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

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

Answer: 3


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

## Clues:
1. The person whose birthday is in March is not in the fifth house.
2. Peter is not in the sixth house.
3. The person whose birthday is in September is not in the sixth house.
4. The person whose birthday is in March is somewhere to the left of the Desert smoothie lover.
5. The person who loves jazz music is Eric.
6. The person who loves swimming is somewhere to the right of the person who loves basketball.
7. The person who loves swimming is the person who loves jazz music.
8. There is one house between the person whose birthday is in September and the Desert smoothie lover.
9. The Watermelon smoothie lover is the person who loves jazz music.
10. There are two houses between the person whose birthday is in January and the Desert smoothie lover.
11. The person whose birthday is in February and the person who drinks Lime smoothies are next to each other.
12. There is one house between Bob and the person who loves jazz music.
13. The Watermelon smoothie lover is somewhere to the right of the person whose birthday is in April.
14. The person who loves rock music is Carol.
15. The person who loves volleyball is directly left of the Dragonfruit smoothie lover.
16. The person who loves rock music is directly left of the person whose birthday is in March.
17. The Desert smoothie lover is the person who loves soccer.
18. The person whose birthday is in January is the person who loves pop music.
19. There are two houses between the person who drinks Blueberry smoothies and the Desert smoothie lover.
20. The person whose birthday is in January is somewhere to the left of the person whose birthday is in March.
21. The Dragonfruit smoothie lover is directly left of the person who loves baseball.
22. Alice is the person who loves hip-hop music.
23. The person who loves classical music is in the sixth house.
 What is the value of attribute Birthday for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: jan


Let $m \circ n = \frac{m+n}{mn+4}$. Compute $((\cdots((2005 \circ 2004) \circ 2003) \circ \cdots \circ 1) \circ 0)$.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 13


In the diagram, there are more than three triangles. If each triangle has the same probability of being selected, what is the probability that a selected triangle has all or part of its interior shaded? Express your answer as a common fraction.

[asy]
draw((0,0)--(1,0)--(0,1)--(0,0)--cycle,linewidth(1));
draw((0,0)--(.5,0)--(.5,.5)--(0,0)--cycle,linewidth(1));

label("A",(0,1),NW);
label("B",(.5,.5),NE);
label("C",(1,0),SE);
label("D",(.5,0),S);
label("E",(0,0),SW);

filldraw((.5,0)--(1,0)--(.5,.5)--(.5,0)--cycle,gray,black);[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 8


If
\[\frac{\sin x}{\cos y} + \frac{\sin y}{\cos x} = 1 \quad \text{and} \quad \frac{\cos x}{\sin y} + \frac{\cos y}{\sin x} = 6,\]then find $\frac{\tan x}{\tan y} + \frac{\tan y}{\tan x}.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 137


Six congruent copies of the parabola $y = x^2$ are arranged in the plane so that each vertex is tangent to a circle, and each parabola is tangent to its two neighbors.  Find the radius of the circle.

[asy]
unitsize(1 cm);

real func (real x) {
  return (x^2 + 3/4);
}

path parab = graph(func,-1.5,1.5);

draw(parab);
draw(rotate(60)*(parab));
draw(rotate(120)*(parab));
draw(rotate(180)*(parab));
draw(rotate(240)*(parab));
draw(rotate(300)*(parab));
draw(Circle((0,0),3/4));
[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 7


Beginning at point $A$ in the diagram below, Dora selects one of the four possible directions with equal probability. Each time she comes to an intersection, she again randomly selects one of the possible directions. What is the probability that, in her first four steps, she will walk completely around the gray square? Express your answer as a common fraction. [asy]size(100);
fill((1,1)--(1,2)--(2,2)--(2,1)--cycle, gray(.6));
draw((0,0)--(0,3)--(3,3)--(3,0)--cycle, linewidth(1.5));
draw((0,1)--(3,1), linewidth(1.5));
draw((0,2)--(3,2), linewidth(1.5));
draw((1,0)--(1,3), linewidth(1.5));
draw((2,0)--(2,3), linewidth(1.5));
dot(MP("A", (1,2), NW));
[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 129


Each of five, standard, six-sided dice is rolled once. Two of the dice come up the same, but the other three are all different from those two and different from each other. The pair is set aside, and the other three dice are re-rolled. The dice are said to show a "full house" if three of the dice show the same value and the other two show the same value (and potentially, but not necessarily, all five dice show the same value). What is the probability that after the second set of rolls, the dice show a full house?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 79


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`, `Eric`, `Arnold`
 - Each person has a unique hobby: `cooking`, `painting`, `gardening`, `photography`
 - The people keep unique animals: `horse`, `fish`, `cat`, `bird`
 - People have unique favorite book genres: `fantasy`, `mystery`, `romance`, `science fiction`
 - Each person has a unique birthday month: `april`, `jan`, `sept`, `feb`
 - People have unique favorite music genres: `pop`, `rock`, `classical`, `jazz`

## Clues:
1. The person who loves cooking is the person who loves romance books.
2. The person whose birthday is in February is the person who loves pop music.
3. Eric is not in the second house.
4. The person who loves romance books is not in the fourth house.
5. The person whose birthday is in February is the fish enthusiast.
6. Alice is somewhere to the right of the person who loves fantasy books.
7. The person who keeps horses is the person who loves rock music.
8. The person who enjoys gardening is the person whose birthday is in April.
9. The person who loves jazz music is the person who loves cooking.
10. The person who loves rock music is the person who loves mystery books.
11. The person who paints as a hobby is directly left of the person who loves romance books.
12. Peter is the person who loves pop music.
13. The person who enjoys gardening is Arnold.
14. The person who loves rock music is directly left of the person whose birthday is in January.
15. The person who loves cooking is not in the third house.
16. The cat lover is somewhere to the right of the person who keeps horses.
 What is the value of attribute BookGenre for the person whose attribute Birthday is jan? Please reason step by step, and put your final answer within \boxed{}

Answer: science fiction


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`, `Arnold`, `Bob`, `Eric`, `Peter`
 - Each person has a unique hobby: `painting`, `photography`, `cooking`, `gardening`, `knitting`
 - Each person has a unique birthday month: `jan`, `sept`, `mar`, `feb`, `april`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `baseball`, `soccer`
 - Each person has an occupation: `engineer`, `lawyer`, `teacher`, `doctor`, `artist`
 - Each person has a unique level of education: `bachelor`, `master`, `high school`, `doctorate`, `associate`

## Clues:
1. The person with a high school diploma is directly left of the person whose birthday is in February.
2. The person who loves soccer is directly left of the photography enthusiast.
3. Arnold is the person whose birthday is in January.
4. The person who is a doctor is the person who loves baseball.
5. The person who is an artist is somewhere to the left of Bob.
6. The photography enthusiast is somewhere to the right of the person with an associate's degree.
7. The photography enthusiast is directly left of the person who paints as a hobby.
8. Eric is in the fifth house.
9. The person whose birthday is in February is directly left of the person who loves tennis.
10. The person who loves baseball is somewhere to the right of the person whose birthday is in January.
11. The person who is an engineer is not in the first house.
12. Alice is not in the second house.
13. The person who enjoys gardening is somewhere to the left of the person whose birthday is in September.
14. The person who loves swimming is not in the first house.
15. The person who is a teacher is Bob.
16. Arnold is the person who enjoys knitting.
17. The person who is an artist is the person whose birthday is in March.
18. Arnold is the person with a master's degree.
19. The person whose birthday is in March is directly left of the person who is a doctor.
20. The person whose birthday is in September is the person with a bachelor's degree.
 What is the value of attribute Birthday for the person whose attribute FavoriteSport is swimming? Please reason step by step, and put your final answer within \boxed{}

Answer: feb


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

## Clues:
1. The person who loves pop music is not in the fourth house.
2. The boba tea drinker is the person whose birthday is in April.
3. The person whose birthday is in January is somewhere to the right of the person whose birthday is in May.
4. Arnold is not in the third house.
5. Eric is in the first house.
6. Carol is the person who loves jazz music.
7. The person who loves classical music is directly left of the root beer lover.
8. The person who loves country music is somewhere to the left of the person who is a teacher.
9. The person who is a teacher is somewhere to the left of the person who is a nurse.
10. The person who is a doctor is somewhere to the left of the person who is a teacher.
11. The root beer lover is the person who loves hip-hop music.
12. Arnold is the person who is a lawyer.
13. Carol is not in the fourth house.
14. The person who is an artist is somewhere to the left of the person who loves pop music.
15. The person whose birthday is in April is Bob.
16. The person who likes milk is not in the second house.
17. The one who only drinks water is somewhere to the right of the person whose birthday is in March.
18. The tea drinker is the person whose birthday is in September.
19. The person whose birthday is in January is the person who loves pop music.
20. The person who loves jazz music is directly left of the person who loves rock music.
21. Alice is the person who is a nurse.
22. The tea drinker is not in the sixth house.
23. The person who is a doctor is the root beer lover.
 What is the value of attribute MusicGenre for the person whose attribute Birthday is feb? 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`, `Alice`, `Arnold`, `Carol`, `Peter`, `Bob`
 - Each person lives in a unique style of house: `mediterranean`, `modern`, `craftsman`, `ranch`, `colonial`, `victorian`
 - People have unique favorite music genres: `country`, `hip hop`, `pop`, `jazz`, `classical`, `rock`
 - Each person has a unique hobby: `cooking`, `painting`, `photography`, `woodworking`, `gardening`, `knitting`

## Clues:
1. The person who loves rock music is in the fifth house.
2. The person who loves classical music and the woodworking hobbyist are next to each other.
3. The person in a Mediterranean-style villa is the person who loves hip-hop music.
4. There are two houses between Arnold and the person residing in a Victorian house.
5. The person who loves jazz music is directly left of Eric.
6. The person who loves hip-hop music is somewhere to the left of the person who enjoys knitting.
7. Carol is the person who loves hip-hop music.
8. The person in a Craftsman-style house is Arnold.
9. The person in a ranch-style home is Eric.
10. The woodworking hobbyist is the person residing in a Victorian house.
11. The person who loves country music is in the first house.
12. There is one house between the person who paints as a hobby and the person living in a colonial-style house.
13. Alice is the photography enthusiast.
14. The person who enjoys gardening is Eric.
15. Bob is in the third house.
 What is the value of attribute MusicGenre for the person whose attribute HouseStyle is mediterranean? Please reason step by step, and put your final answer within \boxed{}

Answer: hip hop


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

## Clues:
1. Alice is somewhere to the right of the person who loves classical music.
2. The person who loves pop music is directly left of the person who loves classical music.
3. Bob is in the fourth house.
4. The person who loves country music is Bob.
5. The person who loves hip-hop music is Peter.
6. Carol is somewhere to the right of the person who loves jazz music.
7. Peter is directly left of Eric.
 What is the value of attribute House for the person whose attribute MusicGenre is country? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Alice`, `Eric`, `Arnold`
 - 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 PhoneModel for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: oneplus 9


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

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

Answer: milk


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`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `lilies`
 - People have unique favorite music genres: `rock`, `classical`, `pop`
 - Each person has a unique favorite drink: `water`, `milk`, `tea`
 - Everyone has a unique favorite cigar: `pall mall`, `blue master`, `prince`
 - Each person has a unique birthday month: `april`, `jan`, `sept`

## Clues:
1. The Prince smoker is not in the first house.
2. The person who likes milk is the person whose birthday is in April.
3. The person whose birthday is in January is somewhere to the right of Eric.
4. The person who loves pop music is directly left of the person who loves a bouquet of daffodils.
5. The one who only drinks water is the person who loves a bouquet of daffodils.
6. The person who likes milk is the person who smokes Blue Master.
7. Arnold is in the second house.
8. The person who loves rock music is the person whose birthday is in September.
9. The person who loves the boquet of lilies is somewhere to the left of the person whose birthday is in April.
 What is the value of attribute House for the person whose attribute Flower is lilies? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Alice`, `Peter`, `Arnold`
 - They all have a unique favorite flower: `carnations`, `daffodils`, `tulips`, `roses`, `lilies`
 - The people are of nationalities: `dane`, `swede`, `brit`, `norwegian`, `german`
 - Each person has an occupation: `artist`, `lawyer`, `engineer`, `teacher`, `doctor`

## Clues:
1. The person who is an engineer is Bob.
2. The person who is a lawyer is not in the second house.
3. The Dane is in the third house.
4. The British person is Eric.
5. The person who is a doctor is the Norwegian.
6. The person who loves the rose bouquet is not in the second house.
7. There is one house between Peter and the person who loves a bouquet of daffodils.
8. The German is the person who is a teacher.
9. The Swedish person is not in the first house.
10. Peter is directly left of the person who loves the boquet of lilies.
11. The person who loves the boquet of lilies is directly left of Bob.
12. The person who loves the rose bouquet is somewhere to the right of the person who is a lawyer.
13. Peter is somewhere to the left of Arnold.
14. There are two houses between the person who loves a carnations arrangement and the person who loves the boquet of lilies.
15. The person who is a teacher is not in the fourth house.
 What is the value of attribute Flower for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: lilies


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Bob`, `Alice`, `Arnold`
 - People have unique heights: `very short`, `average`, `tall`, `short`, `very tall`
 - People own unique car models: `toyota camry`, `bmw 3 series`, `honda civic`, `ford f150`, `tesla model 3`
 - People use unique phone models: `iphone 13`, `oneplus 9`, `samsung galaxy s21`, `google pixel 6`, `huawei p50`

## Clues:
1. Arnold is in the fifth house.
2. The person who is short is the person who owns a Tesla Model 3.
3. The person who has an average height is somewhere to the right of Alice.
4. The person who owns a Honda Civic is not in the third house.
5. The person who owns a BMW 3 Series is directly left of Arnold.
6. The person who uses a OnePlus 9 is directly left of the person who owns a Ford F-150.
7. Alice is directly left of the person who is very short.
8. The person who uses an iPhone 13 is not in the second house.
9. The person who uses a Google Pixel 6 is Arnold.
10. The person who uses a Google Pixel 6 is the person who is tall.
11. Peter is the person who uses a Huawei P50.
12. Peter is directly left of the person who uses a OnePlus 9.
13. Bob is the person who owns a Tesla Model 3.
 What is the value of attribute Height for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: very tall


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Bob`, `Arnold`, `Carol`, `Alice`
 - Everyone has a favorite smoothie: `cherry`, `blueberry`, `dragonfruit`, `watermelon`, `lime`, `desert`
 - The people keep unique animals: `bird`, `cat`, `dog`, `rabbit`, `horse`, `fish`
 - The mothers' names in different houses are unique: `Sarah`, `Holly`, `Janelle`, `Aniya`, `Kailyn`, `Penny`
 - They all have a unique favorite flower: `lilies`, `iris`, `carnations`, `tulips`, `roses`, `daffodils`
 - Each person has an occupation: `lawyer`, `doctor`, `teacher`, `engineer`, `artist`, `nurse`

## Clues:
1. The person who loves a bouquet of daffodils is The person whose mother's name is Holly.
2. The person who likes Cherry smoothies is not in the third house.
3. The person who is a nurse is directly left of Peter.
4. The person who is an artist is the person who loves the rose bouquet.
5. The person who drinks Lime smoothies is not in the fifth house.
6. The cat lover is The person whose mother's name is Sarah.
7. Bob is the Desert smoothie lover.
8. The Dragonfruit smoothie lover is in the sixth house.
9. The person who loves the boquet of lilies and The person whose mother's name is Penny are next to each other.
10. Peter is the bird keeper.
11. The person whose mother's name is Kailyn is the person who loves the boquet of iris.
12. Carol is somewhere to the left of the person who likes Cherry smoothies.
13. The person who is a doctor is directly left of The person whose mother's name is Holly.
14. Bob is The person whose mother's name is Janelle.
15. Arnold is in the third house.
16. Alice is the person who is an engineer.
17. The Dragonfruit smoothie lover is the person who is an artist.
18. The dog owner is directly left of the person who is a teacher.
19. The person who keeps horses is The person whose mother's name is Aniya.
20. The person who loves the boquet of lilies is directly left of the person who loves a carnations arrangement.
21. There is one house between the Desert smoothie lover and the person who loves a carnations arrangement.
22. The person who drinks Lime smoothies is somewhere to the right of The person whose mother's name is Aniya.
23. The person whose mother's name is Kailyn is directly left of the fish enthusiast.
24. The Watermelon smoothie lover is the rabbit owner.
 What is the value of attribute Animal for the person whose attribute Occupation is teacher? Please reason step by step, and put your final answer within \boxed{}

Answer: horse


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Alice`, `Eric`
 - People have unique favorite book genres: `romance`, `mystery`, `fantasy`, `science fiction`
 - People have unique heights: `tall`, `very short`, `short`, `average`
 - Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `desert`, `watermelon`
 - Each person has a favorite color: `red`, `green`, `yellow`, `white`

## Clues:
1. Arnold is not in the fourth house.
2. Peter is the person who loves mystery books.
3. The person who likes Cherry smoothies is somewhere to the right of the Desert smoothie lover.
4. The person who loves science fiction books is directly left of Eric.
5. The Desert smoothie lover is the person who is very short.
6. The person who loves white is the person who loves mystery books.
7. The person who loves fantasy books is the person who loves yellow.
8. The Desert smoothie lover is somewhere to the right of the person who is short.
9. The Dragonfruit smoothie lover and the person whose favorite color is green are next to each other.
10. The Dragonfruit smoothie lover is the person who loves mystery books.
11. The person who loves yellow is in the second house.
12. The person who likes Cherry smoothies is not in the third house.
13. The person who has an average height is the person who likes Cherry smoothies.
 What is the value of attribute Height for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: short


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`, `Arnold`, `Eric`, `Bob`, `Peter`
 - People have unique heights: `average`, `short`, `tall`, `very tall`, `very short`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Kailyn`, `Janelle`, `Aniya`
 - Each person has a unique type of pet: `cat`, `dog`, `hamster`, `fish`, `bird`
 - The people keep unique animals: `bird`, `dog`, `fish`, `horse`, `cat`
 - People have unique favorite sports: `swimming`, `basketball`, `baseball`, `tennis`, `soccer`

## Clues:
1. The person who loves basketball is the person who keeps a pet bird.
2. The person who loves soccer is directly left of the person who is very short.
3. The person whose mother's name is Aniya is directly left of the person who is tall.
4. The person who loves tennis is Eric.
5. The person whose mother's name is Holly is in the first house.
6. The person who owns a dog is somewhere to the right of Bob.
7. The person who owns a dog is the cat lover.
8. The person whose mother's name is Aniya is Peter.
9. The person who is very short is directly left of Alice.
10. The person whose mother's name is Janelle is the bird keeper.
11. The person who has a cat is directly left of the person who has an average height.
12. The person whose mother's name is Penny is the person with an aquarium of fish.
13. The person with an aquarium of fish is directly left of the dog owner.
14. The person who is very tall is somewhere to the right of the fish enthusiast.
15. The person who loves tennis is in the fourth house.
16. The person who loves basketball is directly left of the person who loves baseball.
 What is the value of attribute House for the person whose attribute Animal is dog? Please reason step by step, and put your final answer within \boxed{}

Answer: 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: `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 Color for the person whose attribute Birthday is feb? Please reason step by step, and put your final answer within \boxed{}

Answer: white


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`, `Alice`, `Carol`, `Bob`
 - People have unique favorite book genres: `mystery`, `biography`, `science fiction`, `romance`, `fantasy`, `historical fiction`
 - Each person has a unique favorite drink: `water`, `milk`, `boba tea`, `coffee`, `tea`, `root beer`
 - People have unique hair colors: `blonde`, `red`, `gray`, `black`, `auburn`, `brown`
 - Each person has a unique birthday month: `may`, `mar`, `sept`, `jan`, `april`, `feb`

## Clues:
1. The person who likes milk is the person who has gray hair.
2. Alice is somewhere to the right of the person who has auburn hair.
3. The person whose birthday is in September is not in the sixth house.
4. The tea drinker is somewhere to the left of the person who has auburn hair.
5. Carol is the person who loves biography books.
6. The person who likes milk is in the fourth house.
7. The person who has auburn hair is directly left of Eric.
8. The boba tea drinker is the person who has brown hair.
9. The person whose birthday is in February is the person who has gray hair.
10. Carol is the person who has blonde hair.
11. The person who loves fantasy books is somewhere to the left of Bob.
12. The person who has blonde hair is the person whose birthday is in April.
13. Peter is the coffee drinker.
14. The person who loves historical fiction books is somewhere to the right of the person who loves science fiction books.
15. The person who loves science fiction books is Bob.
16. The root beer lover is not in the first house.
17. There is one house between the person whose birthday is in May and Peter.
18. The person who has black hair is in the second house.
19. The person whose birthday is in February is the person who loves historical fiction books.
20. Arnold is the person whose birthday is in March.
21. The root beer lover is somewhere to the left of the person whose birthday is in May.
22. The person who loves mystery books is in the sixth house.
 What is the value of attribute Drink for the person whose attribute House is 2? 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: `Eric`, `Arnold`, `Alice`, `Bob`, `Peter`
 - Each person has an occupation: `lawyer`, `doctor`, `artist`, `engineer`, `teacher`

## Clues:
1. Bob is the person who is an engineer.
2. The person who is a doctor and the person who is an artist are next to each other.
3. There are two houses between Arnold and the person who is a lawyer.
4. Peter is the person who is an artist.
5. The person who is a lawyer is directly left of Eric.
6. Alice is somewhere to the right of the person who is an artist.
 What is the value of attribute House for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`
 - Everyone has a unique favorite cigar: `blue master`, `prince`, `pall mall`
 - Each mother is accompanied by their child: `Fred`, `Meredith`, `Bella`
 - The mothers' names in different houses are unique: `Holly`, `Aniya`, `Janelle`
 - The people keep unique animals: `cat`, `horse`, `bird`

## Clues:
1. The Prince smoker is The person whose mother's name is Janelle.
2. The person's child is named Bella is somewhere to the left of the person's child is named Meredith.
3. Peter is the person partial to Pall Mall.
4. There is one house between the cat lover and Eric.
5. The person whose mother's name is Aniya is the person's child is named Bella.
6. The person's child is named Meredith is Peter.
7. The bird keeper is somewhere to the right of Peter.
 What is the value of attribute Mother for the person whose attribute Children is Meredith? Please reason step by step, and put your final answer within \boxed{}

Answer: Holly


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Bob`, `Eric`, `Arnold`, `Alice`
 - People have unique hair colors: `brown`, `black`, `gray`, `blonde`, `red`
 - Everyone has a unique favorite cigar: `pall mall`, `dunhill`, `prince`, `blue master`, `blends`
 - Each person has a unique favorite drink: `milk`, `coffee`, `water`, `tea`, `root beer`
 - People have unique favorite music genres: `rock`, `classical`, `pop`, `jazz`, `hip hop`

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

Answer: gray


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Alice`, `Peter`, `Arnold`
 - Each person has a favorite color: `yellow`, `green`, `red`, `white`, `blue`
 - Each mother is accompanied by their child: `Samantha`, `Fred`, `Meredith`, `Bella`, `Timothy`
 - People have unique favorite book genres: `biography`, `mystery`, `romance`, `science fiction`, `fantasy`
 - Each person has a unique birthday month: `april`, `feb`, `sept`, `jan`, `mar`
 - People have unique favorite sports: `soccer`, `basketball`, `swimming`, `baseball`, `tennis`

## Clues:
1. The person who loves swimming is the person who loves mystery books.
2. Eric is somewhere to the left of the person who is the mother of Timothy.
3. The person who loves basketball is somewhere to the left of the person who is the mother of Timothy.
4. The person who loves soccer is the person whose birthday is in September.
5. Eric is the person who loves biography books.
6. The person who loves baseball is the person who loves white.
7. The person who loves science fiction books is the person whose birthday is in March.
8. The person who loves swimming and the person whose favorite color is red are next to each other.
9. The person who loves biography books is not in the first house.
10. The person who loves baseball is somewhere to the left of the person whose favorite color is red.
11. The person who loves fantasy books is Bob.
12. The person's child is named Bella is the person whose birthday is in February.
13. The person who loves tennis and the person's child is named Samantha are next to each other.
14. The person's child is named Bella is not in the first house.
15. The person's child is named Meredith is the person whose birthday is in March.
16. The person who loves blue is directly left of the person who loves yellow.
17. There is one house between Alice and the person whose birthday is in March.
18. The person whose favorite color is red is directly left of the person whose birthday is in January.
19. The person's child is named Meredith is not in the first house.
20. Arnold is somewhere to the right of the person who loves swimming.
21. The person who loves fantasy books is the person who is the mother of Timothy.
 What is the value of attribute Color for the person whose attribute BookGenre is fantasy? Please reason step by step, and put your final answer within \boxed{}

Answer: yellow


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`, `Peter`, `Eric`, `Arnold`
 - Each person has an occupation: `doctor`, `engineer`, `artist`, `teacher`
 - Each person has a unique birthday month: `sept`, `april`, `jan`, `feb`
 - Each person has a favorite color: `red`, `white`, `green`, `yellow`

## Clues:
1. Arnold is in the first house.
2. The person whose birthday is in February and the person who is a teacher are next to each other.
3. The person who is a teacher is the person whose favorite color is red.
4. The person whose favorite color is red is not in the third house.
5. The person who loves yellow is directly left of Peter.
6. The person whose favorite color is green and the person whose favorite color is red are next to each other.
7. The person who is an engineer is the person whose favorite color is green.
8. Eric is the person who is a teacher.
9. The person who loves yellow is not in the second house.
10. The person whose birthday is in September is somewhere to the left of the person whose favorite color is green.
11. The person who is a doctor is somewhere to the right of the person whose birthday is in April.
 What is the value of attribute Occupation for the person whose attribute Birthday is jan? Please reason step by step, and put your final answer within \boxed{}

Answer: teacher


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

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

Answer: 3


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`
 - Each person has a unique favorite drink: `tea`, `water`, `milk`
 - The people are of nationalities: `dane`, `brit`, `swede`
 - Each person has a unique level of education: `high school`, `associate`, `bachelor`
 - Each person lives in a unique style of house: `victorian`, `colonial`, `ranch`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `desert`

## Clues:
1. There is one house between Eric and the tea drinker.
2. The person who likes milk is the person in a ranch-style home.
3. The person with a bachelor's degree is in the second house.
4. The person with a high school diploma is the Dane.
5. The Desert smoothie lover is the Swedish person.
6. The person residing in a Victorian house is not in the first house.
7. The person who likes Cherry smoothies is the person living in a colonial-style house.
8. Arnold is somewhere to the right of the person residing in a Victorian house.
9. The person in a ranch-style home is the person with a high school diploma.
 What is the value of attribute Nationality for the person whose attribute Drink is water? Please reason step by step, and put your final answer within \boxed{}

Answer: swede


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Eric`, `Peter`
 - People have unique favorite music genres: `pop`, `classical`, `rock`, `jazz`
 - Each person has a unique type of pet: `fish`, `dog`, `bird`, `cat`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`, `honda civic`
 - Each person has a unique favorite drink: `water`, `milk`, `tea`, `coffee`

## Clues:
1. The one who only drinks water is the person who owns a dog.
2. The one who only drinks water is Eric.
3. The tea drinker is somewhere to the right of the person who loves rock music.
4. The person who likes milk is the person who owns a Tesla Model 3.
5. The person who loves rock music and the person who loves classical music are next to each other.
6. There is one house between the person who likes milk and the person who loves classical music.
7. The person who owns a Honda Civic is Arnold.
8. The person who has a cat and the coffee drinker are next to each other.
9. Alice is not in the second house.
10. The person who loves classical music is not in the second house.
11. The person with an aquarium of fish is the tea drinker.
12. The person who owns a Ford F-150 is somewhere to the right of the person with an aquarium of fish.
13. The person who owns a Toyota Camry is not in the third house.
14. The tea drinker and the person who loves jazz music are next to each other.
 What is the value of attribute Pet for the person whose attribute MusicGenre is classical? 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: `Alice`, `Peter`, `Bob`, `Eric`, `Arnold`
 - They all have a unique favorite flower: `tulips`, `daffodils`, `lilies`, `roses`, `carnations`
 - Each mother is accompanied by their child: `Fred`, `Bella`, `Timothy`, `Meredith`, `Samantha`
 - Each person has a unique level of education: `high school`, `bachelor`, `doctorate`, `master`, `associate`
 - The people keep unique animals: `cat`, `fish`, `dog`, `horse`, `bird`

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

Answer: lilies


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 book genres: `mystery`, `science fiction`
 - Each person has a unique birthday month: `sept`, `april`
 - Each person has a favorite color: `red`, `yellow`
 - People have unique heights: `very short`, `short`

## Clues:
1. The person who is short is somewhere to the right of the person who loves yellow.
2. The person who loves science fiction books is in the second house.
3. The person whose birthday is in April is the person who loves yellow.
4. Arnold is the person who loves science fiction books.
 What is the value of attribute House for the person whose attribute Height is very short? 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: `Peter`, `Eric`, `Arnold`
 - Each person has a unique birthday month: `jan`, `april`, `sept`
 - Each person has a unique type of pet: `cat`, `dog`, `fish`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `city`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`

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

Answer: 3


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

## Clues:
1. The person who enjoys mountain retreats and the person who prefers city breaks are next to each other.
2. Alice is in the fourth house.
3. The coffee drinker is the person who is a pizza lover.
4. The person who is very tall is directly left of the person who enjoys mountain retreats.
5. Peter is not in the first house.
6. The boba tea drinker is in the fourth house.
7. The person who loves stir fry is somewhere to the left of Eric.
8. The person who likes milk is Bob.
9. The person who loves the soup is not in the second house.
10. The person who loves the spaghetti eater is not in the second house.
11. The person who loves the stew is Eric.
12. The root beer lover is the person who is short.
13. The person who loves beach vacations is in the first house.
14. The person who loves stir fry is the person who enjoys mountain retreats.
15. The person who loves the soup is somewhere to the left of the person who loves the spaghetti eater.
16. The boba tea drinker is somewhere to the left of the person who is a pizza lover.
17. Arnold is the person who likes going on cruises.
18. The person who is very short is somewhere to the right of the person who has an average height.
19. The person who is tall is not in the fourth house.
20. The person who prefers city breaks is the coffee drinker.
21. Bob and the one who only drinks water are next to each other.
22. The person who loves the stew is the person who enjoys camping trips.
23. Arnold is the person who is very short.
24. Carol is in the fifth house.
 What is the value of attribute House for the person whose attribute Vacation is mountain? 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`, `Bob`, `Peter`, `Alice`
 - Everyone has something unique for lunch: `spaghetti`, `stew`, `grilled cheese`, `stir fry`, `pizza`
 - Each person has a favorite color: `red`, `green`, `white`, `blue`, `yellow`

## Clues:
1. The person who loves blue is somewhere to the left of the person who is a pizza lover.
2. Bob is somewhere to the left of the person who loves white.
3. Alice is directly left of the person who loves the stew.
4. The person who loves stir fry is in the fourth house.
5. The person who loves white is Peter.
6. The person who loves the spaghetti eater is Bob.
7. The person whose favorite color is green is Alice.
8. The person who loves blue is in the third house.
9. The person who loves yellow is somewhere to the right of the person who loves stir fry.
10. Alice is in the first house.
11. The person who loves yellow is Arnold.
 What is the value of attribute Food for the person whose attribute House is 5? Please reason step by step, and put your final answer within \boxed{}

Answer: pizza


5-8. In the "Young Photographer" club, 300 children attend. During the class, they divided into 100 groups of 3 people, and in each group, each person took one photo of the other two in their group. No one took any other photos. In total, there were 100 photos of "boy+boy" and 56 photos of "girl+girl". How many "mixed" groups were there, that is, groups that had both a boy and a girl?

Answer: 72


Example 1. Find the integral $\int \cos x \sin ^{2} x d x$.

Answer: \dfrac{\sin^3 x}{3} + C


4. What is the largest number of rooks that can be placed on a $3 n \times 3 n$ chessboard such that each rook is attacked by at most one of the other rooks.

Answer: 4n


Three. (20 points) Let the moving point $P$ be on the line $l_{1}: y=x-4$. Draw two tangents $PA$ and $PB$ from $P$ to the circle $\odot C: x^{2}+y^{2}=1$, where $A$ and $B$ are the points of tangency. Find the equation of the trajectory of the midpoint $M$ of segment $AB$.

Answer: \left(x - \dfrac{1}{8}\right)^2 + \left(y + \dfrac{1}{8}\right)^2 = \dfrac{1}{32}


Example 2: With point $C$ as the center of rotation, rotate $\triangle A B C$ to $\triangle A^{\prime} B^{\prime} C$. Let the midpoints of segments $B A^{\prime}$, $A C$, and $B^{\prime} C$ be $M$, $E$, and $F$ respectively. If $A C \neq B C$ and $E M = F M$, find the degree measure of $\angle E M F$.
(2005, Bulgarian National Mathematical Olympiad)

Answer: 60


The diagonals of a rectangle measure $\sqrt{1993} \mathrm{~cm}$. What are the dimensions of the rectangle, knowing that they are integers?

Answer: 12


A rectangular piece of paper with a length of 12 cm and a width of 8 cm has four identical isosceles right triangles cut from it. The remaining area is at least $\qquad$ square centimeters.

Answer: 24


Suppose you have an equilateral triangle divided into 9 smaller equilateral triangles as shown, with the bottom side horizontal. Starting from the top corner labeled $A$, you must walk to the bottom right corner labeled $B$, and are only allowed to take steps along the edges down to the left, down to the right, or horizontally to the right. Determine the number of possible paths.

![](https://cdn.mathpix.com/cropped/2024_04_17_f3b4fb3fc1d2fb8de6a1g-1.jpg?height=468&width=505&top_left_y=1905&top_left_x=821)

Answer: 22


$2.333 A=\left(\left(z^{2}+\frac{1}{z^{2}}\right)^{2}-4\left(z+\frac{1}{z}\right)^{2}+12\right)^{1 / 4}:(z-1)$.

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

$2.333 A=\left(\left(z^{2}+\frac{1}{z^{2}}\right)^{2}-4\left(z+\frac{1}{z}\right)^{2}+12\right)^{1 / 4}:(z-1)$.

Answer: \frac{z + 1}{z}


6. Given $\overrightarrow{O A}$ and $\overrightarrow{O B}$ are non-zero and non-collinear vectors. Let $\overrightarrow{O C}=\frac{1}{1+r} \overrightarrow{O A}+\frac{r}{1+r} \overrightarrow{O B}$. Define the set of points
$$
M=\left\{K \left\lvert\, \frac{\mid \overrightarrow{K A} \cdot \overrightarrow{K C}}{|\overrightarrow{K A}|}=\frac{\overrightarrow{K B} \cdot \overrightarrow{K C}}{|\overrightarrow{K B}|}\right.\right\} .
$$

When $K_{1} 、 K_{2} \in M$, if for any $r \geqslant 2$, the inequality $\left|\overrightarrow{K_{1} K_{2}}\right| \leqslant c|\overrightarrow{A B}|$ always holds, then the minimum value of the real number $c$ is $\qquad$

Answer: \dfrac{4}{3}


13. Given $\frac{1}{3} \leqslant a \leqslant 1$. If $f(x)=a x^{2}-2 x+1$ has a maximum value $M(a)$ and a minimum value $N(a)$ on $[1,3]$, let $g(a)=M(a)-N(a)$.
(1) Find the function expression for $g(a)$;
(2) Prove that $g(a) \geqslant \frac{1}{2}$ always holds.

Answer: \dfrac{1}{2}


Example 9. As shown in Figure 5, a circle intersects the three sides of an equilateral triangle at six points. Given that $A G=2, G F=13, F C=1, H J=7$. Find $DE$.
(33rd American High School Mathematics Examination)

Answer: 2\sqrt{22}


The distance of a light point from a sphere is equal to three times the radius of the sphere. How does the illuminated part of the sphere's surface relate to the lateral surface of the light cone?

Answer: \dfrac{2}{5}


Problem 10a.1. Find all pairs $(a ; b)$ of integers such that the system

$$
\left\lvert\, \begin{gathered}
x^{2}+2 a x-3 a-1=0 \\
y^{2}-2 b y+x=0
\end{gathered}\right.
$$

has exactly three real solutions.

Answer: (0, 1)


8.3. Find the largest natural number, all digits of which are different, and the product of these digits is a cube of some natural number.

Answer: 984321


For each prime \( p \), there is a kingdom of \( p \)-Landia consisting of \( p \) islands numbered \( 1, 2, \ldots, p \). Two distinct islands numbered \( n \) and \( m \) are connected by a bridge if and only if \( p \) divides \((n^2 - m + 1)(m^2 - n + 1) \). Prove that for infinitely many \( p \), there are two islands in \( p \)-Landia not connected by a chain of bridges.

Answer: \text{For infinitely many primes } p, \text{ the graph is disconnected.}


Example 5 Let the function $f(x)=a x^{2}+8 x+3(a<0)$, for a given negative number $a$, there is a largest positive number $l(a)$, such that the inequality $|f(x)| \leqslant 5$ holds for the entire interval $[0, l(a)]$. Question: For what value of $a$ is $l(a)$ the largest? Find this largest $l(a)$ and prove your conclusion.
(1998 National High School Mathematics Competition Problem)

Answer: -8


10. (10 points) 2012 students line up in a row to report numbers. If a student reports a single-digit number, the next student reports double that number; if a student reports a two-digit number, the next student reports the sum of the unit digit and 5. It is known that the first student reports 1, and by the time it reaches the 100th student, he adds another single-digit natural number to the number reported by the previous student, and no one else notices, continuing to report numbers according to the previous rules, until the last student reports the number 5. What number did the 100th student add to the number reported by the previous student? $\qquad$

Answer: 8


16. Let $O$ be the centre of the circumcircle of $\triangle A B C, P$ and $Q$ the midpoints of $A O$ and $B C$, respectively. Suppose $\angle C B A=4 \angle O P Q$ and $\angle A C B=6 \angle O P Q$. Find the size of $\angle O P Q$ in degrees.

Answer: 12


# Task 4.

In modern conditions, digitalization - the conversion of all information into digital code - is considered relevant. Each letter of the alphabet can be assigned a non-negative integer, called the code of the letter. Then, the weight of a word can be defined as the sum of the codes of all the letters in that word. Is it possible to encode the letters О, П, С, Т, Ь, Я with elementary codes, each consisting of a single digit from 0 to 9, so that the weight of the word "СТО" is not less than the weight of the word "ПЯТЬСОТ"? If such encoding is possible, in how many ways can it be implemented? If such encoding is possible, does it allow for the unambiguous restoration of a word from its code?

Answer: 100


36. Using a protractor with only $0^{\circ}, 4^{\circ}, 10^{\circ}, 16^{\circ}, 43^{\circ}, 89^{\circ}$ and $180^{\circ}$ markings, the maximum number of angles (not greater than $180^{\circ}$) that can be directly measured is $\qquad$.

Answer: 21


Let \(ABCD\) be a cyclic quadrilateral, and denote \(\left(b_{1}\right), \left(b_{2}\right), \left(b_{3}\right), \left(b{4}\right)\) as the respective bisectors of \(\widehat{DAB}, \widehat{ABC}, \widehat{BCD}, \widehat{CDA}\).

Let \(W, X, Y, Z\) be the respective intersection points of \(\left(b_{1}\right), \left(b_{2}\right), \left(b{3}\right)\) and \(\left(b{4}\right)\) with the circumcircle of \(ABCD\).

Show that \(WY\) and \(XZ\) intersect at their midpoint.

Answer: WY \text{ and } XZ \text{ intersect at their midpoint.}


II. (15 points) As shown in Figure 2, in $\triangle ABC$, it is known that $\angle ACB=90^{\circ}, AC=3, BC=4$. With point $B$ as the center, $\triangle ABC$ is rotated clockwise so that point $A$ lands on point $A_{1}$ on the extension of $CB$, and at this time, point $C$ lands on point $C_{1}$. Connect $AA_{1}$ and $CC_{1}$ intersecting at point $O$, $CC_{1}$ intersects $AB$ at point $D$, and $AA_{1}$ intersects $BC_{1}$ at point $E$. Find the area of quadrilateral $BDOE$.

Answer: \dfrac{105}{52}


12. There are $N(N+1) \div 2$ dominoes, each with two numbers, and these numbers are from $1 \sim N$ (the two numbers can be the same). No two dominoes have the same pair of numbers. Now, arrange these dominoes into several "trains," where in each "train," any two adjacent dominoes have the same number on the adjacent sides. The figure below shows a "train" when $N=3$.
When $N=2016$, at least $\qquad$ trains are needed to use up all $2016 \times(2016+1) \div 2$ dominoes.

Answer: 1008


3. In triangle $ABC$, the angle bisectors $BK$ and $CL$ are drawn. A point $N$ is marked on segment $BK$ such that $LN \parallel AC$. It turns out that $NK = LN$. Find the measure of angle $ABC$. (A. Kuznetsov)

Answer: 120^\circ


A cube painted in black and white colors with each face as one square was placed on a square of a chessboard and rolled over it in such a way that the cube landed on every square exactly once. Is it possible to paint the cube and roll it across the board in such a way that each time the color of the square and the face of the cube touching it match?

Answer: \text{No}


6. The base of the pyramid $T A B C D$ is a trapezoid $A B C D (B C \| A D)$. The distances from points $A$ and $B$ to the plane $T C D$ are $r_{1}$ and $r_{2}$, respectively. The area of triangle $T C D$ is $S$. Find the volume of the pyramid $T A B C D$.

Answer: \dfrac{S(r_1 + r_2)}{3}


Given 100 points in space such that for any four of these points, there exists a unit sphere that includes these four points (either inside or on the surface). Prove that there is a unit sphere that includes all 100 points.

Answer: \text{There exists a unit sphere containing all 100 points.}


15-10 A bag of peanuts contains 1988 peanuts. A monkey takes one peanut on the first day, and from the second day onwards, the number of peanuts it takes each day is the total of all the peanuts taken in previous days. If on a certain day the number of peanuts left in the bag is less than the total number of peanuts already taken, on that day it starts over by taking one peanut and follows the original rule for a new cycle. If this continues, on which day will the monkey have taken all the peanuts from the bag?
(Science Experimental Class Mathematics Admission Test, 1989)

Answer: 48


A computer generates all pairs of real numbers $x, y \in (0, 1)$ for which the numbers $a = x+my$ and $b = y+mx$ are both integers, where $m$ is a given positive integer. Finding one such pair $(x, y)$ takes $5$ seconds. Find $m$ if  the computer needs $595$ seconds to find all possible ordered pairs $(x, y)$.

Answer: 11


## Task 3 - 030823

In an auditorium, 300 chairs are arranged in several equally long rows. If 3 chairs are removed from each row to form a central aisle and these chairs are used to create 5 new rows (with a central aisle), the total number of seats remains the same.

How many chairs were originally in each row? Justify your claim!

Answer: 15


4. Xiaoling has two different shapes of cardboard, one is square and the other is rectangular. The ratio of the total number of square cardboard to the total number of rectangular cardboard is $1: 2$. She uses these cardboards to make some vertical and horizontal open-top boxes (as shown in Figure 2-16), using up all the cardboards. In the boxes Xiaoling made, what is the ratio of the total number of vertical boxes to the total number of horizontal boxes?

Answer: 1:2


299. When dividing the polynomial $x^{1051}-1$ by $x^{4}+x^{3}+2 x^{2}+x+1$, a quotient and a remainder are obtained. Find the coefficient of $x^{14}$ in the quotient.

Answer: -1


5. Four friends (from 6th grade, 1 point). Masha, Nina, Lena, and Olya are friends. They are all of different heights, but the difference is very small - it's hard to tell by eye. One day they decided to find out who is taller and who is shorter. It turned out that Nina is shorter than Masha, and Lena is taller than Olya. What is the probability that Nina is taller than Lena under these conditions?

Answer: \dfrac{1}{6}


There are eight identical Black Queens in the first row of a chessboard and eight identical White Queens in the last row. The Queens move one at a time, horizontally, vertically or diagonally by any number of squares as long as no other Queens are in the way. Black and White Queens move alternately. What is the minimal number of moves required for interchanging the Black and White Queens?

*(5 points)*

Answer: 24


Let  $d$  and  $n$  be positive integers such that  $d$  divides  $n$ ,  $n > 1000$ , and  $n$  is not a perfect square.  The minimum possible value of  $\left\lvert d - \sqrt{n} \right\rvert$  can be written in the form  $a\sqrt{b} + c$ , where  $b$  is a positive integer not divisible by the square of any prime, and  $a$  and  $c$  are nonzero integers (not necessarily positive).  Compute  $a+b+c$ .

*Proposed by Matthew Lerner-Brecher*

Answer: 38


Except for the first two terms, each term of the sequence  $1000, x, 1000-x,\ldots$  is obtained by subtracting the preceding term from the one before that.  The last term of the sequence is the first negative term encounted.  What positive integer  $x$  produces a sequence of maximum length?

Answer: 618


Consider a function  $ f(x)\equal{}xe^{\minus{}x^3}$  defined on any real numbers.

(1) Examine the variation and convexity of  $ f(x)$  to draw the garph of  $ f(x)$ .

(2) For a positive number  $ C$ , let  $ D_1$  be the region bounded by  $ y\equal{}f(x)$ , the  $ x$ -axis and  $ x\equal{}C$ . Denote  $ V_1(C)$  the volume obtained by rotation of  $ D_1$  about the  $ x$ -axis. Find  $ \lim_{C\rightarrow \infty} V_1(C)$ .

(3) Let  $ M$  be the maximum value of  $ y\equal{}f(x)$  for  $ x\geq 0$ . Denote  $ D_2$  the region bounded by  $ y\equal{}f(x)$ , the  $ y$ -axis and  $ y\equal{}M$ .
Find the volume  $ V_2$  obtained by rotation of  $ D_2$  about the  $ y$ -axis.

Answer: \pi(e^{-\frac{1}{3}}-\frac{2}{3})


Let  $a_{i}$  and  $b_{i}$ , where  $i \in \{1,2, \dots, 2005 \}$ , be real numbers such that the inequality  $(a_{i}x-b_{i})^{2} \ge \sum_{j=1, j \not= i}^{2005} (a_{j}x-b_{j})$  holds for all  $x \in \mathbb{R}$  and all  $i \in \{1,2, \dots, 2005 \}$ . Find the maximum possible number of positive numbers amongst  $a_{i}$  and  $b_{i}$ ,  $i \in \{1,2, \dots, 2005 \}$ .

Answer: 4009


Tim and Allen are playing a match of *tenus*.  In a match of *tenus*, the two players play a series of games, each of which is won by one of the two players. The match ends when one player has won exactly two more games than the other player, at which point the player who has won more games wins the match. In odd-numbered games, Tim wins with probability  $3/4$ , and in the even-numbered games, Allen wins with probability  $3/4$ .  What is the expected number of games in a match?

Answer: \frac{16}{3}


For positive reals  $p$  and  $q$ , define the *remainder* when  $p$  and  $q$  as the smallest nonnegative real  $r$  such that  $\tfrac{p-r}{q}$  is an integer. For an ordered pair  $(a, b)$  of positive integers, let  $r_1$  and  $r_2$  be the remainder when  $a\sqrt{2} + b\sqrt{3}$  is divided by  $\sqrt{2}$  and  $\sqrt{3}$  respectively. Find the number of pairs  $(a, b)$  such that  $a, b \le 20$  and  $r_1 + r_2 = \sqrt{2}$ .

Answer: 16


Let  $n \geq 3$  be a positive integer. Find the smallest positive real  $k$ , satisfying the following condition: if  $G$  is a connected graph with  $n$  vertices and  $m$  edges, then it is always possible to delete at most  $k(m-\lfloor \frac{n} {2} \rfloor)$  edges, so that the resulting graph has a proper vertex coloring with two colors.

Answer: \frac{1}{2}


Contessa is taking a random lattice walk in the plane, starting at  $(1,1)$ . (In a random lattice walk, one moves up, down, left, or right  $1$  unit with equal probability at each step.) If she lands on a point of the form  $(6m,6n)$  for  $m,n \in \mathbb{Z}$ , she ascends to heaven, but if she lands on a point of the form  $(6m+3,6n+3)$  for  $m,n \in \mathbb{Z}$ , she descends to hell. What is the probability she ascends to heaven?

Answer: \frac{13}{22}


A natural number  $n$  is called *perfect* if it is equal to the sum of all its natural divisors other than  $n$ . For example, the number  $6$  is perfect because  $6 = 1 + 2 + 3$ . Find all even perfect numbers that can be given as the sum of two cubes positive integers.

Answer: 28


Let  $P$  be a square pyramid whose base consists of the four vertices  $(0, 0, 0), (3, 0, 0), (3, 3, 0)$ , and  $(0, 3, 0)$ , and whose apex is the point  $(1, 1, 3)$ . Let  $Q$  be a square pyramid whose base is the same as the base of  $P$ , and whose apex is the point  $(2, 2, 3)$ . Find the volume of the intersection of the interiors of  $P$  and  $Q$ .

Answer: \frac{27}{4}


A teacher wants to divide the  $2010$  questions she asked in the exams during the school year into three folders of  $670$  questions and give each folder to a student who solved all  $670$  questions in that folder. Determine the minimum number of students in the class that makes this possible for all possible situations in which there are at most two students who did not solve any given question.

Answer: 6


Let triangle  $ABC$  be a right triangle in the xy-plane with a right angle at  $C$ . Given that the length of the hypotenuse  $AB$  is 60, and that the medians through  $A$  and  $B$  lie along the lines  $y=x+3$  and  $y=2x+4$  respectively, find the area of triangle  $ABC$ .

Answer: 400


Two disks of radius 1 are drawn so that each disk's circumference passes through the center of the other disk. What is the circumference of the region in which they overlap?

Answer: \frac{4\pi}{3}


Let  $m > 1$  be an integer.  A sequence  $a_1, a_2, a_3, \ldots$  is defined by  $a_1 = a_2 = 1$ ,  $a_3 = 4$ , and for all  $n \ge 4$ ,   $$ a_n = m(a_{n - 1} + a_{n - 2}) - a_{n - 3}. $$  Determine all integers  $m$  such that every term of the sequence is a square.

Answer: 2


On the number line, consider the point $x$ that corresponds to the value $10$. Consider $24$ distinct integer points $y_1, y_2, \ldots, y_{24}$ on the number line such that for all $k$ where $1 \leq k \leq 12$, we have $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of \[\sum_{n=1}^{24} (|y_n-1| + |y_n+1|).\]

Answer: 494


Given three equilateral triangles $\triangle HOW$, $\triangle BOW$, and $\triangle DAH$ in a plane:
- $WO = 7$
- $AH = 2$

It is also given that the points $D$, $A$, and $B$ are collinear in that order.

Find the length of $BA$.

Answer: 11


Let $ABC$ be a triangle with vertices located at the centers of mass of Vincent Huang's house, Tristan Shin's house, and Edward Wan's house; assume the three are not collinear. Let $N = 2017$, and define the $A$-antipodes to be the points $A_1, \dots, A_N$ on segment $BC$ such that $BA_1 = A_1A_2 = \cdots = A_{N-1}A_N = A_NC$. Similarly, define the $B$-antipodes and $C$-antipodes. A line $\ell_A$ through $A$ is called a qevian if it passes through an $A$-antipode. Similarly, define qevians through $B$ and $C$. Compute the number of ordered triples $(\ell_A, \ell_B, \ell_C)$ of concurrent qevians through $A$, $B$, and $C$, respectively.

Answer: 6049


For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$. What is the smallest positive integer $n$ such that \[ \sum_{t \mid n} d(t)^3 \] is divisible by $35$?

Answer: 864


How many ways are there to arrange the array of positive integers such that:
- The number $2012$ is in the first position.
- The number $1$ is in the last position.

Note: Each number following $n$ must be a positive integer less than $\sqrt{n}$.

Answer: 201


Positive integers $a < b < c$ are such that $b + a$ is a multiple of $b - a$ and $c + b$ is a multiple of $c - b$. If $a$ is a $2011$-digit number and $b$ is a $2012$-digit number, exactly how many digits does $c$ have?

Answer: 2013


Jeffrey rolls three fair six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as \( \frac{m}{n} \) for relatively prime positive integers \( m \) and \( n \). Compute \( m+n \).

Answer: 101


A $3 \times 3$ grid is to be painted with three colors: red, green, and blue, such that:
1. No two squares that share an edge are the same color.
2. No two corner squares on the same edge of the grid have the same color.

For example, the upper-left and bottom-left squares cannot both be red, as that would violate condition (ii). In how many ways can this be done? Note that rotations and reflections are considered distinct colorings.

Answer: 24


Portraits of famous scientists hang on a wall. The scientists lived between 1600 and 2008, and none of them lived longer than 80 years. Vasya multiplied the years of birth of these scientists, and Petya multiplied the years of their death. Petya's result is exactly $\frac{5}{4}$ times greater than Vasya's result. What is the minimum number of portraits that can be on the wall?

Answer: 5


\( \triangle ABC \) has \( AB = 4 \) and \( AC = 6 \). Let point \( D \) be on line \( AB \) so that \( A \) is between \( B \) and \( D \). Let the angle bisector of \( \angle BAC \) intersect line \( BC \) at \( E \), and let the angle bisector of \( \angle DAC \) intersect line \( BC \) at \( F \). Given that \( AE = AF \), find the square of the circumcircle's radius' length.

Answer: 13


Let $ABC$ be a triangle with $AB = 4024$, $AC = 4024$, and $BC = 2012$. The reflection of line $AC$ over line $AB$ meets the circumcircle of $\triangle ABC$ at a point $D \neq A$. Find the length of segment $CD$.

Answer: 3521


Suppose 32 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?

Answer: 3616


You have 2020 piles of coins in front of you. The first pile contains 1 coin, the second pile contains 2 coins, the third pile contains 3 coins, and so on. Therefore, the 2020th pile contains 2020 coins. Guess a positive integer $k$, such that the piles contain at least $k$ coins. Take away exactly $k$ coins from these piles. Find the minimum number of turns you need to take away all of these coins.

Answer: 11


Let $M$ be a set of six distinct positive integers whose sum is $60$. These numbers are written on the faces of a cube, one number to each face. A move consists of choosing three faces of the cube that share a common vertex and adding $1$ to the numbers on those faces. Determine the number of sets $M$ for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.

Answer: 84


Passing through the origin of the coordinate plane are $180$ lines, including the coordinate axes, which form $1^\circ$ angles with one another at the origin. Determine the sum of the x-coordinates of the points of intersection of these lines with the line $y = 100-x$.

Answer: 8950


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Carol`, `Alice`, `Peter`, `Eric`
 - The mothers' names in different houses are unique: `Sarah`, `Janelle`, `Aniya`, `Kailyn`, `Holly`, `Penny`
 - Each mother is accompanied by their child: `Fred`, `Samantha`, `Bella`, `Meredith`, `Alice`, `Timothy`
 - Each person prefers a unique type of vacation: `city`, `mountain`, `camping`, `beach`, `cruise`, `cultural`
 - People have unique favorite book genres: `romance`, `mystery`, `historical fiction`, `science fiction`, `biography`, `fantasy`

## Clues:
1. The person who loves beach vacations is not in the second house.
2. The person who loves fantasy books is somewhere to the left of Peter.
3. The person whose mother's name is Sarah is the person who prefers city breaks.
4. The person who enjoys camping trips is somewhere to the right of Peter.
5. The person who likes going on cruises is the person's child is named Meredith.
6. There is one house between the person who is the mother of Timothy and Eric.
7. The person whose mother's name is Janelle is not in the second house.
8. The person's child is named Fred is somewhere to the left of Eric.
9. The person who goes on cultural tours is in the fourth house.
10. The person whose mother's name is Janelle is not in the first house.
11. The person whose mother's name is Holly is somewhere to the right of the person who loves historical fiction books.
12. The person's child is named Bella is somewhere to the left of Alice.
13. Arnold is somewhere to the right of the person who loves fantasy books.
14. The person who loves mystery books is in the fourth house.
15. The person's child is named Alice is the person who enjoys camping trips.
16. The person whose mother's name is Kailyn is the person who likes going on cruises.
17. There are two houses between the person who loves fantasy books and The person whose mother's name is Aniya.
18. The person who loves fantasy books is Carol.
19. The person who likes going on cruises is the person who loves biography books.
20. The person who loves fantasy books is in the third house.
21. The person whose mother's name is Aniya is the person who loves romance books.
22. The person whose mother's name is Janelle is not in the fourth house.
23. The person's child is named Fred is not in the fourth house.
24. The person who loves biography books is not in the second house.
25. There are two houses between The person whose mother's name is Holly and Eric.
 What is the value of attribute Vacation for the person whose attribute BookGenre is science fiction? Please reason step by step, and put your final answer within \boxed{}

Answer: mountain


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`, `Bob`, `Arnold`, `Carol`, `Alice`
 - Each person has a unique birthday month: `may`, `sept`, `jan`, `mar`, `feb`, `april`
 - People have unique hair colors: `brown`, `blonde`, `red`, `auburn`, `black`, `gray`
 - Each mother is accompanied by their child: `Alice`, `Samantha`, `Fred`, `Meredith`, `Timothy`, `Bella`
 - Everyone has a favorite smoothie: `blueberry`, `cherry`, `watermelon`, `lime`, `dragonfruit`, `desert`
 - The mothers' names in different houses are unique: `Sarah`, `Aniya`, `Janelle`, `Holly`, `Kailyn`, `Penny`

## Clues:
1. The person whose birthday is in September and the person's child is named Alice are next to each other.
2. The person whose birthday is in April is the person who is the mother of Timothy.
3. The person who likes Cherry smoothies is the person who has gray hair.
4. Alice is somewhere to the right of Arnold.
5. The person's child is named Alice is somewhere to the right of The person whose mother's name is Sarah.
6. Peter is the person who drinks Blueberry smoothies.
7. The person whose mother's name is Janelle is the person whose birthday is in May.
8. The person whose birthday is in January is somewhere to the left of The person whose mother's name is Sarah.
9. The Watermelon smoothie lover is the person who has brown hair.
10. The person who has red hair is somewhere to the right of the person who has gray hair.
11. The person who drinks Lime smoothies is in the fifth house.
12. The person who has black hair and the person's child is named Meredith are next to each other.
13. The person whose birthday is in March is somewhere to the right of the person who has red hair.
14. The person whose mother's name is Sarah is somewhere to the left of the person who has auburn hair.
15. The person who has black hair is directly left of The person whose mother's name is Penny.
16. The person's child is named Samantha is The person whose mother's name is Holly.
17. The person who has blonde hair and Bob are next to each other.
18. The person who has gray hair is the person whose birthday is in January.
19. The person who has black hair is in the first house.
20. The person whose mother's name is Sarah is the person who has blonde hair.
21. The person whose mother's name is Kailyn is Arnold.
22. The person whose birthday is in January is The person whose mother's name is Kailyn.
23. The person whose birthday is in September is Carol.
24. The person who drinks Lime smoothies is somewhere to the right of the Desert smoothie lover.
25. The person's child is named Fred is not in the third house.
 What is the value of attribute Smoothie for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: watermelon


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`, `Carol`, `Bob`, `Peter`, `Eric`, `Arnold`
 - They all have a unique favorite flower: `lilies`, `tulips`, `roses`, `daffodils`, `iris`, `carnations`
 - The people are of nationalities: `norwegian`, `dane`, `swede`, `german`, `brit`, `chinese`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `desert`, `dragonfruit`, `blueberry`, `lime`
 - Each person has a unique hobby: `photography`, `cooking`, `knitting`, `woodworking`, `painting`, `gardening`
 - Each person has a unique type of pet: `rabbit`, `hamster`, `fish`, `cat`, `bird`, `dog`

## Clues:
1. The Swedish person is the person who loves cooking.
2. The person who drinks Blueberry smoothies is Alice.
3. The person who loves the rose bouquet is not in the sixth house.
4. The photography enthusiast is somewhere to the right of the Dragonfruit smoothie lover.
5. The person who loves cooking is the person who loves a bouquet of daffodils.
6. The person who loves the boquet of lilies and the person who loves the boquet of iris are next to each other.
7. The person who paints as a hobby is Alice.
8. Arnold is somewhere to the right of the Watermelon smoothie lover.
9. The woodworking hobbyist is somewhere to the right of the German.
10. The woodworking hobbyist is somewhere to the left of the person who loves a bouquet of daffodils.
11. The Dane is the Watermelon smoothie lover.
12. The Dragonfruit smoothie lover is Peter.
13. The person who drinks Lime smoothies is the woodworking hobbyist.
14. There is one house between the Watermelon smoothie lover and the person who likes Cherry smoothies.
15. The person who keeps a pet bird is the person who drinks Blueberry smoothies.
16. The Desert smoothie lover is the person who enjoys gardening.
17. The person with a pet hamster is in the fifth house.
18. The person who owns a dog is the Norwegian.
19. The person who has a cat is somewhere to the right of Bob.
20. The British person is directly left of Carol.
21. The person who loves the boquet of iris is Arnold.
22. The German is somewhere to the right of the person who loves a carnations arrangement.
23. Bob is the photography enthusiast.
24. The person with an aquarium of fish is Arnold.
 What is the value of attribute House for the person whose attribute Hobby is cooking? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Peter`, `Eric`, `Arnold`, `Carol`
 - Each mother is accompanied by their child: `Fred`, `Timothy`, `Samantha`, `Alice`, `Meredith`, `Bella`
 - People have unique favorite music genres: `pop`, `hip hop`, `classical`, `jazz`, `rock`, `country`
 - People have unique heights: `average`, `very tall`, `tall`, `super tall`, `very short`, `short`

## Clues:
1. There is one house between the person's child is named Samantha and the person who is short.
2. The person's child is named Alice is Bob.
3. The person who loves country music is directly left of Arnold.
4. Alice is the person who is tall.
5. The person who loves pop music is Eric.
6. Bob is somewhere to the right of the person who is super tall.
7. The person's child is named Fred is Peter.
8. The person's child is named Bella is the person who loves hip-hop music.
9. The person who is the mother of Timothy is not in the sixth house.
10. The person who is super tall is somewhere to the right of the person who has an average height.
11. The person's child is named Alice is somewhere to the right of Arnold.
12. There is one house between the person who is short and the person who is very short.
13. The person who is very short is in the fifth house.
14. The person who loves jazz music is not in the fifth house.
15. Carol is somewhere to the left of the person who is the mother of Timothy.
16. The person who is very tall is not in the sixth house.
17. The person who loves classical music is in the sixth house.
18. The person who loves rock music is in the first house.
 What is the value of attribute House for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - Each person has a unique level of education: `associate`, `bachelor`, `high school`
 - People have unique heights: `short`, `very short`, `average`
 - Each person has a unique hobby: `cooking`, `gardening`, `photography`
 - People have unique favorite book genres: `romance`, `mystery`, `science fiction`
 - Everyone has a favorite smoothie: `cherry`, `desert`, `watermelon`

## Clues:
1. The Watermelon smoothie lover is the person who is very short.
2. The person who likes Cherry smoothies is the person who loves romance books.
3. The Desert smoothie lover is the person who enjoys gardening.
4. The Watermelon smoothie lover is somewhere to the right of the person with a high school diploma.
5. The person with an associate's degree is directly left of the person with a high school diploma.
6. The photography enthusiast is the person with an associate's degree.
7. The person who likes Cherry smoothies is Arnold.
8. The person who loves science fiction books is somewhere to the right of the person who enjoys gardening.
9. The person who is short is not in the first house.
10. The Watermelon smoothie lover is somewhere to the right of Eric.
 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: science fiction


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

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

Answer: very tall


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Carol`, `Alice`, `Eric`, `Bob`
 - People have unique favorite music genres: `pop`, `hip hop`, `rock`, `jazz`, `classical`, `country`
 - The people keep unique animals: `fish`, `dog`, `cat`, `rabbit`, `horse`, `bird`
 - Each person lives in a unique style of house: `modern`, `colonial`, `mediterranean`, `ranch`, `victorian`, `craftsman`
 - Each person has a unique hobby: `gardening`, `woodworking`, `knitting`, `cooking`, `painting`, `photography`
 - The people are of nationalities: `brit`, `german`, `norwegian`, `swede`, `dane`, `chinese`

## Clues:
1. The cat lover is the person in a Mediterranean-style villa.
2. The person who loves country music is somewhere to the right of the rabbit owner.
3. The person who loves pop music is not in the second house.
4. Alice is somewhere to the right of Arnold.
5. The person who loves jazz music and the photography enthusiast are next to each other.
6. The person who loves rock music is the German.
7. The woodworking hobbyist is somewhere to the right of the person in a modern-style house.
8. The person who loves hip-hop music is the Norwegian.
9. The cat lover is somewhere to the left of the woodworking hobbyist.
10. Alice is the person who paints as a hobby.
11. The person in a Craftsman-style house is Carol.
12. Peter is the fish enthusiast.
13. The person who loves hip-hop music is not in the fifth house.
14. The person who loves pop music is not in the sixth house.
15. The person who loves cooking is somewhere to the left of the Swedish person.
16. The Dane is directly left of the person who keeps horses.
17. The rabbit owner is somewhere to the left of the British person.
18. There is one house between Alice and the person who loves country music.
19. The person who paints as a hobby is the person living in a colonial-style house.
20. The German is somewhere to the left of the photography enthusiast.
21. Eric is the Norwegian.
22. The person residing in a Victorian house is directly left of the person who enjoys gardening.
23. There are two houses between the person who loves cooking and the person who loves classical music.
24. The bird keeper is the person who loves cooking.
25. The cat lover is directly left of Bob.
26. Eric is not in the first house.
27. Carol is somewhere to the right of the German.
28. The rabbit owner is directly left of the person in a modern-style house.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 1