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

## Clues:
1. There is one house between the person who has an average height and the person who is short.
2. The person with an aquarium of fish is in the fifth house.
3. The person who has a cat is the person who is tall.
4. The person who is very tall is the person who loves soccer.
5. Peter is in the first house.
6. The person who loves tennis is in the sixth house.
7. Carol is somewhere to the right of the person who is very tall.
8. The person who loves volleyball is directly left of the person who loves soccer.
9. The person who loves baseball is directly left of the person who is super tall.
10. Carol and Bob are next to each other.
11. The person who is super tall is not in the sixth house.
12. The person who is short is not in the fourth house.
13. There is one house between the person who loves swimming and Bob.
14. The person who loves basketball is somewhere to the left of the person who owns a dog.
15. The person who owns a rabbit is somewhere to the left of the person who is short.
16. The person who keeps a pet bird is the person who is short.
17. Arnold is not in the second house.
18. Bob is directly left of Eric.
19. The person who owns a rabbit and the person who is very tall are next to each other.
 What is the value of attribute Name for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Bob`, `Peter`, `Alice`
 - Each person has a unique birthday month: `feb`, `sept`, `april`, `mar`, `jan`

## Clues:
1. The person whose birthday is in March is somewhere to the left of the person whose birthday is in January.
2. Arnold is the person whose birthday is in January.
3. The person whose birthday is in March is directly left of Alice.
4. The person whose birthday is in September and the person whose birthday is in January are next to each other.
5. Eric is not in the fourth house.
6. The person whose birthday is in January is not in the fourth house.
7. Peter is in the first house.
8. Alice is directly left of the person whose birthday is in April.
 What is the value of attribute Birthday for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: jan


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`, `Eric`, `Peter`, `Carol`, `Alice`
 - Each person has a unique hobby: `cooking`, `woodworking`, `gardening`, `knitting`, `photography`, `painting`

## Clues:
1. Alice is directly left of Bob.
2. The photography enthusiast is not in the fourth house.
3. The person who paints as a hobby and the woodworking hobbyist are next to each other.
4. The person who loves cooking is somewhere to the left of the person who enjoys knitting.
5. The woodworking hobbyist is not in the fourth house.
6. Arnold is in the fifth house.
7. There are two houses between Peter and the person who enjoys knitting.
8. The person who loves cooking is in the third house.
9. Bob is not in the third house.
10. There is one house between the woodworking hobbyist and Carol.
 What is the value of attribute Hobby for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: woodworking


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

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

Answer: tennis


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `desert`
 - They all have a unique favorite flower: `carnations`, `lilies`, `daffodils`
 - The people keep unique animals: `cat`, `horse`, `bird`
 - Each person has a unique hobby: `photography`, `cooking`, `gardening`

## Clues:
1. The person who keeps horses and the photography enthusiast are next to each other.
2. The bird keeper is the person who likes Cherry smoothies.
3. The person who loves cooking is the Desert smoothie lover.
4. The person who enjoys gardening is the person who loves a carnations arrangement.
5. The person who loves cooking is directly left of Peter.
6. The person who loves a bouquet of daffodils is the Desert smoothie lover.
7. The Watermelon smoothie lover is the person who keeps horses.
8. The photography enthusiast is Eric.
 What is the value of attribute Animal for the person whose attribute Hobby is gardening? Please reason step by step, and put your final answer within \boxed{}

Answer: horse


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`, `Alice`
 - People have unique favorite book genres: `mystery`, `fantasy`, `romance`, `science fiction`
 - Each person prefers a unique type of vacation: `cruise`, `mountain`, `city`, `beach`
 - They all have a unique favorite flower: `roses`, `daffodils`, `carnations`, `lilies`
 - People have unique favorite music genres: `pop`, `rock`, `jazz`, `classical`

## Clues:
1. Alice is the person who loves jazz music.
2. Eric is the person who loves romance books.
3. The person who loves a carnations arrangement is in the first house.
4. The person who loves a bouquet of daffodils is the person who loves science fiction books.
5. The person who enjoys mountain retreats is in the third house.
6. The person who enjoys mountain retreats is the person who loves mystery books.
7. The person who loves science fiction books is Peter.
8. The person who prefers city breaks is in the second house.
9. The person who prefers city breaks and the person who loves classical music are next to each other.
10. The person who loves beach vacations is not in the first house.
11. The person who loves romance books is the person who loves pop music.
12. The person who loves romance books is somewhere to the left of Alice.
13. The person who loves the boquet of lilies is the person who loves mystery books.
14. Peter is somewhere to the left of Alice.
 What is the value of attribute MusicGenre for the person whose attribute Vacation is cruise? Please reason step by step, and put your final answer within \boxed{}

Answer: pop


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Bob`, `Peter`
 - Each person lives in a unique style of house: `colonial`, `modern`, `ranch`, `victorian`, `craftsman`
 - People have unique favorite sports: `tennis`, `basketball`, `soccer`, `baseball`, `swimming`
 - People have unique favorite music genres: `pop`, `classical`, `jazz`, `rock`, `hip hop`

## Clues:
1. The person in a Craftsman-style house and the person who loves basketball are next to each other.
2. Alice is the person living in a colonial-style house.
3. Peter is somewhere to the right of the person who loves swimming.
4. The person living in a colonial-style house is the person who loves soccer.
5. The person in a ranch-style home is the person who loves baseball.
6. The person who loves pop music is somewhere to the left of the person who loves swimming.
7. The person who loves baseball is not in the second house.
8. The person in a modern-style house is in the fourth house.
9. The person who loves basketball and Peter are next to each other.
10. The person in a Craftsman-style house is somewhere to the left of Arnold.
11. The person who loves jazz music is not in the fourth house.
12. There are two houses between the person who loves tennis and the person in a ranch-style home.
13. Eric is not in the third house.
14. The person who loves hip-hop music is the person in a ranch-style home.
15. The person who loves tennis is the person who loves classical music.
 What is the value of attribute House for the person whose attribute MusicGenre is jazz? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


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`, `Eric`, `Peter`
 - People have unique favorite book genres: `science fiction`, `mystery`, `romance`
 - Each mother is accompanied by their child: `Bella`, `Fred`, `Meredith`
 - People have unique hair colors: `black`, `blonde`, `brown`
 - Each person has a unique level of education: `high school`, `associate`, `bachelor`
 - They all have a unique favorite flower: `daffodils`, `lilies`, `carnations`

## Clues:
1. The person who loves the boquet of lilies is directly left of the person who loves romance books.
2. Peter is the person with a high school diploma.
3. The person who loves the boquet of lilies is the person who has brown hair.
4. The person's child is named Bella is Arnold.
5. The person who loves science fiction books is the person's child is named Fred.
6. The person's child is named Fred is the person who loves the boquet of lilies.
7. The person who has black hair is the person with an associate's degree.
8. The person's child is named Meredith is not in the first house.
9. The person with a high school diploma is the person who loves a carnations arrangement.
10. Peter is not in the third house.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Peter`, `Bob`
 - People use unique phone models: `samsung galaxy s21`, `huawei p50`, `oneplus 9`, `iphone 13`, `google pixel 6`
 - Everyone has a favorite smoothie: `desert`, `dragonfruit`, `watermelon`, `cherry`, `lime`
 - The mothers' names in different houses are unique: `Janelle`, `Holly`, `Penny`, `Aniya`, `Kailyn`
 - Each person has an occupation: `doctor`, `lawyer`, `engineer`, `teacher`, `artist`

## Clues:
1. The person whose mother's name is Kailyn is Peter.
2. The Dragonfruit smoothie lover is not in the fourth house.
3. The person who uses an iPhone 13 is the person who is an artist.
4. The person whose mother's name is Aniya is Arnold.
5. The person whose mother's name is Aniya is the Desert smoothie lover.
6. The person whose mother's name is Aniya is in the second house.
7. The person who drinks Lime smoothies is in the first house.
8. The person who uses a Huawei P50 is the person who is a doctor.
9. Eric is the person who drinks Lime smoothies.
10. There is one house between the person who is a teacher and Bob.
11. The person whose mother's name is Aniya is the person who is a lawyer.
12. The person who uses an iPhone 13 is the person who likes Cherry smoothies.
13. Arnold is the person who uses a OnePlus 9.
14. The person who is a doctor is The person whose mother's name is Penny.
15. Peter is the person who uses a Samsung Galaxy S21.
16. Bob is not in the third house.
17. The person whose mother's name is Holly is directly left of Peter.
 What is the value of attribute House for the person whose attribute PhoneModel is samsung galaxy s21? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Alice`, `Bob`, `Arnold`
 - The people are of nationalities: `norwegian`, `brit`, `swede`, `dane`, `german`
 - Each person prefers a unique type of vacation: `cruise`, `mountain`, `camping`, `beach`, `city`
 - Each person has a unique level of education: `bachelor`, `master`, `associate`, `doctorate`, `high school`
 - Each person has an occupation: `artist`, `doctor`, `engineer`, `teacher`, `lawyer`

## Clues:
1. The person who likes going on cruises is the person who is a lawyer.
2. The person who loves beach vacations is directly left of Arnold.
3. The person with a doctorate is somewhere to the left of Bob.
4. The person with an associate's degree is the person who likes going on cruises.
5. Peter is not in the first house.
6. The person who is an artist is Peter.
7. The person who enjoys camping trips is the person with a master's degree.
8. The Dane is somewhere to the right of the person who is a doctor.
9. The person with an associate's degree is directly left of the person who is an engineer.
10. The person who enjoys camping trips is the British person.
11. The Norwegian and the person with a bachelor's degree are next to each other.
12. The person who is an artist is the Swedish person.
13. Bob is not in the fourth house.
14. The person who enjoys camping trips is Eric.
15. Alice is the German.
16. The person who loves beach vacations is somewhere to the left of the person who prefers city breaks.
17. The person who enjoys mountain retreats is in the fifth house.
18. The person who likes going on cruises is somewhere to the right of the person who loves beach vacations.
19. The person with a bachelor's degree is in the third house.
 What is the value of attribute Name for the person whose attribute Occupation is artist? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


Given the positive integers \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \). Any set of four numbers selected and summed results in the set \(\{44, 45, 46, 47\}\). Determine these five numbers.

Answer: 10, 11, 11, 12, 13


Let \( a, b, c \in \left[\frac{1}{2}, 1\right] \). Define \( s = \frac{a+b}{1+c} + \frac{b+c}{1+a} + \frac{c+a}{1+b} \). What is the range of possible values for \( s \)?

Answer: [2, 3]


We start with the numbers \( a, b, c, d \). We then replace them with \( a' = a - b \), \( b' = b - c \), \( c' = c - d \), \( d' = d - a \). We carry out this process 1996 times. Is it possible to end up with numbers \( A, B, C, D \) such that \( |BC - AD| \), \( |AC - BD| \), \( |AB - CD| \) are all primes?

Answer: \text{No}


Problem 4. Find the number of all natural numbers $n, 4 \leq n \leq$ 1023, such that their binary representations do not contain three consecutive equal digits.

Emil Kolev

Answer: 228


3. As shown in Figure 4, it is known that quadrilateral $ABCD$ is inscribed in $\odot O$, $AB$ is the diameter, $AD=DC$, and $BA$, $CD$ are extended to intersect at point $E$, $BF \perp EC$ intersects the extension of $EC$ at point $F$. If $AE=AO, BC=6$, then the length of $CF$ is $\qquad$.

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