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`, `Eric`, `Carol`, `Bob`, `Arnold`, `Peter`
 - People have unique favorite music genres: `hip hop`, `jazz`, `country`, `classical`, `rock`, `pop`
 - People use unique phone models: `iphone 13`, `oneplus 9`, `xiaomi mi 11`, `huawei p50`, `samsung galaxy s21`, `google pixel 6`
 - People have unique favorite book genres: `biography`, `historical fiction`, `fantasy`, `romance`, `science fiction`, `mystery`

## Clues:
1. Arnold is not in the sixth house.
2. There are two houses between the person who loves country music and the person who loves hip-hop music.
3. The person who loves romance books is Carol.
4. The person who loves biography books is somewhere to the left of the person who loves rock music.
5. The person who loves mystery books is somewhere to the right of Peter.
6. The person who loves romance books is the person who loves classical music.
7. The person who loves mystery books is directly left of the person who loves fantasy books.
8. Bob is the person who loves pop music.
9. The person who loves jazz music is somewhere to the left of the person who loves pop music.
10. Arnold is somewhere to the right of the person who loves hip-hop music.
11. The person who loves hip-hop music and the person who uses a OnePlus 9 are next to each other.
12. The person who loves science fiction books is the person who uses a Xiaomi Mi 11.
13. The person who loves classical music is in the second house.
14. The person who uses a Xiaomi Mi 11 is directly left of Eric.
15. The person who loves historical fiction books is somewhere to the left of the person who loves classical music.
16. The person who uses a Huawei P50 is Peter.
17. The person who uses a Samsung Galaxy S21 is in the fourth house.
18. The person who uses a Samsung Galaxy S21 is somewhere to the right of the person who uses an iPhone 13.
 What is the value of attribute BookGenre for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: science fiction


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 Drink for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: milk


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

## Clues:
1. The person who loves jazz music is the person whose birthday is in April.
2. Alice is the person whose birthday is in April.
3. Peter is directly left of the person who loves cooking.
4. The person whose birthday is in February is Arnold.
5. The Dunhill smoker is not in the third house.
6. The Prince smoker is Arnold.
7. The person who loves jazz music is the person partial to Pall Mall.
8. The person whose birthday is in January is not in the third house.
9. The person who loves rock music is Peter.
10. The person who enjoys gardening is directly left of the photography enthusiast.
11. The person who loves classical music is in the first house.
12. The person who loves pop music is somewhere to the left of the person partial to Pall Mall.
13. The person who smokes Blue Master is the person who loves pop music.
 What is the value of attribute House for the person whose attribute Hobby is gardening? 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`, `Peter`, `Alice`, `Eric`, `Arnold`
 - Each mother is accompanied by their child: `Meredith`, `Bella`, `Samantha`, `Fred`, `Timothy`
 - People have unique favorite music genres: `classical`, `hip hop`, `pop`, `jazz`, `rock`
 - Everyone has a favorite smoothie: `watermelon`, `lime`, `dragonfruit`, `cherry`, `desert`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `city`, `cruise`, `camping`

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

Answer: jazz


Let $A, B, C$ be points in that order along a line, such that $AB = 20$ and $BC = 18$. Let $\omega$ be a circle of nonzero radius centered at $B$, and let $\ell_1$ and $\ell_2$ be tangents to $\omega$ through $A$ and $C$, respectively. Let $K$ be the intersection of $\ell_1$ and $\ell_2$. Let $X$ lie on segment $\overline{KA}$ and $Y$ lie on segment $\overline{KC}$ such that $XY \parallel BC$ and $XY$ is tangent to $\omega$. What is the largest possible integer length for $XY$?

Answer: 35


Let $T = TNFTPP$, and let $S$ be the sum of the digits of $T$. In triangle $ABC$, points $D$, $E$, and $F$ are the feet of the angle bisectors of $\angle A$, $\angle B$, $\angle C$ respectively. Let point $P$ be the intersection of segments $AD$ and $BE$, and let $p$ denote the perimeter of $ABC$. If $AP = 3PD$, $BE = S - 1$, and $CF = 9$, then the value of $\frac{AD}{p}$ can be expressed uniquely as $\frac{\sqrt{m}}{n}$ where $m$ and $n$ are positive integers such that $m$ is not divisible by the square of any prime. Find $m + n$.

Answer: 6


Let $\Omega_1$ be a circle with center $O$, and let $AB$ be the diameter of $\Omega_1$. Let $P$ be a point on the segment $OB$, different from $O$. Suppose another circle $\Omega_2$ with center $P$ lies in the interior of $\Omega_1$. Tangents are drawn from $A$ and $B$ to the circle $\Omega_2$, intersecting $\Omega_1$ again at $A_1$ and $B_1$, respectively, such that $A_1$ and $B_1$ are on opposite sides of $AB$. Given that $A_1B = 5$, $AB_1 = 15$, and $OP = 10$, find the radius of $\Omega_1$.

Answer: 20


The diagram shows an $8 \times 7$ rectangle with a 3-4-5 right triangle drawn in each corner. The lower two triangles have their sides of length 4 along the bottom edge of the rectangle, while the upper two triangles have their sides of length 3 along the top edge of the rectangle. A circle is tangent to the hypotenuse of each triangle. The diameter of the circle is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 47


There is a circle with a circumference of 1 meter and a point marked on it. Two cockroaches start running in the same direction from the marked point at different speeds.

- Whenever the faster cockroach catches up with the slower one, the slower one instantly turns around and starts running in the opposite direction with the same speed.
- Whenever they meet face-to-face, the faster one instantly turns around and starts running in the opposite direction with the same speed.

Determine how far from the marked point their 100th meeting could occur.

Answer: 0


From the digits $1, 2, \ldots, 9$, we write all the numbers formed by these nine digits (the nine digits are all distinct), and we order them in increasing order as follows: $123456789$, $123456798$, $\ldots$, $987654321$. What is the $100000^{th}$ number?

Answer: 358926471


The people in an infinitely long line are numbered $1,2,3,\dots$. Then, each person says either "Karl" or "Lark" independently and at random. Let $S$ be the set of all positive integers $i$ such that people $i$, $i+1$, and $i+2$ all say "Karl," and define $X = \sum_{i \in S} 2^{-i}$. Then the expected value of $X^2$ is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$.

Answer: 1028


Let $\mathbb{Z}_{\ge 0}$ denote the set of nonnegative integers.

Define a function $f:\mathbb{Z}_{\ge 0} \to \mathbb{Z}$ with $f\left(0\right)=1$ and
\[ f\left(n\right)=512^{\left\lfloor n/10 \right\rfloor}f\left(\left\lfloor n/10 \right\rfloor\right) \]
for all $n \ge 1$. Determine the number of nonnegative integers $n$ such that the hexadecimal (base $16$) representation of $f\left(n\right)$ contains no more than $2500$ digits.

Answer: 10010


Mr. Vader gave out a multiple choice test, and every question had an answer that was one of A, B, or C. After the test, he curved the test so that everybody got $+50$ (so a person who got $x\%$ right would get a score of $x+50$). In the class, a score in the range $[90, \infty)$ gets an A, a score in the range $[80, 90)$ gets a B, and a score in the range $[70, 80)$ gets a C. After the curve, Mr. Vader makes this statement: "Guess A, get an A. Guess B, get a B. Guess C, get a C." That is, answering every question with the answer choice $X$ would give, with the curve, a score receiving a grade of $X$, where $X$ is one of A, B, C. Luke, a student in Mr. Vader's class, was told ahead of time that there were either $5$ or $6$ answers as A on the test. Find the sum of all possible values of the number of questions on the test, given this information.

Answer: 64


The diagram below shows a triangle divided into sections by three horizontal lines, which divide the altitude of the triangle into four equal parts, and three lines connecting the top vertex with points that divide the opposite side into four equal parts. If the shaded region has an area of $100$, find the area of the entire triangle.

Answer: 640


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`, `Alice`, `Eric`, `Peter`, `Carol`
 - People have unique favorite sports: `volleyball`, `soccer`, `basketball`, `baseball`, `swimming`, `tennis`
 - Each person has a unique hobby: `gardening`, `cooking`, `painting`, `woodworking`, `knitting`, `photography`
 - Each mother is accompanied by their child: `Bella`, `Meredith`, `Alice`, `Samantha`, `Fred`, `Timothy`
 - Each person has a unique birthday month: `feb`, `jan`, `mar`, `may`, `sept`, `april`
 - Everyone has something unique for lunch: `stir fry`, `grilled cheese`, `pizza`, `spaghetti`, `stew`, `soup`

## Clues:
1. The person who loves soccer is somewhere to the right of Eric.
2. The person who paints as a hobby is somewhere to the left of the woodworking hobbyist.
3. The person whose birthday is in February is not in the third house.
4. The person who is the mother of Timothy is directly left of Arnold.
5. The person who loves cooking is somewhere to the left of the person's child is named Fred.
6. Alice is the person who paints as a hobby.
7. The person's child is named Alice is the person who loves cooking.
8. The person whose birthday is in March is the person's child is named Samantha.
9. The person who loves the soup is the person whose birthday is in May.
10. Carol is somewhere to the right of the person whose birthday is in May.
11. The person whose birthday is in March and the person who is a pizza lover are next to each other.
12. The person whose birthday is in February is not in the fourth house.
13. There are two houses between the person who loves baseball and the person who loves the stew.
14. The person whose birthday is in May is somewhere to the left of the person who loves volleyball.
15. The person whose birthday is in January is somewhere to the right of Arnold.
16. The person who loves cooking is not in the third house.
17. The person who enjoys knitting is the person whose birthday is in September.
18. The person who enjoys gardening is the person's child is named Meredith.
19. The person who is a pizza lover is the person who loves swimming.
20. There is one house between the person who loves basketball and the person whose birthday is in May.
21. The person who loves the soup is directly left of the person who loves the spaghetti eater.
22. The person who loves the stew is the person's child is named Bella.
23. The person who enjoys knitting is somewhere to the left of the person who loves the soup.
24. Bob is somewhere to the left of the person who loves the stew.
25. Arnold is the person who loves stir fry.
26. The woodworking hobbyist is directly left of the person who loves the soup.
 What is the value of attribute Hobby for the person whose attribute FavoriteSport is swimming? Please reason step by step, and put your final answer within \boxed{}

Answer: knitting


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Peter`
 - Each person has a unique favorite drink: `milk`, `tea`, `coffee`, `water`
 - The people keep unique animals: `horse`, `bird`, `fish`, `cat`
 - Each person prefers a unique type of vacation: `cruise`, `city`, `mountain`, `beach`

## Clues:
1. Peter is the cat lover.
2. The fish enthusiast is not in the third house.
3. The person who loves beach vacations is the person who keeps horses.
4. The fish enthusiast is not in the fourth house.
5. The coffee drinker is directly left of the bird keeper.
6. The one who only drinks water is Eric.
7. The tea drinker is somewhere to the right of Peter.
8. Eric is not in the second house.
9. The person who loves beach vacations is somewhere to the right of Peter.
10. The person who loves beach vacations is Alice.
11. The cat lover is somewhere to the left of the person who likes going on cruises.
12. The person who keeps horses is directly left of the person who enjoys mountain retreats.
 What is the value of attribute Name for the person whose attribute Animal is cat? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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

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

Answer: knitting


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Eric`, `Bob`
 - The mothers' names in different houses are unique: `Aniya`, `Penny`, `Holly`, `Kailyn`, `Janelle`
 - The people are of nationalities: `swede`, `brit`, `dane`, `norwegian`, `german`
 - Each person has a unique birthday month: `mar`, `april`, `feb`, `sept`, `jan`

## Clues:
1. The Dane is the person whose birthday is in January.
2. Alice is the British person.
3. Eric is not in the second house.
4. The person whose birthday is in January is somewhere to the left of Arnold.
5. The British person is somewhere to the left of The person whose mother's name is Janelle.
6. Peter is the German.
7. The person whose mother's name is Aniya is Peter.
8. The person whose birthday is in March is The person whose mother's name is Janelle.
9. The person whose mother's name is Kailyn is the Dane.
10. The British person is in the third house.
11. The person whose birthday is in January is somewhere to the left of The person whose mother's name is Penny.
12. The British person is directly left of the person whose birthday is in February.
13. The Dane is somewhere to the left of the Norwegian.
14. Eric is the person whose birthday is in April.
 What is the value of attribute House for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


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

## Clues:
1. Alice is directly left of the person whose favorite color is red.
2. The person who loves a bouquet of daffodils is The person whose mother's name is Janelle.
3. Eric is not in the first house.
4. The person who loves the boquet of lilies is directly left of the person living in a colonial-style house.
5. The person residing in a Victorian house is in the second house.
6. The person who loves the rose bouquet is the person who loves yellow.
7. The person whose mother's name is Aniya is the person who loves yellow.
8. The person who enjoys mountain retreats is in the first house.
9. The person who likes going on cruises is the person whose favorite color is red.
10. The person who loves yellow is in the second house.
11. The person who loves white is the person who prefers city breaks.
12. The person whose mother's name is Holly is not in the fourth house.
13. The person whose favorite color is red is directly left of Peter.
14. The person who loves a bouquet of daffodils is the person in a ranch-style home.
 What is the value of attribute Mother for the person whose attribute Vacation is cruise? Please reason step by step, and put your final answer within \boxed{}

Answer: Holly


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`, `Peter`, `Eric`, `Alice`
 - People have unique favorite music genres: `classical`, `hip hop`, `jazz`, `pop`, `rock`, `country`
 - Everyone has something unique for lunch: `soup`, `stir fry`, `grilled cheese`, `spaghetti`, `stew`, `pizza`
 - Each person lives in a unique style of house: `craftsman`, `ranch`, `mediterranean`, `modern`, `colonial`, `victorian`
 - Each person has a unique birthday month: `feb`, `sept`, `april`, `may`, `jan`, `mar`
 - People have unique hair colors: `red`, `black`, `blonde`, `brown`, `gray`, `auburn`

## Clues:
1. The person whose birthday is in April is Bob.
2. The person residing in a Victorian house is the person who loves stir fry.
3. There are two houses between the person who loves jazz music and the person who has auburn hair.
4. The person who loves the stew is in the second house.
5. The person who loves stir fry is the person whose birthday is in February.
6. The person who loves classical music is in the fifth house.
7. The person who loves jazz music is Alice.
8. The person living in a colonial-style house is in the fifth house.
9. The person who has gray hair is the person who loves the stew.
10. The person whose birthday is in March is the person who has red hair.
11. The person who loves country music is directly left of the person whose birthday is in September.
12. The person whose birthday is in May is somewhere to the left of the person who loves country music.
13. The person in a ranch-style home is the person who loves the soup.
14. The person whose birthday is in January is the person who loves the stew.
15. The person who has blonde hair is the person in a Mediterranean-style villa.
16. Arnold is somewhere to the right of the person whose birthday is in April.
17. The person whose birthday is in February is the person who loves hip-hop music.
18. The person who has blonde hair is somewhere to the left of Eric.
19. The person residing in a Victorian house is Carol.
20. The person who has black hair is the person in a ranch-style home.
21. The person in a modern-style house is not in the second house.
22. The person who loves the spaghetti eater is somewhere to the left of the person who loves country music.
23. The person who is a pizza lover is Bob.
24. The person who loves pop music is somewhere to the left of the person whose birthday is in September.
 What is the value of attribute Name for the person whose attribute Birthday is may? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Arnold`, `Eric`
 - The people keep unique animals: `cat`, `fish`, `bird`, `horse`
 - People have unique favorite sports: `soccer`, `swimming`, `tennis`, `basketball`
 - People have unique hair colors: `blonde`, `brown`, `black`, `red`
 - Each person has a unique type of pet: `bird`, `fish`, `dog`, `cat`

## Clues:
1. The person who loves tennis is in the second house.
2. The person who has brown hair and the person who loves basketball are next to each other.
3. Alice is the person who keeps a pet bird.
4. Arnold is the person who loves soccer.
5. The person who has blonde hair is the person with an aquarium of fish.
6. The person who keeps a pet bird is in the second house.
7. Arnold is in the third house.
8. The person who loves swimming is in the first house.
9. The person who has red hair is not in the first house.
10. The person who keeps horses is in the fourth house.
11. Eric is somewhere to the left of the fish enthusiast.
12. The person who has a cat is directly left of the bird keeper.
 What is the value of attribute HairColor for the person whose attribute Pet is bird? 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`, `Arnold`, `Carol`, `Peter`, `Bob`, `Eric`
 - People use unique phone models: `huawei p50`, `samsung galaxy s21`, `iphone 13`, `xiaomi mi 11`, `google pixel 6`, `oneplus 9`
 - Each person has a unique hobby: `photography`, `painting`, `gardening`, `woodworking`, `cooking`, `knitting`

## Clues:
1. The woodworking hobbyist is somewhere to the left of Bob.
2. The person who paints as a hobby is in the fifth house.
3. Arnold is directly left of the person who uses a Google Pixel 6.
4. The person who uses a Samsung Galaxy S21 is Arnold.
5. Carol is somewhere to the right of the person who uses a Samsung Galaxy S21.
6. The person who uses a Google Pixel 6 is directly left of the person who uses a OnePlus 9.
7. The person who uses a OnePlus 9 is somewhere to the right of Eric.
8. There are two houses between the person who loves cooking and Peter.
9. The person who enjoys knitting is Eric.
10. Peter is the person who paints as a hobby.
11. The person who enjoys gardening is somewhere to the right of Bob.
12. There is one house between Bob and the person who uses a Huawei P50.
13. The person who uses a Xiaomi Mi 11 is not in the first house.
14. Alice is in the sixth house.
 What is the value of attribute Hobby for the person whose attribute PhoneModel is oneplus 9? Please reason step by step, and put your final answer within \boxed{}

Answer: photography


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 Birthday for the person whose attribute Children is Bella? Please reason step by step, and put your final answer within \boxed{}

Answer: may


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Eric`, `Peter`, `Arnold`
 - People have unique heights: `average`, `short`, `very tall`, `very short`, `tall`
 - Each person has an occupation: `engineer`, `artist`, `doctor`, `lawyer`, `teacher`
 - The mothers' names in different houses are unique: `Janelle`, `Aniya`, `Holly`, `Penny`, `Kailyn`
 - Each person has a unique birthday month: `mar`, `april`, `jan`, `feb`, `sept`
 - Each person has a unique type of pet: `hamster`, `fish`, `bird`, `dog`, `cat`

## Clues:
1. The person who is a doctor is somewhere to the left of the person with a pet hamster.
2. Arnold is in the first house.
3. The person who is an engineer is the person who owns a dog.
4. The person whose birthday is in March is somewhere to the left of Peter.
5. The person who is an artist is in the third house.
6. The person who is a lawyer is in the fifth house.
7. The person who has a cat is The person whose mother's name is Kailyn.
8. Alice is directly left of Eric.
9. The person whose birthday is in February is directly left of The person whose mother's name is Holly.
10. The person whose birthday is in April is somewhere to the right of the person who is a teacher.
11. The person whose birthday is in February is The person whose mother's name is Kailyn.
12. Peter is in the fourth house.
13. The person with an aquarium of fish is the person who is very short.
14. The person whose mother's name is Kailyn and the person with an aquarium of fish are next to each other.
15. The person who is short is directly left of the person who is a lawyer.
16. The person whose mother's name is Aniya and the person who is a lawyer are next to each other.
17. The person whose birthday is in January is The person whose mother's name is Penny.
18. The person who has an average height is the person who keeps a pet bird.
19. The person who is tall is the person who is an engineer.
 What is the value of attribute Birthday for the person whose attribute Height is short? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Peter`, `Arnold`, `Bob`, `Carol`
 - Each mother is accompanied by their child: `Meredith`, `Fred`, `Bella`, `Samantha`, `Alice`, `Timothy`
 - Each person has a unique birthday month: `jan`, `feb`, `mar`, `may`, `sept`, `april`
 - Everyone has a unique favorite cigar: `dunhill`, `blends`, `blue master`, `prince`, `yellow monster`, `pall mall`

## Clues:
1. The person's child is named Meredith and the person who smokes Yellow Monster are next to each other.
2. Carol is directly left of Arnold.
3. The person's child is named Meredith is the person who smokes many unique blends.
4. The person's child is named Samantha is the person whose birthday is in March.
5. The person partial to Pall Mall is somewhere to the right of the Dunhill smoker.
6. Eric is the Dunhill smoker.
7. There is one house between the person whose birthday is in May and the person who is the mother of Timothy.
8. The person who smokes Blue Master is the person's child is named Bella.
9. The person's child is named Samantha is somewhere to the right of the person whose birthday is in January.
10. The person's child is named Samantha is in the fifth house.
11. The person whose birthday is in May is the person's child is named Meredith.
12. The person's child is named Alice is the person whose birthday is in February.
13. There is one house between Alice and Arnold.
14. Bob is the person's child is named Meredith.
15. Carol is the person whose birthday is in April.
16. The person partial to Pall Mall is the person whose birthday is in April.
17. The person whose birthday is in May is somewhere to the left of the person whose birthday is in January.
 What is the value of attribute Children for the person whose attribute Cigar is yellow monster? Please reason step by step, and put your final answer within \boxed{}

Answer: Fred


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`
 - People use unique phone models: `iphone 13`, `samsung galaxy s21`, `google pixel 6`
 - People have unique heights: `very short`, `average`, `short`
 - Each person lives in a unique style of house: `colonial`, `ranch`, `victorian`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`

## Clues:
1. Peter is somewhere to the right of Eric.
2. The person living in a colonial-style house is in the second house.
3. The person who owns a Tesla Model 3 is the person who is very short.
4. The person who is short is directly left of the person who uses a Samsung Galaxy S21.
5. The person who uses an iPhone 13 is directly left of the person who uses a Google Pixel 6.
6. The person living in a colonial-style house is somewhere to the right of the person in a ranch-style home.
7. Arnold is in the second house.
8. The person who owns a Ford F-150 is somewhere to the right of the person who owns a Toyota Camry.
9. The person who has an average height is in the first house.
 What is the value of attribute CarModel for the person whose attribute HouseStyle is victorian? Please reason step by step, and put your final answer within \boxed{}

Answer: tesla model 3


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Eric`
 - People use unique phone models: `iphone 13`, `oneplus 9`, `google pixel 6`, `samsung galaxy s21`
 - Each person prefers a unique type of vacation: `mountain`, `cruise`, `beach`, `city`
 - They all have a unique favorite flower: `carnations`, `daffodils`, `lilies`, `roses`

## Clues:
1. The person who uses an iPhone 13 is the person who enjoys mountain retreats.
2. Arnold is not in the second house.
3. The person who loves a bouquet of daffodils is directly left of the person who loves a carnations arrangement.
4. The person who uses a OnePlus 9 is not in the first house.
5. The person who uses an iPhone 13 is somewhere to the right of Eric.
6. The person who loves the rose bouquet is the person who uses a Samsung Galaxy S21.
7. The person who enjoys mountain retreats is in the third house.
8. The person who loves a carnations arrangement is directly left of the person who loves the boquet of lilies.
9. The person who prefers city breaks is not in the second house.
10. The person who uses a OnePlus 9 is Arnold.
11. The person who prefers city breaks is somewhere to the left of the person who loves the boquet of lilies.
12. The person who likes going on cruises is Peter.
 What is the value of attribute House for the person whose attribute Flower is lilies? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Eric`
 - Each person has an occupation: `teacher`, `doctor`, `engineer`, `artist`
 - The people are of nationalities: `swede`, `dane`, `norwegian`, `brit`
 - The people keep unique animals: `horse`, `bird`, `cat`, `fish`
 - Each person has a favorite color: `green`, `white`, `red`, `yellow`
 - People have unique favorite sports: `tennis`, `soccer`, `swimming`, `basketball`

## Clues:
1. The person who is an artist is directly left of the person who loves swimming.
2. Alice is the fish enthusiast.
3. The person who is a teacher is in the second house.
4. The Norwegian is the person who loves white.
5. Arnold is directly left of the Swedish person.
6. The person who loves white and the person who loves tennis are next to each other.
7. The person who keeps horses is Eric.
8. The person whose favorite color is green is the person who loves tennis.
9. The person who keeps horses is directly left of the person who loves tennis.
10. The person who loves yellow is Peter.
11. The Dane is in the first house.
12. Alice is somewhere to the right of Peter.
13. The person who loves white is not in the second house.
14. There is one house between the cat lover and the person who loves soccer.
15. The person who loves tennis is somewhere to the right of the person who is a doctor.
 What is the value of attribute FavoriteSport for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: soccer


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`, `Bob`, `Peter`, `Arnold`
 - Each person has a unique birthday month: `mar`, `april`, `sept`, `feb`, `jan`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Kailyn`, `Penny`, `Aniya`
 - Each person has an occupation: `engineer`, `doctor`, `lawyer`, `artist`, `teacher`
 - People have unique hair colors: `red`, `blonde`, `black`, `gray`, `brown`

## Clues:
1. The person whose birthday is in March is in the fifth house.
2. The person whose birthday is in February is in the first house.
3. The person who is a doctor is Eric.
4. The person whose mother's name is Janelle is in the third house.
5. The person who is an artist is the person who has brown hair.
6. The person who is an artist is in the fourth house.
7. The person whose mother's name is Penny is somewhere to the left of the person who has black hair.
8. Peter is the person who has black hair.
9. The person who has gray hair is the person who is a teacher.
10. Alice is The person whose mother's name is Kailyn.
11. Arnold is somewhere to the right of the person whose birthday is in September.
12. The person who has brown hair is the person whose birthday is in January.
13. Arnold is the person who has blonde hair.
14. The person whose mother's name is Holly is the person who has black hair.
15. Peter is the person who is a lawyer.
16. The person whose birthday is in September is somewhere to the left of The person whose mother's name is Kailyn.
17. Alice is the person who has gray hair.
 What is the value of attribute Birthday for the person whose attribute Mother is Kailyn? 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: `Peter`, `Eric`, `Arnold`, `Bob`, `Alice`, `Carol`
 - The people keep unique animals: `rabbit`, `dog`, `fish`, `bird`, `cat`, `horse`
 - Each person has a unique hobby: `cooking`, `knitting`, `painting`, `woodworking`, `photography`, `gardening`
 - People have unique favorite sports: `basketball`, `soccer`, `swimming`, `tennis`, `baseball`, `volleyball`
 - Each person prefers a unique type of vacation: `cruise`, `cultural`, `city`, `camping`, `mountain`, `beach`
 - Each person has a unique favorite drink: `water`, `root beer`, `boba tea`, `tea`, `milk`, `coffee`

## Clues:
1. The person who enjoys camping trips is the person who loves cooking.
2. The person who loves baseball is the photography enthusiast.
3. The person who loves volleyball is somewhere to the left of the person who prefers city breaks.
4. Carol is the person who loves tennis.
5. The dog owner is the person who loves swimming.
6. The photography enthusiast is the tea drinker.
7. The person who loves baseball is directly left of the person who goes on cultural tours.
8. Peter is the person who enjoys camping trips.
9. The person who enjoys knitting is the person who keeps horses.
10. The coffee drinker is directly left of the person who loves soccer.
11. The person who enjoys mountain retreats is the person who loves soccer.
12. The boba tea drinker is not in the fourth house.
13. The cat lover is somewhere to the left of the dog owner.
14. The person who loves beach vacations is Bob.
15. The coffee drinker is somewhere to the right of Alice.
16. The tea drinker is the fish enthusiast.
17. The person who likes milk is Carol.
18. There is one house between the rabbit owner and the woodworking hobbyist.
19. The person who loves cooking is not in the third house.
20. The person who loves tennis is the rabbit owner.
21. The dog owner is directly left of the person who likes going on cruises.
22. The woodworking hobbyist is the dog owner.
23. There are two houses between the person who paints as a hobby and the person who keeps horses.
24. Arnold is the root beer lover.
 What is the value of attribute FavoriteSport for the person whose attribute Vacation is camping? Please reason step by step, and put your final answer within \boxed{}

Answer: volleyball


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 Drink for the person whose attribute Flower is carnations? Please reason step by step, and put your final answer within \boxed{}

Answer: water


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Eric`
 - People have unique heights: `tall`, `short`, `very short`, `average`
 - Each person has a unique hobby: `photography`, `painting`, `cooking`, `gardening`

## Clues:
1. Alice is somewhere to the right of Peter.
2. There are two houses between the person who paints as a hobby and the person who has an average height.
3. Peter is somewhere to the left of the person who loves cooking.
4. Arnold is the person who is very short.
5. The person who is tall and Alice are next to each other.
6. The person who loves cooking is not in the third house.
7. The photography enthusiast is Peter.
8. The photography enthusiast and the person who enjoys gardening are next to each other.
9. The person who loves cooking is Eric.
 What is the value of attribute Hobby for the person whose attribute Height is short? Please reason step by step, and put your final answer within \boxed{}

Answer: gardening


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 horse? 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`, `Carol`, `Alice`, `Eric`, `Arnold`, `Peter`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `soup`, `grilled cheese`, `stew`, `stir fry`
 - Each person has a unique level of education: `associate`, `doctorate`, `trade school`, `high school`, `master`, `bachelor`
 - The people keep unique animals: `horse`, `fish`, `rabbit`, `cat`, `bird`, `dog`
 - Each person lives in a unique style of house: `ranch`, `victorian`, `modern`, `craftsman`, `mediterranean`, `colonial`

## Clues:
1. The person residing in a Victorian house is in the fourth house.
2. The fish enthusiast is in the first house.
3. The person in a ranch-style home is the bird keeper.
4. The person residing in a Victorian house is somewhere to the left of the person who loves the soup.
5. Carol is directly left of the person who keeps horses.
6. The person who loves stir fry is the person with a doctorate.
7. The person with a master's degree is somewhere to the left of the person with a doctorate.
8. There is one house between the person in a Mediterranean-style villa and the person with a bachelor's degree.
9. The person who attended trade school is the cat lover.
10. The person with a master's degree is the person who is a pizza lover.
11. The person who is a pizza lover is somewhere to the right of Alice.
12. The person living in a colonial-style house is the person who loves the stew.
13. Arnold is the person with a master's degree.
14. There is one house between Arnold and the person who loves eating grilled cheese.
15. The person in a modern-style house and the person who attended trade school are next to each other.
16. Eric is directly left of the person with a doctorate.
17. Alice and the person in a Mediterranean-style villa are next to each other.
18. The person with an associate's degree is directly left of Eric.
19. The dog owner is Bob.
 What is the value of attribute HouseStyle for the person whose attribute Food is stir fry? Please reason step by step, and put your final answer within \boxed{}

Answer: modern


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`
 - Each person prefers a unique type of vacation: `cruise`, `city`, `camping`, `beach`, `mountain`
 - Each mother is accompanied by their child: `Bella`, `Samantha`, `Fred`, `Meredith`, `Timothy`
 - The people are of nationalities: `dane`, `norwegian`, `brit`, `german`, `swede`

## Clues:
1. The Norwegian is Peter.
2. The Swedish person is the person's child is named Bella.
3. The person who loves beach vacations is directly left of the person's child is named Samantha.
4. The person's child is named Bella is not in the second house.
5. Alice is the British person.
6. The person who likes going on cruises is in the first house.
7. The person's child is named Meredith is in the fourth house.
8. Eric is not in the fifth house.
9. The Swedish person is somewhere to the right of the Norwegian.
10. There is one house between the person's child is named Fred and the person who prefers city breaks.
11. Bob is the person who enjoys camping trips.
12. The Dane is in the fifth house.
13. The person who enjoys camping trips is not in the fifth house.
 What is the value of attribute Name for the person whose attribute Children is Fred? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Eric`, `Peter`, `Arnold`
 - People have unique heights: `average`, `short`, `very tall`, `very short`, `tall`
 - Each person has an occupation: `engineer`, `artist`, `doctor`, `lawyer`, `teacher`
 - The mothers' names in different houses are unique: `Janelle`, `Aniya`, `Holly`, `Penny`, `Kailyn`
 - Each person has a unique birthday month: `mar`, `april`, `jan`, `feb`, `sept`
 - Each person has a unique type of pet: `hamster`, `fish`, `bird`, `dog`, `cat`

## Clues:
1. The person who is a doctor is somewhere to the left of the person with a pet hamster.
2. Arnold is in the first house.
3. The person who is an engineer is the person who owns a dog.
4. The person whose birthday is in March is somewhere to the left of Peter.
5. The person who is an artist is in the third house.
6. The person who is a lawyer is in the fifth house.
7. The person who has a cat is The person whose mother's name is Kailyn.
8. Alice is directly left of Eric.
9. The person whose birthday is in February is directly left of The person whose mother's name is Holly.
10. The person whose birthday is in April is somewhere to the right of the person who is a teacher.
11. The person whose birthday is in February is The person whose mother's name is Kailyn.
12. Peter is in the fourth house.
13. The person with an aquarium of fish is the person who is very short.
14. The person whose mother's name is Kailyn and the person with an aquarium of fish are next to each other.
15. The person who is short is directly left of the person who is a lawyer.
16. The person whose mother's name is Aniya and the person who is a lawyer are next to each other.
17. The person whose birthday is in January is The person whose mother's name is Penny.
18. The person who has an average height is the person who keeps a pet bird.
19. The person who is tall is the person who is an engineer.
 What is the value of attribute House for the person whose attribute Mother is Aniya? 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`, `Eric`, `Peter`, `Alice`, `Carol`, `Bob`
 - People have unique favorite music genres: `jazz`, `pop`, `classical`, `rock`, `hip hop`, `country`

## Clues:
1. Bob is directly left of the person who loves jazz music.
2. Eric is somewhere to the left of the person who loves hip-hop music.
3. Carol is in the sixth house.
4. Eric and the person who loves hip-hop music are next to each other.
5. The person who loves country music is Carol.
6. Arnold is not in the fifth house.
7. Arnold is somewhere to the right of the person who loves pop music.
8. The person who loves pop music is Peter.
9. The person who loves hip-hop music is in the third house.
10. There is one house between Peter and Bob.
11. The person who loves rock music is not in the fifth house.
 What is the value of attribute MusicGenre for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: pop


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Alice`
 - Each person has a unique birthday month: `april`, `sept`, `jan`, `feb`
 - Everyone has a unique favorite cigar: `dunhill`, `prince`, `pall mall`, `blue master`
 - People use unique phone models: `google pixel 6`, `samsung galaxy s21`, `oneplus 9`, `iphone 13`
 - They all have a unique favorite flower: `lilies`, `roses`, `carnations`, `daffodils`
 - People have unique heights: `average`, `short`, `tall`, `very short`

## Clues:
1. The person who loves a bouquet of daffodils is Arnold.
2. The person whose birthday is in April is directly left of the person who smokes Blue Master.
3. The person who is short is the person who uses an iPhone 13.
4. The person who uses a Samsung Galaxy S21 is the person who is tall.
5. Eric is the person who uses a OnePlus 9.
6. The person who uses a Samsung Galaxy S21 is directly left of the person who uses a Google Pixel 6.
7. The person whose birthday is in February is Alice.
8. The Prince smoker is somewhere to the right of the person who is short.
9. The Prince smoker is the person whose birthday is in February.
10. The person who uses a OnePlus 9 is directly left of the person who loves the rose bouquet.
11. The person partial to Pall Mall is somewhere to the left of the person who loves the boquet of lilies.
12. The person who has an average height and the person whose birthday is in September are next to each other.
13. The person who loves a bouquet of daffodils is in the fourth house.
 What is the value of attribute PhoneModel for the person whose attribute Cigar is dunhill? Please reason step by step, and put your final answer within \boxed{}

Answer: google pixel 6


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`, `Bob`
 - Each person has a unique level of education: `associate`, `bachelor`, `high school`, `master`, `doctorate`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `baseball`, `soccer`
 - Each person prefers a unique type of vacation: `camping`, `cruise`, `city`, `mountain`, `beach`
 - People use unique phone models: `google pixel 6`, `huawei p50`, `oneplus 9`, `samsung galaxy s21`, `iphone 13`

## Clues:
1. The person who uses a Huawei P50 is not in the second house.
2. The person with a master's degree is the person who uses a Google Pixel 6.
3. The person who loves swimming is the person with a high school diploma.
4. The person with a bachelor's degree is directly left of the person who loves tennis.
5. The person who enjoys camping trips is the person who loves tennis.
6. Bob is the person who uses a Samsung Galaxy S21.
7. The person with a high school diploma is not in the third house.
8. The person who loves swimming is directly left of the person who loves beach vacations.
9. The person who likes going on cruises is Arnold.
10. Bob is the person with an associate's degree.
11. The person who uses a OnePlus 9 is directly left of the person who loves basketball.
12. The person who enjoys camping trips and the person who prefers city breaks are next to each other.
13. Eric is the person who loves swimming.
14. The person who loves basketball is the person who enjoys mountain retreats.
15. The person who loves tennis is somewhere to the left of Alice.
16. Arnold is the person who loves soccer.
 What is the value of attribute House for the person whose attribute Name is Eric? 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`, `Peter`, `Alice`, `Carol`, `Bob`, `Eric`
 - Each person has an occupation: `artist`, `nurse`, `engineer`, `lawyer`, `teacher`, `doctor`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Aniya`, `Janelle`, `Sarah`, `Kailyn`
 - People have unique favorite music genres: `pop`, `rock`, `classical`, `jazz`, `hip hop`, `country`

## Clues:
1. The person who is an engineer is directly left of the person who is an artist.
2. The person whose mother's name is Janelle is the person who is an artist.
3. The person who loves classical music is directly left of Peter.
4. The person whose mother's name is Janelle and the person who loves pop music are next to each other.
5. The person whose mother's name is Sarah is somewhere to the left of Arnold.
6. The person who loves country music is The person whose mother's name is Aniya.
7. The person whose mother's name is Penny is not in the second house.
8. Bob is the person who loves jazz music.
9. Carol is not in the sixth house.
10. The person who is an engineer is the person who loves classical music.
11. The person who is a nurse is somewhere to the right of the person who loves rock music.
12. There are two houses between Bob and The person whose mother's name is Penny.
13. The person who is a lawyer is directly left of the person who is a doctor.
14. The person whose mother's name is Penny is not in the third house.
15. The person who is an engineer is not in the fourth house.
16. The person who is a doctor is The person whose mother's name is Kailyn.
17. Carol is somewhere to the right of the person who loves country music.
18. The person who loves classical music is Alice.
19. The person who is a nurse is The person whose mother's name is Holly.
 What is the value of attribute House for the person whose attribute Mother is Janelle? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


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

## Clues:
1. Eric is directly left of the Prince smoker.
2. The person who has an average height is somewhere to the right of Peter.
3. The person who uses a Samsung Galaxy S21 is in the second house.
4. The person who smokes Blue Master is in the third house.
5. The person who uses a Google Pixel 6 is Eric.
6. Arnold is somewhere to the right of the person who loves science fiction books.
7. The person who uses an iPhone 13 is the person's child is named Bella.
8. The person who is short is the Prince smoker.
9. The person who loves mystery books is the person who uses a Google Pixel 6.
10. The person's child is named Meredith and the person who is short are next to each other.
 What is the value of attribute House for the person whose attribute Children is Meredith? 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: `Carol`, `Peter`, `Bob`, `Eric`, `Arnold`, `Alice`
 - Each person has a unique birthday month: `jan`, `may`, `mar`, `feb`, `april`, `sept`
 - Each person has a unique type of pet: `rabbit`, `bird`, `cat`, `dog`, `fish`, `hamster`
 - People have unique favorite music genres: `hip hop`, `rock`, `pop`, `country`, `classical`, `jazz`
 - People have unique favorite book genres: `fantasy`, `romance`, `historical fiction`, `mystery`, `biography`, `science fiction`

## Clues:
1. The person with an aquarium of fish is not in the first house.
2. The person who loves country music is the person whose birthday is in February.
3. Arnold is directly left of the person whose birthday is in May.
4. There are two houses between the person who loves science fiction books and Alice.
5. The person who loves biography books is Eric.
6. The person with a pet hamster is Arnold.
7. The person who loves biography books is the person whose birthday is in May.
8. The person who loves pop music is the person who loves fantasy books.
9. The person who loves historical fiction books is the person whose birthday is in February.
10. The person who has a cat is the person whose birthday is in January.
11. Bob is in the sixth house.
12. The person who loves hip-hop music is the person whose birthday is in January.
13. The person who loves pop music is directly left of the person who loves jazz music.
14. The person who loves classical music is not in the third house.
15. The person whose birthday is in March is somewhere to the left of Carol.
16. Alice is the person whose birthday is in September.
17. The person with an aquarium of fish is the person who loves country music.
18. The person who owns a dog is not in the fifth house.
19. The person who loves pop music is in the fifth house.
20. Bob is the person who loves mystery books.
21. The person who keeps a pet bird is the person whose birthday is in April.
 What is the value of attribute BookGenre for the person whose attribute Pet is cat? Please reason step by step, and put your final answer within \boxed{}

Answer: romance


On a $1000 \times 1000$-board, we place dominoes such that each domino covers exactly two squares on the board. Additionally, two dominoes are not allowed to be adjacent but are allowed to touch at a vertex. Determine the maximum number of dominoes that can be placed on the board in this way.

Answer: 250000


Sergei chooses two different natural numbers $a$ and $b$. He writes four numbers in a notebook: $a$, $a+2$, $b$, and $b+2$. He then writes all six pairwise products of the numbers in the notebook on the blackboard. Let $S$ be the number of perfect squares on the blackboard. Find the maximum value of $S$.

Answer: 2


There are $100$ countries participating in an olympiad. Suppose $n$ is a positive integer such that each of the $100$ countries is willing to communicate in exactly $n$ languages. If each set of $20$ countries can communicate in exactly one common language, and no language is common to all $100$ countries, what is the minimum possible value of $n$?

Answer: 20


Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him $1$ Joule of energy to hop one step north or one step south, and $1$ Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with $100$ Joules of energy and hops till he falls asleep with $0$ energy. How many different places could he have gone to sleep?

Answer: 10201


There is a table with $n$ rows and $18$ columns. Each cell contains either a $0$ or a $1$. The table satisfies the following properties:

1. Every two rows are different.
2. Each row contains exactly $6$ cells that contain $1$.
3. For every three rows, there exists a column such that the intersection of the column with the three rows (the three cells) all contain $0$.

What is the greatest possible value of $n$?

Answer: 12376


Initially, all edges of the $K_{2024}$ are painted with $13$ different colors. If there exist $k$ colors such that the subgraph constructed by the edges which are colored with these $k$ colors is connected no matter how the initial coloring was, find the minimum value of $k$.

Answer: 7


Let $a, b, c, d$ be the four roots of the polynomial:
\[
x^4 + 3x^3 - x^2 + x - 2.
\]
Given that:
\[
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = \frac{1}{2}
\]
and
\[
\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} + \frac{1}{d^2} = -\frac{3}{4},
\]
the value of:
\[
\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} + \frac{1}{d^3}
\]
can be expressed as \(\frac{m}{n}\) for relatively prime positive integers \(m\) and \(n\). Compute \(m+n\).

Answer: 39


Two semicircles are tangent to a middle circle, and both semicircles and the middle circle are tangent to a horizontal line as shown. If $PQ = QR = RS = 24$, find the length of the radius $r$.

Answer: 15


Two circles, both with the same radius $r$, are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A, B$ and the other at points $C, D$, so that $|AB| = |BC| = |CD| = 14\text{ cm}$. Another line intersects the circles at $E, F$, respectively $G, H$ so that $|EF| = |FG| = |GH| = 6\text{ cm}$. Find the radius $r$.

Answer: 13


A table tennis club hosts a series of doubles matches following several rules:
1. Each player belongs to two pairs at most.
2. Every two distinct pairs play one game against each other at most.
3. Players in the same pair do not play against each other when they pair with others respectively.

Every player plays a certain number of games in this series. All these distinct numbers make up a set called the "set of games".

Consider a set $A=\{a_1,a_2,\ldots ,a_k\}$ of positive integers such that every element in $A$ is divisible by $6$. Determine the minimum number of players needed to participate in this series so that a schedule for which the corresponding "set of games" is equal to set $A$ exists.

Answer: 6


$n$ consecutive positive integers are arranged in a row (not necessarily in order) such that the sum of any three successive integers in the row is divisible by the leftmost number in the triple. Determine the largest possible value of $n$ if the last number in the row is odd.

Answer: 5


Three identical circles are packed into a unit square. Each of the three circles is tangent to each other and tangent to at least one side of the square. If $r$ is the maximum possible radius of the circle, what is $(2-\frac{1}{r})^2$?

Answer: 2


In an abandoned chemistry lab, Gerome found a two-pan balance scale and three 1-gram weights, three 5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile. Find the number of different positive weights of chemicals that Gerome could measure.

Answer: 129


For integers $0 \le m,n \le 2^{2017}-1$, let $\alpha(m,n)$ be the number of nonnegative integers $k$ for which $\left\lfloor \frac{m}{2^k} \right\rfloor$ and $\left\lfloor \frac{n}{2^k} \right\rfloor$ are both odd integers. Consider a $2^{2017} \times 2^{2017}$ matrix $M$ whose $(i,j)$th entry (for $1 \le i, j \le 2^{2017}$) is $(-1)^{\alpha(i-1, j-1)}$. For $1 \le i, j \le 2^{2017}$, let $M_{i,j}$ be the matrix with the same entries as $M$ except for the $(i,j)$th entry, denoted by $a_{i,j}$, and such that $\det M_{i,j}=0$. Suppose that $A$ is the $2^{2017} \times 2^{2017}$ matrix whose $(i,j)$th entry is $a_{i,j}$ for all $1 \le i, j \le 2^{2017}$. Compute the remainder when $\det A$ is divided by $2017$.

Answer: 1382


Define a \textit{subsequence} of a string \( \mathcal{S} \) of letters to be a positive-length string using any number of the letters in \( \mathcal{S} \) in order. For example, a subsequence of \( HARRISON \) is \( ARRON \). Compute the number of subsequences in \( HARRISON \).

Answer: 255


Let $O$ be the center and let $F$ be one of the foci of the ellipse $25x^2 +16 y^2 = 400$. A second ellipse, lying inside and tangent to the first ellipse, has its foci at $O$ and $F$. What is the length of the minor axis of this second ellipse?The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 12


The parabola $y = x^2$ is tangent to the graph of $y = x^4 + ax^3 + x^2 + bx + 1$ at two points.  Find the positive difference between the $x$-coordinates of the points of tangency.The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 7


Let $\tau = \frac{1 + \sqrt{5}}{2}.$  Find
\[\sum_{n = 0}^\infty \frac{\lfloor \tau^n \rceil}{2^n}.\]Note: For a real number $x,$ $\lfloor x \rceil$ denotes the integer closest to $x.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 13


4. Let $M$ be a set of six mutually different positive integers which sum up to 60 . We write these numbers on faces of a cube (on each face one). In a move we choose three faces with a common vertex and we increase each number on these faces by one. Find the number of all sets $M$, whose elements (numbers) can be written on the faces of the cube in such a way that we can even up the numbers on the faces in finitely many moves.

(Peter Novotný)

Answer: 84


10.5. An increasing geometric progression consists of four different positive numbers, three of which form an arithmetic progression. What can the denominator of this progression be? Provide all possible answers and prove that there are no others.

Answer: \dfrac{1 + \sqrt{5}}{2}


G2.3 In Figure 1 , the square $P Q R S$ is inscribed in $\triangle A B C$. The areas of $\triangle A P Q, \triangle P B S$ and $\triangle Q R C$ are 4,4 and 12 respectively. If the area of the square is $c$, find the value of $c$.

Answer: 16


2. Simplify $\sqrt[3]{10+7 \sqrt{2}}+\sqrt[3]{10-7 \sqrt{2}}$ to get the result $\qquad$ .

Answer: \sqrt[3]{32}


There are 6 moments, $6: 30, 6: 31, 6: 32, 6: 33, 6: 34, 6: 35$. At $\qquad$ moment, the hour hand and the minute hand are closest to each other, and at $\qquad$ moment, the hour hand and the minute hand are farthest from each other.

Answer: 6:33


Consider functions $f : [0, 1] \rightarrow \mathbb{R}$ which satisfy

     (i)$f(x)\ge0$ for all $x$ in $[0, 1]$,

     (ii)$f(1) = 1$,

     (iii)     $f(x) + f(y) \le f(x + y)$ whenever $x$, $y$, and $x + y$ are all in $[0, 1]$.
Find, with proof, the smallest constant $c$ such that

$f(x) \le cx$
for every function $f$ satisfying (i)-(iii) and every $x$ in $[0, 1]$.

Answer: 2


A circle of radius  $4$  is inscribed in a triangle  $ABC$ . We call  $D$  the touchpoint between the circle and side BC. Let  $CD =8$ ,  $DB= 10$ . What is the length of the sides  $AB$  and  $AC$ ?

Answer: 12.5


Every of  $n$  guests invited to a dinner has got an invitation denoted by a number from  $1$  to  $n$ . The guests will be sitting around a round table with  $n$  seats. The waiter has decided to derve them according to the following rule. At first, he selects one guest and serves him/her at any place. Thereafter, he selects the guests one by one: having chosen a guest, he goes around the table for the number of seats equal to the preceeding guest's invitation number (starting from the seat of the preceeding guest), and serves the guest there.
Find all  $n$  for which he can select the guests in such an order to serve all the guests.

Answer: n


$n$  is a natural number. Given   $3n \cdot 3n$  table, the unit cells are  colored white and black such that starting from the left up corner diagonals are colored in pure white or black in ratio of 2:1 respectively. ( See the picture below). In one step any chosen  $2 \cdot 2$  square's white cells are colored orange, orange are colored black and black are colored white. Find all  $n$   such that with finite steps, all the white cells in the table turns to black, and all black cells in the table turns to white. ( From starting point)

Answer: n


For a positive integer  $N>1$  with unique factorization  $N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}$ , we define
\[\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.\]
Let  $a_1,a_2,\dotsc, a_n$  be positive integers and  $p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)$  such that for all positive integers  $k$ ,  $\Omega(P(k))$  is even. Show that  $n$  is an even number.

Answer: n


We define a sequence  $a_n$  so that  $a_0=1$  and
\[a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2,  a_n + d & \textrm{ otherwise. } \end{cases}  \]
for all postive integers  $n$ .
Find all positive integers  $d$  such that there is some positive integer  $i$  for which  $a_i=1$ .

Answer: d


A rectangular pool table has a hole at each of three of its corners. The lengths of sides of the table are the real numbers  $a$  and  $b$ . A billiard ball is shot from the fourth corner along its angle bisector. The ball falls in one of the holes. What should the relation between  $a$  and  $b$  be for this to happen?

Answer: \frac{a}{b}


A town-planner has built an isolated city whose road network consists of  $2N$  roundabouts, each connecting exactly three roads. A series of tunnels and bridges ensure that all roads in the town meet only at roundabouts. All roads are two-way, and each roundabout is oriented clockwise.
Vlad has recently passed his driving test, and is nervous about roundabouts. He starts driving from his house, and always takes the first edit at each roundabout he encounters. It turns out his journey incluldes every road in the town in both directions before he arrives back at the starting point in the starting direction. For what values of  $N$  is this possible?

Answer: N


9.5. Let $M$ - be a finite set of numbers (distinct). It is known that among any three of its elements, there will be two whose sum belongs to $M$. What is the maximum number of elements that can be in $M$?

Answer: 7


3. Given two sets of numbers, set $A$ is: $1,2, \cdots, 100$; set $B$ is: $1^{2}, 2^{2}, \cdots, 100^{2}$. For a number $x$ in set $A$, if there is a number $y$ in set $B$ such that $x+y$ is also a number in set $B$, then $x$ is called an "associated number". Therefore, the number of such associated numbers in set $A$ is $\qquad$.

Answer: 73


20. Let $\lfloor x\rfloor$ be the greatest integer not exceeding $x$. For instance, $\lfloor 3.4\rfloor=3,\lfloor 2\rfloor=2$, and $\lfloor-2.7\rfloor=-3$. Determine the value of the constant $\lambda>0$ so that $2\lfloor\lambda n\rfloor=1-n+\lfloor\lambda\lfloor\lambda n\rfloor\rfloor$ for all positive integers $n$.

Answer: 1 + \sqrt{2}


1. The pond has a rectangular shape. On the first frosty day, the part of the pond within 10 meters of the nearest shore froze. On the second day, the part within 20 meters froze, on the third day, the part within 30 meters, and so on. On the first day, the area of open water decreased by 20.2%, and on the second day, it decreased by 18.6% of the original area. On which day will the pond be completely frozen?

Answer: 7


6. There are infinitely many cards, each with a real number written on it. For each real number $x$, there is exactly one card with the number $x$ written on it. Two players each select a set of 100 cards, denoted as $A$ and $B$, such that the sets are disjoint. Formulate a rule to determine which of the two players wins, satisfying the following conditions:
(1) The winner depends only on the relative order of these 200 cards: if these 200 cards are placed face down in increasing order, and the audience is informed which card belongs to which player, but not what number is written on each card, the audience can still determine who will win;
(2) If the elements of the two sets are written in increasing order as
$$
A=\left\{a_{1}, a_{2}, \cdots, a_{100}\right\}, B=\left\{b_{1}, b_{2}, \cdots, b_{100}\right\},
$$

where, for all $i \in\{1,2, \cdots, 100\}$, $a_{i}>b_{i}$, then $A$ defeats $B$;
(3) If three players each select a set of 100 cards, denoted as $A$, $B$, and $C$, and $A$ defeats $B$, and $B$ defeats $C$, then $A$ defeats $C$.
Question: How many such rules are there?
[Note] Two different rules mean that there exist two sets $A$ and $B$ such that in one rule, $A$ defeats $B$, and in the other rule, $B$ defeats $A$.

Answer: 100


In a roulette wheel, any number from 0 to 2007 can come up with equal probability. The roulette wheel is spun repeatedly. Let $P_{k}$ denote the probability that at some point the sum of the numbers that have come up in all the spins equals $k$. Which number is greater: $P_{2007}$ or $P_{2008} ?$

Answer: P_{2007}


1. For a regular triangular prism with base edge length and side edge length both 1, a section is made through one side of the base and the midpoint of the line connecting the centers of the top and bottom bases. What is the area of the section? $\qquad$ .

Answer: \dfrac{4\sqrt{3}}{9}


An excursion group of 6 tourists is sightseeing. At each attraction, three people take photos while the others photograph them. After visiting the minimal number of attractions, how many attractions will it take for each tourist to have photos of all the other participants?

Answer: 4


Kanel-Belov A.Y.

The periods of two sequences are 7 and 13. What is the maximum length of the initial segment that can coincide?

Answer: 18


9. In a convex quadrilateral $A B C D, \angle B A C=\angle C A D, \angle A B C=\angle A C D$, the extensions of $A D$ and $B C$ meet at $E$, and the extensions of $A B$ and $D C$ meet at $F$. Determine the value of
$$
\frac{A B \cdot D E}{B C \cdot C E} .
$$

Answer: 1


5. A semicircle of radius 1 is drawn inside a semicircle of radius 2, as shown in the diagram, where $O A=O B=2$.
A circle is drawn so that it touches each of the semicircles and their common diameter, as shown.
What is the radius of the circle?

Answer: \dfrac{8}{9}


Exercise 7. Let $n$ be a strictly positive integer. Domitille has a rectangular grid divided into unit squares. Inside each unit square is written a strictly positive integer. She can perform the following operations as many times as she wishes:

- Choose a row and multiply each number in the row by $n$.
- Choose a column and subtract $n$ from each integer in the column.

Determine all values of $n$ for which the following property is satisfied:

Regardless of the dimensions of the rectangle and the integers written in the cells, Domitille can end up with a rectangle containing only 0s after a finite number of operations.

Answer: 2


4. How many numbers at most can be chosen from the set $M=\{1,2, \ldots, 2018\}$ such that the difference of any two chosen numbers is not equal to a prime number?

The regional round of category B takes place

## on Tuesday, April 10, 2018

so that it starts no later than 10:00 AM and the contestants have 4 hours of pure time to solve the problems. Allowed aids are writing and drawing supplies and school MF tables. Calculators, laptops, and any other electronic aids are not allowed. Each problem can earn the contestant 6 points; the logical correctness and completeness of the written solution are also evaluated. The point threshold (higher than 7 points) for determining successful solvers will be centrally set after evaluating the statistics of results from all regions. This information will be communicated to the students before the competition begins.

Answer: 1009


In triangle $ABC$ , $AC=13$ , $BC=14$ , and $AB=15$ . Points $M$ and $D$ lie on $AC$ with $AM=MC$ and $\angle ABD = \angle DBC$ . Points $N$ and $E$ lie on $AB$ with $AN=NB$ and $\angle ACE = \angle ECB$ . Let $P$ be the point, other than $A$ , of intersection of the circumcircles of $\triangle AMN$ and $\triangle ADE$ . Ray $AP$ meets $BC$ at $Q$ . The ratio $\frac{BQ}{CQ}$ can be written in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m-n$ .
Please give the answer directly without any intermediate steps.

Answer: 218


1. Find all five-digit numbers $\overline{a b c d e}$ that are divisible by 9 and for which $\overline{a c e}-\overline{b d a}=760$.

Answer: 81828


5. A subset $M$ of the set $S=\{1928,1929, \cdots, 1949\}$ is called "red" if and only if the sum of any two distinct elements in $M$ is not divisible by 4. Let $x$ and $y$ represent the number of red four-element subsets and red five-element subsets of $S$, respectively. Try to compare the sizes of $x$ and $y$, and explain the reason.

---

The translation preserves the original text's formatting and structure.

Answer: y > x


We define the *Fibonacci sequence*  $\{F_n\}_{n\ge0}$  by  $F_0=0$ ,  $F_1=1$ , and for  $n\ge2$ ,  $F_n=F_{n-1}+F_{n-2}$ ; we define the *Stirling number of the second kind*  $S(n,k)$  as the number of ways to partition a set of  $n\ge1$  distinguishable elements into  $k\ge1$  indistinguishable nonempty subsets.

For every positive integer  $n$ , let  $t_n = \sum_{k=1}^{n} S(n,k) F_k$ . Let  $p\ge7$  be a prime. Prove that \[ t_{n+p^{2p}-1} \equiv t_n \pmod{p} \] for all  $n\ge1$ .

*Proposed by Victor Wang*

Answer: t_{n+p^{2p}-1} \equiv t_n \pmod{p}


Given \( n (\geqslant 4) \) points in a plane, where no three points are collinear, connected by \( l \left( \geqslant \left[\frac{n^{2}}{4} \right] + 1\right) \) line segments. If two triangles share a common edge, they are called an (unordered) triangle pair. Let \( T \) be the number of triangle pairs formed by these \( l \) line segments. Prove that \( T \geqslant \frac{l\left(4 l - n^{2}\right)\left(4 l - n^{2} - n\right)}{2 n^{2}} \).

Answer: T \geqslant \frac{l\left(4 l - n^{2}\right)\left(4 l - n^{2} - n\right)}{2 n^{2}}


6. Let $a_{1}, a_{2}, \cdots, a_{2014}$ be a permutation of the positive integers $1,2, \cdots$, 2014. Denote
$$
S_{k}=a_{1}+a_{2}+\cdots+a_{k}(k=1,2, \cdots, 2014) \text {. }
$$

Then the maximum number of odd numbers in $S_{1}, S_{2}, \cdots, S_{2014}$ is $\qquad$

Answer: 1511


In regular octagon $ABCDEFGH$, $M$ and $N$ are midpoints of $\overline{BC}$ and $\overline{FG}$ respectively.  Compute $[ABMO]/[EDCMO]$. ($[ABCD]$ denotes the area of polygon $ABCD$.) [asy]
pair A,B,C,D,E,F,G,H;
F=(0,0); E=(2,0); D=(2+sqrt(2),sqrt(2)); C=(2+sqrt(2),2+sqrt(2));
B=(2,2+2sqrt(2)); A=(0,2+2*sqrt(2)); H=(-sqrt(2),2+sqrt(2)); G=(-sqrt(2),sqrt(2));
draw(A--B--C--D--E--F--G--H--cycle);
draw(A--E);
pair M=(B+C)/2; pair N=(F+G)/2;
draw(M--N);

label("$A$",A,N); label("$B$",B,NE); label("$C$",C,E); label("$D$",D,E);

label("$E$",E,S); label("$F$",F,S); label("$G$",G,W); label("$H$",H,W);
label("$M$",M,NE); label("$N$",N,SW);

label("$O$",(1,2.4),E);

[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 8


A few (at least $5$) integers are placed on a circle such that each integer is divisible by the sum of its neighboring integers. If the sum of all the integers is positive, what is the minimal value of this sum?

Answer: 2


Let $S = \{1, 2, \dots, 9\}$. Compute the number of functions $f : S \rightarrow S$ such that, for all $s \in S$, $f(f(f(s))) = s$ and $f(s) - s$ is not divisible by $3$.

Answer: 288


An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M \in \overline{AB}$, $Q \in \overline{AC}$, and $N, P \in \overline{BC}$.

Suppose that $ABC$ is an equilateral triangle with side length $2$, and that $AMNPQ$ has a line of symmetry perpendicular to $BC$. The area of $AMNPQ$ is given by $n - p \sqrt{q}$, where $n, p, q$ are positive integers and $q$ is not divisible by the square of a prime. Compute $100n + 10p + q$.

Answer: 5073


A group of 6 students decided to form "study groups" and "service activity groups" based on the following principle:

- Each group must have exactly 3 members.
- For any pair of students, there are the same number of study groups and service activity groups that both students are members of.

Given that there is at least one group and no three students belong to both the same study group and service activity group, find the minimum number of groups.

Answer: 8


A rectangle-shaped puzzle is assembled with $2000$ pieces that are all equal rectangles and similar to the large rectangle. The sides of the small rectangles are parallel to those of the large one. The shortest side of each piece measures $1$. Determine the minimum possible value of the area of the large rectangle.

Answer: 2000


Define the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$ for all positive real numbers $x$. How many integers from $1$ to $2023$ inclusive are in the range of $f(x)$? Note that $\lfloor x \rfloor$ is known as the $\textit{floor}$ function, which returns the greatest integer less than or equal to $x$.

Answer: 1973


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

Answer: 47


Let $N = \prod_{k=1}^{1000} (4^k - 1)$. Determine the largest positive integer $n$ such that $5^n$ divides evenly into $N$.

Answer: 624


Felix chooses a positive integer as the starting number and writes it on the board. He then repeats the following step:
- If the number $n$ on the board is even, he replaces it with $\frac{1}{2}n$.
- If the number $n$ on the board is odd, he replaces it with $n^2 + 3$.

Determine for how many choices of starting numbers below $2023$ Felix will never write a number of more than four digits on the board.

Answer: 21


Integers $a$, $b$, $c$ are selected independently and at random from the set $\{ 1, 2, \cdots, 10 \}$, with replacement. If $p$ is the probability that $a^{b-1}b^{c-1}c^{a-1}$ is a power of two, compute $1000p$.

Answer: 136


Two congruent equilateral triangles $\triangle ABC$ and $\triangle DEF$ lie on the same side of line $BC$ so that $B$, $C$, $E$, and $F$ are collinear as shown. A line intersects $\overline{AB}$, $\overline{AC}$, $\overline{DE}$, and $\overline{EF}$ at $W$, $X$, $Y$, and $Z$, respectively, such that $\frac{AW}{BW} = \frac{2}{9}$, $\frac{AX}{CX} = \frac{5}{6}$, and $\frac{DY}{EY} = \frac{9}{2}$. The ratio $\frac{EZ}{FZ}$ can then be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 33


An arithmetic sequence of exactly $10$ positive integers has the property that any two elements are relatively prime. Compute the smallest possible sum of the $10$ numbers.

Answer: 1360


Let $N$ be the number of positive divisors of $2010^{2010}$ that end in the digit 2. What is the remainder when $N$ is divided by 2010?

Answer: 503


Let $a > 0$. If the inequality $22 < ax < 222$ holds for precisely $10$ positive integers $x$, find how many positive integers satisfy the inequality $222 < ax < 2022$?

Answer: 90


For a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes, respectively. A positive integer $n$ is called fair if $s(n) = c(n) > 1$. Find the number of fair integers less than $100$.

Answer: 7


The vertices of a rhombus are located on the sides of a parallelogram, and the sides of the rhombus are parallel to the diagonals of the parallelogram. Find the ratio of the areas of the rhombus and the parallelogram if the ratio of the diagonals of the parallelogram is \( k \).

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


Point \( K \) lies on edge \( AB \) of pyramid \( ABCD \). Construct the cross-section of the pyramid with a plane passing through point \( K \) parallel to lines \( BC \) and \( AD \).

Answer: K L M N


Five children and a monkey want to share a bag of walnuts. One of the children—unbeknownst to the others—divided the walnuts in the bag into five equal parts. He found one extra walnut, gave it to the monkey, and left with one-fifth of the walnuts. The second child then divided the remaining walnuts into five equal parts, also found one extra walnut, gave it to the monkey, and took one-fifth of the portion. The remaining three children did the same. Finally, the five children gathered the remaining walnuts, divided them into five equal parts, found one extra walnut, and gave it to the monkey.

What is the minimum number of walnuts that were originally in the bag?

Answer: 15621


The height of a triangle divides an angle of the triangle in the ratio \(2: 1\), and the base into segments whose ratio (of the larger to the smaller) is equal to \(k\). Find the sine of the smaller angle at the base and the permissible values of \(k\).

Answer: \frac{1}{k-1}


Given the angles of triangle ABC and the radius R of the circumscribed circle passing through the vertices, connect the feet of the altitudes of the triangle. Calculate the sides, angles, and the circumradius of the triangle formed by these feet (the orthic triangle).

Answer: \frac{R}{2}


There are 11 kilograms of cereal. How can one measure out 1 kilogram of cereal using a balance scale and a single 3-kilogram weight in just two weighings?

Answer: 1 \text{ kg}


Through every three vertices of a cube, located at the ends of each trio of edges meeting at one vertex, a plane is drawn. Find the volume of the solid bounded by these planes if the edge length of the cube is $a$.

Answer: \frac{a^3}{6}


Let \( n \) be a positive integer. Let \( V_{n} \) be the set of all sequences of 0's and 1's of length \( n \). Define \( G_{n} \) to be the graph having vertex set \( V_{n} \), such that two sequences are adjacent in \( G_{n} \) if and only if they differ in either 1 or 2 places. For instance, if \( n=3 \), the sequences \((1,0,0)\), \((1,1,0)\), and \((1,1,1)\) are mutually adjacent, but \((1,0,0)\) is not adjacent to \((0,1,1)\).

Show that, if \( n+1 \) is not a power of 2, then the chromatic number of \( G_{n} \) is at least \( n+2 \).

Answer: n+2


There are 100 light bulbs arranged in a row on a Christmas tree. The bulbs are then toggled according to the following algorithm: initially, all the bulbs are turned on; after one second, every second bulb is turned off; after another second, every third bulb is toggled (turned off if it was on, turned on if it was off); after another second, every fourth bulb is toggled, and so on. After 100 seconds, the process stops. Find the probability that a randomly selected bulb is on after this process (the bulbs do not burn out or break).

Answer: 0.1


Consider the numbers \( k \) and \( n \) such that they are relatively prime, with \( n \geq 5 \) and \( k < \frac{n}{2} \). A regular \((n ; k)\)-star is defined as a closed broken line obtained by replacing every \( k \) consecutive sides of a regular \( n \)-gon with a diagonal connecting those endpoints. For example, a \((5 ; 2)\)-star has 5 self-intersection points as illustrated by the bold points in the figure. How many self-intersections does a \((2018 ; 25)\)-star have?

Answer: 48432


The length of the road is 300 km. Car $A$ starts at one end of the road at noon and moves at a constant speed of 50 km/h. At the same time, car $B$ starts at the other end of the road with a constant speed of 100 km/h, and a fly starts with a speed of 150 km/h. Upon meeting car $A$, the fly turns around and heads towards car $B$.

1) When does the fly meet car $B$?
2) If upon meeting car $B$, the fly turns around, heads towards car $A$, meets it, turns around again, and continues flying back and forth between $A$ and $B$ until they collide, when do the cars crush the fly?

Answer: 2 \ \text{hours}


The microcalculator MK-97 can perform only three operations on numbers stored in memory:

1) Check if two chosen numbers are equal,
2) Add the chosen numbers,
3) For chosen numbers \( a \) and \( b \), find the roots of the equation \( x^2 + ax + b = 0 \), and if there are no roots, display a message indicating so.

All results of actions are stored in memory. Initially, one number \( x \) is stored in the memory. Using the MK-97, how can you determine if this number is equal to one?

Answer: x = 1


Let \( a \otimes b = a \times \sqrt{b + \sqrt{b + 1 + \sqrt{b + 2 + \sqrt{b + 3 + ...}}}} \). If \( 3 \otimes h = 15 \), find the value of \( h \).

Answer: 20


Three congruent isosceles triangles \( EFO \), \( OFG \), and \( OGH \) have \( EF=EO=OG=GH=15 \) and \( FG=EO=OH=18 \). These triangles are arranged to form trapezoid \( EFGH \), as shown. Point \( Q \) is on side \( FG \) such that \( OQ \) is perpendicular to \( FG \).

Point \( Z \) is the midpoint of \( EF \) and point \( W \) is the midpoint of \( GH \). When \( Z \) and \( W \) are joined, the trapezoid is divided into two smaller trapezoids. Find the ratio of the area of trapezoid \( EFZW \) to the area of trapezoid \( ZWGH \) in simplified form, and calculate \( r+s \) where the ratio is \( r:s \).

Answer: 2


The numbers $\sqrt{3u-2}$, $\sqrt{3u+2}$, and $2\sqrt{u}$ are given as the side lengths of a triangle. What is the measure of the largest angle in this triangle?

Answer: 90


Consider the function $y = g(x)$ where $g(x) = \frac{x^2}{Dx^2 + Ex + F}$, and $D, E, F$ are integers. The graph has vertical asymptotes at $x = -3$ and $x = 2$. It is also given that for all $x > 5$, $g(x) > 0.5$. If additionally, $g(0) = \frac{1}{4}$, find $D + E + F$.

Answer: -4


Marguerite drove 100 miles in 2.4 hours. If Sam drove for 4 hours at the same average rate as Marguerite but rested for 1 hour during this period, how many miles did he drive in total?

Answer: 125 \text{ miles}


In a square layout, the vertices of the square are the centers of four circles. Each side of the square is 10 cm, and the radius of each circle is 3 cm. Calculate the area in square centimeters of the shaded region formed by the exclusion of the parts of the circles that overlap with the square.

Answer: 100 - 9\pi \text{ cm}^2


Each triangle is a 45-45-90 triangle, and the hypotenuse of one triangle is the leg of an adjacent triangle. The hypotenuse of the largest triangle is 10 centimeters. What is the length of the leg of the smallest triangle? Express your answer as a common fraction.

Answer: \frac{5}{2}


A bowling ball is redesigned with a spherical surface and a diameter of 24 cm. For custom fitting, three holes are drilled for a new bowler, each 6 cm deep. The diameters of the holes are 2 cm, 2.5 cm, and 4 cm. Assuming the holes are right circular cylinders, calculate the volume of the fitted bowling ball in cubic centimeters, expressed in terms of \(\pi\).

Answer: 2264.625\pi \text{ cubic cm}


Let
\[ g(x) = x^3 + 5x^2 + 15x + 35. \]
The graphs of \( y = g(x) \) and \( y = g^{-1}(x) \) intersect at exactly one point \((c,d)\). Enter the ordered pair \((c,d)\).

Answer: (-5, -5)


In $\Delta ABC$, $AC = BC$, $m\angle DCB = 50^{\circ}$, and $CD \parallel AB$. Point $E$ is on extension of $CD$ such that $DE \parallel BC$. Determine $m\angle ECD$.

Answer: 50^\circ


Jamie paid for a $2.40 meal using exactly 50 coins consisting of pennies, nickels, and dimes. How many dimes did Jamie use if no change was received?

Answer: 10