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`, `Bob`, `Alice`, `Arnold`, `Carol`, `Eric`
 - The people are of nationalities: `swede`, `chinese`, `norwegian`, `german`, `brit`, `dane`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`, `chevrolet silverado`, `honda civic`, `bmw 3 series`
 - Each person has a unique favorite drink: `root beer`, `milk`, `tea`, `water`, `coffee`, `boba tea`

## Clues:
1. The boba tea drinker is the person who owns a Ford F-150.
2. The British person is directly left of Eric.
3. Alice is the person who owns a Honda Civic.
4. The tea drinker is not in the first house.
5. Eric is the Chinese.
6. The tea drinker is the person who owns a BMW 3 Series.
7. The German is in the second house.
8. The person who owns a Chevrolet Silverado is somewhere to the left of the person who likes milk.
9. Bob is in the fifth house.
10. The person who owns a Toyota Camry is directly left of the one who only drinks water.
11. The root beer lover is directly left of the Norwegian.
12. The Dane is somewhere to the right of the person who owns a BMW 3 Series.
13. Arnold is in the second house.
14. The person who owns a Honda Civic is directly left of the person who owns a Ford F-150.
15. Peter is not in the third house.
16. The person who owns a Toyota Camry is in the third house.
 What is the value of attribute House for the person whose attribute CarModel is bmw 3 series? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Carol`, `Bob`, `Peter`, `Alice`
 - Each person lives in a unique style of house: `ranch`, `victorian`, `modern`, `colonial`, `craftsman`, `mediterranean`
 - Each person has a unique favorite drink: `coffee`, `water`, `root beer`, `tea`, `milk`, `boba tea`
 - Each person has an occupation: `engineer`, `nurse`, `doctor`, `artist`, `lawyer`, `teacher`
 - The people keep unique animals: `horse`, `fish`, `dog`, `cat`, `rabbit`, `bird`
 - Each person has a favorite color: `red`, `purple`, `white`, `green`, `yellow`, `blue`

## Clues:
1. The fish enthusiast is in the first house.
2. Carol is somewhere to the left of the person in a Mediterranean-style villa.
3. The bird keeper is the person who loves purple.
4. The person in a ranch-style home is the one who only drinks water.
5. The coffee drinker is not in the fourth house.
6. The person living in a colonial-style house and the boba tea drinker are next to each other.
7. The cat lover is the person who is a teacher.
8. The person in a modern-style house is in the third house.
9. The person who is an engineer and Peter are next to each other.
10. Arnold is the person who is an artist.
11. The person whose favorite color is red is Arnold.
12. The coffee drinker is somewhere to the right of the person who is a teacher.
13. The bird keeper is the person in a ranch-style home.
14. The rabbit owner is in the sixth house.
15. The boba tea drinker is Peter.
16. Eric is the cat lover.
17. The person who is a nurse is directly left of the person in a modern-style house.
18. The bird keeper is Carol.
19. There are two houses between the cat lover and the boba tea drinker.
20. Bob is the tea drinker.
21. The person residing in a Victorian house is the dog owner.
22. The person whose favorite color is green is in the fourth house.
23. The person who loves blue is the root beer lover.
24. The person who loves white is not in the sixth house.
25. Peter is the person who is a doctor.
 What is the value of attribute Drink for the person whose attribute Occupation is nurse? Please reason step by step, and put your final answer within \boxed{}

Answer: water


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`, `Alice`, `Arnold`, `Eric`
 - The people keep unique animals: `cat`, `fish`, `bird`, `horse`
 - People have unique favorite sports: `soccer`, `swimming`, `tennis`, `basketball`
 - People have unique hair colors: `blonde`, `brown`, `black`, `red`
 - Each person has a unique type of pet: `bird`, `fish`, `dog`, `cat`

## Clues:
1. The person who loves tennis is in the second house.
2. The person who has brown hair and the person who loves basketball are next to each other.
3. Alice is the person who keeps a pet bird.
4. Arnold is the person who loves soccer.
5. The person who has blonde hair is the person with an aquarium of fish.
6. The person who keeps a pet bird is in the second house.
7. Arnold is in the third house.
8. The person who loves swimming is in the first house.
9. The person who has red hair is not in the first house.
10. The person who keeps horses is in the fourth house.
11. Eric is somewhere to the left of the fish enthusiast.
12. The person who has a cat is directly left of the bird keeper.
 What is the value of attribute FavoriteSport for the person whose attribute Pet is cat? Please reason step by step, and put your final answer within \boxed{}

Answer: swimming


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`
 - They all have a unique favorite flower: `lilies`, `roses`, `daffodils`, `carnations`
 - People have unique favorite sports: `swimming`, `tennis`, `soccer`, `basketball`
 - People have unique hair colors: `black`, `brown`, `blonde`, `red`
 - Each person has a unique birthday month: `feb`, `jan`, `april`, `sept`
 - Each person prefers a unique type of vacation: `cruise`, `city`, `mountain`, `beach`

## Clues:
1. The person who enjoys mountain retreats is in the fourth house.
2. The person whose birthday is in April and the person who loves a bouquet of daffodils are next to each other.
3. The person who likes going on cruises is the person who loves a carnations arrangement.
4. The person whose birthday is in September is the person who loves soccer.
5. Arnold is the person who loves a bouquet of daffodils.
6. Alice is in the third house.
7. The person who loves basketball is the person whose birthday is in April.
8. The person whose birthday is in February is the person who has brown hair.
9. The person who has black hair is somewhere to the left of the person who likes going on cruises.
10. The person who prefers city breaks is Eric.
11. The person who has brown hair is Arnold.
12. The person who has blonde hair is the person who loves tennis.
13. The person who loves tennis is the person who likes going on cruises.
14. The person who loves beach vacations is not in the third house.
15. The person who loves the boquet of lilies is in the first house.
 What is the value of attribute Birthday for the person whose attribute House is 4? 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: `Arnold`, `Bob`, `Peter`, `Eric`, `Alice`, `Carol`
 - Each person has a unique birthday month: `april`, `feb`, `mar`, `jan`, `may`, `sept`
 - Each person prefers a unique type of vacation: `city`, `beach`, `mountain`, `cruise`, `cultural`, `camping`
 - Each person has a favorite color: `blue`, `white`, `green`, `purple`, `red`, `yellow`

## Clues:
1. The person whose birthday is in January is the person who loves blue.
2. The person whose birthday is in January is directly left of the person who loves white.
3. The person who enjoys camping trips is in the fourth house.
4. Eric is in the sixth house.
5. There are two houses between the person who likes going on cruises and the person whose birthday is in February.
6. The person whose birthday is in May is not in the sixth house.
7. Peter is directly left of the person who loves yellow.
8. The person who goes on cultural tours is Eric.
9. The person whose birthday is in September is directly left of the person whose favorite color is green.
10. Arnold and the person who loves purple are next to each other.
11. The person who loves beach vacations is the person whose birthday is in March.
12. The person whose favorite color is green is somewhere to the left of Carol.
13. The person whose birthday is in May is somewhere to the right of Peter.
14. The person who loves blue is Bob.
15. The person who enjoys mountain retreats is in the fifth house.
16. The person who loves purple is in the third house.
17. Carol is in the fourth house.
 What is the value of attribute Birthday for the person whose attribute Vacation is beach? Please reason step by step, and put your final answer within \boxed{}

Answer: mar


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

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

Answer: Timothy


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`, `Peter`, `Eric`, `Arnold`, `Carol`, `Bob`
 - People have unique heights: `very short`, `very tall`, `short`, `super tall`, `tall`, `average`
 - Everyone has something unique for lunch: `grilled cheese`, `stir fry`, `pizza`, `spaghetti`, `soup`, `stew`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Penny`, `Aniya`, `Kailyn`, `Sarah`
 - Each person has a unique favorite drink: `tea`, `water`, `milk`, `coffee`, `boba tea`, `root beer`
 - Each person has a unique type of pet: `rabbit`, `fish`, `bird`, `hamster`, `cat`, `dog`

## Clues:
1. Arnold is the person who loves the stew.
2. The person who owns a dog is directly left of the person who has a cat.
3. The person whose mother's name is Holly is in the sixth house.
4. The tea drinker is somewhere to the left of the boba tea drinker.
5. The tea drinker is in the second house.
6. The person who is very tall is Bob.
7. The person who has a cat is The person whose mother's name is Penny.
8. The person who is very short is directly left of Alice.
9. Carol is the person who is a pizza lover.
10. The person who has an average height is not in the sixth house.
11. The person who loves stir fry is Bob.
12. The person with a pet hamster is not in the third house.
13. The person whose mother's name is Janelle is somewhere to the left of the person with a pet hamster.
14. Eric is the person who loves the soup.
15. The person whose mother's name is Kailyn is directly left of the person who keeps a pet bird.
16. The one who only drinks water is somewhere to the right of the person who likes milk.
17. The person who owns a rabbit is the person who loves the stew.
18. The person who is tall is The person whose mother's name is Aniya.
19. The person who is very short is somewhere to the left of the person who likes milk.
20. The person who is tall is not in the fifth house.
21. The person who is very short is in the second house.
22. The person who loves the stew is the root beer lover.
23. The person whose mother's name is Kailyn is the boba tea drinker.
24. The person who is tall is not in the fourth house.
25. The person who is super tall is directly left of the person who loves eating grilled cheese.
26. Carol is directly left of the root beer lover.
 What is the value of attribute Pet for the person whose attribute Height is tall? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Bob`, `Alice`
 - Each person lives in a unique style of house: `modern`, `craftsman`, `ranch`, `victorian`, `colonial`
 - The mothers' names in different houses are unique: `Penny`, `Kailyn`, `Holly`, `Janelle`, `Aniya`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `huawei p50`, `iphone 13`, `samsung galaxy s21`
 - Each person has a unique favorite drink: `coffee`, `water`, `root beer`, `tea`, `milk`
 - The people keep unique animals: `fish`, `dog`, `horse`, `bird`, `cat`

## Clues:
1. The person who uses a Google Pixel 6 is not in the first house.
2. The one who only drinks water is Alice.
3. The person living in a colonial-style house is somewhere to the right of the person who uses a Huawei P50.
4. The person who keeps horses is the person who uses a OnePlus 9.
5. The person in a ranch-style home is The person whose mother's name is Kailyn.
6. The root beer lover is the cat lover.
7. The person living in a colonial-style house is not in the fourth house.
8. The bird keeper is in the fourth house.
9. The tea drinker is Bob.
10. The tea drinker is somewhere to the right of The person whose mother's name is Kailyn.
11. The root beer lover is somewhere to the left of The person whose mother's name is Kailyn.
12. The person who keeps horses is the person in a modern-style house.
13. The person who uses an iPhone 13 is the person who likes milk.
14. The dog owner is the person who likes milk.
15. The person who uses a Google Pixel 6 is the person in a Craftsman-style house.
16. Eric is not in the second house.
17. The tea drinker is in the fourth house.
18. The person who keeps horses is in the third house.
19. The person in a modern-style house is The person whose mother's name is Penny.
20. The root beer lover is Peter.
21. The person whose mother's name is Aniya is not in the fourth house.
22. The person whose mother's name is Janelle is the one who only drinks water.
 What is the value of attribute PhoneModel for the person whose attribute HouseStyle is modern? Please reason step by step, and put your final answer within \boxed{}

Answer: oneplus 9


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Eric`
 - Each person has a unique type of pet: `bird`, `fish`, `dog`, `cat`
 - Each person lives in a unique style of house: `colonial`, `craftsman`, `ranch`, `victorian`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Kailyn`, `Aniya`
 - Each mother is accompanied by their child: `Fred`, `Samantha`, `Meredith`, `Bella`

## Clues:
1. The person whose mother's name is Kailyn is in the second house.
2. The person living in a colonial-style house is the person's child is named Meredith.
3. There is one house between the person's child is named Samantha and The person whose mother's name is Janelle.
4. The person in a ranch-style home is the person's child is named Fred.
5. Eric is the person's child is named Bella.
6. The person whose mother's name is Janelle is the person with an aquarium of fish.
7. Arnold is the person in a Craftsman-style house.
8. The person who keeps a pet bird and the person with an aquarium of fish are next to each other.
9. The person whose mother's name is Holly is directly left of Peter.
10. The person who has a cat is the person living in a colonial-style house.
11. The person's child is named Fred is somewhere to the left of The person whose mother's name is Janelle.
 What is the value of attribute Name for the person whose attribute Children is Fred? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Peter`, `Eric`
 - Each person lives in a unique style of house: `victorian`, `craftsman`, `ranch`, `colonial`
 - The people keep unique animals: `fish`, `cat`, `horse`, `bird`
 - Each person has a unique hobby: `cooking`, `painting`, `gardening`, `photography`

## Clues:
1. The person who loves cooking is in the second house.
2. The fish enthusiast is not in the first house.
3. Alice is in the second house.
4. The cat lover is the person residing in a Victorian house.
5. The person who enjoys gardening is the person residing in a Victorian house.
6. The person in a ranch-style home is not in the first house.
7. The person who keeps horses is somewhere to the right of the person residing in a Victorian house.
8. Peter is somewhere to the left of the person who paints as a hobby.
9. The person in a Craftsman-style house is Eric.
10. The person residing in a Victorian house is in the third house.
11. Peter is directly left of the person living in a colonial-style house.
 What is the value of attribute House for the person whose attribute HouseStyle is colonial? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Peter`, `Arnold`
 - Each person has a unique level of education: `master`, `bachelor`, `associate`, `high school`
 - People have unique hair colors: `black`, `red`, `brown`, `blonde`
 - Each person has a unique type of pet: `bird`, `cat`, `fish`, `dog`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `pizza`, `stew`
 - The people keep unique animals: `horse`, `cat`, `fish`, `bird`

## Clues:
1. Eric is the person with an associate's degree.
2. The person who keeps a pet bird is in the third house.
3. The person with a master's degree is not in the first house.
4. The person who has a cat is the bird keeper.
5. There is one house between the person who keeps horses and the person with an aquarium of fish.
6. The person who loves eating grilled cheese is Arnold.
7. The person who loves the stew is somewhere to the left of the person with a high school diploma.
8. The person who loves eating grilled cheese is the person with a master's degree.
9. The person who has blonde hair is somewhere to the left of the fish enthusiast.
10. The person who has black hair is directly left of the fish enthusiast.
11. The person who has black hair is the person with an associate's degree.
12. The person who has red hair is directly left of Alice.
13. Eric is the person who loves the stew.
14. The person with an aquarium of fish is the person who loves the spaghetti eater.
 What is the value of attribute Food for the person whose attribute Education is associate? Please reason step by step, and put your final answer within \boxed{}

Answer: stew


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - Each person lives in a unique style of house: `victorian`, `ranch`, `colonial`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `lilies`
 - People have unique favorite book genres: `romance`, `mystery`, `science fiction`
 - People own unique car models: `ford f150`, `toyota camry`, `tesla model 3`
 - Each person has a unique level of education: `associate`, `high school`, `bachelor`

## Clues:
1. The person who loves science fiction books is in the second house.
2. The person with a high school diploma is the person residing in a Victorian house.
3. The person who loves a bouquet of daffodils is in the second house.
4. The person with an associate's degree is the person in a ranch-style home.
5. The person who owns a Tesla Model 3 is not in the third house.
6. The person who loves the boquet of lilies is the person residing in a Victorian house.
7. Arnold is the person in a ranch-style home.
8. The person who owns a Tesla Model 3 and Eric are next to each other.
9. The person who owns a Ford F-150 is the person who loves romance books.
10. The person who owns a Tesla Model 3 is the person residing in a Victorian house.
 What is the value of attribute House for the person whose attribute HouseStyle is victorian? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


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`
 - Everyone has a unique favorite cigar: `pall mall`, `blue master`, `prince`
 - The people keep unique animals: `horse`, `cat`, `bird`
 - Each mother is accompanied by their child: `Bella`, `Fred`, `Meredith`
 - People have unique favorite book genres: `science fiction`, `romance`, `mystery`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `samsung galaxy s21`

## Clues:
1. The person who loves mystery books is the person's child is named Fred.
2. The cat lover is Eric.
3. The person partial to Pall Mall is in the second house.
4. The person who keeps horses is the person's child is named Meredith.
5. The person's child is named Bella is the Prince smoker.
6. The person who uses an iPhone 13 is directly left of the person who uses a Samsung Galaxy S21.
7. The person's child is named Fred is directly left of Arnold.
8. Peter is somewhere to the left of Eric.
9. The person who loves science fiction books is the person who uses a Samsung Galaxy S21.
10. The person who loves science fiction books is in the third house.
11. The person who loves mystery books is not in the second house.
 What is the value of attribute BookGenre for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: mystery


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`, `blue master`, `prince`, `dunhill`
 - People have unique favorite sports: `soccer`, `swimming`, `tennis`, `basketball`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `craftsman`, `victorian`
 - Everyone has a favorite smoothie: `watermelon`, `desert`, `cherry`, `dragonfruit`

## Clues:
1. The Dunhill smoker is in the first house.
2. The person who smokes Blue Master is the Watermelon smoothie lover.
3. The person in a ranch-style home is the person who loves basketball.
4. The person who loves soccer is in the second house.
5. The Dunhill smoker is the person who likes Cherry smoothies.
6. The person who loves swimming is somewhere to the right of the person who smokes Blue Master.
7. The Dragonfruit smoothie lover is Alice.
8. The Prince smoker is the Dragonfruit smoothie lover.
9. The Watermelon smoothie lover is not in the second house.
10. Arnold is the person who smokes Blue Master.
11. The person who loves swimming is the person living in a colonial-style house.
12. The person who loves swimming is Peter.
13. The person in a Craftsman-style house is directly left of the person in a ranch-style home.
 What is the value of attribute House for the person whose attribute Smoothie is cherry? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


7.98 Two people are playing a math game. Player A selects a set of 1-digit integers $x_{1}, x_{2}, \cdots, x_{n}$ as the answer, which can be positive or negative. Player B can ask: what is the sum $a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}$? Here, $a_{1}, a_{2}, \cdots, a_{n}$ can be any set of numbers. Determine the minimum number of questions Player B needs to ask to guess the answer:

Answer: 1


G7 Let $A B C$ be an isosceles triangle with $A C=B C$. The point $D$ lies on the side $A B$ such that the semicircle with diameter $[B D]$ and center $O$ is tangent to the side $A C$ in the point $P$ and intersects the side $B C$ at the point $Q$. The radius $O P$ intersects the chord $D Q$ at the point $E$ such that $5 \cdot P E=3 \cdot D E$. Find the ratio $\frac{A B}{B C}$.

Answer: \dfrac{6}{5}


## Task B-3.7.

In a right-angled triangle, the sum of the lengths of the legs and three times the length of the height on the hypotenuse is 12. The triangle rotates around the line on which the hypotenuse lies, thus forming a rotational body of maximum surface area. What is the volume of the resulting body?

Answer: \dfrac{8\pi}{3} (\sqrt{10} - 1)


Points \( O, A, B, \) and \( C \) do not lie on the same plane. Prove that the point \( X \) lies on the plane \( ABC \) if and only if \(\overrightarrow{OX} = p \overrightarrow{OA} + q \overrightarrow{OB} + r \overrightarrow{OC}\), where \( p + q + r = 1 \). Additionally, if the point \( X \) belongs to the triangle \( ABC \), then \( p : q : r = S_{BXC} : S_{CXA} : S_{AXB} \).

Answer: p : q : r = S_{BXC} : S_{CXA} : S_{AXB}


Problem 8.7.1. Given trapezoid $A B C D (B C \| A D)$. Point $H$ on side $A B$ is such that $\angle D H A=$ $90^{\circ}$. It is known that $C H=C D=13$ and $A D=19$. Find the length of segment $B C$.

![](https://cdn.mathpix.com/cropped/2024_05_06_1ea0b100610baa73554bg-10.jpg?height=412&width=579&top_left_y=309&top_left_x=437)

Answer: \dfrac{19}{2}


58. From the 205 positive integers $1,2,3, \cdots \cdots, 205$, the maximum number of integers that can be selected such that for any three selected numbers $a, b, c(a<b<c)$, we have $a b \neq c$ is $\qquad$.

Answer: 193


Let $F$ be the left focus of the hyperbola $\Gamma: x^{2}-y^{2}=1$. A line passing through $F$ intersects $\Gamma$ at points $M$ and $N$.
(1) If both $M$ and $N$ are on the left branch of the hyperbola, find the minimum area of the triangle $\triangle OMN$;
(2) Does there exist a point $P$ on the $x$-axis such that $\overrightarrow{PM} \cdot \overrightarrow{PN}$ is constant? If such a point $P$ exists, find its coordinates; if not, explain why.

Answer: \sqrt{2}


A driver is transporting milk in a cylindrical tank. The diameter of the base is $180 \mathrm{~cm}$, and the length of the tank is $4 \mathrm{~m}$. How many hectoliters of milk are in the tank if it is filled to three-quarters of the diameter?

![](https://cdn.mathpix.com/cropped/2024_04_17_2a0500418bf8c778174dg-5.jpg?height=337&width=920&top_left_y=1051&top_left_x=608)

(M. Krejčová)

Answer: 82


In the following figure, we have a $2 \times 2 \times 2$ cube made from wire pieces. The edge of each small cube is a piece of $1 \text{ cm}$ wire, and a total of 54 such pieces were used. For constructing a $10 \times 10 \times 10$ cube, how many pieces of wire will be used?

Answer: 3630


381. Given a triangle $A B C$ and a point $O$ inside it. Construct a segment with its midpoint at point $O$, the ends of which lie on the boundary of triangle $A B C$. What is the maximum number of solutions this problem can have?

Answer: 3


8.5. The numbers from 1 to 10 were written in some order and resulted in the numbers \(a_{1}, a_{2}, a_{3}, \ldots, a_{10}\), and then the sums \(S_{1}=a_{1}\), \(S_{2}=a_{1}+a_{2}\), \(S_{3}=a_{1}+a_{2}+a_{3}\), \ldots, \(S_{10}=a_{1}+a_{2}+a_{3}+\ldots+a_{10}\) were calculated. What is the maximum number of prime numbers that could be among the numbers \(S_{1}, S_{2}\), \(S_{3}, \ldots, S_{10}\)?

Answer: 7


2.2. $A L, B M, C N$ - medians of triangle $A B C$, intersecting at point $K$. It is known that quadrilateral $C L K M$ is cyclic, and $A B=2$. Find the length of median $C N$.

Answer: \sqrt{3}


4. Let $Z$ be a complex number, $A, B, C$ be real numbers, $A<0$, and let $f(Z)=A|Z|^{2}+2 B \cdot \operatorname{Re} Z+C$ (where $\operatorname{Re} Z$ denotes the real part of $Z$), then when $Z=$ $\qquad$, $f(Z)$ has the most $\qquad$ value $\qquad$ .

Answer: -\dfrac{B}{A}


Five identical circles are placed in a line inside a larger one as shown. If the shown chord has length $16,$ find the radius of the large circle.

Answer: 8


7. Given that the 6027-digit number $\frac{a b c a b c \cdots a b c}{2000 \uparrow a b c}$ is a multiple of 91. Find the sum of the minimum and maximum values of the three-digit number $\overline{a b c}$.

Answer: 1092


A rectangular pyramid has a base with a length of 20 cm and a width of 10 cm, and its altitude is 12 cm. A cut is made parallel to the base of the pyramid 6 cm from the apex, creating a smaller pyramid and a frustum. The area of the top rectangle of the frustum is one-fourth the area of the base of the original pyramid. Calculate the volume of the frustum.

Answer: 700 \text{ cm}^3


Let $g(x) = \cos{x} + 3\sin{x} + 4\cot{x}$, using radian measure for the variable $x$. Let $s$ be the smallest positive value of $x$ for which $g(x) = 0$. Find $\lfloor s \rfloor$.

Answer: 2


After Jack makes 30 shots, he has scored $60\%$ of them. He then takes 10 more shots, and his shooting percentage increases to $62\%$. How many of the last 10 shots did Jack make?

Answer: 7


Let $Q(x) = x^4 - 4x^3 - 20x^2 + 44x + 120$. A real number $x$ is chosen at random from the interval $3 \le x \le 10$. Calculate the probability that $\lfloor \sqrt[4]{Q(x)} \rfloor = \sqrt[4]{Q(\lfloor x \rfloor)}$.

Answer: 0


A butcher needs to know the weight in pounds of a steer that weighs 350 kg for record-keeping, using the conversion rate of 0.454 kilograms per pound. Round the result to the nearest whole pound.

Answer: 771 \text{ pounds}


The modified Fibonacci sequence $F_n$ is defined by $F_0 = 4, F_1 = 3$, and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 2$. Find the units digit of $F_{F_{10}}$.

Answer: 3


The local theater has two ticket windows operating. In how many ways can six people line up and choose exactly one ticket window each to buy a ticket?

Answer: 46080


Cara is sitting at a circular table with her seven friends. How many different possible pairs of people could Cara be sitting between if one specific friend, named Alex, must always be one of the neighbors?

Answer: 6


A cube is painted so that one side is black, two sides are yellow, and three sides are white. Determine how many different ways this cube can be painted, considering two cubes identical if one can be rotated to match the other.

Answer: 3


At the school cafeteria, Michael wants to buy a lunch consisting of one main course, one beverage, and one snack. The table below lists Michael's options in the cafeteria. However, due to dietary restrictions, if Michael chooses a meat-based main course, he cannot select a soda as his beverage. How many distinct possible lunches can he buy from these options?

\begin{tabular}{ |c | c | c | }
\hline \textbf{Main Courses} & \textbf{Beverages}&\textbf{Snacks} \\ \hline
Hamburger & Water & Apple Pie \\ \hline
Veggie Burger & Soda & Fruit Cup \\ \hline
Chicken Sandwich& & \\ \hline
Pasta & & \\ \hline
\end{tabular}

Answer: 12


Let $a$, $b$, $c$, $d$, $e$, and $f$ be integers such that $729x^3 + 64 = (ax^2 + bx + c)(dx^2 + ex + f)$ holds for all $x$. Compute $a^2 + b^2 + c^2 + d^2 + e^2 + f^2$.

Answer: 8210


Let triangle $PQR$ be a right triangle in the xy-plane with a right angle at $R$. The lengths of legs $PR$ and $RQ$ are given as 24 and 73 respectively, and $PQ$ (hypotenuse) becomes 75. The median through $P$ lies along the line $y = 3x + 4$ and the median through $Q$ lies along the line $y = -x + 5$. Find the area of triangle $PQR$.

Answer: 876


2. Each of the 10 people is either a knight, who always tells the truth, or a liar, who always lies. Each of them thought of some natural number. Then the first said: “My number is greater than 1”, the second said: “My number is greater than 2”, ..., the tenth said: “My number is greater than 10”. After that, they, in a different order, said (each one phrase): “My number is less than 1”, “My number is less than 2”, ..., “My number is less than 10”. What is the maximum number of knights that could have been among these 10 people? (O. Podlipsky)

Answer: 8


18. As shown in the figure, $A B C D$ is a square with a side length of $\mathbf{6}$, $A D G H$ is a trapezoid, points $E$ and $F$ are the midpoints of $A D$ and $G H$ respectively, $H F=6, E F=4, E F \perp G H$. Connect $H E$ and extend it to intersect $C D$ at point $I$, draw $I J \perp H A$, then $I J=$ $\qquad$ -

Answer: \dfrac{18}{5}


2A. For the positive real numbers $x$ and $y$, the equality

$$
x^{2}+y^{2}=6 x y
$$

is satisfied. Calculate the value of the expression $\frac{x+y}{x-y}$.

Answer: \sqrt{2}


14.5.6 * 8 cents and 15 cents stamps can be used without limit. Some postage amounts, such as 7 cents, 29 cents, cannot be made up exactly. Find the largest amount $n$ that cannot be made up, and prove your conclusion.

Answer: 97


In a triangle $ABC$, the median $AD$ (with $D$ on $BC$) and the angle bisector $BE$ (with $E$ on $AC$) are perpedicular to each other. If $AD = 7$ and $BE = 9$, find the integer nearest to the area of triangle $ABC$.

Answer: 47


【Question 11】
Wang Wei walks from place A to place B, while at the same time, Zhang Ming rides a bicycle from place B to place A. Half an hour later, the two meet on the way. After Zhang Ming arrives at place A, he immediately returns to place B, and 20 minutes after the first meeting, he catches up with Wang Wei again. After Zhang Ming reaches place B, he turns back again, and the two meet for the third time $\qquad$ minutes after the second meeting.

Answer: 40


## Zadatak B-1.7.

Dva pravokutna trokuta $P S T$ i $R S T$ imaju zajedničku hipotenuzu $\overline{S T}$. Vrhovi $P$ i $R$ nalaze se s iste strane hipotenuze $\overline{S T}$. Katete $\overline{P T}$ i $\overline{R S}$ sijeku se u točki $Q$. Ako je $|P S|=6 \mathrm{~cm},|P T|=17 \mathrm{~cm}$ i $|R T|=1 \mathrm{~cm}$, kolika je površina trokuta $Q S T$ ?

Answer: \dfrac{39}{5}


Let \(n \geq 4\) and \(k, d \geq 2\) be natural numbers with \(k \cdot d \leq n\). The \(n\) participants in the Mathematical Olympiad are sitting around a round table. Patrick needs to select \(k\) of the \(n\) participants to remain, ensuring there are at least \(d-1\) empty seats between every two of the remaining \(k\) participants. How many ways can Patrick do this, assuming all seats were initially occupied?

Answer: \dfrac{n}{k} \dbinom{n - k(d - 1) - 1}{k - 1}


4 $[\quad$ Equifacetal Tetrahedron $\quad]$

In the triangular pyramid $A B C D$, the sums of the three dihedral angles at each of the vertices $B$ and $C$ are $180^{\circ}$ and $A D=B C$. Find the volume of the pyramid if the area of the face $B C D$ is 100, and the distance from the center of the circumscribed sphere to the plane of the base $A B C$ is 3.

Answer: 400


Robot "Mag-o-matic" manipulates $101$ glasses, displaced in a row whose positions are numbered from $1$ to $101$. In each glass you can find a ball or not. Mag-o-matic only accepts elementary instructions of the form $(a;b,c)$, which it interprets as
"consider the glass in position $a$: if it contains a ball, then switch the glasses in positions $b$ and $c$ (together with their own content), otherwise move on to the following instruction"
(it means that $a,\,b,\,c$ are integers between $1$ and $101$, with $b$ and $c$ different from each other but not necessarily different from $a$). A $\emph{programme}$ is a finite sequence of elementary instructions, assigned at the beginning, that Mag-o-matic does one by one.
A subset $S\subseteq \{0,\,1,\,2,\dots,\,101\}$ is said to be $\emph{identifiable}$ if there exists a programme which, starting from any initial configuration, produces a final configuration in which the glass in position $1$ contains a ball if and only if the number of glasses containing a ball is an element of $S$.
(a) Prove that the subset $S$ of the odd numbers is identifiable.
(b) Determine all subsets $S$ that are identifiable.

Answer: S \text{ is the set of odd numbers}


A positive integer $n$ is called $\textit{good}$ if $2 \mid \tau(n)$ and if its divisors are  $$1=d_1<d_2<\ldots<d_{2k-1}<d_{2k}=n, $$ then $d_{k+1}-d_k=2$ and $d_{k+2}-d_{k-1}=65$. Find the smallest $\textit{good}$ number.

Answer: 2024


2. Two circles with radii $R$ and $r$ touch at the point $M$. On the circle with radius $r$, there is a point $N$, diametrically opposite to $M$, and a tangent is drawn at this point. Find the radius of the circle that touches both circles and the tangent passing through point $N$.

Answer: \dfrac{r(R + r)}{R}


## Task 4 - 221214

Determine all real numbers $x$ that have the property that of the following statements (A) to (F), four are true and two are false!

(A) $x$ is a positive rational number.

(B) $x$ is a natural number, or $x$ can be represented in the form $x=\frac{1}{g}$ with an integer $g \neq 0$.

(C) $x^{2}$ is an integer, but $x$ itself is not an integer.

(D) $7 < x^{2} < 9$.

(E) $x$ is a positive real number, but not a natural number.

(F) If $x$ is rational, then $x$ is an integer.

Hint: A statement of the form "If $p$, then $q$ " is true if and only if the statement "(not $p$) or $q$ " is true. A statement of the form " $u$ or $v$ " is true if and only if at least one of the two sub-statements $u$ and $v$ is true.

Answer: 2\sqrt{2}


The altitudes of the acute-angled triangle $ABC$, drawn from points $B$ and $C$, were extended to intersect the circumscribed circle at points $B_1$ and $C_1$. It turned out that the segment $B_1 C_1$ passes through the center of the circumscribed circle. Find the angle $BAC$.

#

Answer: 45^\circ


In a $12\times 12$ square table some stones are placed in the cells with at most one stone per cell. If the number of stones on each line, column, and diagonal is even, what is the maximum number of the stones?

[b]Note[/b]. Each diagonal is parallel to one of two main diagonals of the table and consists of $1,2\ldots,11$ or $12$ cells.

Answer: 120


18. Find all integers $p$ for which there exist rational numbers $a$ and $b$, such that the polynomial $x^{5}-p x-1$ has at least one common root with a polynomial $x^{2}-a x+b$.

Answer: -1


8-8. A chess tournament is held according to the following system: each of the 15 students from the "White Rook" school must play one game with each of the 20 students from the "Black Bishop" school, i.e., a total of 300 games should be played. At any given time, no more than one game is played.

After $n$ games, a spectator named Sasha, who has watched all the games and knows all the participants, exclaims: "I can definitely name one of the participants of the next game!" What is the smallest $n$ for which this could happen?

Answer: 280


2. In the cells of an $11 \times 11$ square, zeros and ones are arranged in such a way that in any figure of four cells in the shape of $\square$, the sum of the numbers is odd. (The figure can be rotated and flipped). What is the minimum number of ones that can be in such an arrangement?

Answer: 25


[

The Law of Cosines

[Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part

A tangent $A M$ (where $M$ is the point of tangency) is drawn from point $A$ to a circle with radius $R$. A secant passing through point $A$ intersects the circle at points $K$ and $L$, with $L$ being the midpoint of segment $A K$, and the angle $A M K$ is $60^{\circ}$. Find the area of triangle $A M K$.

#

Answer: \dfrac{3R^2}{8} \left( \sqrt{15} - \sqrt{3} \right)


Problem 8.3. In triangle $ABC$, the sides $AC=14$ and $AB=6$ are known. A circle with center $O$, constructed on side $AC$ as the diameter, intersects side $BC$ at point $K$. It turns out that $\angle BAK = \angle ACB$. Find the area of triangle $BOC$.

![](https://cdn.mathpix.com/cropped/2024_05_06_899cf5197845501d962eg-24.jpg?height=378&width=422&top_left_y=1434&top_left_x=512)

Answer: 21


6.31 Team A and Team B each send out 7 players to participate in a Go relay match according to a pre-arranged order. Both sides start with the No. 1 player competing, the loser is eliminated, and the winner then competes with the No. 2 player from the losing side, $\cdots$, until all players from one side are eliminated, and the other side wins, forming a match process. How many different possible match processes are there?

Answer: 3432


Example 4 Solve the equation $y^{4}-6 y^{3}+10 y^{2}-4 y=0$.

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

Example 4 Solve the equation $y^{4}-6 y^{3}+10 y^{2}-4 y=0$.

Answer: y = 0, \quad y = 2, \quad y = 2 + \sqrt{2}, \quad y = 2 - \sqrt{2}


The points \(A(-5,0), B(-5,7), C(-1,7)\), and \(D(-1,4)\) are graphed, as shown. The line \(L\) has equation \(x=3\) and point \(Q\) is the reflection of point \(C\) in \(L\).

1. Point \(P\) is the reflection of point \(D\) in \(L\). Point \(R\) is the reflection of point \(B\) in \(L\). Point \(S\) is the reflection of point \(A\) in \(L\). State the coordinates of \(P, R\) and \(S\).
2. Calculate the perimeter of the figure \(ABCDPQRS\).
3. Figure \(ABCDPQRS\) is rotated around the line \(L\) to create a three-dimensional solid. This solid is a larger cylinder with a smaller cylinder removed. Determine the volume and the surface area of this solid. (Note: A cylinder with radius \(r\) and height \(h\) has volume \(\pi r^{2} h\) and surface area \(2 \pi r^{2} + 2 \pi r h\)).

Answer: 52


26th Putnam 1965 Problem B1 X is the unit n-cube, [0, 1] n . Let k n = ∫ X cos 2 ( π(x 1 + x 2 + ... + x n )/(2n) ) dx 1 ... dx n . What is lim n→∞ k n ?

Answer: \dfrac{1}{2}


Given the parabola $C: y^{2}-2 p x(p>0)$, and a fixed point $A\left(\frac{p}{2}, p\right)$, does there exist a point $T$ in the coordinate plane such that any line $l$ passing through $T$ intersects the parabola $C$ at points $B$ and $C$, and as long as $A$, $B$, and $C$ are not collinear, $\triangle A B C$ can always form a right triangle with $\angle B A C$ as the right angle? If such a point exists, find all such points.

Answer: \left( \dfrac{5p}{2}, -p \right)


4. If 4 dice are thrown at random, the probability that the sum of the numbers on two of them is 9 is $\qquad$ .

Answer: \dfrac{145}{324}


Let $p, q \in \mathbf{R}_{+}, x \in\left(0, \frac{\pi}{2}\right)$. Find the minimum value of the function $f(x)=\frac{p}{\sqrt{\sin x}}+\frac{q}{\sqrt{\cos x}}$.

Answer: \left( p^{4/5} + q^{4/5} \right)^{5/4}


(12) (20 points) Given the ellipse $\frac{x^{2}}{4}+y^{2}=1$, and $P$ is any point on the circle $x^{2}+y^{2}=16$. Draw the tangents $P A$ and $P B$ to the ellipse from $P$, with the points of tangency being $A$ and $B$, respectively. Find the maximum and minimum values of $\overrightarrow{P A} \cdot \overrightarrow{P B}$.

Answer: \dfrac{165}{16}


Problem 6.3. A country has the shape of a square and is divided into 25 identical square counties. In each county, either a knight-count, who always tells the truth, or a liar-count, who always lies, rules. One day, each count said: "Among my neighbors, there are an equal number of knights and liars." What is the maximum number of knights that could have been? (Counts are neighbors if their counties share a common side.)

Answer: 8


Answer: 2


53. A five-digit positive odd number $x$, replace all 2s in $x$ with 5s, and all 5s with 2s, keeping other digits unchanged, to get a new five-digit number $y$. If $x, y$ satisfy $y=2(x+1)$, then $x$ is $\qquad$.

Answer: 29995


Ex. 103. A diameter divides a circle into two parts, one of which contains a smaller inscribed circle touching the larger circle at point $M$, and the diameter at point $K$. The ray $MK$ intersects the larger circle a second time at point

![](https://cdn.mathpix.com/cropped/2024_05_21_91ff4e46083e03d62f1eg-17.jpg?height=414&width=423&top_left_y=1278&top_left_x=1276)
$N$. Find the length of $MN$, if the sum of the distances from point $M$ to the ends of the diameter is 6.

Answer: 3\sqrt{2}


Three, (25 points) Given that $a$ and $b$ are integers, and satisfy $a-b$ is a prime number, $ab$ is a perfect square. If $a \geqslant 2011$, find the minimum value of $a$.

Answer: 2025


LI OM - III - Task 5

For a given natural number $ n \geq 2 $, find the smallest number $ k $ with the following property. From any $ k $-element set of fields of the $ n \times n $ chessboard, one can select a non-empty subset such that the number of fields of this subset in each row and each column of the chessboard is even.

Answer: 2n


Aurumban the value of worked gold items is proportional to the square of their mass. Thieves steal a gold item worth 100 petaks, and from this they make identical medals whose total value is 10 petaks. A jeweler buys the medals and from them (not necessarily of the same mass) makes necklaces in such a way that an integer number of medals are used for each necklace. The total value of the necklaces is 46 petaks. How much are the individual necklaces worth?

Answer: 1


1. Inside triangle $A B C$, a point $M$ is taken such that $\angle C M B=100^{\circ}$. The perpendicular bisectors of $B M$ and $C M$ intersect sides $A B$ and $A C$ at points $P$ and $Q$, respectively. Points $P, Q$, and $M$ lie on the same line. Find the measure of $\angle C A B$.

Answer: 20^\circ


8. In an $m$ row by 10 column grid, fill each cell with either 0 or 1, such that each column contains exactly three 1s. Let the sum of the numbers in the $i$-th row ($i=1,2, \cdots, m$) be denoted as $x_{i}$, and for any two columns, there always exists a row where the cells intersecting with these two columns are both filled with 1. Let $x=\max _{1 \leqslant i \leqslant m}\left\{x_{i}\right\}$. Then $x_{\text {min }}=$ $\qquad$

Answer: 5


81. Solve the equation $y^{\prime}+y \tan x=\cos ^{2} x$.

Answer: y = \cos x (\sin x + C)


2. Circles $k(S ; 6 \mathrm{~cm})$ and $l(O ; 4 \mathrm{~cm})$ have an internal tangency at point $B$. Determine the lengths of the sides of triangle $A B C$, where point $A$ is the intersection of line $O B$ with circle $k$ and point $C$ is the intersection of circle $k$ with the tangent from point $A$ to circle $l$.

Answer: 12


The number 1089 has an interesting property. When we multiply this number by 9, as shown below,

| 1 | 0 | 8 | 9 |
| :--- | :--- | :--- | :--- |
|  |  | $\times$ | 9 |
| 9 | 8 | 0 | 1 |

we get the number 9801, which is the number 1089 with its digits written from left to right!

a) Find a five-digit number $A B C D E$ such that its multiplication by 9 is equal to the number that has the digits of $A B C D E$ written from right to left, that is,

| $A$ | $B$ | $C$ | $D$ | $E$ |
| :---: | :---: | :---: | :---: | :---: |
|  |  |  | $\times$ | 9 |
| $E$ | $D$ | $C$ | $B$ | $A$ |

b) Find all the seven-digit numbers whose multiplication by 9, as before, reverses the position of their digits.

Answer: 10989


A bubble jumps between the vertices of the square $A B C D$, starting from vertex $A$. With each jump, it has a $\frac{1}{2}-\frac{1}{2}$ probability of landing on one of the adjacent vertices. The bubble stops when it reaches the last vertex it hasn't visited yet. Determine the probability that each vertex will be the last one visited.

Answer: \dfrac{1}{3}


We take turns rolling a fair die until the sum $S$ of the numbers obtained exceeds 100. What is the most likely value of $S$?

Answer: 101


5. (3 points) Three circles with radii 1, 1, and \(2 \sqrt{\frac{13-6 \sqrt{3}}{13}}\) are arranged such that the triangle formed by their centers is equilateral with a side length of \(\sqrt{3}\). Find the radius of the circumcircle of the triangle, each vertex of which is the point of intersection of two of these circles, farthest from the center of the third circle.

Answer: \(4 \sqrt{3}-6\)

Answer: 4\sqrt{3} - 6


There are \( n \) boys and \( n \) girls in a class (\( n \geq 3 \)). They are seated around a circular table in such a way that no two boys and no two girls sit next to each other. The teacher has \( 2n \) cards, numbered \( 1, 2, 3, \ldots, 2n \), each number appearing exactly once. The teacher distributes one card to each student such that the number on any girl's card is greater than the number on any boy's card. Then, each girl writes down the sum of the numbers on three cards: her own and those of the boys sitting adjacent to her. For which values of \( n \) could all the sums written down by the \( n \) girls be equal?

Answer: n \text{ is odd}


Yang has a standard $6$-sided die, a standard $8$-sided die, and a standard $10$-sided die. He tosses these three dice simultaneously. The probability that the three numbers that show up form the side lengths of a right triangle can be expressed as $\frac{m}{n}$, for relatively prime positive integers $m$ and $n$. Find $100m+n$.

[i]Proposed by Yannick Yao[/i]

Answer: 1180


6. Four spheres are placed inside a cone, each touching each other (externally) and the lateral surface of the cone. Three of the spheres have a radius of 3 and also touch the base of the cone. Find the radius of the fourth sphere if the angle between the slant height and the base of the cone is $\frac{\pi}{3}$.

Answer: 9 - 4\sqrt{2}


4. In $\triangle A B C$, it is known that the three interior angles $\angle A, \angle B, \angle C$ form an arithmetic sequence, with their opposite sides being $a, b, c$ respectively, and $c-a$ equals the height $h$ from vertex $A$ to side $AC$. Then $\sin \frac{C-A}{2}=$

Answer: \dfrac{1}{2}


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`
 - The people are of nationalities: `brit`, `swede`, `norwegian`, `dane`
 - They all have a unique favorite flower: `carnations`, `lilies`, `roses`, `daffodils`
 - Each person has a unique level of education: `high school`, `bachelor`, `master`, `associate`

## Clues:
1. The Norwegian is directly left of the person who loves the rose bouquet.
2. The person with a high school diploma is Eric.
3. Arnold is not in the first house.
4. The person with an associate's degree is the person who loves a carnations arrangement.
5. The Dane is not in the third house.
6. Peter is the person who loves the rose bouquet.
7. The Swedish person and the Norwegian are next to each other.
8. Eric is in the second house.
9. The person who loves the boquet of lilies is the person with a master's degree.
10. The person who loves the boquet of lilies is the Swedish person.
 What is the value of attribute House for the person whose attribute Flower is carnations? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 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`, `Peter`, `Bob`
 - The people are of nationalities: `german`, `norwegian`, `brit`, `dane`, `swede`
 - People have unique favorite sports: `swimming`, `soccer`, `baseball`, `basketball`, `tennis`
 - People have unique hair colors: `blonde`, `brown`, `red`, `black`, `gray`

## Clues:
1. Arnold is the person who loves baseball.
2. The Swedish person is somewhere to the left of Arnold.
3. There are two houses between the person who loves basketball and the person who has black hair.
4. The Norwegian is in the fourth house.
5. Peter is the person who has black hair.
6. The person who has red hair is in the second house.
7. Bob is the German.
8. The person who has blonde hair is not in the third house.
9. The person who loves swimming is the person who has red hair.
10. The German is not in the first house.
11. The person who loves basketball is directly left of the Dane.
12. The person who loves soccer is directly left of Arnold.
13. Alice is the person who has gray hair.
 What is the value of attribute Nationality for the person whose attribute HairColor is brown? Please reason step by step, and put your final answer within \boxed{}

Answer: german


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Arnold`, `Peter`, `Eric`
 - The mothers' names in different houses are unique: `Penny`, `Aniya`, `Kailyn`, `Janelle`, `Holly`
 - The people keep unique animals: `cat`, `bird`, `dog`, `fish`, `horse`
 - Everyone has a favorite smoothie: `desert`, `dragonfruit`, `cherry`, `watermelon`, `lime`
 - Each person has an occupation: `doctor`, `engineer`, `artist`, `teacher`, `lawyer`

## Clues:
1. Bob is the Desert smoothie lover.
2. The cat lover is the person who is an engineer.
3. The person whose mother's name is Penny is the bird keeper.
4. The person who is a lawyer is the fish enthusiast.
5. The person who likes Cherry smoothies is directly left of Alice.
6. The Watermelon smoothie lover and the person who is an engineer are next to each other.
7. The person whose mother's name is Holly is the dog owner.
8. The person who is a teacher is in the first house.
9. The person whose mother's name is Penny is in the second house.
10. The person who drinks Lime smoothies is the person who is a doctor.
11. The person whose mother's name is Janelle is the person who is a lawyer.
12. Arnold is not in the third house.
13. The Dragonfruit smoothie lover is directly left of Peter.
14. The person whose mother's name is Kailyn is the Dragonfruit smoothie lover.
15. Arnold is somewhere to the right of Peter.
16. The dog owner is not in the fifth house.
17. The Watermelon smoothie lover is somewhere to the left of the Desert smoothie lover.
 What is the value of attribute Name for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Carol`, `Peter`, `Bob`, `Eric`
 - The people keep unique animals: `fish`, `bird`, `rabbit`, `cat`, `horse`, `dog`
 - Each person has a unique favorite drink: `tea`, `root beer`, `water`, `milk`, `boba tea`, `coffee`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `huawei p50`, `xiaomi mi 11`, `oneplus 9`
 - Each mother is accompanied by their child: `Alice`, `Fred`, `Meredith`, `Samantha`, `Timothy`, `Bella`

## Clues:
1. The tea drinker is the person who uses an iPhone 13.
2. There is one house between the dog owner and the tea drinker.
3. The fish enthusiast is somewhere to the left of the tea drinker.
4. Eric and the person's child is named Alice are next to each other.
5. The one who only drinks water is not in the third house.
6. The root beer lover is directly left of the person's child is named Samantha.
7. The person who is the mother of Timothy is the coffee drinker.
8. The fish enthusiast is directly left of the person who likes milk.
9. Peter is the person who likes milk.
10. The person who uses a OnePlus 9 is directly left of the person's child is named Alice.
11. The person who keeps horses is not in the third house.
12. The person's child is named Meredith is directly left of Bob.
13. The person's child is named Bella is the person who uses a Google Pixel 6.
14. The bird keeper is directly left of the person's child is named Meredith.
15. The person who uses a Samsung Galaxy S21 is directly left of the person's child is named Bella.
16. The one who only drinks water is somewhere to the left of the person's child is named Fred.
17. Arnold is the tea drinker.
18. The person who keeps horses is somewhere to the left of the cat lover.
19. The person who uses a Huawei P50 and Alice are next to each other.
20. Bob is directly left of the root beer lover.
21. The person who is the mother of Timothy is Eric.
 What is the value of attribute Drink for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}

Answer: root beer


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`, `Alice`, `Bob`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `cruise`, `city`, `camping`
 - Each person lives in a unique style of house: `ranch`, `modern`, `craftsman`, `colonial`, `victorian`
 - Each person has a unique level of education: `bachelor`, `doctorate`, `master`, `associate`, `high school`
 - The mothers' names in different houses are unique: `Aniya`, `Kailyn`, `Janelle`, `Penny`, `Holly`

## Clues:
1. The person whose mother's name is Aniya is the person in a ranch-style home.
2. The person in a Craftsman-style house is Bob.
3. The person with a high school diploma is not in the fourth house.
4. Arnold and the person in a Craftsman-style house are next to each other.
5. Arnold is The person whose mother's name is Penny.
6. The person with an associate's degree is somewhere to the left of the person with a master's degree.
7. Bob and Peter are next to each other.
8. The person living in a colonial-style house is The person whose mother's name is Penny.
9. The person whose mother's name is Holly is the person who enjoys camping trips.
10. The person with a doctorate is the person who prefers city breaks.
11. The person with a bachelor's degree is The person whose mother's name is Aniya.
12. The person with an associate's degree is somewhere to the right of Eric.
13. Arnold is directly left of the person who likes going on cruises.
14. The person who enjoys camping trips is somewhere to the right of the person residing in a Victorian house.
15. The person who enjoys mountain retreats is in the first house.
16. The person whose mother's name is Janelle is the person with an associate's degree.
17. The person with an associate's degree is directly left of Arnold.
 What is the value of attribute Mother for the person whose attribute Education is associate? Please reason step by step, and put your final answer within \boxed{}

Answer: Janelle


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Peter`
 - People have unique favorite sports: `swimming`, `soccer`, `tennis`, `basketball`
 - Each person has a unique type of pet: `bird`, `fish`, `cat`, `dog`
 - Everyone has a favorite smoothie: `dragonfruit`, `desert`, `watermelon`, `cherry`
 - People have unique hair colors: `brown`, `black`, `blonde`, `red`
 - Each person has a unique birthday month: `jan`, `feb`, `sept`, `april`

## Clues:
1. The person whose birthday is in September is the person who likes Cherry smoothies.
2. The person whose birthday is in February is not in the first house.
3. The Desert smoothie lover is the person who loves basketball.
4. Peter is the person who has black hair.
5. The person who loves tennis is somewhere to the right of the person who has a cat.
6. The person with an aquarium of fish is not in the fourth house.
7. Arnold is directly left of the person who has blonde hair.
8. There is one house between the person who has red hair and the person whose birthday is in January.
9. The person whose birthday is in September is directly left of the person who loves soccer.
10. Peter is directly left of the person who loves basketball.
11. The person who has blonde hair is the person whose birthday is in September.
12. Arnold is the person who keeps a pet bird.
13. The person whose birthday is in January is the Watermelon smoothie lover.
14. The person who has blonde hair is Eric.
 What is the value of attribute Birthday for the person whose attribute FavoriteSport is tennis? 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: `Arnold`, `Eric`, `Carol`, `Bob`, `Alice`, `Peter`
 - People have unique heights: `short`, `very short`, `very tall`, `tall`, `average`, `super tall`
 - Each person has a unique type of pet: `rabbit`, `bird`, `cat`, `fish`, `hamster`, `dog`
 - People use unique phone models: `huawei p50`, `google pixel 6`, `iphone 13`, `samsung galaxy s21`, `xiaomi mi 11`, `oneplus 9`

## Clues:
1. The person who uses a Google Pixel 6 is somewhere to the right of the person who is very short.
2. The person who has a cat is the person who uses a Google Pixel 6.
3. The person with a pet hamster is Arnold.
4. Bob is the person who has a cat.
5. There is one house between Eric and the person who is tall.
6. The person who owns a rabbit is the person who is super tall.
7. The person who is very tall is Alice.
8. The person who is very tall is somewhere to the right of the person who has an average height.
9. There is one house between the person who is super tall and Carol.
10. The person who owns a dog is not in the sixth house.
11. The person who is short is the person who uses a Samsung Galaxy S21.
12. The person who is tall and the person who uses a Huawei P50 are next to each other.
13. The person who uses a OnePlus 9 is not in the fourth house.
14. The person who uses a Xiaomi Mi 11 is somewhere to the left of the person who uses an iPhone 13.
15. The person who is tall is not in the fourth house.
16. Peter is the person who uses a OnePlus 9.
17. The person who uses a Samsung Galaxy S21 is directly left of Peter.
18. Peter is directly left of the person who is tall.
19. Carol is the person with an aquarium of fish.
 What is the value of attribute PhoneModel for the person whose attribute Height is super tall? Please reason step by step, and put your final answer within \boxed{}

Answer: oneplus 9


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`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`, `honda civic`, `bmw 3 series`
 - The people are of nationalities: `brit`, `norwegian`, `german`, `swede`, `dane`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `lilies`, `tulips`, `roses`

## Clues:
1. The person who owns a Tesla Model 3 is the person who loves a carnations arrangement.
2. The person who owns a Toyota Camry is Alice.
3. The person who owns a BMW 3 Series is somewhere to the right of Arnold.
4. Arnold is not in the third house.
5. The Swedish person is in the first house.
6. The German is the person who loves a bouquet of daffodils.
7. Arnold is not in the second house.
8. The Norwegian is the person who owns a Tesla Model 3.
9. The Dane is somewhere to the left of the person who owns a Ford F-150.
10. The person who owns a BMW 3 Series is somewhere to the right of the person who loves a carnations arrangement.
11. Alice is directly left of the person who loves the rose bouquet.
12. Peter and the person who loves the vase of tulips are next to each other.
13. The British person is somewhere to the right of the Norwegian.
14. The person who loves the vase of tulips is Eric.
15. The person who owns a Ford F-150 is Eric.
 What is the value of attribute House for the person whose attribute Flower is roses? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Carol`, `Bob`, `Alice`, `Arnold`
 - People have unique favorite music genres: `country`, `classical`, `jazz`, `hip hop`, `rock`, `pop`
 - Each person has a unique birthday month: `jan`, `feb`, `april`, `mar`, `sept`, `may`

## Clues:
1. Peter is in the first house.
2. The person who loves rock music is the person whose birthday is in January.
3. Arnold is in the fourth house.
4. The person whose birthday is in January is somewhere to the right of the person who loves classical music.
5. Eric is not in the sixth house.
6. The person whose birthday is in February is somewhere to the left of Eric.
7. Bob is the person whose birthday is in March.
8. There is one house between Bob and the person who loves jazz music.
9. The person who loves rock music and Carol are next to each other.
10. The person who loves country music is not in the sixth house.
11. Alice is directly left of the person whose birthday is in February.
12. The person who loves pop music is somewhere to the left of the person who loves country music.
13. The person whose birthday is in April is not in the first house.
14. The person who loves rock music and the person whose birthday is in September are next to each other.
15. The person who loves classical music is the person whose birthday is in March.
 What is the value of attribute MusicGenre for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: rock


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`, `Peter`, `Carol`, `Alice`, `Arnold`, `Bob`
 - Each mother is accompanied by their child: `Samantha`, `Alice`, `Timothy`, `Fred`, `Bella`, `Meredith`

## Clues:
1. Arnold is not in the sixth house.
2. The person's child is named Alice is somewhere to the left of Peter.
3. The person's child is named Bella is in the fourth house.
4. Carol is in the third house.
5. The person's child is named Meredith is not in the sixth house.
6. The person's child is named Alice is somewhere to the left of Arnold.
7. Bob and Carol are next to each other.
8. Eric is the person's child is named Bella.
9. The person's child is named Meredith is somewhere to the right of the person's child is named Alice.
10. The person's child is named Fred is in the fifth house.
11. The person who is the mother of Timothy is Carol.
 What is the value of attribute Children for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: Fred


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`, `Arnold`, `Carol`, `Bob`
 - People have unique heights: `very tall`, `very short`, `tall`, `super tall`, `short`, `average`
 - Each person prefers a unique type of vacation: `cruise`, `mountain`, `camping`, `beach`, `city`, `cultural`
 - People have unique hair colors: `blonde`, `red`, `gray`, `brown`, `auburn`, `black`
 - Each person has a unique favorite drink: `tea`, `boba tea`, `milk`, `water`, `coffee`, `root beer`
 - They all have a unique favorite flower: `lilies`, `carnations`, `iris`, `tulips`, `roses`, `daffodils`

## Clues:
1. The person who loves the vase of tulips is the person who has auburn hair.
2. The person who is short is somewhere to the right of Arnold.
3. The person who goes on cultural tours is Arnold.
4. The person who has auburn hair is Bob.
5. The person who is very short is the person who loves a bouquet of daffodils.
6. The person who loves the boquet of iris is the person who is very tall.
7. Peter is somewhere to the right of the root beer lover.
8. The person who loves beach vacations is directly left of the person who prefers city breaks.
9. Bob is directly left of the coffee drinker.
10. The person who is very short is the person who has gray hair.
11. Eric is somewhere to the right of the tea drinker.
12. The person who has red hair is the person who is tall.
13. The person who is tall is somewhere to the right of the person who is short.
14. The person who has black hair is the person who enjoys mountain retreats.
15. The one who only drinks water is the person who prefers city breaks.
16. The person who has brown hair is Carol.
17. The person who has an average height is Arnold.
18. The person who has brown hair is directly left of the person who likes milk.
19. The root beer lover is the person who is very short.
20. The person who is short is somewhere to the left of the coffee drinker.
21. The person who enjoys camping trips is the person who loves a carnations arrangement.
22. The person who is super tall and Arnold are next to each other.
23. There is one house between the person who loves the boquet of lilies and Peter.
24. The person who loves the vase of tulips is directly left of the person who has black hair.
 What is the value of attribute Height for the person whose attribute Vacation is camping? Please reason step by step, and put your final answer within \boxed{}

Answer: tall


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`
 - The people are of nationalities: `dane`, `swede`, `german`, `brit`, `norwegian`
 - Each person has a unique hobby: `gardening`, `painting`, `knitting`, `photography`, `cooking`
 - Everyone has a favorite smoothie: `dragonfruit`, `lime`, `watermelon`, `desert`, `cherry`
 - People have unique hair colors: `brown`, `gray`, `black`, `blonde`, `red`
 - People have unique favorite sports: `baseball`, `soccer`, `swimming`, `tennis`, `basketball`

## Clues:
1. The person who loves cooking is not in the second house.
2. The Watermelon smoothie lover is not in the fourth house.
3. The person who loves baseball is somewhere to the right of the Desert smoothie lover.
4. The person who drinks Lime smoothies is the person who has brown hair.
5. The person who has gray hair is somewhere to the right of the Swedish person.
6. The photography enthusiast is not in the fifth house.
7. The Swedish person is not in the first house.
8. The German is the person who has blonde hair.
9. There is one house between the Watermelon smoothie lover and the photography enthusiast.
10. Alice and the person who drinks Lime smoothies are next to each other.
11. The British person and the person who enjoys knitting are next to each other.
12. The person who has black hair is the Swedish person.
13. The British person is the person who enjoys gardening.
14. The person who loves swimming is the person who loves cooking.
15. The Dragonfruit smoothie lover is not in the third house.
16. The Watermelon smoothie lover is not in the second house.
17. Eric is somewhere to the left of the person who drinks Lime smoothies.
18. The person who loves soccer is Alice.
19. Peter is the photography enthusiast.
20. The person who enjoys gardening is directly left of the person who loves tennis.
21. The Norwegian and the photography enthusiast are next to each other.
22. The British person is Alice.
23. Bob is the person who has blonde hair.
 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: 5


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Bob`, `Eric`, `Arnold`
 - Each person has a unique type of pet: `hamster`, `bird`, `fish`, `cat`, `dog`
 - Each person has an occupation: `doctor`, `engineer`, `lawyer`, `artist`, `teacher`
 - Each person lives in a unique style of house: `colonial`, `modern`, `ranch`, `craftsman`, `victorian`
 - Each person has a unique level of education: `high school`, `bachelor`, `master`, `doctorate`, `associate`

## Clues:
1. There are two houses between the person with a master's degree and the person in a modern-style house.
2. The person with an aquarium of fish is the person who is an artist.
3. The person who has a cat is the person who is an engineer.
4. The person with a bachelor's degree is the person with an aquarium of fish.
5. The person who has a cat is somewhere to the right of the person with an associate's degree.
6. The person with a master's degree is the person who is a lawyer.
7. The person who is a teacher is somewhere to the right of the person who keeps a pet bird.
8. The person with an associate's degree is the person living in a colonial-style house.
9. Peter is the person in a Craftsman-style house.
10. The person residing in a Victorian house is in the second house.
11. The person who is a teacher is Bob.
12. The person with an associate's degree is in the fourth house.
13. The person with a bachelor's degree is somewhere to the right of the person with a high school diploma.
14. There is one house between the person who owns a dog and the person in a ranch-style home.
15. Bob is somewhere to the right of Alice.
16. Eric is the person with a bachelor's degree.
 What is the value of attribute Occupation for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: lawyer


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

## Clues:
1. The person in a ranch-style home is in the fifth house.
2. The person in a modern-style house is not in the first house.
3. The person who loves stir fry is directly left of Peter.
4. There is one house between the person in a Craftsman-style house and Arnold.
5. The person who loves eating grilled cheese is the person residing in a Victorian house.
6. The person who loves the stew is the person living in a colonial-style house.
7. The person who loves the soup is directly left of the person who is a pizza lover.
8. Alice is the person who loves the stew.
9. There are two houses between the person in a modern-style house and Arnold.
10. The person who loves the stew is not in the sixth house.
11. The person who loves the spaghetti eater is Bob.
12. The person who loves eating grilled cheese is in the second house.
13. Carol is not in the fourth house.
 What is the value of attribute HouseStyle for the person whose attribute Name is Peter? 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`, `Peter`, `Alice`, `Arnold`, `Bob`, `Carol`
 - Each person has a unique hobby: `cooking`, `knitting`, `painting`, `gardening`, `woodworking`, `photography`
 - Everyone has a favorite smoothie: `lime`, `cherry`, `desert`, `blueberry`, `watermelon`, `dragonfruit`
 - The people are of nationalities: `chinese`, `dane`, `german`, `norwegian`, `swede`, `brit`
 - People have unique favorite music genres: `hip hop`, `country`, `rock`, `jazz`, `classical`, `pop`

## Clues:
1. The person who loves rock music is Bob.
2. The person who enjoys knitting is the German.
3. The woodworking hobbyist is in the sixth house.
4. The person who loves rock music is directly left of the woodworking hobbyist.
5. There is one house between the Watermelon smoothie lover and the person who loves hip-hop music.
6. The Chinese is the Watermelon smoothie lover.
7. The Desert smoothie lover is the Norwegian.
8. The person who drinks Blueberry smoothies is not in the sixth house.
9. The person who loves pop music is somewhere to the left of the person who enjoys gardening.
10. Carol is somewhere to the left of the Swedish person.
11. Peter is somewhere to the right of the person who loves pop music.
12. The Norwegian is somewhere to the right of the Dragonfruit smoothie lover.
13. The German is the person who loves country music.
14. The person who paints as a hobby is directly left of the person who loves hip-hop music.
15. The Dane is Arnold.
16. The person who loves jazz music is the Chinese.
17. The British person is not in the third house.
18. The Swedish person is the person who loves pop music.
19. The person who drinks Lime smoothies is the photography enthusiast.
20. The person who loves hip-hop music is directly left of Alice.
21. The Dane is the woodworking hobbyist.
 What is the value of attribute MusicGenre for the person whose attribute Hobby is photography? Please reason step by step, and put your final answer within \boxed{}

Answer: rock


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 Smoothie for the person whose attribute Hobby is gardening? Please reason step by step, and put your final answer within \boxed{}

Answer: watermelon


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

## Clues:
1. The person who loves mystery books is somewhere to the left of the person who is the mother of Timothy.
2. The person whose mother's name is Janelle is in the third house.
3. The person who loves science fiction books is directly left of the person's child is named Bella.
4. Arnold is somewhere to the right of the person's child is named Meredith.
5. The person's child is named Fred is Peter.
6. Alice is directly left of Eric.
7. The person whose mother's name is Aniya is in the second house.
8. The person's child is named Fred is not in the fifth house.
9. The person who loves romance books is Alice.
10. The person whose mother's name is Janelle is directly left of the person who is the mother of Timothy.
11. The person whose mother's name is Aniya is somewhere to the left of The person whose mother's name is Kailyn.
12. The person who loves science fiction books is Bob.
13. The person whose mother's name is Janelle is somewhere to the left of the person who loves fantasy books.
14. The person who loves fantasy books is somewhere to the left of The person whose mother's name is Penny.
 What is the value of attribute BookGenre for the person whose attribute Mother is Janelle? Please reason step by step, and put your final answer within \boxed{}

Answer: romance


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

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

Answer: lime


There are 2 houses, numbered 1 to 2 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`
 - People have unique favorite music genres: `rock`, `pop`
 - Each person lives in a unique style of house: `colonial`, `victorian`
 - Everyone has something unique for lunch: `grilled cheese`, `pizza`
 - People have unique favorite sports: `basketball`, `soccer`
 - Each mother is accompanied by their child: `Bella`, `Fred`

## Clues:
1. The person living in a colonial-style house is the person who is a pizza lover.
2. Arnold is the person living in a colonial-style house.
3. The person who loves pop music is the person living in a colonial-style house.
4. The person who is a pizza lover is not in the second house.
5. Eric is the person who loves soccer.
6. The person who loves eating grilled cheese is the person's child is named Bella.
 What is the value of attribute Children for the person whose attribute HouseStyle is colonial? Please reason step by step, and put your final answer within \boxed{}

Answer: Fred


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Eric`, `Peter`
 - Everyone has something unique for lunch: `spaghetti`, `stew`, `pizza`, `grilled cheese`
 - The people keep unique animals: `fish`, `horse`, `bird`, `cat`

## Clues:
1. The fish enthusiast is Alice.
2. Eric is in the second house.
3. The person who is a pizza lover is directly left of the fish enthusiast.
4. Arnold is the bird keeper.
5. The person who loves eating grilled cheese is Peter.
6. The cat lover and the person who loves the stew are next to each other.
7. The person who is a pizza lover is somewhere to the right of Eric.
 What is the value of attribute Animal for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


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`
 - They all have a unique favorite flower: `lilies`, `roses`, `daffodils`, `carnations`
 - People have unique favorite sports: `swimming`, `tennis`, `soccer`, `basketball`
 - People have unique hair colors: `black`, `brown`, `blonde`, `red`
 - Each person has a unique birthday month: `feb`, `jan`, `april`, `sept`
 - Each person prefers a unique type of vacation: `cruise`, `city`, `mountain`, `beach`

## Clues:
1. The person who enjoys mountain retreats is in the fourth house.
2. The person whose birthday is in April and the person who loves a bouquet of daffodils are next to each other.
3. The person who likes going on cruises is the person who loves a carnations arrangement.
4. The person whose birthday is in September is the person who loves soccer.
5. Arnold is the person who loves a bouquet of daffodils.
6. Alice is in the third house.
7. The person who loves basketball is the person whose birthday is in April.
8. The person whose birthday is in February is the person who has brown hair.
9. The person who has black hair is somewhere to the left of the person who likes going on cruises.
10. The person who prefers city breaks is Eric.
11. The person who has brown hair is Arnold.
12. The person who has blonde hair is the person who loves tennis.
13. The person who loves tennis is the person who likes going on cruises.
14. The person who loves beach vacations is not in the third house.
15. The person who loves the boquet of lilies is in the first house.
 What is the value of attribute Flower for the person whose attribute HairColor is red? Please reason step by step, and put your final answer within \boxed{}

Answer: roses


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`, `Arnold`, `Alice`, `Bob`
 - Each person has a unique level of education: `associate`, `bachelor`, `high school`, `master`, `doctorate`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `baseball`, `soccer`
 - Each person prefers a unique type of vacation: `camping`, `cruise`, `city`, `mountain`, `beach`
 - People use unique phone models: `google pixel 6`, `huawei p50`, `oneplus 9`, `samsung galaxy s21`, `iphone 13`

## Clues:
1. The person who uses a Huawei P50 is not in the second house.
2. The person with a master's degree is the person who uses a Google Pixel 6.
3. The person who loves swimming is the person with a high school diploma.
4. The person with a bachelor's degree is directly left of the person who loves tennis.
5. The person who enjoys camping trips is the person who loves tennis.
6. Bob is the person who uses a Samsung Galaxy S21.
7. The person with a high school diploma is not in the third house.
8. The person who loves swimming is directly left of the person who loves beach vacations.
9. The person who likes going on cruises is Arnold.
10. Bob is the person with an associate's degree.
11. The person who uses a OnePlus 9 is directly left of the person who loves basketball.
12. The person who enjoys camping trips and the person who prefers city breaks are next to each other.
13. Eric is the person who loves swimming.
14. The person who loves basketball is the person who enjoys mountain retreats.
15. The person who loves tennis is somewhere to the left of Alice.
16. Arnold is the person who loves soccer.
 What is the value of attribute House for the person whose attribute Vacation is beach? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


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`, `Arnold`, `Alice`, `Bob`
 - Everyone has something unique for lunch: `stir fry`, `spaghetti`, `stew`, `grilled cheese`, `pizza`
 - People own unique car models: `ford f150`, `tesla model 3`, `bmw 3 series`, `toyota camry`, `honda civic`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `oneplus 9`, `huawei p50`
 - Each person has an occupation: `teacher`, `lawyer`, `doctor`, `artist`, `engineer`
 - Each person has a unique favorite drink: `tea`, `milk`, `water`, `root beer`, `coffee`

## Clues:
1. The root beer lover is the person who owns a Honda Civic.
2. The person who likes milk is directly left of the person who loves eating grilled cheese.
3. Alice is the person who uses a Samsung Galaxy S21.
4. Alice is the person who loves stir fry.
5. The tea drinker is not in the fifth house.
6. The person who owns a BMW 3 Series is somewhere to the left of the tea drinker.
7. The person who is a doctor is Arnold.
8. The person who uses an iPhone 13 is the coffee drinker.
9. The person who is an engineer is the person who owns a BMW 3 Series.
10. The person who loves the stew is the person who uses an iPhone 13.
11. The person who is a doctor is directly left of the person who uses a OnePlus 9.
12. The person who owns a Honda Civic is directly left of the person who loves the spaghetti eater.
13. The person who uses a Google Pixel 6 is the tea drinker.
14. Alice is the person who is an artist.
15. There is one house between Alice and the person who owns a Ford F-150.
16. Arnold is the person who owns a Toyota Camry.
17. Eric is in the fourth house.
18. The person who uses a OnePlus 9 is the person who is a lawyer.
19. The person who loves eating grilled cheese is Peter.
 What is the value of attribute CarModel for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: honda civic


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`
 - They all have a unique favorite flower: `carnations`, `tulips`, `lilies`, `roses`, `daffodils`, `iris`
 - People have unique favorite book genres: `science fiction`, `fantasy`, `romance`, `mystery`, `biography`, `historical fiction`
 - People use unique phone models: `huawei p50`, `oneplus 9`, `iphone 13`, `xiaomi mi 11`, `samsung galaxy s21`, `google pixel 6`
 - People have unique hair colors: `black`, `gray`, `red`, `auburn`, `blonde`, `brown`
 - Each person has a favorite color: `yellow`, `green`, `red`, `white`, `blue`, `purple`

## Clues:
1. There is one house between the person who has gray hair and the person who uses an iPhone 13.
2. The person who loves the rose bouquet is the person who has blonde hair.
3. There is one house between the person who has gray hair and the person who loves yellow.
4. Carol is somewhere to the right of the person who loves a bouquet of daffodils.
5. The person who loves white is somewhere to the left of Peter.
6. The person who loves white is somewhere to the left of the person who loves science fiction books.
7. Arnold is directly left of the person who loves historical fiction books.
8. The person who uses a Huawei P50 is the person who loves a bouquet of daffodils.
9. The person who loves purple is somewhere to the left of the person who loves fantasy books.
10. Eric is the person who loves science fiction books.
11. The person who loves a bouquet of daffodils is the person who has auburn hair.
12. Alice is the person whose favorite color is green.
13. The person who loves fantasy books is in the second house.
14. The person who uses a Samsung Galaxy S21 is Carol.
15. The person who uses a Xiaomi Mi 11 is not in the fourth house.
16. Eric is the person who has auburn hair.
17. The person who loves the rose bouquet is somewhere to the left of the person who has red hair.
18. The person who loves historical fiction books is the person who loves the rose bouquet.
19. The person who uses a Xiaomi Mi 11 is somewhere to the right of the person who loves white.
20. The person who has black hair is in the fourth house.
21. The person whose favorite color is red is the person who loves mystery books.
22. The person who loves historical fiction books and the person who loves the vase of tulips are next to each other.
23. The person who has brown hair is the person who uses a OnePlus 9.
24. The person who loves romance books is somewhere to the left of the person who loves fantasy books.
25. The person who loves the boquet of lilies is in the second house.
26. The person who loves the boquet of iris is somewhere to the right of the person whose favorite color is green.
 What is the value of attribute HairColor for the person whose attribute BookGenre is mystery? Please reason step by step, and put your final answer within \boxed{}

Answer: red


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`, `Eric`, `Arnold`, `Peter`
 - Each person has an occupation: `artist`, `engineer`, `teacher`, `doctor`
 - People have unique favorite book genres: `fantasy`, `science fiction`, `mystery`, `romance`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `oneplus 9`, `samsung galaxy s21`

## Clues:
1. The person who is an engineer is directly left of the person who uses a Samsung Galaxy S21.
2. The person who loves fantasy books is in the second house.
3. Alice is not in the second house.
4. Eric is the person who is a teacher.
5. The person who uses a Samsung Galaxy S21 is the person who loves fantasy books.
6. The person who uses an iPhone 13 is the person who loves science fiction books.
7. The person who loves science fiction books is somewhere to the left of the person who uses a OnePlus 9.
8. The person who uses a OnePlus 9 is Arnold.
9. The person who is a doctor is the person who loves mystery books.
10. The person who uses an iPhone 13 is the person who is a teacher.
 What is the value of attribute BookGenre for the person whose attribute Occupation is artist? Please reason step by step, and put your final answer within \boxed{}

Answer: fantasy