left of the bird keeper.
5. The person who keeps horses is directly left of the person who owns a Honda Civic.
6. The person who owns a Ford F-150 is in the second house.
7. The person who owns a Ford F-150 is Peter.
8. The person who keeps horses is directly left of the fish enthusiast.
9. The bird keeper is the person who owns a Tesla Model 3.
 What is the value of attribute House for the person whose attribute Animal is dog? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Alice`, `Eric`, `Peter`, `Arnold`, `Bob`
 - The mothers' names in different houses are unique: `Penny`, `Janelle`, `Aniya`, `Sarah`, `Holly`, `Kailyn`
 - Everyone has a unique favorite cigar: `blends`, `dunhill`, `blue master`, `prince`, `yellow monster`, `pall mall`
 - Each person has a unique type of pet: `cat`, `dog`, `rabbit`, `hamster`, `bird`, `fish`
 - The people keep unique animals: `bird`, `horse`, `cat`, `rabbit`, `dog`, `fish`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `lime`, `cherry`, `blueberry`, `dragonfruit`

## Clues:
1. The dog owner is not in the third house.
2. The fish enthusiast is not in the fourth house.
3. The person who owns a dog is Eric.
4. The cat lover is not in the second house.
5. The person who smokes Blue Master is somewhere to the right of The person whose mother's name is Sarah.
6. The cat lover is not in the fourth house.
7. There are two houses between Carol and the person who owns a rabbit.
8. The person whose mother's name is Penny is the person with a pet hamster.
9. The person whose mother's name is Kailyn is somewhere to the right of the person with a pet hamster.
10. The dog owner and the person who drinks Lime smoothies are next to each other.
11. The Prince smoker is somewhere to the right of The person whose mother's name is Janelle.
12. There is one house between The person whose mother's name is Penny and Bob.
13. There are two houses between the person partial to Pall Mall and The person whose mother's name is Penny.
14. Peter is somewhere to the left of the person who drinks Lime smoothies.
15. The bird keeper is the Watermelon smoothie lover.
16. The person with a pet hamster is the Prince smoker.
17. The person who smokes Yellow Monster is the person who drinks Lime smoothies.
18. The person whose mother's name is Sarah is the person who smokes Yellow Monster.
19. The Dunhill smoker is somewhere to the left of the Watermelon smoothie lover.
20. The person whose mother's name is Holly is directly left of the person who owns a dog.
21. The bird keeper is The person whose mother's name is Kailyn.
22. The person whose mother's name is Sarah is not in the third house.
23. The person who smokes many unique blends is the rabbit owner.
24. The person who has a cat is somewhere to the right of the person with an aquarium of fish.
25. The person who smokes Yellow Monster is Eric.
26. The person who keeps a pet bird is the Desert smoothie lover.
27. Alice is The person whose mother's name is Kailyn.
28. The Dragonfruit smoothie lover is Carol.
29. The person who keeps horses is the person who likes Cherry smoothies.
 What is the value of attribute Mother for the person whose attribute Smoothie is watermelon? Please reason step by step, and put your final answer within \boxed{}

Answer: Kailyn


## Task B-1.6.

The sum of the fractions $\frac{12}{23}+\frac{1212}{2323}+\frac{121212}{232323}+\cdots+\frac{1212 \ldots 12}{2323 \ldots 23}$ is 528. The number of digits 1 and 2 in the numerator and the number of digits 2 and 3 in the denominator of these fractions increase by one. How many times does the digit 2 appear in the last fraction?

Answer: 2024


## Task B-1.5.

On the way to the sports hall, Marin asked Ivan for the number of his seat. Wanting to reduce the tension before the important sports event and entertain his friend, Ivan enigmatically replied: "All the digits of that number are different. If you add up all six two-digit numbers that can be formed from the digits of that number, so that each of those two-digit numbers also has different digits, and then divide the obtained sum by two, you will get the number of my seat." What is Ivan's seat number?

Answer: 198


42nd Putnam 1981 Problem A6 A, B, C are lattice points in the plane. The triangle ABC contains exactly one lattice point, X, in its interior. The line AX meets BC at E. What is the largest possible value of AX/XE? Solution

Answer: 5


## Task B-1.2.

Ivo, Alen, Vanja, Marko, and Saša are chefs in a hotel. Alen and Marko are responsible for preparing breakfast and lunch, Ivo and Vanja work on preparing lunch and dinner, while Saša is available for all three meals. In how many ways can their daily cooking schedule be arranged if each meal is prepared by exactly two chefs, and a chef who works must be assigned to exactly two meals? Can each chef have at least one day off in the week with such a daily schedule? Explain.

Answer: 4


8.5. On a 10 by 10 cell board, some 10 cells are marked. For what largest $n$ is it always possible, regardless of which cells are marked, to find a rectangle consisting of several cells, the perimeter of which will be at least $n$? The length or width of the rectangle can be equal to one cell.

Answer: 20


12. Let $\left.f(m)=2^{2^{2^{2}}}\right\} m$. Find the least $m$ so that $\log _{10} f(m)$ exceeds 6 .

Answer: 5


An open organ pipe has its first overtone $N_{1}=512$, if it is blown with air (the speed of sound in air is $V=340 \frac{\mathrm{m}}{\mathrm{sec}}$). How high will the fundamental frequency of this pipe be if it is blown with hydrogen, in which the speed of sound is $V^{\prime}=1200 \overline{\mathrm{m}} \sec$?

Answer: 904


13. Given quadrilateral $A B C D$ satisfies $A D \| B C, B A=A D=D C=\frac{1}{2} B C=a, E$ is the midpoint of $B C$, and $\triangle B A E$ is folded along $A E$ to form $\triangle B_{1} A E$, such that plane $B_{1} A E \perp$ plane $A E C D$, and $F$ is the midpoint of $B_{1} D$.
(1) Find the volume of the pyramid $B_{1}-A E C D$;
(2) Find the sine of the angle between plane $A D B_{1}$ and plane $E C B_{1}$.

Answer: \dfrac{a^3}{4}


Gy. 1729. Solve the system of equations,

$$
\begin{aligned}
& a_{11} x+a_{12} y+a_{13} z=0 \\
& a_{21} x+a_{22} y+a_{23} z=0 \\
& a_{31} x+a_{32} y+a_{33} z=0
\end{aligned}
$$

if the coefficients satisfy the following conditions:

a) each one is positive

b) the sum of the coefficients in each row and each column is 1,
c) $a_{11}=a_{22}=a_{33}=1 / 2$.

Answer: x = 0


Example 15. Map the sector $0<\arg z<\frac{\pi}{4}$ onto the unit disk $|w|<1$ such that the point $z_{1}=e^{i \pi / 8}$ is mapped to the center $w_{1}=0$, and the point $z_{2}=0$ is mapped to the point $w_{2}=1$.

Answer: w = \dfrac{i - z^4}{i + z^4}


## 29. Apples

Amelie picked apples, ate a quarter of them herself, and gave the rest to her little sisters Bertha, Charlotte, and Dorothy. Bertha ate a quarter of her apples and gave the rest to her three sisters. Charlotte ate one apple and gave the rest to her three sisters. As for little Dorothy, she also ate a quarter of her share before giving the rest to her three sisters.

"Look," said Bertha to Charlotte, "I have twice as many apples left as you do."

How many apples did Amelie pick?

Answer: 32


Question 173, Given that a square has three vertices on the parabola $y=x^{2}$, find the minimum value of the area of such a square.

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

(Note: The note itself is not part of the translation task and should not be included in the final output.)

Question 173, Given that a square has three vertices on the parabola $y=x^{2}$, find the minimum value of the area of such a square.

Answer: 2


A2. Two ants start at the same point in the plane. Each minute they choose whether to walk due north, east, south or west. They each walk 1 meter in the first minute. In each subsequent minute the distance they walk is multiplied by a rational number $q>0$. They meet after a whole number of minutes, but have not taken exactly the same route within that time. Determine all possible values of $q$.

(United Kingdom)

Answer: 1


In parallelogram $A B C D(A B \| C D)$, diagonal $B D=a$, and $O$ is the point of intersection of the diagonals. Find the area of the parallelogram if $\angle D B A=45^{\circ}, \angle A O B=105^{\circ}$.

## Answer

$S=\frac{1+\sqrt{3}}{4} a^{2}$.

Circles with radii 3 and 6 have centers at points $O_{1}$ and $O_{2}$, respectively, and touch externally at point $A$. A common external tangent and a common internal tangent are drawn to the circles. These tangents intersect at point $B$, and $L$ is the common point of the external tangent and the circle with radius 6. Find the radius of the circle inscribed in quadrilateral $A B L O_{2}$.

Answer: 6(\sqrt{2} -1)


In a regular triangular pyramid $S A B C$ ( $S$ - vertex) points $D$ and $E$ are the midpoints of edges $A C$ and $B C$ respectively. Through point $E$, a plane $\beta$ is drawn, intersecting edges $A B$ and $S B$ and being equidistant from points $D$ and $B$ by a distance of $\frac{1}{2}$. Find the lengths of the segments into which the plane divides edge $S B$, if $B C=4$

$$
\text { , } S C=3 \text {. }
$$

Answer: \dfrac{12}{5}


91. On a long strip of paper, consecutive natural numbers $1, 2, 3, \cdots \cdots n$ are written in sequence. The long strip is cut into five segments, each containing some numbers (the digits of a single number will not be cut into different segments). The average of the numbers in these five segments are $1234, 345, 128, 19$, and $9.5$ (the order of these five numbers is scrambled). Therefore, $n=$

Answer: 2014


3. Determine all prime numbers $p$, such that the number

$$
\left(\begin{array}{l}
p \\
1
\end{array}\right)^{2}+\left(\begin{array}{l}
p \\
2
\end{array}\right)^{2}+\ldots+\left(\begin{array}{c}
p \\
p-1
\end{array}\right)^{2}
$$

is divisible by $p^{3}$.

Answer: 5


13.352. A girl, an acquaintance of a young man sitting by the window of a tram, was walking towards the moving tram. 8 seconds after she passed by the window, the young man got off the tram and followed her. How much time passed from that moment until he caught up with the girl? The young man's speed is twice the girl's speed and five times less than the tram's speed.

Answer: 88


$$
\text { II. (40 points) Let } a_{i} \in(0,1](1 \leqslant i \leqslant 45) \text {. }
$$

Find the maximum value of $\lambda$ such that the inequality
$$
\sqrt{\frac{45}{\sum_{i=1}^{45} a_{i}}} \geqslant 1+\lambda \prod_{i=1}^{45}\left(1-a_{i}\right)
$$

holds.

Answer: 8 \left( \dfrac{81}{80} \right)^{45}


5. In a convex quadrilateral $A B C D$, $A B+A C+C D=16$. When the diagonals $A C$ and $B D$ are what values, will the area of quadrilateral $A B C D$ be maximized? What is the maximum area?

Answer: 32


Shapovalov A.V.

Several natural numbers are written in a row with a sum of 2019. No number and no sum of several consecutive numbers equals 40. What is the maximum number of numbers that could have been written?

Answer: 1019


5.17. In triangle $A B C$, point $N$ lies on side $A B$ and $A N=$ $=3 N B$; median $A M$ intersects $C N$ at point $O$. Find $A B$, if $A M=C N=7$ cm and $\angle N O M=60^{\circ}$.

Answer: 4\sqrt{7}


5. Given $P$ as a moving point on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, draw the perpendiculars from the foci $F_{1}$ and $F_{2}$ to the external angle bisector of $\angle F_{1} P F_{2}$. Then the locus of the feet of these perpendiculars is $\qquad$ .

Answer: x^2 + y^2 = a^2


4. Given a tetrahedron $ABCD$ satisfying: $AB=2, CD=2\sqrt{5}, AC=BD=3, AD=BC=\sqrt{5}$, then the volume of the tetrahedron is $\qquad$.

Answer: \dfrac{4}{3}


2. (15 points) Find the range of the similarity coefficient of triangles with side lengths $x, y, z$ and $y, z, p$. In your answer, specify the nearest integers between which the found interval lies.

Answer: 1


52. On the side $AB$ of the rectangular trapezoid $ABCD (AB \perp BC)$, a semicircle (with the side as the diameter) is constructed, which touches the side $CD$ at point $K$. The diagonals of the trapezoid intersect at point $O$. Find the length of the segment $OK$, if the lengths of the bases of the trapezoid $ABCD$ are 2 and 3.

Answer: \dfrac{6}{5}


Let $n \ge 3$ be a fixed integer.  The numbers $1,2,3, \cdots , n$ are written on a board.  In every move one chooses two numbers and replaces them by their arithmetic mean.  This is done until only a single number remains on the board.

Determine the least integer that can be reached at the end by an appropriate sequence of moves.

(Theresia Eisenkölbl)

Answer: \left\lceil \dfrac{n+1}{2} \right\rceil


A shipping company charges $.30l+.40w+.50h$ dollars to process a right rectangular prism-shaped box with dimensions $l,w,h$ in inches.  The customers themselves are allowed to label the three dimensions of their box with $l,w,h$ for the purpose of calculating the processing fee.  A customer finds that there are two different ways to label the dimensions of their box $B$ to get a fee of $\$8.10$, and two different ways to label $B$ to get a fee of $\$8.70$.  None of the faces of $B$ are squares.  Find the surface area of $B$, in square inches.

[i]Proposed by Justin Hsieh[/i]

Answer: 276


In the diagram, a go-kart track layout is shown. The start and finish are at point $A$, and the go-kart driver can return to point $A$ and re-enter the track as many times as desired.

On the way from $A$ to $B$ or back, the young racer Yura spends one minute. Yura also spends one minute on the loop. The loop can only be driven counterclockwise (the arrows indicate the possible directions of travel). Yura does not turn back halfway and does not stop. The total duration of the race is 10 minutes. Find the number of possible different routes (sequences of sections traveled).

Answer: 34


A square-based right pyramid has its lateral faces forming a $60^{\circ}$ angle with the base. The plane that bisects the angle between the base and one of the lateral faces divides the pyramid into two parts. Determine the ratio of the volumes of the two resulting solids.

Answer: \dfrac{3}{5}


## Task A-2.7.

In a basketball tournament, each team plays exactly twice against each of the other teams. A win brings 2 points, a loss 0 points, and there are no draws. Determine all natural numbers $n$ for which there exists a basketball tournament with $n$ teams where one team, the tournament winner, has 26 points, and exactly two teams have the lowest number of points, which is 20 points.

Answer: 12


3. A chess piece moves along the edges of a regular hexagon with a side length of $1 \mathrm{~cm}$, starting from point $A$. On the first move, it travels $1 \mathrm{~cm}$ to reach point $B$, on the second move, it travels $2 \mathrm{~cm}$ to reach point $D$, $\cdots \cdots$, on the $n$-th move, it travels $n \mathrm{~cm}$, until the 100th move is completed. At the point where the chess piece arrives the most times (the initial $A$ point also counts as one arrival), it has arrived $\qquad$ times.

Answer: 34


2A. Let $A B C D$ be a cyclic quadrilateral such that: $\overline{A B}=\overline{B C}=\sqrt{3}$, $\overline{C D}=2$ and $\measuredangle B A D=30^{\circ}$. Calculate the area of the quadrilateral $A B C D$.

Answer: \dfrac{7\sqrt{3}}{4}


8 Given $\triangle A B C$, $\tan ^{2} B=\tan A \tan C$, find the range of $B$.

In $\triangle A B C$, it is known that $\tan ^{2} B=\tan A \tan C$. Find the range of $B$.

Answer: \left[ \dfrac{\pi}{3}, \dfrac{\pi}{2} \right)


Let $p$ be a prime number and let $\mathbb{F}_p$ be the finite field with $p$ elements. Consider an automorphism $\tau$ of the polynomial ring $\mathbb{F}_p[x]$ given by
\[\tau(f)(x)=f(x+1).\]
Let $R$ denote the subring of $\mathbb{F}_p[x]$ consisting of those polynomials $f$ with $\tau(f)=f$. Find a polynomial $g \in \mathbb{F}_p[x]$ such that $\mathbb{F}_p[x]$ is a free module over $R$ with basis $g,\tau(g),\dots,\tau^{p-1}(g)$.

Answer: x^{p-1}


Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. At the beginning of a turn there are n ≥ 1 marbles on the table, then the player whose turn is removes k marbles, where k ≥ 1 either is an even number with $k \le \frac{n}{2}$ or an odd number with $ \frac{n}{2}\le k \le n$. A player wins the game if she removes the last marble from the table.
Determine the smallest number $N\ge100000$ which Berta has wining strategy.

[i]proposed by Gerhard Woeginger[/i]

Answer: 131070


Consider 2017 lines in the plane, which intersect pairwise at distinct points. Let $E$ be the set of these intersection points.
We want to assign a color to each point in $E$ such that any two points on the same line, whose segment connecting them contains no other point of $E$, are of different colors.
What is the minimum number of colors needed to achieve such a coloring?

Answer: 3


1. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x \in \mathbb{R}$, $f(x) \leqslant x$ and for all $x, y \in \mathbb{R}$, $f(x+y) \leqslant f(x)+f(y)$.

Answer: f(x) = x


8. Correct the error. In the incorrect equation $101=10^{2}-1$, move one digit to make it correct.

Answer: 101 = 102 - 1


14. In the sequence $\left\{a_{n} \mid\right.$, $a_{7}=\frac{16}{3}$, when $n \geqslant 2$, $a_{n}=\frac{3 a_{n-1}-1}{7-a_{n-1}+4}$,
(1) Does there exist a natural number $m$, such that when $n>m$, $a_{n}<2$? If it exists, find the value of $m$; if not, explain the reason.
(2) When $n \geqslant 10$, prove that $\frac{a_{n-1}+a_{n}+1}{2}<a_{n}$.

Answer: 8


15. Fill in the numbers $1, 2, 3, 4, 5$ in the following table (the letters and numbers are used to label rows, columns, or small cells, for example, D3 refers to the white small cell in row D and column 3), ensuring that the five numbers in each row and each column are all different. The following are some clues:
(1) The sum of the numbers in the gray cells in the 4th column is 9;
(2) The sum of the numbers in the gray cells in row C is 7;
(3) The sum of the numbers in the white cells in the 2nd column is 8;
(4) The sum of the numbers in the gray cells in row B is less than the sum of the numbers in the gray cells in row D. Therefore, the number in D4 is $\qquad$ -

Answer: 5


8. In the sequence $\left\{a_{n}\right\}_{n=1}^{+\infty}$, $x_{1}=1$, if for any positive integer $n$ there is $x_{n+1}=x_{n}+\left[\sqrt{x_{n}}\right]$, where the symbol $[x]$ denotes the greatest integer not exceeding $x$, then the smallest positive integer $n$ such that $x_{n} \geqslant 2019$ is $\qquad$.

Answer: 95


B3. In triangle ABC, the angle $\alpha=58^{\circ}$ and the angle $\beta=84^{\circ}$. What is the measure of angle $x$ between the angle bisector of $\gamma$ and the altitude to side $c$?

Answer: 13


Bakayev E.V.

In each cell of a $1000 \times 1000$ square, a number is inscribed such that in any rectangle of area $s$ that does not extend beyond the square and whose sides lie along the cell boundaries, the sum of the numbers is the same. For which $s$ will the numbers in all cells necessarily be the same?

Answer: s


10,11

During an interview, ten people were offered a test consisting of several questions. It is known that any five people together answered all the questions (that is, at least one of the five gave the correct answer to each question), but any four did not. What is the minimum number of questions for which this could have been the case?

#

Answer: 210


Let $k$ and $n$ be natural numbers such that $k \leq n$. Arrange the first $n^{2}$ natural numbers in an $n \times n$ table so that the numbers in each row are in ascending order and the sum of the numbers in the $k$-th column is a) the smallest; b) the largest.

Answer: \frac{k n (n + 1)}{2}


We are given sufficiently many stones of the forms of a rectangle $2\times 1$ and square $1\times 1$. Let $n > 3$ be a natural number. In how many ways can one tile a rectangle $3 \times n$ using these stones, so that no two $2 \times 1$ rectangles have a common point, and each of them has the longer side parallel to the shorter side of the big rectangle?

Answer: \dfrac{2^{n+2} - (-1)^n}{3}


Let \( AL \) be the angle bisector of triangle \( ABC \). The perpendicular bisector of segment \( AL \) intersects the circumcircle of triangle \( ABC \) at points \( P \) and \( Q \). Prove that the circumcircle of triangle \( PLQ \) is tangent to side \( BC \).

Answer: \text{The circumcircle of triangle } PLQ \text{ is tangent to side } BC


Task 3. (15 points) At the research institute, a scientific employee, Ivan Ivanovich, received an object for research containing about 300 oil samples (a container designed for 300 samples, which was almost completely filled). Each sample has certain characteristics in terms of sulfur content - either low-sulfur or high-sulfur, and density - either light or heavy. The relative frequency (statistical probability) that a randomly selected sample will be a heavy oil sample is $\frac{1}{8}$. The relative frequency that a randomly selected sample will be a light low-sulfur oil sample is $\frac{22}{37}$. How many high-sulfur oil samples does the object contain, if there were no low-sulfur samples among the heavy oil samples?

Answer: 120


There is a schoolbag containing 12 cards labeled $1, 1, 2, 2, \cdots, 6, 6$. A person draws one card at a time without replacement. If a card is drawn that has the same number as a previously drawn card, both cards are discarded. The process ends when the person has 3 single cards in hand or all cards in the schoolbag have been drawn. Find the probability that all cards in the schoolbag are drawn.

Answer: \dfrac{9}{385}


Oliver has a training loop for his weekend cycling. He starts by cycling due north for 3 miles. He then cycles northeast, making a 30° angle with the north for 2 miles, followed by cycling southeast, making a 60° angle with the south for 2 miles. He completes his loop by cycling directly back to the starting point. Find the distance of this final segment of his ride.

Answer: \sqrt{11 + 6\sqrt{3}}


Two boys start at the same point A on an elliptical track and move in opposite directions. The speeds of the boys are 7 ft. per second and 11 ft. per second, respectively. The elliptical track has a major axis of 100 ft and a minor axis of 60 ft. Calculate the number of times they meet, excluding the start and finish, assuming they start at the same time and finish when they first meet at point A again.

Answer: 16


Lisa celebrated her 20th birthday on Monday, August 15, 2010, which was not a leap year. Find the next year when her birthday will fall on a Thursday.

Answer: 2012


During a journey from the United States to France, Lucas took $d$ U.S. dollars. At the airport, he exchanged all his money to Euros at a rate where $5$ U.S. dollars yielded $8$ Euros. After spending $80$ Euros on souvenirs, Lucas found that he had exactly $d$ Euros left. Find the sum of the digits of $d$.

Answer: 7


In a circle, diameter $\overline{EB}$ is parallel to chord $\overline{DC}$. If the angles $AEB$ and $ABE$ are in the ratio of $2 : 3$, determine the degree measure of angle $BCD$.

Answer: 36^\circ


Cagney can frost a cupcake in 15 seconds and Lacey can frost a cupcake in 25 seconds. After a 1-minute warm-up, their frosting speeds increase by 20%. Calculate the total number of cupcakes they can frost together in 10 minutes.

Answer: 76


Given a three-digit positive integer and a two-digit positive integer with all five digits in these numbers being different, the sum of the two numbers is a three-digit number. Find the smallest possible value for the sum of the digits of this three-digit sum.

Answer: 3


The number $2024$ needs to be expressed in the form $\frac{a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$, where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_m + b_n$ is as small as possible. Find the value of $|a_m - b_n|$.

Answer: 1


The product of the two 99-digit numbers 707,070,707,...,070,707 and 909,090,909,...,090,909 has a thousands digit A and a units digit B. Calculate the sum of A and B.

Answer: 5


A rectangle $ABCD$ has its diagonal $DB$ divided into three segments by two lines $L$ and $L'$, which are parallel to each other and perpendicular to $DB$. Line $L$ passes through $A$, and $L'$ passes through $C$. The segments along $DB$ created by $L$ and $L'$ have lengths $DE = 2$, $EF = 1$, and $FB = 1$. Find the area of rectangle $ABCD$, rounded to one decimal place.
A) $6.8$
B) $6.9$
C) $7.0$
D) $7.1$

Answer: 6.9


Let there be $k$ red balls and $N$ green balls arranged in a line, where $k$ is a positive integer and $N$ is a positive multiple of 10. Determine the least value of $N$ such that the probability that at least $\frac{3}{5}$ of the green balls are on the same side of all the red balls combined is less than $\frac{8}{10}$ when there are $2$ red balls, and calculate the sum of its digits.

Answer: 1


A circular disc with a diameter of $2D$ is placed on a $6 \times 6$ checkerboard, aiming for the centers to coincide. Calculate the number of checkerboard squares which are completely covered by the disc.

Answer: 16


Let $ABCD$ be a square with side length 3. Points $E$ and $F$ are on sides $AB$ and $AD$ respectively such that $AE=2AF$. Determine the maximum area of quadrilateral $CDFE$.
A) $\frac{1}{2}$
B) $\frac{3}{2}$
C) $\frac{5}{2}$
D) $\frac{7}{2}$
E) $\frac{9}{2}$

Answer: \frac{9}{2}


A rectangular indoor football field is being designed with a length of $100 \text{m}$ and a width of $80 \text{m}$. The field must be illuminated by four spotlights, each hanging at some point on the ceiling. Each spotlight illuminates a circle with a radius equal to the height at which the spotlight is hanging. Determine the minimum possible ceiling height such that the following conditions are met: every point on the football field is illuminated by at least one spotlight; the ceiling height must be a multiple of $0.1 \text{m}$ (e.g., $19.2 \text{m}$, $28 \text{m}$, $33.9 \text{m}$, etc.).

Answer: 32.1


29. There are $n$ numbers arranged in a row, starting from the second one, each is 7 greater than the one immediately preceding it: $1,8,15,22,29, \ldots, 7 n-6$
Their product ends with exactly 10 zeros. Then the maximum value of $n$ is $\qquad$

Answer: 42


Let $\triangle ABC$ have median $CM$ ($M\in AB$) and circumcenter $O$. The circumcircle of $\triangle AMO$ bisects $CM$. Determine the least possible perimeter of $\triangle ABC$ if it has integer side lengths.

Answer: 24


8. For natural numbers $x, y$, $(x, y)$ is called a pair. It is stipulated that the pair $(x, y)$ is the same as $(y, x)$. If two pairs satisfy the following condition: the sum of the two numbers in one pair equals the product of the two numbers in the other pair, then these two pairs are called "companion pairs". Except for the pair $(2,2)$ and $(2,2)$, the "companion pairs" are

Answer: (1, 5)


8. There are 9 students participating in a math competition in the same classroom, with seats arranged in 3 rows and 3 columns, represented by a $3 \times 3$ grid, where each cell represents a seat. To prevent cheating, three types of exams, $A$, $B$, and $C$, are used, and it is required that any two adjacent seats (cells sharing a common edge) receive different types of exams. The number of ways to distribute the exams that meet the conditions is $\qquad$ kinds.

Answer: 246


Bakayev E.V.

A mathematician with five children walked into a pizzeria.

Masha: I want one with tomatoes and no sausage.

Vanya: And I want one with mushrooms.

Dasha: I'll have one without tomatoes.

Nikita: I want one with tomatoes. But no mushrooms!

Igor: And I want one without mushrooms. But with sausage!

Dad: Well, with such picky eaters, one pizza definitely won't be enough...

Will the mathematician be able to order two pizzas and treat each child to the one they want, or will he have to order three pizzas?

Answer: 3


7. Arrange $\{1,2, \cdots, 7\}$ randomly into $\left\{a_{1}, a_{2}, \cdots\right.$, $\left.a_{7}\right\}$. The probability that $a_{1}+a_{2}+a_{3} \geqslant a_{5}+a_{6}+a_{7}$ is $\qquad$

Answer: \dfrac{73}{140}


3. Four balls with equal radii $r$ are given, and they touch each other pairwise. A fifth ball is described around them. Find the radius of the fifth ball.

Answer: r\left(1 + \dfrac{\sqrt{6}}{2}\right)


Problem 2. Does there exist a fraction equal to $\frac{7}{13}$, the difference between the denominator and the numerator of which is 24?

Answer: \dfrac{28}{52}


One, (20 points) Given $m>0$, and $m$ is a rational number, $\sqrt{m}$ is an irrational number. Find all positive integers $a, b, c$, such that $\frac{\sqrt{m} a+b}{\sqrt{m} b+c}$ is a rational number, and $\frac{a^{2}+b^{2}+c^{2}}{a+b+c}=3$.

Answer: (1, 2, 4)


In a set \( X \) of 17 elements, a family of \( N \) distinct non-empty subsets is selected such that each element of set \( X \) is contained in exactly two subsets from this family. What is the maximum value of \( N \)? Find the number of all possible distinct types of such families for the maximum \( N \). Two families of subsets have different types if they cannot be obtained from each other by permuting the elements of \( X \).

Answer: 25


6. (10 points) As shown in the figure, a rectangular block with dimensions $15 \mathrm{~cm}, 5 \mathrm{~cm}, 4 \mathrm{~cm}$ has a smaller rectangular block with dimensions $y \mathrm{~cm}, 5 \mathrm{~cm}, x \mathrm{~cm}$ cut out from it $\left(x, y\right.$ are integers). The remaining volume is $120 \mathrm{~cm}^{3}$. Therefore, $x+y=$ . $\qquad$

Answer: 15


2. Anton, Boris, Vasya, and Grisha met. It is known that each of them is either from the tribe of knights (who always tell the truth) or from the tribe of liars (who always lie). Anton said that he and Grisha are from different tribes. Boris and Vasya called each other liars. And Grisha claimed that there are at least two knights among them. How many of them are there really?

Answer: 1


2. It is known that the equation $3 \sin x+4 \cos x=a$ has exactly two equal real roots $\alpha, \beta$ in the interval $(0,2 \pi)$, $\alpha+\beta=$ $\qquad$ .

Answer: 2 \arctan \frac{3}{4}


Jacob likes to watch Mickey Mouse Clubhouse! One day, he decides to create his own Mickey Mouse head with two circles $\omega_1$ and $\omega_2$ and a circle $\omega$, with centers $O_1$, $O_2$, and $O$, respectively. Let $\omega_1$ and $\omega$ meet at points $P_1$ and $Q_1$, and let $\omega_2$ and $\omega$ meet at points $P_2$ and $Q_2$. Point $P_1$ is closer to $O_2$ than $Q_1$, and point $P_2$ is closer to $O_1$ than $Q_2$. Given that $P_1$ and $P_2$ lie on $O_1O_2$ such that $O_1P_1 = P_1P_2 = P_2O_2 = 2$, and $Q_1O_1 \parallel Q_2O_2$, the area of $\omega$ can be written as $n \pi$. Find $n$.

Answer: 4


A list of positive integers is called *good* if the maximum element of the list appears exactly once. A *sublist* is a list formed by one or more consecutive elements of a list. For example, in the list $10, 34, 34, 22, 30, 22$, the sublist $22, 30, 22$ is good, while $10, 34, 34, 22$ is not.

A list is *very good* if all its sublists are good. Find the minimum value of $k$ such that there exists a very good list of length $2019$ with $k$ different values in it.

Answer: 11


In a right triangle $ABC$ (where $\angle C = 90^\circ$), it is given that $AC = 4$ cm and $BC = 3$ cm. The points $A_1$, $B_1$, and $C_1$ are positioned such that $AA_1 \parallel BC$, $BB_1 \parallel A_1C$, and $CC_1 \parallel A_1B_1$. Additionally, $\angle A_1B_1C_1 = 90^\circ$ and $A_1B_1 = 1$ cm. Find the length of $B_1C_1$.

Answer: 12


In the figure, points $A$, $B$, $C$, and $D$ are on the same line and are the centers of four tangent circles at the same point. Segment $AB$ measures $8$, and segment $CD$ measures $4$. The circumferences with centers $A$ and $C$ are of equal size. We know that the sum of the areas of the two medium circles is equivalent to the sum of the areas of the small and large circles. What is the length of segment $AD$?

Answer: 24


A number is considered *interesting* if it meets the following criteria:

1. It is a $6$-digit integer.
2. It contains no zeros.
3. The first $3$ digits are strictly increasing.
4. The last $3$ digits are non-increasing.

What is the average of all interesting numbers?

Answer: 308253


Mustafa bought a big rug. The seller measured the rug with a ruler that was supposed to measure one meter. As it turned out to be $30$ meters long by $20$ meters wide, he charged Rs $120,000$. When Mustafa arrived home, he measured the rug again and realized that the seller had overcharged him by $9,408$ Rs. How many centimeters long is the ruler used by the seller?

Answer: 96


Tyler has two calculators, both of which initially display zero. The first calculator has only two buttons, $[+1]$ and $[\times 2]$. The second has only the buttons $[+1]$ and $[\times 4]$. Both calculators update their displays immediately after each keystroke.

A positive integer $n$ is called \textit{ambivalent} if the minimum number of keystrokes needed to display $n$ on the first calculator equals the minimum number of keystrokes needed to display $n$ on the second calculator. Find the sum of all ambivalent integers between $256$ and $1024$ inclusive.

Answer: 34776


A square $ABCD$ has a side length of $1$. A circle passes through the vertices of the square. Let $P, Q, R, S$ be the midpoints of the arcs which are symmetrical to the arcs $AB$, $BC$, $CD$, $DA$ when reflected on sides $AB$, $BC$, $CD$, $DA$, respectively. The area of square $PQRS$ is $a + b\sqrt{2}$, where $a$ and $b$ are integers. Find the value of $a + b$.

Answer: 1


Euhan and Minjune are playing a game. They choose a number $N$ so that they can only say integers up to $N$. Euhan starts by saying $1$, and each player takes turns saying either $n+1$ or $4n$ (if possible), where $n$ is the last number said. The player who says $N$ wins. What is the smallest number larger than $2019$ for which Minjune has a winning strategy?

Answer: 2020


\( \triangle ABC \) has an area of \( 240 \). Points \( X, Y, Z \) lie on sides \( AB, BC, \) and \( CA, \) respectively. Given that \( \frac{AX}{BX} = 3 \), \( \frac{BY}{CY} = 4 \), and \( \frac{CZ}{AZ} = 5 \), find the area of \( \triangle XYZ \).

Answer: 122


Seven people of seven different ages are attending a meeting. The seven people leave the meeting one at a time in random order. Given that the youngest person leaves the meeting sometime before the oldest person leaves the meeting, determine the probability that the third, fourth, and fifth people to leave the meeting do so in order of their ages (youngest to oldest). The probability is given by $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 25


Amir enters Fine Hall and sees the number $2$ written on the blackboard. Amir can perform the following operation: he flips a coin, and if it is heads, he replaces the number $x$ on the blackboard with $3x + 1$; otherwise, he replaces $x$ with $\lfloor x/3 \rfloor$. If Amir performs the operation four times, let $\frac{m}{n}$ denote the expected number of times that he writes the digit $1$ on the blackboard, where $m, n$ are relatively prime positive integers. Find $m + n$.

Answer: 27


The points $A$, $B$, $C$, $D$, $E$ lie on a line $\ell$ in this order. Suppose $T$ is a point not on $\ell$ such that $\angle BTC = \angle DTE$, and $\overline{AT}$ is tangent to the circumcircle of triangle $BTE$. If $AB = 2$, $BC = 36$, and $CD = 15$, compute $DE$.

Answer: 954


In the plane, there are $10$ lines in general position, meaning that no two are parallel and no three are concurrent. At each intersection of two lines, we measure the smaller of the two angles formed between them. What is the maximum value of the sum of the measures of these $45$ angles?

Answer: 4050


Example 7 Let the complex number $z$ satisfy $\left|\arg \left(\frac{z+1}{z+2}\right)\right|=\frac{\pi}{6}$, find the range of $\arg z$.

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

Answer: \left[ \frac{5\pi}{6} - \arcsin\left(\frac{1}{\sqrt{3}}\right), \frac{7\pi}{6} + \arcsin\left(\frac{1}{\sqrt{3}}\right) \right]


A model car is tested on some closed circuit $600$ meters long, consisting of flat stretches, uphill and downhill. All uphill and downhill have the same slope. The test highlights the following facts:

[list]
(a) The velocity of the car depends only on the fact that the car is driving along a stretch of uphill, plane or downhill; calling these three velocities $v_s, v_p$ and $v_d$ respectively, we have $v_s <v_p <v_d$;

(b) $v_s,v_p$ and $v_d$, expressed in meter per second, are integers.

(c) Whatever may be the structure of the circuit, the time taken to complete the circuit is always $50$ seconds.
[/list]

Find all possible values of $v_s, v_p$ and $v_d$.

Answer: (7, 12, 42)


5. In the store "Third is Not Excessive," there is a promotion: if a customer presents three items at the cash register, the cheapest of them is free. Ivan wants to buy 11 items costing $100, 200, 300, \ldots, 1100$ rubles. For what minimum amount of money can he buy these items?

Answer: 4800


Vertices $A$ and $B$ of the prism $A B C A 1 B 1 C 1$ lie on the axis of the cylinder, while the other vertices lie on the lateral surface of the cylinder. Find the dihedral angle in this prism with edge $A B$.

Answer: 90^\circ


7. As shown in the figure, the beads on the bracelet are numbered from 1 to 22 in a counterclockwise direction starting from the pendant bead. Xiao Ming is playing a bead counting game, with the rule being: starting from bead 1, count natural numbers in a clockwise direction, but skip any number that contains the digit 7 or is a multiple of 7, and directly count the next number. For example: after counting to 6, the next number is 8; after counting to 13, the next number is 15, and so on. So, when counting to 100, which bead number $\qquad$ will it land on?

Answer: 20


1. Find all values of $p$, for each of which the numbers $9 p+10, 3 p$ and $|p-8|$ are respectively the first, second, and third terms of some geometric progression.

Answer: -1


In the diagram, two circles are centred at $O$. The smaller circle has a radius of 1 and the larger circle has a radius of 3. Points $P$ and $Q$ are placed on the larger circle so that the areas of the two shaded regions are equal. If $\angle P O Q=x^{\circ}$, what is the value of $x$ ?

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-011.jpg?height=368&width=357&top_left_y=697&top_left_x=1315)

Answer: 40


1. Find the largest three-digit number from which, after erasing any digit, we get a prime number.

Answer: 731


Exercise 6. In 5 boxes are found respectively $402, 403, 404, 405$ and 406 stones. The only allowed operation is to take 4 stones from a pile that has at least 4 stones and to put one in each of the other piles. What is the maximum number of stones that can be in a single pile?

Answer: 2014


12. On the highway between $\mathrm{A}$ and $\mathrm{B}$, the speed limit for vehicles traveling from $\mathrm{A}$ to $\mathrm{B}$ is 50 kilometers per hour, and for vehicles traveling from $\mathrm{B}$ to $\mathrm{A}$ is 60 kilometers per hour. Now, two cars, Car A and Car B, start simultaneously from $\mathrm{A}$ and $\mathrm{B}$ respectively, and travel back and forth between the two places. When Car A reaches $\mathrm{B}$ and is on its way back to $\mathrm{A}$, it stops due to a malfunction, and the stopping point is 30 kilometers from $\mathrm{B}$. The two cars meet for the second time at this location, which is 1 hour and 12 minutes later than the originally scheduled second meeting time. What is the distance between the two places? $\qquad$ kilometers.

Answer: 198


157. Secret Code. Two conspirators had a secret code. Sometimes their correspondence contained simple arithmetic operations that looked completely innocent. However, in the code, each of the ten digits represented its own letter of the alphabet. Once, a sum was encountered which, after substituting the corresponding letters for the digits, took the form

$$
\begin{gathered}
F L Y \\
F O R \\
Y O U R \\
\hline L I F E^{* *}
\end{gathered}
$$

It would be interesting to restore this sum, knowing that $I$ and $O$ represent the digits 1 and 0, respectively.

Answer: 9152


A nearsighted rook attacks all cells in its row and column that can be reached within 60 steps, moving from one cell to an adjacent cell. What is the maximum number of non-attacking nearsighted rooks that can be placed on a $100 \times 100$ grid?

Answer: 178


13. (10 points) A class of 30 students goes to the movies, and their student IDs are $1, 2, \cdots, 30$; the movie tickets they hold are exactly the 1st, 2nd, ..., 30th seats in a row. Now, the movie tickets are to be distributed to these students according to the following requirements: for any two students, A and B, if A's student ID can be divided by B's student ID, then A's movie ticket number can also be divided by B's movie ticket number. How many different ways are there to distribute the movie tickets? $\qquad$

Answer: 48


Four. (50 points) Color each cell of a $5 \times 5$ grid with one of five colors, such that the number of cells of each color is the same. If two adjacent cells have different colors, their common edge is called a "separating edge." Find the minimum number of separating edges.

Color each cell of a $5 \times 5$ grid with one of five colors, such that the number of cells of each color is the same. If two adjacent cells have different colors, their common edge is called a "separating edge." Find the minimum number of separating edges.

Answer: 16


1. We have 50 chips numbered from 1 to 50, and they need to be colored red or blue. We know that chip 5 is blue. For the coloring of the rest of the chips, the following rules are followed:

a) If the chip with the number $x$ and the chip with the number $y$ are of different colors, then the chip with the number $|x-y|$ is painted red.

b) If the chip with the number $x$ and the chip with the number $y$ are of different colors and $x \cdot y$ is a number between 1 and 50 (inclusive), then the chip with the number $x \cdot y$ is painted blue.

Determine how many different colorings can be performed on the set of chips.

Answer: 2


18. The figure shows a track scheme for karting. The start and finish are at point $A$, and the kart driver can make as many laps as they want, returning to the starting point.

![](https://cdn.mathpix.com/cropped/2024_05_06_5bc45e45226f411effd0g-14.jpg?height=402&width=1288&top_left_y=867&top_left_x=384)

The young driver Yura spends one minute on the path from $A$ to $B$ or back. Yura also spends one minute on the loop. The loop can only be driven counterclockwise (arrows indicate possible directions of movement). Yura does not turn back halfway and does not stop. The race duration is 10 minutes. Find the number of possible different routes (sequences of passing sections). #

Answer: 34


33rd Swedish 1993 Problem 2 A railway line passes through (in order) the 11 stations A, B, ... , K. The distance from A to K is 56. The distances AC, BD, CE, ... , IK are each ≤ 12. The distances AD, BE, CF, ... , HK are each ≥ 17. Find the distance between B and G.

Answer: 29


Let \( K(x) \) be the number of such irreducible fractions \(\frac{a}{b}\) such that \( a < x \) and \( b < x \) ( \( a \) and \( b \) are natural numbers). For example, compute the sum \( K(100) + K\left( \frac{100}{2} \right) + K\left( \frac{100}{3} \right) + \ldots + K\left( \frac{100}{99} \right) + K\left( \frac{100}{100} \right) \).

Answer: 9801


10.124 $* *$ Find the largest square number, such that after removing its last two digits, you still get a square number. Assume the two removed digits are not both 0.

Answer: 1681


13. There are two four-digit numbers, satisfying:
(1) Their digit sums differ by 1;
(2) Their difference is 710;
(3) The last four digits of their squares are the same.
Then the larger four-digit number has $\qquad$ different possible cases.

Answer: 8


13. (i) (Grade 11) In the arithmetic sequence $\left\{a_{n}\right\}: a_{n}=4 n -1\left(n \in \mathbf{N}_{+}\right)$, after deleting all numbers that can be divided by 3 or 5, the remaining numbers are arranged in ascending order to form a sequence $\left\{b_{n}\right\}$. Find the value of $b_{2006}$.
(ii) (Grade 12) Given $\odot P: x^{2}+y^{2}=2 x$ and the parabola $S: y^{2}=4 x$, a line $l$ is drawn through the center $P$ intersecting the two curves at four points, which are sequentially labeled as $A, B, C, D$ from top to bottom. If the lengths of segments $A B, B C, C D$ form an arithmetic sequence in this order, find the equation of the line $l$.

Answer: 15043


3-3. The height of a truncated cone is equal to the radius of its larger base; the perimeter of a regular hexagon circumscribed around the smaller base is equal to the perimeter of an equilateral triangle inscribed in the larger base. Determine the angle of inclination of the cone's generatrix to the base plane.

Answer: \arctan 4


Problem 3. In each of the 16 cells of a $4 \times 4$ square, one of the numbers $1,2,3, \ldots, 16$ is written. The sum of the numbers in each column is calculated. If one of the obtained sums is strictly greater than the other three, it is denoted by $S$.

a) Give an example of a completion of the square where $S=40$.

b) What is the smallest possible value of $S$?

Answer: 40


2. Find all values of the real parameter \(a\) for which some two distinct solutions of the equation

\[
x^{4} - a x^{3} + x^{2} + a = 0
\]

(in the set \(\mathbb{C}\)) have a sum equal to 1.

Answer: \frac{1}{2}


7. In a cyclic quadrilateral $ABCD$, there is a point $X$ on side $AB$ such that diagonal $BD$ bisects $CX$ and $AC$ bisects $DX$. Find the minimum value of $\frac{AB}{CD}$.

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

Answer: 2


NT3 Let $s(a)$ denote the sum of digits of a given positive integer $a$. The sequence $a_{1}, a_{2}, \ldots a_{n}, \ldots$ of positive integers is such that $a_{n+1}=a_{n}+s\left(a_{n}\right)$ for each positive integer $n$. Find the greatest possible $n$ for which it is possible to have $a_{n}=2008$.

Answer: 6


5. On the plane, there is a set of points $M$, the coordinates $X$ and $y$ of which are related by the equation

$$
\sin (x+2 y)=\sin x+\sin 2 y
$$

A circle of radius $R$, located on the same plane, does not intersect with the set $M$.

What values can the radius of such a circle take

Answer: \left(0, \dfrac{\pi(3 - \sqrt{5})}{2}\right)


6. A racing robot has the following characteristics:
(1) The step length can be manually set to 0.1 meters, 0.2 meters, $\cdots$, 1.8 meters, or 1.9 meters:
(2) After the start command, the robot immediately takes the first step of the set length, and each step is completed instantaneously;
(3) When the step length is set to $a$ meters, the robot needs exactly $a$ seconds between each consecutive step.

Then the minimum time required for the robot to run 50 meters (allowing to exceed 50 meters) is

Answer: 48.6


G3 The vertices $A$ and $B$ of an equilateral $\triangle A B C$ lie on a circle $k$ of radius 1 , and the vertex $C$ is inside $k$. The point $D \neq B$ lies on $k, A D=A B$ and the line $D C$ intersects $k$ for the second time in point $E$. Find the length of the segment $C E$.

Answer: 1


Reflect a cube in space in the planes of each of its six faces in some order. How many different transformations are produced by the succession of these six reflections?

Answer: 8


## Task A-2.4.

Let $\overline{A B}$ and $\overline{C D}$ be diameters of the circle $k$ with center $S$ and let $\varangle B A D=28^{\circ}$. A circle with center $A$ passing through the point $S$ intersects the circle $k$ at points $E$ and $F$, with $D$ and $F$ on the same side of the line $A B$. Determine $\varangle C F S$.

Answer: 32^\circ


4. Out of 43 coins arranged in a row, 8 coins are facing "tails" up, and 35 are facing "heads" up. In one step, any 20 coins can be flipped. Is it possible after a finite number of steps for all the coins to be facing "heads" up? In the minimum number of steps, how is this possible? Explain your answer!

Answer: 2


3. In tetrahedron $ABCD$, the lengths of edges $AB$ and $CD$ are $a$ and $b$ respectively, and the distance between the midpoints of these two edges is $d$. Then the maximum volume of tetrahedron $ABCD$ is $\qquad$ (1994, Shanghai High School Mathematics Competition for Grade 10)

Answer: \dfrac{1}{6} a b d


## Task B-3.2.

Determine two natural numbers that are divisible by four and for which the difference of their cubes is a four-digit number divisible by 91.

Answer: 20


3. Given $\frac{\sin (\beta+\gamma) \sin (\gamma+\alpha)}{\cos \alpha \cos \gamma}=\frac{4}{9}$, then the value of $\frac{\sin (\beta+\gamma) \sin (\gamma+\alpha)}{\cos (\alpha+\beta+\gamma) \cos \gamma}$ is $\qquad$ .

Answer: \dfrac{4}{5}


In a $25 \times n$ grid, each square is colored with a color chosen among $8$ different colors. Let $n$ be as minimal as possible such that, independently from the coloration used, it is always possible to select $4$ coloumns and $4$ rows such that the $16$ squares of the interesections are all of the same color. Find the remainder when $n$ is divided by $1000$.

[i]Proposed by [b]FedeX333X[/b][/i]

Answer: 601


【Example 3】A party needs to arrange 10 singing programs and 4 dance programs, hoping that at least two singing programs are inserted between every two dances. How many different performance orders can be arranged?

Answer: 70 \times 10! \times 4!


Let $\{a_n\}$ be a sequence of integers satisfying $(n-1)a_{n+1}=(n+1)a_n-2(n-1) \forall n\ge 1$. If $2000|a_{1999}$, find the smallest $n\ge 2$ such that $2000|a_n$.

Answer: 249


Suppose two circles $\Omega_1$ and $\Omega_2$ with centers $O_1$ and $O_2$ have radii $3$ and $4$, respectively. Suppose that points $A$ and $B$ lie on circles $\Omega_1$ and $\Omega_2$, respectively, such that segments $AB$ and $O_1O_2$ intersect and that $AB$ is tangent to $\Omega_1$ and $\Omega_2$. If $O_1O_2=25$, find the area of quadrilateral $O_1AO_2B$.

Answer: 84


The natural number $n > 1$ is called "heavy" if it is coprime with the sum of its divisors. What is the maximal number of consecutive "heavy" numbers?

Answer: 4


We say that a polynomial $p$ is respectful if $\forall x, y \in \mathbb{Z}$, $y - x$ divides $p(y) - p(x)$, and $\forall x \in \mathbb{Z}$, $p(x)  \in \mathbb{Z}$. We say that a respectful polynomial is disguising if it is nonzero, and all of its non-zero coefficients lie between $0$ and $1$, exclusive. Determine $\sum \deg(f)\cdot  f(2)$, where the sum includes all disguising polynomials $f$ of degree at most $5$.

Answer: 0


Let $ S(n) $ be the sum of the squares of the positive integers less than and coprime to $ n $. For example, $ S(5) = 1^2 + 2^2 + 3^2 + 4^2 $, but $ S(4) = 1^2 + 3^2 $. Let $ p = 2^7 - 1 = 127 $ and $ q = 2^5 - 1 = 31 $ be primes. The quantity $ S(pq) $ can be written in the form $$ \frac{p^2q^2}{6}\left(a - \frac{b}{c} \right) $$ where $ a $, $ b $, and $ c $ are positive integers, with $ b $ and $ c $ coprime and $ b < c $. Find $ a $.

Answer: 7561


Three. (25 points) Given that $a, b, c$ are real numbers. The functions are $y_{1} = a x^{2} + b x + c, y_{2} = a x + b (a > 0)$. When $-1 \leqslant x \leqslant 1$, it is given that $-1 \leqslant y_{1} \leqslant 1$ and $y_{2}$ has a maximum value of 2. Try to find the area of the figure formed by connecting in sequence all the lattice points (points with integer coordinates) within and on the boundary of the closed region enclosed by the parabola $y_{1} = a x^{2} + b x + c$ and the line $y_{2} = a x + b$.

Answer: 1


Given 25 containers, their capacities are $1, 2, \ldots, 25$ liters. In how many ways can ten of them be chosen such that any amount of 1 liter of water can be measured using any two of the selected containers? (An unlimited amount of water is available, and water can be transferred from one container to another.)

Translating the text into English while preserving the original line breaks and format, the result is as above.

Answer: 16


5. There are 2009 piles, each containing 2 stones. It is allowed to take the largest pile from those in which the number of stones is even (if there are several, then any of them), and move exactly half of the stones from it to any other pile. What is the maximum number of stones that can be obtained in one pile using such operations?

Answer: 2010


2. [4] 16 progamers are playing in another single elimination tournament. Each round, each of the remaining progamers plays against another and the loser is eliminated. Additionally, each time a progamer wins, he will have a ceremony to celebrate. A player's first ceremony is ten seconds long, and afterward each ceremony is ten seconds longer than the last. What is the total length in seconds of all the ceremonies over the entire tournament?

Answer: 260


Problem 10.5. Greedy Vovochka has 25 classmates. For his birthday, he brought 200 candies to the class. Vovochka's mother, to prevent him from eating them all himself, told him to distribute the candies so that any 16 of his classmates would collectively have at least 100 candies. What is the maximum number of candies Vovochka can keep for himself while fulfilling his mother's request?

Answer: 37


5. Given the function $f:\{1,2, \cdots, 10\} \rightarrow\{1,2,3,4,5\}$, and for all $k=1,2, \cdots, 9$, we have $|f(k+1)-f(k)| \geqslant 3$. Then the number of functions $f$ that satisfy this condition is $\qquad$.

Answer: 288


7. There is a figure composed of 4 regular hexagons, each with an area of 6. Using the vertices of these 4 hexagons as vertices, the number of equilateral triangles that can be formed with an area of $\mathbf{4}$ is $\qquad$.

Answer: 8


40th Putnam 1979 Problem A2 R is the reals. For what real k can we find a continuous function f : R → R such that f(f(x)) = k x 9 for all x.

Answer: k \geq 0


Example 3 Find all triples $(x, y, n), x, y, n \in \mathbf{N}^{\cdot}$ satisfying $\frac{x!+y!}{n!}=3^{n}$.

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

Answer: (1, 2, 1)


5. In the game "set," all possible four-digit numbers consisting of the digits $1,2,3$ (each digit appearing exactly once) are used. It is said that a triplet of numbers forms a set if, in each digit place, either all three numbers contain the same digit, or all three numbers contain different digits.

For example, the numbers 1232, 2213, 3221 form a set (in the first place, all three digits appear, in the second place, only the digit two appears, in the third place, all three digits appear, and in the fourth place, all three digits appear). The numbers $1123,2231,3311$ do not form a set (in the last place, two ones and a three appear).

How many sets exist in the game?

(Permuting the numbers does not create a new set: $1232,2213,3221$ and $2213,1232,3221$ are the same set.)

Answer: 1080


Anton and Britta play a game with the set $M=\left \{ 1,2,\dots,n-1 \right \}$ where $n \geq 5$ is an odd integer. In each step Anton removes a number from $M$ and puts it in his set $A$, and Britta removes a number from $M$ and puts it in her set $B$ (both $A$ and $B$ are empty to begin with). When $M$ is empty, Anton picks two distinct numbers $x_1, x_2$ from $A$ and shows them to Britta. Britta then picks two distinct numbers $y_1, y_2$ from $B$. Britta wins if
$(x_1x_2(x_1-y_1)(x_2-y_2))^{\frac{n-1}{2}}\equiv 1\mod n$
otherwise Anton wins. Find all $n$ for which Britta has a winning strategy.

Answer: n \text{ is a prime number}


Task 1. We call a collection of lines in the plane beautiful if each line in the collection intersects an odd number of the other lines in the collection.

Determine the smallest integer $k \geq 0$ with the following property: for any 2018 different lines $\ell_{1}, \ell_{2}, \ldots, \ell_{2018}$ in the plane, there exist lines $\ell_{2018+1}, \ell_{2018+2}, \ldots$, $\ell_{2018+k}$ such that the lines $\ell_{1}, \ell_{2}, \ldots, \ell_{2018+k}$ are all different and form a beautiful collection.

Answer: 1010


Let $A B C D E$ be a convex pentagon satisfying property $(P): \operatorname{area}(A B C)=\operatorname{area}(B C D)=$ area $(C D E)=\operatorname{area}(D E A)=\operatorname{area}(E A B)=1$. Calculate the area of the pentagon $A B C D E$. Is this pentagon necessarily regular?

## - Solutions of the exercises -

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


19. Circles with radii $R$ and $r$ touch each other internally. Find the side of a regular triangle, one vertex of which coincides with the point of tangency, and the other two lie on different given circles.

Answer: \dfrac{R r \sqrt{3}}{\sqrt{R^2 + r^2 - R r}}


7.3 In a pond with carp, 30 pikes were released, which began to gradually eat each other. A pike is considered full if it has eaten three other pikes (hungry or full), and each full pike ate exactly one carp for dessert (hungry pikes did not eat carp). What is the maximum number of carp that could have been eaten? Justify your answer.

Answer: 9


Let  $x = \left( 1 + \frac{1}{n}\right)^n$ and $y = \left( 1 + \frac{1}{n}\right)^{n+1}$ where $n \in \mathbb{N}$. Which one of the numbers  $x^y$, $y^x$ is bigger ?

Answer: y^x


Let $A, B$ be two variable points on a parabola $P_{0}$, such that the tangents at $A$ and $B$ are perpendicular to each other. Show that the locus of the centroid of the triangle formed by $A,B$ and the vertex of $P_0$ is a parabola $P_1 .$ Apply the same process to $P_1$ and repeat the process, obtaining the sequence of parabolas $P_1, P_2 , \ldots, P_n$. If the equation of $P_0$ is $y=m x^2$, find the equation of $P_n .$

Answer: y = 3^n m x^2 + \dfrac{1 - 3^{-n}}{4m}


The number $395$ is expressed in the form $395 = \frac{a_1!a_2!}{b_1!b_2!b_3!}$, where $a_1 \ge a_2$ and $b_1 \ge b_2 \ge b_3$ are positive integers, and $a_1 + b_1$ is as small as possible. Calculate the value of $|a_1 - b_1|$.

Answer: 1


Consider an expanded diamond-shaped grid where you need to spell the word "DIAMOND" by moving rightward, downward, or diagonally downward to the right between adjacent letters in the given diagram. For how many paths, consisting of a sequence of horizontal, vertical, or diagonal line segments, with each segment connecting a pair of adjacent letters, is the word DIAMOND spelled out as the path is traversed from beginning to end?

Answer: 64


6. (10 points) A water pool contains one-eighteenth of its capacity. Two water pipes start filling the pool simultaneously. When the water level reaches two-ninths of the pool's capacity, the first pipe continues to fill the pool alone for 81 minutes, during which it injects an amount of water equal to what the second pipe has already injected. Then, the second pipe fills the pool alone for 49 minutes, at which point the total amount of water injected by both pipes is the same. After that, both pipes continue to fill the pool. How many more minutes do the two pipes need to fill the pool together?

Answer: 231


# 6.1. Condition:

Petya thought of a natural number and wrote down the sums of each pair of its digits on the board. After that, he erased some of the sums, and the numbers $2,0,2,2$ remained on the board. What is the smallest number Petya could have thought of?

Answer: 2000


Given $H$ is the orthocenter of the acute triangle $\triangle ABC$, where $AB \neq AC$. Point $F$ is on the circumcircle of $\triangle ABC$ and is not the same as point $A$, such that $\angle AFH = 90^{\circ}$. Point $K$ is the reflection of $H$ over $B$, and point $P$ satisfies $\angle PHB = \angle PBC = 90^{\circ}$. A perpendicular line to $CP$ is drawn through point $B$, with the foot of the perpendicular being $Q$. Prove that $HQ$ is tangent to the circumcircle of $\triangle FHK$.

Answer: HQ \text{ is tangent to the circumcircle of } \triangle FHK


3. In the tetrahedron $S-ABC$, $BC=3, AC=4$, the distance from point $C$ to the midpoint of $AB$ is $\frac{5}{2}$, and the dihedral angles between the lateral faces $SAB$, $SBC$, and $SAC$ and the base are all $60^{\circ}$. Then the volume of the tetrahedron is $\qquad$

Answer: 2\sqrt{3}


3. As shown in the figure, points $M, N$ are on the sides $A B, A C$ of $\triangle A B C$, respectively, and $\overrightarrow{A M}=$ $x \overrightarrow{A B}, \overrightarrow{A N}=y \overrightarrow{A C}, D$ is the midpoint of segment $B C$, and $G$ is the intersection of segment $M N$ and $A D$. If $\overrightarrow{A G}=\frac{3}{5} \overrightarrow{A D}$, then the minimum value of $\frac{1}{x^{2}}+\frac{4}{y^{2}}$ is $\qquad$

Answer: \dfrac{85}{9}


【Question 9】
$A$ and $B$ start running around the track from the same point at the same time. If they run in the same direction, $A$ catches up with $B$ after 3 hours; if they run in opposite directions, they meet after 2 hours. The ratio of the running speeds of $A$ and $B$ is $\qquad$.

Answer: \dfrac{5}{1}


In an equilateral triangle $A B C$, an equilateral triangle $D E F$ is inscribed, see the figure. The vertices $D, E$, and $F$ lie on the sides $A B, B C$, and $A C$ such that the sides of triangle $D E F$ are perpendicular to the sides of triangle $A B C$. Furthermore, the segment $D G$ is a median in triangle $D E F$, and point $H$ is the intersection of the lines $D G$ and $B C$.

Determine the ratio of the areas of triangles $H G C$ and $B E D$.

(E. Patáková)

![](https://cdn.mathpix.com/cropped/2024_04_17_60411719b4f3e7daffe2g-2.jpg?height=617&width=646&top_left_y=548&top_left_x=705)

Answer: \dfrac{1}{4}


22. As shown in the figure, extend the sides $A B, B C, C D, D A$ of quadrilateral $A B C D$ to $E, F, G, H$, respectively, such that $\frac{B E}{A B}=\frac{C F}{B C}=\frac{D G}{C D}=\frac{A H}{D A}=\frac{4}{5}$. If the area of quadrilateral $A B C D$ is 100, then the area of quadrilateral $E F G H$ is $\qquad$

Answer: 388


$n$ players participate in a chess tournament where each player plays a game with every other player. The winner of a game receives 1 point, the loser receives 0 points, and a draw awards each player 0.5 points. A tournament is said to have property $P(m)$ if, among any $m$ players, there is one player who has defeated the other $m-1$ players, and there is one player who has lost to the other $m-1$ players. For a given $m$ ($m \geq 4$), find the minimum value of $n$, denoted by $f(m)$, such that in any tournament with property $P(m)$, all $n$ players have different scores.

Answer: 2m - 3


11. (12 points) In a newly built 5-story building, each floor of a unit has 2 apartments on the east and west sides: the room numbers on each floor are as shown in the figure. Now, five families, Zhao, Qian, Sun, Li, and Zhou, have moved in. One day, the five of them were chatting in the garden:
Zhao said: “My family is the 3rd to move in, and the 1st to move in lives right across from me.”
Qian said: “Only my family lives on the top floor.”
Sun said: “When my family moved in, the same side of the floor above and below us were already occupied.”
Li said: “My family is the last of the five to move in, and the floor below us is completely empty.”
Zhou said: “My family lives in room 106, room 104 is empty, and room 108 is also empty.”
All of what they said is true. Let the unit digits of the room numbers of the 1st, 2nd, 3rd, 4th, and 5th families to move in be $A$, $B$, $C$, $D$, and $E$ respectively. Then the five-digit number $\overline{\mathrm{ABCDE}}=$ . $\qquad$

Answer: 69573


A laboratory has flasks of two different sizes (volume $V$ and volume $V/3$) with a total of 100 flasks, with at least 2 flasks of each size. A technician randomly selects two flasks in sequence, filling the first one with a 70% salt solution and the second with a 40% salt solution. The contents of these two flasks are then mixed in one dish, and the percentage of salt in the mixture is determined. What is the minimum number of large flasks $N$ needed to ensure that the event "the percentage of salt in the dish is between 50% and 60% inclusive" occurs less frequently than the event "when two fair coins are flipped, one shows heads and the other shows tails (in any order)"? Justify your answer.

Answer: 46


Regular tetrahedron $ABCD$ has center $O$ and side length $1$. Points $A'$, $B'$, $C'$, and $D'$ are defined by reflecting $A$, $B$, $C$, and $D$ about $O$. Compute the volume of the polyhedron with vertices $ABCDA'B'C'D'$.

Answer: \dfrac{\sqrt{2}}{4}


Problem 8.6. In a $3 \times 3$ table, real numbers are arranged. It turns out that the product of the numbers in any row and any column is 10, and the product of the numbers in any $2 \times 2$ square is 3. Find the number in the central cell.

Answer: \dfrac{81}{100000}


There are $2^{2n+1}$ towns with $2n+1$ companies and each two towns are connected with airlines from one of the companies. What’s the greatest number $k$ with the following property:
We can close $k$ of the companies and their airlines in such way that we can still reach each town from any other (connected graph).

Answer: n


If a natural number has an even number of even digits, it is called a "Hope Number." For example: 26, 201, 533 are Hope Numbers, while 8, 36, 208 are not Hope Numbers. If all Hope Numbers are arranged in ascending order, what is the 2010th Hope Number?

Answer: 4019


30. There are 111 lamps, and each lamp has its own switch. It is allowed to simultaneously switch 13 of them. At the initial moment, some lamps are on, and some are off.

a) Is it possible to turn off all the lamps?

b) How many switches will be required for this if all the lamps were initially on?

Answer: Yes


Given an integer \( n > 2 \), let \( a_{1}, a_{2}, \cdots, a_{n} \) be positive real numbers satisfying \( a_{k} \leq 1 \) for \( k = 1, 2, \cdots, n \). Define \( A_{k} = \frac{a_{1} + a_{2} + \cdots + a_{k}}{k} \) for \( k = 1, 2, \cdots, n \). Prove that \( \left| \sum_{k=1}^{n} a_{k} - \sum_{k=1}^{n} A_{k} \right| < \frac{n-1}{2} \).

Answer: \left| \sum_{k=1}^{n} a_{k} - \sum_{k=1}^{n} A_{k} \right| < \frac{n-1}{2}


In a quadrilateral $A B C D$, a circle of radius 2 is inscribed. The angle $\angle D A B$ is a right angle. Side $A B$ is 5, and side $B C$ is 6. Find the area of the quadrilateral $A B C D$.

Answer: \dfrac{300}{17}


The average of the five numbers in a list is $54$ . The average of the first two
numbers is $48$ . What is the average of the last three numbers?
$\textbf{(A)}\ 55 \qquad \textbf{(B)}\ 56 \qquad \textbf{(C)}\ 57 \qquad \textbf{(D)}\ 58 \qquad \textbf{(E)}\ 59$

Answer: 58


Problem 2. A group of adventurers is showing off their loot. It is known that exactly 5 adventurers have rubies; exactly 11 have emeralds; exactly 10 have sapphires; exactly 6 have diamonds. In addition, it is known that

- if an adventurer has diamonds, then they have either emeralds or sapphires (but not both at the same time);
- if an adventurer has emeralds, then they have either rubies or diamonds (but not both at the same time).

What is the minimum number of adventurers that can be in such a group?

Answer: 16


$\underline{\text { Kanel-Belov A.Y. }}$

There is one red and $k(k>1)$ blue cells, as well as a deck of $2 n$ cards numbered from 1 to $2 n$. Initially, the entire deck lies in a random order in the red cell. From any cell, you can take the top card and move it either to an empty cell or on top of a card with a number that is one greater. For what largest $n$ can the entire deck be moved to one of the blue cells using such operations?

Answer: k-1


Task 3. On the ceiling of a room, there are 15 lamps, numbered from 1 to 15. Initially, all the lamps are off. In another room, there are 15 switches: one switch for lamp 1 and 2, one switch for lamp 2 and 3, one switch for lamp 3 and 4, and so on, up to a switch for lamp 15 and 1. When the switch for such a pair of lamps is flipped, the status of each of those two lamps changes (from on to off or vice versa). The switches are arranged in a random order and all look identical. Raymond wants to find out which switch corresponds to which pair of lamps. From the room with the switches, he cannot see the lamps. However, he can flip a number of switches and then go to the other room to see which lamps are on. He can do this multiple times. How many times does he need to go to the other room at a minimum to be certain which switch corresponds to which pair of lamps?

Answer: 4


5. In several packages, there are 20 candies, and there are no two packages with the same number of candies and no empty packages. Some packages may be inside other packages (then it is considered that a candy lying in the inner package also lies in the outer package). However, it is forbidden to place a package inside another package that already contains a package. What is the maximum possible number of packages?

Answer: 8


3. (25 points) In the sum $0^{2}+1^{2}+2^{2}+3^{2}+\cdots+$ $2005^{2}$, some of the “+” signs can be changed to “-” signs. If the resulting algebraic sum is $n$, the number $n$ is called “representable”. Which of the first 10 positive integers 1, 2, $3, \cdots, 10$ are representable? Explain your reasoning.

Answer: 1,3,5,7,9


10. (15 points) Kangaroo A and B are jumping in the area shown in the figure. A jumps in the order of $A B C D E F G H I A B C \cdots$, and B jumps in the order of $A B D E G H A B D \cdots$. If both kangaroos start from $A$ and this counts as the first time they jump together, how many times have they jumped together after 2017 jumps?

Answer: 226


A4. Twenty students go abseiling during a school trip. In each round, one student gets a turn to abseil, so after twenty rounds, everyone has safely descended. To determine who goes first in round 1, cards with numbers 1 to 20 are distributed to the students. The one who gets the 1 starts. In round 2, cards with numbers 1 to 19 are distributed to the remaining students; the student who gets the 1 is next. This continues for each round, until in round 20, only one student is left, who automatically gets a card with 1. By a bizarre coincidence, no student gets a card with the same number twice. Sara gets the card with 11 in the first round.

What result does Sara get if she adds up all the numbers on her cards?

Answer: 66


9. On the table, there are param 1 externally identical coins. It is known that among them, there are exactly param 2 counterfeit ones. You are allowed to point to any two coins and ask whether it is true that both these coins are counterfeit. What is the minimum number of questions needed to guarantee getting at least one "Yes" answer?

| param1 | param2 |  |
| :---: | :---: | :---: |
| 120 | 60 |  |

Answer: 63


2. 12 friends agree to go out for a meal once a week, each time evenly divided into 3 tables. Each table seats 4 people, until any two people have eaten at the same table at least once. How many weeks are needed at minimum?

Answer: 5


A line contains 40 characters: 20 crosses and 20 zeros. In one move, you can swap any two adjacent characters. What is the minimum number of moves required to ensure that 20 consecutive characters are crosses?

#

Answer: 200


19. The figure on the right is a square composed of Jia, Yi, Bing, and Ding, where Jia and Ding are squares with side lengths in whole centimeters, and Yi and Bing are rectangles. The sum of the areas of Jia and Ding is 100 square centimeters. Therefore, the sum of the areas of the rectangles Yi and Bing is ( ) square centimeters.

Answer: 96


10. (12 points) In the Sheep Sheep Sports Meet, Happy Sheep, Boiling Sheep, Lazy Sheep, Warm Sheep, and Big Bad Wolf participated in a 400-meter race. After the race, the five of them discussed the results.
First place said: “Happy Sheep ran faster than Lazy Sheep.”
Second place said: “I ran faster than Warm Sheep.”
Third place said: “I ran faster than Big Bad Wolf.”
Fourth place said: “Happy Sheep ran faster than Boiling Sheep.”
Fifth place said: “Warm Sheep ran faster than Big Bad Wolf.”
If only Big Bad Wolf lied among the five, then Happy Sheep got the $\qquad$th place.

Answer: 2


# 5.1. Condition:

In the warehouse, there are 8 cabinets, each containing 4 boxes, each with 10 mobile phones. The warehouse, each cabinet, and each box are locked. The manager has been tasked with retrieving 52 mobile phones. What is the minimum number of keys the manager should take with them?

Answer: 9