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

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

Answer: Peter


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`
 - The people keep unique animals: `horse`, `bird`, `cat`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `desert`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`
 - People have unique favorite music genres: `pop`, `classical`, `rock`
 - Each person has a favorite color: `yellow`, `white`, `red`

## Clues:
1. The cat lover is in the third house.
2. There is one house between the person who loves yellow and the cat lover.
3. The person who loves classical music is directly left of Eric.
4. Arnold is the Desert smoothie lover.
5. The bird keeper is the Watermelon smoothie lover.
6. The person whose mother's name is Janelle is somewhere to the left of the person who loves white.
7. The person who loves yellow is the person who loves rock music.
8. The person whose mother's name is Holly is directly left of Peter.
 What is the value of attribute Mother for the person whose attribute Smoothie is desert? Please reason step by step, and put your final answer within \boxed{}

Answer: Holly


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 lives in a unique style of house: `colonial`, `victorian`, `ranch`
 - People have unique favorite book genres: `science fiction`, `mystery`, `romance`
 - Each person has a unique type of pet: `fish`, `cat`, `dog`
 - Everyone has a unique favorite cigar: `prince`, `blue master`, `pall mall`
 - Each person has a unique hobby: `cooking`, `photography`, `gardening`

## Clues:
1. The person who loves cooking is in the second house.
2. The person who loves romance books is the person who smokes Blue Master.
3. The person who loves mystery books is the person partial to Pall Mall.
4. The person who smokes Blue Master and the person who loves cooking are next to each other.
5. The person who owns a dog is the person in a ranch-style home.
6. The person residing in a Victorian house is the person with an aquarium of fish.
7. The person who has a cat is the Prince smoker.
8. Arnold and the person who has a cat are next to each other.
9. The photography enthusiast is directly left of Arnold.
10. Eric is not in the third house.
11. Eric is the person with an aquarium of fish.
 What is the value of attribute House for the person whose attribute HouseStyle is colonial? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


Let us consider all rectangles with sides of length $a$ and $b$, both of which are whole numbers. Do more of these rectangles have a perimeter of $2000$ or a perimeter of $2002$?

Answer: 2002


There are $n \ge 3$ puddings in a room. If a pudding $A$ hates a pudding $B$, then $B$ hates $A$ as well. Suppose the following two conditions hold:

1. Given any four puddings, there are two puddings who like each other.
2. For any positive integer $m$, if there are $m$ puddings who like each other, then there exist 3 puddings (from the other $n-m$ puddings) that hate each other.

Find the smallest possible value of $n$.

Answer: 7


Julia adds up the numbers from $1$ to $2016$ in a calculator. However, every time she inputs a $2$, the calculator malfunctions and inputs a $3$ instead. For example, when Julia inputs $202$, the calculator inputs $303$ instead. How much larger is the total sum returned by the broken calculator?

Answer: 39202


Of the $9$ people who reached the final stage of the competition, only $4$ should receive a prize. The candidates were renumbered and lined up in a circle. Then a certain number $m$ (possibly greater than $9$) and the direction of reference were determined. People began to be counted, starting from the first. Each one became a winner and was eliminated from the drawing, and counting, starting from the next, continued until four winners were identified. The first three prizes were awarded to three people who had numbers $2$, $7$, and $5$ in the original lineup (they were eliminated in that order). What number did the fourth winner of the competition have in the initial lineup?

Answer: 8


Centuries ago, the pirate Captain Blackboard buried a vast amount of treasure in a single cell of a $2 \times 4$ grid-structured island. The treasure was buried in a single cell of an $M \times N$ ($2 \le M$, $N$) grid. You and your crew have reached the island and have brought special treasure detectors to find the cell with the treasure.

For each detector, you can set it up to scan a specific subgrid $[a,b] \times [c,d]$ with $1 \le a \le b \le 2$ and $1 \le c \le d \le 4$. Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected.

You plan on setting up $Q$ detectors, which may only be run simultaneously after all $Q$ detectors are ready. What is the minimum $Q$ required to guarantee to determine the location of the Blackboard’s legendary treasure?

Answer: 4


The equation $AB \times CD = EFGH$, where each letter $A$, $B$, $C$, $D$, $E$, $F$, $G$, and $H$ represents a different digit, and the values of $A$, $C$, and $E$ are all nonzero, has many solutions. For example, $46 \times 85 = 3910$. Find the smallest value of the four-digit number $EFGH$ for which there is a solution.

Answer: 1058


The first $510$ positive integers are written on a blackboard: $1, 2, 3, \ldots, 510$. An operation consists of erasing two numbers whose sum is a prime number. What is the maximum number of operations that can be performed in a row? Show how it is accomplished and explain why it cannot be done in more operations.

Answer: 255


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

Answer: 81


A semicircular paper is folded along a chord such that the folded circular arc is tangent to the diameter of the semicircle. The radius of the semicircle is $4$ units, and the point of tangency divides the diameter in the ratio $7:1$. If the length of the crease (the dotted line segment in the figure) is $\ell$, then determine $\ell^2$.

Answer: 39


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

Answer: 2024


Define a permutation of the set $\{1,2,3,...,n\}$ to be \textit{sortable} if, upon cancelling an appropriate term of such permutation, the remaining $n-1$ terms are in increasing order. If $f(n)$ is the number of sortable permutations of $\{1,2,3,...,n\}$, find the remainder when \(\sum\limits_{i=1}^{2018} (-1)^i \cdot f(i)\) is divided by $1000$. Note that the empty set is considered sortable.

Answer: 153


A number is called "cool" if the sum of its digits is a multiple of $17$, and the sum of the digits of its successor is also a multiple of $17$. What is the smallest cool number?

Answer: 8899


Let $ABCD$ be a trapezoid such that $AB \parallel CD$ and let $P = AC \cap BD$, with $AB = 21$, $CD = 7$, $AD = 13$, and $[ABCD] = 168$. Let the line parallel to $AB$ through $P$ intersect the circumcircle of $\triangle BCP$ at $X$. The circumcircles of $\triangle BCP$ and $\triangle APD$ intersect at $P$ and $Y$. Let $XY \cap BC = Z$. If $\angle ADC$ is obtuse, then $BZ = \frac{a}{b}$, where $a$ and $b$ are coprime positive integers. Compute $a + b$.

Answer: 17


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Arnold`, `Eric`
 - Each person lives in a unique style of house: `victorian`, `ranch`, `craftsman`, `colonial`
 - People have unique favorite music genres: `pop`, `classical`, `rock`, `jazz`
 - They all have a unique favorite flower: `roses`, `lilies`, `carnations`, `daffodils`
 - Each person prefers a unique type of vacation: `city`, `beach`, `mountain`, `cruise`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`

## Clues:
1. The person who loves classical music is Peter.
2. The person in a ranch-style home is not in the fourth house.
3. The person who loves a bouquet of daffodils is the person living in a colonial-style house.
4. The person living in a colonial-style house is the person who loves white.
5. The person who prefers city breaks and the person whose favorite color is red are next to each other.
6. The person who loves the rose bouquet is not in the fourth house.
7. The person who likes going on cruises is in the second house.
8. The person who prefers city breaks is the person who loves rock music.
9. The person who loves pop music is Alice.
10. The person who loves the rose bouquet is the person who prefers city breaks.
11. Eric is the person who enjoys mountain retreats.
12. The person who enjoys mountain retreats is directly left of the person who loves a bouquet of daffodils.
13. The person who loves yellow is somewhere to the left of the person who likes going on cruises.
14. The person who enjoys mountain retreats is the person residing in a Victorian house.
15. The person who loves classical music is the person who likes going on cruises.
16. Peter is somewhere to the right of the person who loves the boquet of lilies.
 What is the value of attribute Flower for the person whose attribute House is 1? 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: `Peter`, `Alice`, `Bob`, `Eric`, `Arnold`
 - People have unique heights: `very tall`, `average`, `tall`, `very short`, `short`

## Clues:
1. The person who is short is in the second house.
2. Peter is directly left of Bob.
3. Eric is somewhere to the left of Peter.
4. The person who is very tall is directly left of Peter.
5. Alice is directly left of the person who has an average height.
6. The person who is short and the person who is very short are next to each other.
7. The person who has an average height is in the fifth house.
 What is the value of attribute House for the person whose attribute Height is tall? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Alice`, `Arnold`
 - People have unique hair colors: `brown`, `black`, `blonde`, `red`
 - People have unique favorite sports: `tennis`, `basketball`, `soccer`, `swimming`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `dragonfruit`, `desert`
 - Each person has a unique birthday month: `april`, `sept`, `feb`, `jan`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `iphone 13`, `google pixel 6`

## Clues:
1. The person whose birthday is in January is somewhere to the left of the person who loves basketball.
2. The person whose birthday is in February is directly left of the person who has red hair.
3. Eric is the person who uses a OnePlus 9.
4. The person whose birthday is in April is Arnold.
5. The person who uses a Samsung Galaxy S21 and the person who likes Cherry smoothies are next to each other.
6. The person who has blonde hair and Peter are next to each other.
7. The person who has black hair is somewhere to the right of the person who has brown hair.
8. The person who has black hair is the person whose birthday is in February.
9. The Dragonfruit smoothie lover is in the second house.
10. The person who loves soccer is somewhere to the right of the person who uses a OnePlus 9.
11. There are two houses between the person who loves tennis and the person who loves basketball.
12. The person who has black hair is the person who uses a Samsung Galaxy S21.
13. Alice is the person who likes Cherry smoothies.
14. The Desert smoothie lover is the person who uses a Google Pixel 6.
 What is the value of attribute HairColor for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: red


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Bob`, `Eric`, `Carol`
 - Each person has a unique hobby: `cooking`, `gardening`, `painting`, `knitting`, `photography`, `woodworking`
 - Each person has a unique birthday month: `april`, `sept`, `mar`, `jan`, `feb`, `may`
 - Everyone has a unique favorite cigar: `blends`, `prince`, `pall mall`, `dunhill`, `yellow monster`, `blue master`

## Clues:
1. The person whose birthday is in April is Eric.
2. Arnold is in the first house.
3. The person whose birthday is in January is the person who smokes many unique blends.
4. The Dunhill smoker is Alice.
5. Bob is the person whose birthday is in May.
6. The person whose birthday is in September and the photography enthusiast are next to each other.
7. Bob is the person who smokes Blue Master.
8. The person who loves cooking is the person whose birthday is in May.
9. The woodworking hobbyist and the person whose birthday is in March are next to each other.
10. The person who enjoys knitting is directly left of the person who smokes Yellow Monster.
11. Peter and the person who paints as a hobby are next to each other.
12. The person whose birthday is in March is somewhere to the left of the Prince smoker.
13. The person whose birthday is in February is somewhere to the left of the person who smokes many unique blends.
14. The person whose birthday is in March is somewhere to the right of the person whose birthday is in September.
15. The person who smokes many unique blends is directly left of the person whose birthday is in May.
16. The photography enthusiast and Peter are next to each other.
 What is the value of attribute Birthday for the person whose attribute Cigar is dunhill? Please reason step by step, and put your final answer within \boxed{}

Answer: mar


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

## Clues:
1. The fish enthusiast is in the first house.
2. Carol is somewhere to the left of the person in a Mediterranean-style villa.
3. The bird keeper is the person who loves purple.
4. The person in a ranch-style home is the one who only drinks water.
5. The coffee drinker is not in the fourth house.
6. The person living in a colonial-style house and the boba tea drinker are next to each other.
7. The cat lover is the person who is a teacher.
8. The person in a modern-style house is in the third house.
9. The person who is an engineer and Peter are next to each other.
10. Arnold is the person who is an artist.
11. The person whose favorite color is red is Arnold.
12. The coffee drinker is somewhere to the right of the person who is a teacher.
13. The bird keeper is the person in a ranch-style home.
14. The rabbit owner is in the sixth house.
15. The boba tea drinker is Peter.
16. Eric is the cat lover.
17. The person who is a nurse is directly left of the person in a modern-style house.
18. The bird keeper is Carol.
19. There are two houses between the cat lover and the boba tea drinker.
20. Bob is the tea drinker.
21. The person residing in a Victorian house is the dog owner.
22. The person whose favorite color is green is in the fourth house.
23. The person who loves blue is the root beer lover.
24. The person who loves white is not in the sixth house.
25. Peter is the person who is a doctor.
 What is the value of attribute House for the person whose attribute Drink is milk? 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: `Bob`, `Eric`, `Alice`, `Arnold`, `Peter`
 - The people keep unique animals: `horse`, `dog`, `cat`, `fish`, `bird`
 - People use unique phone models: `oneplus 9`, `iphone 13`, `huawei p50`, `google pixel 6`, `samsung galaxy s21`

## Clues:
1. Bob is not in the fourth house.
2. The person who uses an iPhone 13 is somewhere to the left of the person who uses a OnePlus 9.
3. The person who uses a Huawei P50 is somewhere to the left of Alice.
4. Arnold is the dog owner.
5. Peter is somewhere to the left of the cat lover.
6. Arnold is not in the fourth house.
7. Peter is somewhere to the left of the fish enthusiast.
8. Peter is the person who uses a Huawei P50.
9. The person who uses a OnePlus 9 is directly left of the person who uses a Google Pixel 6.
10. The dog owner is somewhere to the right of the person who keeps horses.
11. The person who keeps horses is not in the first house.
12. There is one house between Bob and the person who uses an iPhone 13.
13. Eric is the person who uses a Google Pixel 6.
14. The person who uses a Google Pixel 6 is somewhere to the right of the fish enthusiast.
 What is the value of attribute PhoneModel for the person whose attribute Animal is dog? Please reason step by step, and put your final answer within \boxed{}

Answer: iphone 13


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Bob`, `Peter`, `Carol`, `Alice`
 - Each mother is accompanied by their child: `Fred`, `Bella`, `Timothy`, `Meredith`, `Samantha`, `Alice`
 - Everyone has something unique for lunch: `pizza`, `stir fry`, `grilled cheese`, `spaghetti`, `soup`, `stew`
 - Each person has a unique type of pet: `rabbit`, `fish`, `cat`, `dog`, `bird`, `hamster`
 - Each person has a unique birthday month: `may`, `sept`, `mar`, `jan`, `april`, `feb`
 - Each person has a unique favorite drink: `water`, `milk`, `coffee`, `tea`, `boba tea`, `root beer`

## Clues:
1. The person whose birthday is in May is the person who loves the spaghetti eater.
2. The person's child is named Fred is the person who loves the stew.
3. The person who likes milk is the person with a pet hamster.
4. There is one house between the person who loves the spaghetti eater and the person who has a cat.
5. The person who is the mother of Timothy is Bob.
6. The person whose birthday is in January is directly left of the one who only drinks water.
7. Carol is the coffee drinker.
8. Peter is not in the fifth house.
9. The person whose birthday is in April is the person's child is named Alice.
10. The person who loves eating grilled cheese is somewhere to the right of the person whose birthday is in September.
11. The person's child is named Bella and Bob are next to each other.
12. The person who owns a rabbit is Peter.
13. The person who keeps a pet bird is the person who loves the soup.
14. The person who owns a rabbit is somewhere to the right of the person who loves stir fry.
15. The person who loves eating grilled cheese is somewhere to the left of the person who owns a rabbit.
16. The person whose birthday is in March is the tea drinker.
17. Peter is the person's child is named Bella.
18. The person who is the mother of Timothy is somewhere to the right of the person's child is named Bella.
19. The person with an aquarium of fish and the person's child is named Bella are next to each other.
20. The one who only drinks water is the person's child is named Fred.
21. The person whose birthday is in May is somewhere to the right of the person with an aquarium of fish.
22. The person with a pet hamster is the person's child is named Meredith.
23. The root beer lover is the person who loves the spaghetti eater.
24. Carol is directly left of Alice.
25. Eric is in the third house.
26. The person who is a pizza lover is Bob.
 What is the value of attribute Food for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: pizza


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - Each person prefers a unique type of vacation: `mountain`, `city`, `beach`
 - People have unique heights: `very short`, `average`, `short`
 - They all have a unique favorite flower: `carnations`, `daffodils`, `lilies`
 - People have unique hair colors: `brown`, `black`, `blonde`
 - Each person has a unique level of education: `associate`, `bachelor`, `high school`

## Clues:
1. Peter is the person who has an average height.
2. The person who loves a bouquet of daffodils is Arnold.
3. The person who is very short is not in the second house.
4. The person who loves beach vacations is in the first house.
5. The person with a high school diploma is in the third house.
6. The person who is short is somewhere to the right of the person who is very short.
7. The person who loves the boquet of lilies is Eric.
8. The person who loves the boquet of lilies is the person with a bachelor's degree.
9. The person who prefers city breaks is somewhere to the right of Peter.
10. The person who has blonde hair is in the third house.
11. The person who loves beach vacations is the person who has brown hair.
 What is the value of attribute House for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


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

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

Answer: prince


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`, `Eric`, `Carol`, `Alice`, `Peter`
 - Each person has a unique level of education: `doctorate`, `trade school`, `associate`, `high school`, `bachelor`, `master`
 - People own unique car models: `chevrolet silverado`, `ford f150`, `honda civic`, `tesla model 3`, `bmw 3 series`, `toyota camry`
 - Each person has a unique favorite drink: `tea`, `water`, `boba tea`, `coffee`, `root beer`, `milk`
 - They all have a unique favorite flower: `carnations`, `iris`, `lilies`, `tulips`, `daffodils`, `roses`

## Clues:
1. Eric is the one who only drinks water.
2. The person with an associate's degree is the person who owns a Toyota Camry.
3. The boba tea drinker is in the fourth house.
4. The person who loves the vase of tulips is directly left of the person who owns a Tesla Model 3.
5. The person with a master's degree is somewhere to the left of the person who owns a BMW 3 Series.
6. The person who owns a Toyota Camry is directly left of the person who loves a bouquet of daffodils.
7. The person who loves the boquet of iris is somewhere to the left of the root beer lover.
8. The person who likes milk is somewhere to the right of the person who owns a Honda Civic.
9. Carol is the person who owns a Honda Civic.
10. The person with a high school diploma is in the second house.
11. The person with a master's degree is the boba tea drinker.
12. The person who owns a Tesla Model 3 is Arnold.
13. The person who loves the boquet of lilies is the person who owns a Tesla Model 3.
14. The person who loves a carnations arrangement is somewhere to the left of Alice.
15. The person with a bachelor's degree is somewhere to the right of the person who likes milk.
16. The one who only drinks water is the person who owns a Ford F-150.
17. Carol is somewhere to the left of the person with a doctorate.
18. There are two houses between the person who owns a Toyota Camry and the person who loves the vase of tulips.
19. The person who owns a Tesla Model 3 is somewhere to the left of the coffee drinker.
20. Bob is in the third house.
 What is the value of attribute CarModel for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}

Answer: honda civic


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Eric`, `Arnold`, `Carol`, `Bob`
 - People have unique heights: `very short`, `very tall`, `short`, `super tall`, `tall`, `average`
 - Everyone has something unique for lunch: `grilled cheese`, `stir fry`, `pizza`, `spaghetti`, `soup`, `stew`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Penny`, `Aniya`, `Kailyn`, `Sarah`
 - Each person has a unique favorite drink: `tea`, `water`, `milk`, `coffee`, `boba tea`, `root beer`
 - Each person has a unique type of pet: `rabbit`, `fish`, `bird`, `hamster`, `cat`, `dog`

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

Answer: stir fry


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`
 - People have unique favorite sports: `basketball`, `soccer`, `tennis`, `baseball`, `swimming`
 - The people keep unique animals: `dog`, `fish`, `horse`, `bird`, `cat`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `samsung galaxy s21`, `iphone 13`, `huawei p50`
 - People have unique hair colors: `black`, `red`, `brown`, `blonde`, `gray`
 - Each person lives in a unique style of house: `modern`, `craftsman`, `victorian`, `ranch`, `colonial`

## Clues:
1. The person who has gray hair is the person who loves tennis.
2. The person in a ranch-style home is in the fourth house.
3. The person living in a colonial-style house is the person who loves swimming.
4. The dog owner is somewhere to the right of the person who has gray hair.
5. The fish enthusiast is the person who has gray hair.
6. The person who has black hair is not in the fifth house.
7. The person who uses a Samsung Galaxy S21 is the bird keeper.
8. There is one house between the fish enthusiast and the person who uses a Samsung Galaxy S21.
9. The person residing in a Victorian house is somewhere to the left of the person in a Craftsman-style house.
10. The person who has red hair is not in the fifth house.
11. There is one house between the person who uses a Samsung Galaxy S21 and the person who uses a Huawei P50.
12. The person who uses an iPhone 13 is the person in a Craftsman-style house.
13. Eric is the person who has red hair.
14. The person who has red hair is not in the fourth house.
15. The person who uses a OnePlus 9 is in the first house.
16. The person who loves soccer is the dog owner.
17. The person who keeps horses is Bob.
18. Arnold is the fish enthusiast.
19. The person who loves baseball is Peter.
20. The person who loves soccer is in the fourth house.
21. The person who has blonde hair is directly left of the person who has black hair.
 What is the value of attribute PhoneModel for the person whose attribute FavoriteSport is swimming? Please reason step by step, and put your final answer within \boxed{}

Answer: huawei p50


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`, `Arnold`, `Bob`
 - Each person lives in a unique style of house: `craftsman`, `modern`, `victorian`, `ranch`, `colonial`
 - Each person has a unique type of pet: `cat`, `bird`, `dog`, `hamster`, `fish`
 - Each person has a unique favorite drink: `coffee`, `milk`, `tea`, `water`, `root beer`

## Clues:
1. The person who has a cat is in the third house.
2. The person with an aquarium of fish is Arnold.
3. The root beer lover is in the fourth house.
4. There is one house between the person in a Craftsman-style house and the person residing in a Victorian house.
5. The person who has a cat is Bob.
6. Peter is the person with a pet hamster.
7. The person in a modern-style house is the person who has a cat.
8. The coffee drinker is not in the fifth house.
9. The one who only drinks water is the person who owns a dog.
10. The person living in a colonial-style house and the person with an aquarium of fish are next to each other.
11. The person with an aquarium of fish is the person in a Craftsman-style house.
12. The tea drinker is in the third house.
13. The person residing in a Victorian house is somewhere to the right of Alice.
14. The person in a ranch-style home is somewhere to the right of Eric.
 What is the value of attribute House for the person whose attribute HouseStyle is craftsman? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Peter`, `Carol`, `Alice`, `Eric`
 - Everyone has something unique for lunch: `stir fry`, `stew`, `soup`, `pizza`, `grilled cheese`, `spaghetti`
 - Each person has an occupation: `doctor`, `artist`, `engineer`, `teacher`, `lawyer`, `nurse`
 - Each person has a unique type of pet: `dog`, `bird`, `hamster`, `cat`, `fish`, `rabbit`
 - People have unique favorite music genres: `pop`, `country`, `jazz`, `classical`, `rock`, `hip hop`
 - Everyone has a favorite smoothie: `blueberry`, `desert`, `watermelon`, `cherry`, `lime`, `dragonfruit`

## Clues:
1. The person who loves the spaghetti eater is the Dragonfruit smoothie lover.
2. The person who drinks Lime smoothies is in the second house.
3. Carol is somewhere to the left of the Desert smoothie lover.
4. The person who is an engineer is not in the second house.
5. The person who loves the soup is in the second house.
6. The person who owns a dog is not in the second house.
7. The person who is a pizza lover is Alice.
8. Arnold is somewhere to the left of Eric.
9. The person who loves classical music is the Dragonfruit smoothie lover.
10. The person who owns a rabbit is somewhere to the left of the Desert smoothie lover.
11. The person who is a nurse is not in the sixth house.
12. There are two houses between the person who is a nurse and the person who loves the spaghetti eater.
13. The person who likes Cherry smoothies is the person with an aquarium of fish.
14. The person who has a cat is Alice.
15. The person who likes Cherry smoothies is somewhere to the right of the Desert smoothie lover.
16. There is one house between the person who drinks Blueberry smoothies and the person who loves eating grilled cheese.
17. The person who loves classical music is somewhere to the left of the person who loves the stew.
18. The person who is a lawyer is the person who loves classical music.
19. The person who loves jazz music is Bob.
20. Bob is the person who drinks Lime smoothies.
21. The person who loves pop music is the person who is a teacher.
22. The person who is an engineer is directly left of the person who loves rock music.
23. The person who loves hip-hop music is not in the first house.
24. The person with an aquarium of fish is Peter.
25. The person who drinks Blueberry smoothies is the person who keeps a pet bird.
26. The person who has a cat is in the first house.
27. The person who is a doctor is not in the sixth house.
28. The person who owns a rabbit is not in the second house.
29. Carol is not in the fourth 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: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Carol`, `Alice`, `Bob`, `Eric`
 - Each person has a unique level of education: `bachelor`, `high school`, `doctorate`, `trade school`, `associate`, `master`
 - Each person has an occupation: `teacher`, `artist`, `nurse`, `lawyer`, `engineer`, `doctor`
 - People own unique car models: `ford f150`, `toyota camry`, `honda civic`, `chevrolet silverado`, `bmw 3 series`, `tesla model 3`
 - Each person has a unique birthday month: `sept`, `jan`, `mar`, `april`, `feb`, `may`

## Clues:
1. The person with a bachelor's degree is somewhere to the right of the person who owns a Toyota Camry.
2. The person whose birthday is in February is the person with an associate's degree.
3. The person whose birthday is in September is not in the fifth house.
4. The person who owns a BMW 3 Series is somewhere to the right of the person who is an engineer.
5. The person whose birthday is in January is Arnold.
6. The person who is an artist is directly left of the person with a doctorate.
7. The person who owns a Ford F-150 is the person who is an engineer.
8. The person who attended trade school and Eric are next to each other.
9. There is one house between the person who attended trade school and the person with a bachelor's degree.
10. The person whose birthday is in May is the person who owns a BMW 3 Series.
11. The person who owns a Chevrolet Silverado is somewhere to the left of Alice.
12. The person who is an engineer is directly left of Bob.
13. The person who owns a Honda Civic is the person whose birthday is in April.
14. The person who is an artist is the person with a high school diploma.
15. Arnold is the person who owns a Toyota Camry.
16. The person who owns a Tesla Model 3 is in the first house.
17. The person who is a doctor is somewhere to the left of the person whose birthday is in April.
18. The person who is a nurse is directly left of the person whose birthday is in September.
19. The person who is a teacher is the person with a doctorate.
20. There is one house between the person who owns a Toyota Camry and the person who owns a Honda Civic.
21. The person who owns a Tesla Model 3 is Peter.
 What is the value of attribute Name for the person whose attribute Occupation is doctor? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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 Animal is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: Janelle


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Bob`, `Peter`, `Carol`, `Alice`
 - Everyone has a favorite smoothie: `lime`, `desert`, `blueberry`, `cherry`, `watermelon`, `dragonfruit`

## Clues:
1. Alice is in the fifth house.
2. Bob is somewhere to the left of the Watermelon smoothie lover.
3. The person who drinks Lime smoothies is in the fifth house.
4. Eric is the Desert smoothie lover.
5. The person who drinks Blueberry smoothies is somewhere to the right of the Desert smoothie lover.
6. Bob is somewhere to the right of Arnold.
7. There are two houses between the Dragonfruit smoothie lover and the person who likes Cherry smoothies.
8. The person who likes Cherry smoothies is somewhere to the right of the person who drinks Lime smoothies.
9. Peter is somewhere to the right of Carol.
 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: 6


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 favorite color: `white`, `red`, `yellow`
 - People use unique phone models: `google pixel 6`, `samsung galaxy s21`, `iphone 13`
 - People have unique favorite music genres: `classical`, `rock`, `pop`
 - Each person has a unique level of education: `high school`, `bachelor`, `associate`

## Clues:
1. The person who loves pop music is Peter.
2. Arnold is the person who loves classical music.
3. The person with a bachelor's degree is not in the first house.
4. The person who loves white is in the second house.
5. The person who loves pop music is in the first house.
6. The person who uses a Samsung Galaxy S21 is the person with an associate's degree.
7. The person with a bachelor's degree is the person whose favorite color is red.
8. The person with a high school diploma is the person who uses a Google Pixel 6.
9. The person who uses a Samsung Galaxy S21 is the person who loves rock music.
 What is the value of attribute Color for the person whose attribute MusicGenre is classical? Please reason step by step, and put your final answer within \boxed{}

Answer: red


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 birthday month: `april`, `jan`, `sept`
 - People have unique favorite book genres: `mystery`, `romance`, `science fiction`
 - Each person lives in a unique style of house: `ranch`, `victorian`, `colonial`
 - They all have a unique favorite flower: `carnations`, `daffodils`, `lilies`

## Clues:
1. The person who loves romance books is in the third house.
2. Arnold is the person who loves a carnations arrangement.
3. The person living in a colonial-style house is the person whose birthday is in September.
4. The person who loves a carnations arrangement is in the second house.
5. The person whose birthday is in April is the person in a ranch-style home.
6. The person who loves science fiction books is Arnold.
7. The person in a ranch-style home is the person who loves a carnations arrangement.
8. The person whose birthday is in January is somewhere to the left of Peter.
9. The person whose birthday is in April is somewhere to the left of the person who loves the boquet of lilies.
 What is the value of attribute Birthday for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: jan


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Eric`, `Peter`, `Arnold`
 - People have unique favorite music genres: `rock`, `pop`, `classical`, `jazz`, `hip hop`
 - People have unique hair colors: `gray`, `black`, `red`, `blonde`, `brown`
 - The people are of nationalities: `norwegian`, `german`, `dane`, `brit`, `swede`

## Clues:
1. The German is somewhere to the left of the person who loves rock music.
2. There are two houses between Alice and the British person.
3. The person who loves jazz music is the person who has gray hair.
4. There are two houses between the person who has black hair and Peter.
5. The Swedish person is the person who has red hair.
6. The person who loves pop music is somewhere to the left of Peter.
7. The person who has gray hair is the German.
8. Eric is the person who loves rock music.
9. Arnold is in the second house.
10. The person who has blonde hair is in the third house.
11. Alice is the Dane.
12. The person who loves classical music is somewhere to the left of the German.
 What is the value of attribute House for the person whose attribute MusicGenre is rock? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Alice`, `Bob`, `Peter`
 - Each person prefers a unique type of vacation: `mountain`, `city`, `cruise`, `beach`, `camping`
 - Each person has a unique level of education: `doctorate`, `high school`, `bachelor`, `associate`, `master`
 - Each person has a favorite color: `blue`, `red`, `white`, `yellow`, `green`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `oneplus 9`, `huawei p50`, `samsung galaxy s21`
 - Everyone has something unique for lunch: `grilled cheese`, `stir fry`, `pizza`, `spaghetti`, `stew`

## Clues:
1. The person who loves the stew is not in the first house.
2. There are two houses between the person who loves stir fry and the person with an associate's degree.
3. The person who enjoys mountain retreats is the person with a bachelor's degree.
4. The person with a doctorate is somewhere to the right of Bob.
5. The person who uses a Samsung Galaxy S21 is in the third house.
6. Eric is the person with a doctorate.
7. The person with a doctorate is in the third house.
8. The person who loves stir fry is the person with a bachelor's degree.
9. The person with a doctorate is the person who is a pizza lover.
10. The person whose favorite color is green is somewhere to the right of Peter.
11. The person who enjoys camping trips is the person who uses an iPhone 13.
12. The person who likes going on cruises is Alice.
13. There is one house between the person with a high school diploma and the person who uses a Samsung Galaxy S21.
14. The person who uses a Google Pixel 6 is Arnold.
15. The person who uses a OnePlus 9 is somewhere to the right of the person who uses a Huawei P50.
16. Arnold is the person who loves eating grilled cheese.
17. The person who loves eating grilled cheese is not in the fourth house.
18. There are two houses between the person with a bachelor's degree and the person whose favorite color is red.
19. The person who loves beach vacations is somewhere to the right of the person who prefers city breaks.
20. The person whose favorite color is green is not in the second house.
21. The person who loves blue is somewhere to the right of Peter.
22. There is one house between the person who enjoys camping trips and the person who loves yellow.
 What is the value of attribute House for the person whose attribute Color is blue? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Peter`, `Bob`, `Eric`, `Carol`
 - Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `watermelon`, `lime`, `desert`, `blueberry`
 - Each person has a unique birthday month: `mar`, `april`, `jan`, `may`, `sept`, `feb`
 - The people keep unique animals: `cat`, `rabbit`, `horse`, `bird`, `fish`, `dog`

## Clues:
1. The bird keeper is in the third house.
2. The person whose birthday is in May is somewhere to the left of Eric.
3. Eric is the Desert smoothie lover.
4. The rabbit owner is the person whose birthday is in January.
5. The fish enthusiast is directly left of the person who likes Cherry smoothies.
6. The person whose birthday is in September is Peter.
7. There is one house between Peter and the rabbit owner.
8. The person who drinks Lime smoothies is not in the sixth house.
9. The Watermelon smoothie lover is the dog owner.
10. The person whose birthday is in April is somewhere to the right of Carol.
11. The bird keeper is somewhere to the left of the person who keeps horses.
12. The person who drinks Lime smoothies is Alice.
13. The Dragonfruit smoothie lover is directly left of Carol.
14. The person who drinks Blueberry smoothies is the person whose birthday is in February.
15. The person whose birthday is in September is in the third house.
16. Bob is the Dragonfruit smoothie lover.
17. The person who likes Cherry smoothies is somewhere to the left of the person whose birthday is in May.
18. The rabbit owner is the Desert smoothie lover.
 What is the value of attribute House for the person whose attribute Animal is fish? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Alice`, `Peter`
 - Each person has a unique birthday month: `feb`, `jan`, `april`, `sept`
 - People have unique hair colors: `black`, `blonde`, `brown`, `red`
 - Each person lives in a unique style of house: `colonial`, `ranch`, `victorian`, `craftsman`
 - Each person has an occupation: `engineer`, `artist`, `doctor`, `teacher`
 - Everyone has a favorite smoothie: `cherry`, `dragonfruit`, `desert`, `watermelon`

## Clues:
1. The person who is an engineer is the person whose birthday is in February.
2. The person who likes Cherry smoothies is somewhere to the left of the person who has red hair.
3. The person who is a doctor is the person who has blonde hair.
4. The person whose birthday is in January is the person living in a colonial-style house.
5. Alice is the person residing in a Victorian house.
6. Peter is the Desert smoothie lover.
7. Peter is the person whose birthday is in September.
8. The Dragonfruit smoothie lover is the person who is a teacher.
9. The Watermelon smoothie lover is the person residing in a Victorian house.
10. The person living in a colonial-style house is directly left of the person who has black hair.
11. The person whose birthday is in February is in the fourth house.
12. The person who has black hair is not in the third house.
13. Eric is the person in a ranch-style home.
14. The person who has blonde hair is directly left of the person who likes Cherry smoothies.
 What is the value of attribute Birthday for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: feb


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`
 - Each person has a unique hobby: `cooking`, `painting`, `photography`, `gardening`
 - Each person has a unique birthday month: `april`, `jan`, `sept`, `feb`
 - Each person has a unique level of education: `master`, `bachelor`, `associate`, `high school`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `desert`, `dragonfruit`

## Clues:
1. The Desert smoothie lover is the person whose birthday is in January.
2. Eric is the person with a bachelor's degree.
3. The person whose birthday is in January is the person with a bachelor's degree.
4. The person with a high school diploma is in the third house.
5. The Watermelon smoothie lover is not in the third house.
6. The person with an associate's degree is Arnold.
7. The person with a master's degree is the person who paints as a hobby.
8. There is one house between the Dragonfruit smoothie lover and the person whose birthday is in September.
9. The person with a high school diploma is the person whose birthday is in September.
10. The person who loves cooking is Alice.
11. The person whose birthday is in April and the person who enjoys gardening are next to each other.
12. The person who paints as a hobby is the person whose birthday is in February.
 What is the value of attribute House for the person whose attribute Birthday is feb? 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: `Carol`, `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`
 - They all have a unique favorite flower: `carnations`, `tulips`, `lilies`, `roses`, `daffodils`, `iris`
 - People have unique favorite book genres: `science fiction`, `fantasy`, `romance`, `mystery`, `biography`, `historical fiction`
 - People use unique phone models: `huawei p50`, `oneplus 9`, `iphone 13`, `xiaomi mi 11`, `samsung galaxy s21`, `google pixel 6`
 - People have unique hair colors: `black`, `gray`, `red`, `auburn`, `blonde`, `brown`
 - Each person has a favorite color: `yellow`, `green`, `red`, `white`, `blue`, `purple`

## Clues:
1. There is one house between the person who has gray hair and the person who uses an iPhone 13.
2. The person who loves the rose bouquet is the person who has blonde hair.
3. There is one house between the person who has gray hair and the person who loves yellow.
4. Carol is somewhere to the right of the person who loves a bouquet of daffodils.
5. The person who loves white is somewhere to the left of Peter.
6. The person who loves white is somewhere to the left of the person who loves science fiction books.
7. Arnold is directly left of the person who loves historical fiction books.
8. The person who uses a Huawei P50 is the person who loves a bouquet of daffodils.
9. The person who loves purple is somewhere to the left of the person who loves fantasy books.
10. Eric is the person who loves science fiction books.
11. The person who loves a bouquet of daffodils is the person who has auburn hair.
12. Alice is the person whose favorite color is green.
13. The person who loves fantasy books is in the second house.
14. The person who uses a Samsung Galaxy S21 is Carol.
15. The person who uses a Xiaomi Mi 11 is not in the fourth house.
16. Eric is the person who has auburn hair.
17. The person who loves the rose bouquet is somewhere to the left of the person who has red hair.
18. The person who loves historical fiction books is the person who loves the rose bouquet.
19. The person who uses a Xiaomi Mi 11 is somewhere to the right of the person who loves white.
20. The person who has black hair is in the fourth house.
21. The person whose favorite color is red is the person who loves mystery books.
22. The person who loves historical fiction books and the person who loves the vase of tulips are next to each other.
23. The person who has brown hair is the person who uses a OnePlus 9.
24. The person who loves romance books is somewhere to the left of the person who loves fantasy books.
25. The person who loves the boquet of lilies is in the second house.
26. The person who loves the boquet of iris is somewhere to the right of the person whose favorite color is green.
 What is the value of attribute Color for the person whose attribute PhoneModel is samsung galaxy s21? Please reason step by step, and put your final answer within \boxed{}

Answer: red


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

## Clues:
1. There is one house between the person who has gray hair and the person who uses an iPhone 13.
2. The person who loves the rose bouquet is the person who has blonde hair.
3. There is one house between the person who has gray hair and the person who loves yellow.
4. Carol is somewhere to the right of the person who loves a bouquet of daffodils.
5. The person who loves white is somewhere to the left of Peter.
6. The person who loves white is somewhere to the left of the person who loves science fiction books.
7. Arnold is directly left of the person who loves historical fiction books.
8. The person who uses a Huawei P50 is the person who loves a bouquet of daffodils.
9. The person who loves purple is somewhere to the left of the person who loves fantasy books.
10. Eric is the person who loves science fiction books.
11. The person who loves a bouquet of daffodils is the person who has auburn hair.
12. Alice is the person whose favorite color is green.
13. The person who loves fantasy books is in the second house.
14. The person who uses a Samsung Galaxy S21 is Carol.
15. The person who uses a Xiaomi Mi 11 is not in the fourth house.
16. Eric is the person who has auburn hair.
17. The person who loves the rose bouquet is somewhere to the left of the person who has red hair.
18. The person who loves historical fiction books is the person who loves the rose bouquet.
19. The person who uses a Xiaomi Mi 11 is somewhere to the right of the person who loves white.
20. The person who has black hair is in the fourth house.
21. The person whose favorite color is red is the person who loves mystery books.
22. The person who loves historical fiction books and the person who loves the vase of tulips are next to each other.
23. The person who has brown hair is the person who uses a OnePlus 9.
24. The person who loves romance books is somewhere to the left of the person who loves fantasy books.
25. The person who loves the boquet of lilies is in the second house.
26. The person who loves the boquet of iris is somewhere to the right of the person whose favorite color is green.
 What is the value of attribute House for the person whose attribute BookGenre is biography? 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`, `Carol`, `Eric`, `Alice`, `Peter`
 - Each person lives in a unique style of house: `craftsman`, `modern`, `ranch`, `mediterranean`, `victorian`, `colonial`

## Clues:
1. The person in a ranch-style home is directly left of Carol.
2. The person in a Mediterranean-style villa is in the fifth house.
3. The person in a ranch-style home is Arnold.
4. The person living in a colonial-style house is not in the fourth house.
5. Alice is somewhere to the left of the person residing in a Victorian house.
6. The person residing in a Victorian house is in the third house.
7. Bob is the person in a Mediterranean-style villa.
8. Eric is somewhere to the left of the person in a modern-style house.
 What is the value of attribute House for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


The spokesperson for an oil and gas company is preparing a presentation to highlight the industry's efforts in reducing emissions and improving efficiency. They mention that last year, the company reduced its carbon emissions by 15% compared to the previous year, resulting in a total reduction of 450,000 tons of carbon dioxide.

If the company aims to achieve an additional reduction of 10% this year, how many tons of carbon dioxide emissions should the company aim to reduce this year?

Answer: 255000


Javiera and Claudio play on a board consisting of a row with $2019$ cells. Claudio starts by placing a token anywhere on the board. Next, Javiera says a natural number $k$, where $1 \leq k \leq n$, and Claudio must move the token to the right or to the left by $k$ squares, at his choice, and so on. Javiera wins if she manages to remove the piece that Claudio moves from the board. Determine the smallest $n$ such that Javiera always wins after a finite number of moves.

Answer: 1010


Given that $x$, $y$ are positive integers such that $x(x+1)$ divides $y(y+1)$, but neither $x$ nor $x+1$ divides either $y$ or $y+1$. Find the smallest possible value of $x^2 + y^2$.

Answer: 596


In a room, there is a series of bulbs on a wall and corresponding switches on the opposite wall. If you turn on the $n$-th switch, the $n$-th bulb will light up. There is a group of men who are operating the switches according to the following rule:

- They enter one by one and start flipping the switches starting from the first switch until they have to turn on a bulb.
- As soon as they turn a bulb on, they leave the room.

For example:
- The first person enters, turns the first switch on, and leaves.
- The second person enters, sees the first switch on, turns it off, and then lights the second bulb.
- The third person enters, finds the first switch off, turns it on, and leaves the room.
- The fourth person enters, switches off the first and second bulbs, and switches on the third.

This process continues in this manner. Finally, we find out that the first 10 bulbs are off and the 11th bulb is on. How many people were involved in the entire process?

Answer: 1024


The kid cut out a rectangle from grid paper, with each cell having a side length of 1, and calculated its area and perimeter. Carlson then took the scissors and cut out a square from this rectangle, along the grid lines, that was adjacent to the boundary of the rectangle.

- "My rectangle..." the kid sobbed. "There is something strange about this figure!"
- "Nonsense, do not mention it," Carlson said, waving his hand carelessly. "Here you see, in this figure, the perimeter is the same as the area of the rectangle was, and the area is the same as the perimeter was!"

What size square did Carlson cut out?

Answer: 2


Consider a graph $G$ with 2019 nodes. Define "inverting" a vertex $v$ as follows: for every other vertex $u$, if there was an edge between $v$ and $u$, it is deleted; if there wasn't, it is added. We aim to minimize the number of edges in the graph through several inverting operations (we can invert the same vertex multiple times). Determine the smallest number $M$ such that we can always reduce the number of edges in the graph to be no greater than $M$, regardless of the initial configuration of $G$.

Answer: 1018081


The number $125$ can be expressed as a sum of pairwise coprime integers, each greater than $1$. Determine the maximum number of terms possible in such a sum.

Answer: 8


Does there exist a polynomial of one variable that is irreducible over the field of integers, has degree $60$, and is a multiple of the form $X^n - 1$? If so, how many such polynomials exist?

Answer: 9


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 expressed as the sum of two cubes of positive integers.

Answer: 28


There are $10001$ students at a university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold:

1. Each pair of students is in exactly one club.
2. For each student and each society, the student is in exactly one club of the society.
3. Each club has an odd number of students. In addition, a club with $2m+1$ students ($m$ is a positive integer) is in exactly $m$ societies.

Find all possible values of $k$.

Answer: 5000


In a circle with radius $10$, point $M$ is located on the chord $PQ$ such that $PM = 5$ and $MQ = 10$. Through point $M$, we draw chords $AB$ and $CD$. Points $X$ and $Y$ are the intersection points of chords $AD$ and $BC$ with chord $PQ$, respectively. Given that $XM = 3$, find the length of $MY$.

Answer: 2


Let $a_0, a_1, a_2, \ldots$ be a sequence of real numbers such that $a_0=0, a_1=1$, and for every $n \geqslant 2$ there exists $1 \leqslant k \leqslant n$ satisfying
$$
a_n=\frac{a_{n-1}+\cdots+a_{n-k}}{k}
$$
Find the maximal possible value of $a_{2018}-a_{2017}$. The original answer is in the form \frac{m}{n^2}, please give the value of $m + n$.

Answer: 4033


A circle $\omega$ of radius 1 is given. A collection $T$ of triangles is called good, if the following conditions hold: (i) each triangle from $T$ is inscribed in $\omega$; (ii) no two triangles from $T$ have a common interior point. Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$. Please provide the maximum value of $t$.

Answer: 4


Version 1. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, $$\sqrt[N]{\frac{x^{2N}+1}{2}}\leqslant a_n(x-1)^2+x.$$ Please provide the value of N when $a_n = N/2$.\n\nVersion 2. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, $$\sqrt[N]{\frac{x^{2N}+1}{2}}\leqslant b_N(x-1)^2+x.$$ Please provide the sum of  $a_{4}$$b_{6}$.

Answer: 5


Let $n$ be an integer with $n \geqslant 2$. On a slope of a mountain, $n^2$ checkpoints are marked, numbered from 1 to $n^2$ from the bottom to the top. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars numbered from 1 to $k$; each cable car provides a transfer from some checkpoint to a higher one. For each company, and for any $i$ and $j$ with $1 \leqslant i<j \leqslant k$, the starting point of car $j$ is higher than the starting point of car $i$; similarly, the finishing point of car $j$ is higher than the finishing point of car $i$. Say that two checkpoints are linked by some company if one can start from the lower checkpoint and reach the higher one by using one or more cars of that company (no movement on foot is allowed). Determine the smallest $k$ for which one can guarantee that there are two checkpoints that are linked by each of the two companies. Please provide the value of $k$ when $n=3$.

Answer: 7


Determine the largest $N$ for which there exists a table $T$ of integers with $N$ rows and 100 columns that has the following properties:
(i) Every row contains the numbers $1,2, \ldots, 100$ in some order.
(ii) For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T(r, c)-T(s, c)| \geqslant 2$. Here $T(r, c)$ means the number at the intersection of the row $r$ and the column $c$.The largest such integer is $N=a!/ 2^{b}$.求$a+b$.

Answer: 150


In each square of a garden shaped like a $2022\times2022$ board, there is initially a tree of height 0.A gardener and a lumberjack alternate turns playing the following game,with the gardener taking the first turn:

• The gardener chooses a square in the garden.Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller.
• The lumberjack then chooses four different squares on the board.Each tree of positive height on those squares then becomes one unit shorter.

We say that a tree is majestic if its height is at least $10^6$ .Determine the largest number $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board,no matter how the lumberjack plays.

Answer: 2271380


Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. For positive integer $n=8$, ﬁnd the smallest number of non-empty piles that one can obtain by performing a ﬁnite sequence of moves of this form.

Answer: 1


Let $\mathbb{Z}_{\geqslant 0}$ be the set of non-negative integers, and let $f: \mathbb{Z}_{\geqslant 0} \times \mathbb{Z}_{\geqslant 0} \rightarrow \mathbb{Z}_{\geqslant 0}$ be a bijection such that whenever $f\left(x_1, y_1\right)>f\left(x_2, y_2\right)$, we have $f\left(x_1+1, y_1\right)>f\left(x_2+1, y_2\right)$ and $f\left(x_1, y_1+1\right)>f\left(x_2, y_2+1\right)$.\n\nLet $N$ be the number of pairs of integers $(x, y)$, with $0 \leqslant x, y<100$, such that $f(x, y)$ is odd. Find the smallest and largest possible value of $N$. Please provide the sum of these two values.

Answer: 10000


Let $n$ be a positive integer. We arrange $1+2+\cdots+n$ circles in a triangle with $n$ rows, such that the $i^{\text {th }}$ row contains exactly $i$ circles. The following figure shows the case $n=6$. In this triangle, a ninja-path is a sequence of circles obtained by repeatedly going from a circle to one of the two circles directly below it. In terms of $n$, find the largest value of $k$ such that if one circle from every row is coloured red, we can always find a ninja-path in which at least $k$ of the circles are red. Please provide the value of $k$ when $n = 6$.

Answer: 3


Let $n \geqslant 2$ be a positive integer. Paul has a $1 \times n^{2}$ rectangular strip consisting of $n^{2}$ unit squares, where the $i^{\text {th }}$ square is labelled with $i$ for all $1 \leqslant i \leqslant n^{2}$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then translate (without rotating or flipping) the pieces to obtain an $n \times n$ square satisfying the following property: if the unit square in the $i^{\text {th }}$ row and $j^{\text {th }}$ column is labelled with $a_{i j}$, then $a_{i j}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this. Please provide the value of the smallest number of pieces when $n=3$.

Answer: 5


Determine all pairs $(a, p)$ of positive integers with $p$ prime such that $p^{a}+a^{4}$ is a perfect square. Please provide the sum of all $a + p$ for the solutions.

Answer: 30


Let $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $m = CQ$. The original answer is in the form 4\frac{a}{b}m￥, please give the value of $a + b$ .

Answer: 5


Let $n>3$ be a positive integer. Equilateral triangle $ABC$ is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Find the difference $m-d$ in terms of $n$. Please provide the value of $m-d$ when $n = 5$. The answer should be an integer.

Answer: 21


Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$, and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$, evaluate the angle $NMB$. The original answer is in the format of an angle in degrees. Please provide the integer part of the degrees measure.

Answer: 45


The height of a truncated cone is equal to the radius of its base. The perimeter of a regular hexagon circumscribing its top is equal to the perimeter of an equilateral triangle inscribed in its base. Find the angle $\varphi$ between the cone's generating line and its base. Please provide the value of the parameter used in the arctan function, given that the original answer is in the format $\varphi=\arctan x$.Write down the value of $x$.

Answer: 4


All rectangles that can be inscribed in an isosceles triangle with two of their vertices on the triangle’s base have the same perimeter. Construct the triangle such that the condition can be expressed as a simple integer calculation, e.g., height = base. Provide the sum of base and height if both are integers.

Answer: 10


Let $ \mathcal{C}$ be the hyperbola $ y^2 \minus{} x^2 \equal{} 1$.  Given a point $ P_0$ on the $ x$-axis, we construct a sequence of points $ (P_n)$ on the $ x$-axis in the following manner: let $ \ell_n$ be the line with slope $ 1$ passing passing through $ P_n$, then $ P_{n\plus{}1}$ is the orthogonal projection of the point of intersection of $ \ell_n$ and $ \mathcal C$ onto the $ x$-axis. (If $ P_n \equal{} 0$, then the sequence simply terminates.)

Let $ N$ be the number of starting positions $ P_0$ on the $ x$-axis such that $ P_0 \equal{} P_{2008}$. Determine the remainder of $ N$ when divided by $ 2008$.

Answer: 254


There are $5$ students on a team for a math competition. The math competition has $5$ subject tests. Each student on the team must choose $2$ distinct tests, and each test must be taken by exactly two people. In how many ways can this be done?

Answer: 2040


Points $A$ and $B$ lie on circle $\omega$.  Point $P$ lies on the extension of segment $AB$ past $B$.  Line $\ell$ passes through $P$ and is tangent to $\omega$.  The tangents to $\omega$ at points $A$ and $B$ intersect $\ell$ at points $D$ and $C$ respectively.  Given that $AB=7$, $BC=2$, and $AD=3$, compute $BP$.

Answer: 9


A sphere is the set of points at a fixed positive distance $r$ from its center. Let $\mathcal{S}$ be a set of $2010$-dimensional spheres. Suppose that the number of points lying on every element of $\mathcal{S}$ is a finite number $n$. Find the maximal possible value of $n$.

Answer: 2


Suppose that a polynomial of the form $p(x)=x^{2010}\pm x^{2009}\pm \cdots \pm x \pm 1$ has no real roots.  What is the maximum possible number of coefficients of $-1$ in $p$?

Answer: 1005


Let $f(x)=cx(x-1)$, where $c$ is a positive real number.  We use $f^n(x)$ to denote the polynomial obtained by composing $f$ with itself $n$ times.  For every positive integer $n$, all the roots of $f^n(x)$ are real.  What is the smallest possible value of $c$?

Answer: 2


Suppose that $x$ and $y$ are complex numbers such that $x+y=1$ and $x^{20}+y^{20}=20$.  Find the sum of all possible values of $x^2+y^2$.

Answer: -90


Let $ABC$ be an obtuse triangle with circumcenter $O$ such that $\angle ABC = 15^o$ and $\angle  BAC > 90^o$. Suppose that $AO$ meets $BC$ at $D$, and that $OD^2 + OC \cdot DC = OC^2$. Find $\angle  C$.

Answer: 35


Let $S$ be the set of integers of the form $2^x+2^y+2^z$, where $x,y,z$ are pairwise distinct non-negative integers.  Determine the $100$th smallest element of $S$.

Answer: 577


Let $ABCD$ be a quadrilateral such that $\angle ABC = \angle  CDA = 90^o$, and $BC = 7$. Let $E$ and $F$ be on $BD$ such that $AE$ and $CF$ are perpendicular to BD. Suppose that $BE = 3$. Determine the product of the smallest and largest possible lengths of $DF$.

Answer: 9


Let $f:\mathbb{N}\to\mathbb{N}$ be a function satisfying the following conditions:

(a) $f(1)=1$.
(b) $f(a)\leq f(b)$ whenever $a$ and $b$ are positive integers with $a\leq b$.
(c) $f(2a)=f(a)+1$ for all positive integers $a$.

How many possible values can the $2014$-tuple $(f(1),f(2),\ldots,f(2014))$ take?

Answer: 1007


We have a calculator with two buttons that displays and integer $x$. Pressing the first button replaces $x$ by $\lfloor \frac{x}{2} \rfloor$, and pressing the second button replaces $x$ by $4x+1$. Initially, the calculator displays $0$. How many integers less than or equal to $2014$ can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, $\lfloor y \rfloor$ denotes the greatest integer less than or equal to the real number $y$).

Answer: 233


There are 60 chips in a box. Each chip is either small or large. If the number of small chips exceeds the number of large chips by a prime number, and this prime number also divides the number of large chips, what is the greatest possible number of large chips?

Answer: 29


Let \( f : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ f(f(x) + y) = f(x^2 - y) + kf(x)y \]
for all real numbers \( x \) and \( y \), and for some constant \( k \). Determine the number of possible values of \( f(2) \), and let \( s \) be the sum of all possible values of \( f(2) \). Find the product of the number of possible values and the sum, i.e., \( n \times s \), where \( n \) is the number of possibilities for \( f(2) \).

Answer: 8


In Brenda's garden, there are three trees: a pine tree that is $14\frac{1}{4}$ feet tall, a birch tree that is $18\frac{1}{2}$ feet tall, and a cedar tree that is $20\frac{5}{8}$ feet tall. How much taller is the cedar tree compared to the pine tree and the birch tree, expressed as simplified mixed numbers?

Answer: 2\frac{1}{8}\text{ feet}


Bobby takes 30 free throws and makes 60% of them. After an additional 10 throws, his overall shooting percentage increases to 64%. How many of the last 10 throws did he make?

Answer: 8


A student receives an 80 points score on a teacher's 100-point test after 4 hours of preparation. Given that the score varies directly with the time spent preparing, determine the score the student would receive if they spent 6 hours preparing for another test.

Answer: 120 \text{ points}


A stock investment appreciated by 40% in one year. Assuming it starts the following year at this increased value, what percent decrease is required in the second year to return the stock’s value to its original price at the beginning of the first year?

Answer: 28.57\%


Piravena needs to travel from city $X$ to city $Y$, from $Y$ to city $Z$, and then back to $X$. The cities form a right triangle with $Z$ a distance of 3600 km from $X$ and $Y$ a distance of 3900 km from $X$. The cost structure for travel includes:
- Bus: $\$0.20$ per kilometer.
- Airplane: $\$120$ booking fee plus $\$0.12$ per kilometer.

Calculate the least expensive total cost for Piravena to complete her journey.

Answer: \$1440


In a configuration where triangle $PAB$ is formed by three tangents to circle $O$ and $\angle APB = 30^\circ$, there is another smaller circle $O'$ inside triangle $PAB$ tangent to all three sides. Find the $\angle AOB$.

Answer: 150^\circ


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - Each person has a unique type of pet: `dog`, `fish`, `cat`
 - People have unique favorite book genres: `mystery`, `romance`, `science fiction`
 - People own unique car models: `ford f150`, `tesla model 3`, `toyota camry`
 - Each person has a unique birthday month: `april`, `jan`, `sept`
 - People have unique favorite music genres: `rock`, `classical`, `pop`

## Clues:
1. The person who loves romance books is directly left of the person who owns a Ford F-150.
2. The person whose birthday is in September is the person who owns a Toyota Camry.
3. The person whose birthday is in September is Peter.
4. The person who owns a Tesla Model 3 is the person whose birthday is in January.
5. The person with an aquarium of fish is not in the second house.
6. Arnold is the person who has a cat.
7. The person who owns a Tesla Model 3 is the person who loves science fiction books.
8. The person who loves science fiction books is not in the third house.
9. The person who owns a Ford F-150 is the person who loves pop music.
10. The person who loves rock music is directly left of Eric.
 What is the value of attribute MusicGenre for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{}

Answer: rock


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`, `Peter`, `Eric`
 - Everyone has a unique favorite cigar: `dunhill`, `prince`, `blue master`, `pall mall`
 - People have unique favorite book genres: `mystery`, `romance`, `science fiction`, `fantasy`

## Clues:
1. There is one house between the person partial to Pall Mall and the person who loves mystery books.
2. There is one house between Eric and the person who loves mystery books.
3. The person who loves romance books is the Dunhill smoker.
4. Peter is not in the fourth house.
5. The person who loves science fiction books is Eric.
6. The person who loves fantasy books is not in the fourth house.
7. The Prince smoker and the Dunhill smoker are next to each other.
8. Arnold is the person who loves mystery books.
9. The Prince smoker is not in the fourth house.
10. The person who loves science fiction books is somewhere to the left of Arnold.
 What is the value of attribute House for the person whose attribute Cigar is dunhill? 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: `Bob`, `Arnold`, `Carol`, `Alice`, `Peter`, `Eric`
 - People have unique favorite book genres: `romance`, `historical fiction`, `biography`, `mystery`, `fantasy`, `science fiction`
 - Each person has an occupation: `artist`, `doctor`, `nurse`, `engineer`, `teacher`, `lawyer`

## Clues:
1. Alice is the person who loves fantasy books.
2. The person who loves mystery books and Bob are next to each other.
3. Carol is the person who loves mystery books.
4. The person who is a lawyer is the person who loves fantasy books.
5. Bob is not in the fifth house.
6. Arnold is somewhere to the left of the person who is an engineer.
7. The person who is a nurse is directly left of Alice.
8. The person who loves biography books is the person who is a teacher.
9. The person who loves historical fiction books is somewhere to the left of the person who is a teacher.
10. The person who is a doctor is in the first house.
11. The person who loves science fiction books is the person who is an artist.
12. Eric is in the third house.
13. The person who loves mystery books is not in the fifth house.
 What is the value of attribute BookGenre for the person whose attribute Occupation is artist? Please reason step by step, and put your final answer within \boxed{}

Answer: 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: `Bob`, `Eric`, `Arnold`, `Peter`, `Alice`
 - Everyone has a unique favorite cigar: `blends`, `prince`, `pall mall`, `blue master`, `dunhill`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `stew`, `pizza`, `stir fry`
 - The people are of nationalities: `norwegian`, `swede`, `brit`, `dane`, `german`
 - Each person has an occupation: `teacher`, `lawyer`, `doctor`, `artist`, `engineer`
 - People have unique heights: `average`, `very tall`, `short`, `very short`, `tall`

## Clues:
1. The person who smokes Blue Master is the person who is a lawyer.
2. Alice is the person partial to Pall Mall.
3. The person partial to Pall Mall is somewhere to the left of the Dunhill smoker.
4. The German is the person who loves stir fry.
5. The person who is tall is the person who smokes many unique blends.
6. The person who is an artist is the person who loves the spaghetti eater.
7. Peter is the person who is short.
8. The Dunhill smoker is not in the fourth house.
9. The person who is very short is the person who smokes Blue Master.
10. There are two houses between the person who loves the spaghetti eater and the person who loves stir fry.
11. The person who has an average height is in the second house.
12. The Prince smoker is the person who is a doctor.
13. The Swedish person is the person who is an engineer.
14. Eric is the person who loves stir fry.
15. There is one house between the British person and the person who loves the stew.
16. The person who is very tall is the Norwegian.
17. The Dane is the person who is short.
18. The person who is very tall is Arnold.
19. There are two houses between the person who is a pizza lover and the person who is very tall.
20. The person who is a teacher is not in the third house.
 What is the value of attribute Nationality for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: norwegian


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`
 - Everyone has a unique favorite cigar: `pall mall`, `blue master`, `prince`
 - The people keep unique animals: `horse`, `cat`, `bird`
 - Each mother is accompanied by their child: `Bella`, `Fred`, `Meredith`
 - People have unique favorite book genres: `science fiction`, `romance`, `mystery`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `samsung galaxy s21`

## Clues:
1. The person who loves mystery books is the person's child is named Fred.
2. The cat lover is Eric.
3. The person partial to Pall Mall is in the second house.
4. The person who keeps horses is the person's child is named Meredith.
5. The person's child is named Bella is the Prince smoker.
6. The person who uses an iPhone 13 is directly left of the person who uses a Samsung Galaxy S21.
7. The person's child is named Fred is directly left of Arnold.
8. Peter is somewhere to the left of Eric.
9. The person who loves science fiction books is the person who uses a Samsung Galaxy S21.
10. The person who loves science fiction books is in the third house.
11. The person who loves mystery books is not in the second house.
 What is the value of attribute Animal for the person whose attribute House is 2? 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 heights: `tall`, `short`, `very short`, `average`
 - Each person has a unique birthday month: `april`, `jan`, `sept`, `feb`
 - People have unique hair colors: `red`, `black`, `brown`, `blonde`
 - People have unique favorite book genres: `science fiction`, `fantasy`, `romance`, `mystery`

## Clues:
1. Eric is the person who has blonde hair.
2. The person who is short is somewhere to the left of the person who has brown hair.
3. Eric is the person who is tall.
4. Arnold is the person who loves fantasy books.
5. The person who loves science fiction books is Eric.
6. The person who is very short is the person who has black hair.
7. The person whose birthday is in April is the person who has brown hair.
8. The person who has blonde hair and the person who has black hair are next to each other.
9. The person who has black hair is the person who loves fantasy books.
10. The person who has red hair is in the first house.
11. The person whose birthday is in September is in the second house.
12. The person whose birthday is in January is the person who has black hair.
13. The person who loves romance books is somewhere to the right of Alice.
 What is the value of attribute House for the person whose attribute Height is average? 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: `Alice`, `Bob`, `Arnold`, `Eric`, `Peter`
 - The people keep unique animals: `fish`, `dog`, `cat`, `horse`, `bird`
 - People own unique car models: `tesla model 3`, `ford f150`, `honda civic`, `bmw 3 series`, `toyota camry`

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

Answer: ford f150


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: `water`, `milk`, `tea`
 - They all have a unique favorite flower: `carnations`, `lilies`, `daffodils`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`, `blue master`

## Clues:
1. The person who smokes Blue Master is the person who loves a bouquet of daffodils.
2. Arnold is somewhere to the right of the person who loves a carnations arrangement.
3. Eric is somewhere to the right of the Prince smoker.
4. The one who only drinks water is the person partial to Pall Mall.
5. The person who loves the boquet of lilies is Peter.
6. The tea drinker is not in the third house.
 What is the value of attribute Flower for the person whose attribute Cigar is prince? Please reason step by step, and put your final answer within \boxed{}

Answer: lilies


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Peter`
 - People have unique heights: `tall`, `very short`, `average`, `short`
 - Each person has a unique favorite drink: `tea`, `coffee`, `water`, `milk`
 - Each mother is accompanied by their child: `Fred`, `Meredith`, `Bella`, `Samantha`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`, `Kailyn`

## Clues:
1. The person who is tall is Arnold.
2. The person's child is named Samantha is the tea drinker.
3. The person whose mother's name is Holly is the coffee drinker.
4. The person's child is named Samantha and The person whose mother's name is Holly are next to each other.
5. The person's child is named Meredith and the person who is very short are next to each other.
6. The person's child is named Samantha is The person whose mother's name is Aniya.
7. The person who likes milk is the person who is short.
8. Alice is the person's child is named Fred.
9. Peter and the person who is very short are next to each other.
10. The coffee drinker is somewhere to the right of the person's child is named Samantha.
11. Eric is the person who is short.
12. The person whose mother's name is Janelle is somewhere to the left of The person whose mother's name is Kailyn.
13. Alice is The person whose mother's name is Kailyn.
14. The person's child is named Meredith is not in the first house.
 What is the value of attribute Children for the person whose attribute Drink is tea? Please reason step by step, and put your final answer within \boxed{}

Answer: Samantha