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`, `Arnold`, `Peter`
 - Each mother is accompanied by their child: `Samantha`, `Meredith`, `Timothy`, `Bella`, `Fred`
 - People have unique hair colors: `red`, `blonde`, `black`, `brown`, `gray`
 - Each person has a unique hobby: `gardening`, `cooking`, `painting`, `knitting`, `photography`
 - Each person has a unique type of pet: `hamster`, `bird`, `cat`, `fish`, `dog`
 - They all have a unique favorite flower: `tulips`, `roses`, `carnations`, `lilies`, `daffodils`

## Clues:
1. The person's child is named Samantha is somewhere to the right of the person's child is named Meredith.
2. The person who loves the boquet of lilies is the person who keeps a pet bird.
3. The person who has blonde hair is the person who loves the vase of tulips.
4. Bob is the person who has brown hair.
5. The person who paints as a hobby is the person with an aquarium of fish.
6. The person's child is named Fred is Peter.
7. The person with an aquarium of fish is somewhere to the left of the person who has a cat.
8. The person with a pet hamster is the person's child is named Samantha.
9. The person's child is named Bella is Arnold.
10. The person's child is named Bella is in the third house.
11. Peter is not in the first house.
12. Alice is the person who has blonde hair.
13. The person who loves the rose bouquet is not in the third house.
14. The person who owns a dog is the person's child is named Fred.
15. The person who has red hair is directly left of the person who has black hair.
16. The photography enthusiast is directly left of Eric.
17. The person who has black hair is the person who loves the rose bouquet.
18. The person who loves a carnations arrangement is not in the third house.
19. The person who enjoys knitting is somewhere to the right of the person's child is named Bella.
20. The person who loves the boquet of lilies is the person who enjoys gardening.
21. The person who keeps a pet bird is in the fifth house.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 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`, `Bob`, `Eric`, `Carol`
 - Each person lives in a unique style of house: `ranch`, `mediterranean`, `victorian`, `craftsman`, `colonial`, `modern`

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

Answer: Peter


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Eric`, `Arnold`
 - Each person has a unique level of education: `high school`, `bachelor`, `associate`, `master`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`, `fantasy`

## Clues:
1. The person who loves mystery books is the person with a high school diploma.
2. The person who loves science fiction books is Alice.
3. The person with a master's degree is directly left of the person who loves mystery books.
4. The person with a master's degree is in the first house.
5. The person with a bachelor's degree is not in the third house.
6. Peter is directly left of the person who loves romance books.
7. The person who loves romance books is directly left of Arnold.
 What is the value of attribute Name for the person whose attribute BookGenre is mystery? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Arnold`, `Eric`, `Alice`, `Bob`, `Peter`
 - Each person has a unique level of education: `bachelor`, `high school`, `master`, `associate`, `trade school`, `doctorate`
 - Each person has a favorite color: `purple`, `yellow`, `white`, `red`, `blue`, `green`
 - The people are of nationalities: `swede`, `dane`, `norwegian`, `chinese`, `brit`, `german`
 - Each person has a unique birthday month: `feb`, `may`, `sept`, `april`, `jan`, `mar`

## Clues:
1. The person who loves yellow is somewhere to the right of the person whose birthday is in January.
2. Bob is the person who loves white.
3. The person who loves blue is the person whose birthday is in April.
4. The person with an associate's degree is Eric.
5. The person whose favorite color is green is the person whose birthday is in May.
6. The person who loves yellow is not in the fifth house.
7. The person with an associate's degree is directly left of the Swedish person.
8. The person with a doctorate is the Dane.
9. The person whose birthday is in January is somewhere to the right of the person whose favorite color is green.
10. There are two houses between the British person and the person whose birthday is in February.
11. The person whose birthday is in February is in the fourth house.
12. Alice is the person with a high school diploma.
13. Carol is in the first house.
14. Bob is the German.
15. The person whose birthday is in May is directly left of the Norwegian.
16. The person whose favorite color is red is not in the fifth house.
17. The person whose birthday is in September is in the second house.
18. There are two houses between the person whose birthday is in May and the person with a master's degree.
19. The Norwegian is Peter.
20. The person who attended trade school is the person who loves purple.
 What is the value of attribute Color for the person whose attribute Education is high school? Please reason step by step, and put your final answer within \boxed{}

Answer: yellow


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`, `Arnold`, `Alice`, `Bob`, `Eric`
 - Each person has an occupation: `lawyer`, `teacher`, `artist`, `engineer`, `nurse`, `doctor`
 - Each person has a unique type of pet: `dog`, `hamster`, `rabbit`, `cat`, `fish`, `bird`

## Clues:
1. Carol is in the third house.
2. Carol is the person who owns a rabbit.
3. Arnold is in the fifth house.
4. The person with a pet hamster is not in the fourth house.
5. The person with an aquarium of fish is not in the fourth house.
6. The person who owns a dog is Peter.
7. Eric is the person who is a nurse.
8. The person who is a teacher is in the second house.
9. Alice is somewhere to the left of the person who has a cat.
10. The person with a pet hamster is somewhere to the left of Eric.
11. The person with an aquarium of fish is not in the second house.
12. The person who owns a dog is directly left of the person who is an artist.
13. The person who is a doctor and the person who is a teacher are next to each other.
14. Bob is the person with an aquarium of fish.
15. The person who is a lawyer is the person with a pet hamster.
 What is the value of attribute House for the person whose attribute Pet is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Bob`, `Peter`, `Alice`, `Carol`
 - People have unique heights: `very tall`, `tall`, `average`, `short`, `super tall`, `very short`
 - Each person has a unique birthday month: `sept`, `may`, `jan`, `april`, `feb`, `mar`
 - People have unique hair colors: `blonde`, `red`, `black`, `auburn`, `gray`, `brown`
 - People have unique favorite book genres: `science fiction`, `romance`, `biography`, `mystery`, `fantasy`, `historical fiction`
 - Each person has a unique type of pet: `bird`, `hamster`, `fish`, `dog`, `cat`, `rabbit`

## Clues:
1. The person with an aquarium of fish is the person who loves mystery books.
2. The person who has black hair is the person who owns a dog.
3. The person who has gray hair is the person who loves science fiction books.
4. The person who loves historical fiction books is somewhere to the left of the person who has black hair.
5. The person who loves historical fiction books is directly left of the person who owns a rabbit.
6. The person whose birthday is in February is the person who owns a dog.
7. Bob is the person who keeps a pet bird.
8. The person who keeps a pet bird is the person who loves romance books.
9. Alice is directly left of the person who loves fantasy books.
10. The person who owns a rabbit is directly left of the person who is very short.
11. The person who is short is somewhere to the left of the person who is super tall.
12. The person who has an average height is in the sixth house.
13. The person who has red hair is not in the first house.
14. The person who is very tall is somewhere to the right of the person who is super tall.
15. There are two houses between the person who loves romance books and Eric.
16. The person with an aquarium of fish is in the sixth house.
17. The person who has blonde hair is the person who loves historical fiction books.
18. The person who is tall is somewhere to the right of the person whose birthday is in February.
19. The person who has auburn hair is Carol.
20. The person with an aquarium of fish is the person whose birthday is in March.
21. The person who loves science fiction books is directly left of the person with an aquarium of fish.
22. The person whose birthday is in September is not in the first house.
23. The person who loves biography books is directly left of the person who has a cat.
24. The person whose birthday is in January is the person who loves science fiction books.
25. Arnold is the person whose birthday is in May.
 What is the value of attribute BookGenre for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: science fiction


There are 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`, `Eric`, `Peter`, `Alice`, `Bob`, `Arnold`
 - Everyone has something unique for lunch: `spaghetti`, `soup`, `grilled cheese`, `stir fry`, `stew`, `pizza`
 - Each person has a favorite color: `green`, `purple`, `blue`, `red`, `white`, `yellow`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Aniya`, `Kailyn`, `Sarah`, `Janelle`
 - Each person lives in a unique style of house: `colonial`, `craftsman`, `modern`, `ranch`, `victorian`, `mediterranean`

## Clues:
1. The person whose mother's name is Janelle is the person living in a colonial-style house.
2. Peter is the person whose favorite color is green.
3. The person who loves blue is directly left of the person who loves the spaghetti eater.
4. The person whose mother's name is Penny is directly left of the person whose favorite color is red.
5. The person who loves blue is The person whose mother's name is Penny.
6. The person whose mother's name is Aniya is directly left of the person whose favorite color is green.
7. The person who loves white is somewhere to the left of The person whose mother's name is Janelle.
8. Eric is the person who loves eating grilled cheese.
9. The person whose mother's name is Holly is directly left of Alice.
10. The person whose mother's name is Holly and the person who loves the stew are next to each other.
11. Alice is the person in a ranch-style home.
12. Carol is not in the sixth house.
13. The person who loves stir fry is the person who loves purple.
14. The person whose favorite color is red and the person living in a colonial-style house are next to each other.
15. The person who loves the soup is Peter.
16. The person who is a pizza lover is somewhere to the left of The person whose mother's name is Sarah.
17. The person residing in a Victorian house is the person who loves eating grilled cheese.
18. The person who loves the spaghetti eater is not in the fifth house.
19. There are two houses between the person who loves eating grilled cheese and The person whose mother's name is Aniya.
20. Arnold is the person whose favorite color is red.
21. The person in a modern-style house is Arnold.
22. The person who loves yellow is the person in a Mediterranean-style villa.
 What is the value of attribute Color for the person whose attribute Food is spaghetti? Please reason step by step, and put your final answer within \boxed{}

Answer: red


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Bob`, `Eric`, `Carol`
 - Everyone has something unique for lunch: `stir fry`, `grilled cheese`, `spaghetti`, `pizza`, `soup`, `stew`
 - People use unique phone models: `xiaomi mi 11`, `iphone 13`, `oneplus 9`, `google pixel 6`, `samsung galaxy s21`, `huawei p50`
 - People have unique favorite music genres: `pop`, `country`, `classical`, `jazz`, `rock`, `hip hop`
 - Each person has a unique favorite drink: `water`, `boba tea`, `milk`, `root beer`, `tea`, `coffee`
 - Each person has an occupation: `nurse`, `lawyer`, `artist`, `doctor`, `teacher`, `engineer`

## Clues:
1. The tea drinker is somewhere to the right of the person who is a teacher.
2. Eric is the person who uses a Google Pixel 6.
3. The person who loves the spaghetti eater and the person who loves hip-hop music are next to each other.
4. The boba tea drinker is somewhere to the left of the person who uses a Google Pixel 6.
5. The person who uses a Huawei P50 is somewhere to the left of the person who uses a OnePlus 9.
6. The person who is a pizza lover is the person who loves rock music.
7. The person who is a pizza lover is somewhere to the right of the person who loves the soup.
8. The person who likes milk is somewhere to the left of the person who loves hip-hop music.
9. The person who uses a OnePlus 9 is Arnold.
10. The person who is a lawyer is the person who loves stir fry.
11. The person who is an artist is Alice.
12. Bob is the person who loves classical music.
13. The person who loves rock music is directly left of the person who loves classical music.
14. The person who is a doctor is the person who uses a Huawei P50.
15. Bob is the root beer lover.
16. There are two houses between the person who is an engineer and the person who uses a OnePlus 9.
17. Eric is the person who loves jazz music.
18. The person who uses a Xiaomi Mi 11 is the person who is an artist.
19. Carol is directly left of the person who uses a Huawei P50.
20. The person who is a lawyer is the coffee drinker.
21. The person who loves eating grilled cheese is the person who loves pop music.
22. The person who loves hip-hop music is the person who uses a Samsung Galaxy S21.
23. The coffee drinker is somewhere to the left of the person who likes milk.
 What is the value of attribute House for the person whose attribute PhoneModel is iphone 13? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Carol`, `Alice`, `Eric`, `Bob`
 - People have unique favorite book genres: `mystery`, `science fiction`, `fantasy`, `romance`, `biography`, `historical fiction`
 - Everyone has a unique favorite cigar: `dunhill`, `blends`, `yellow monster`, `prince`, `blue master`, `pall mall`
 - The people are of nationalities: `brit`, `swede`, `chinese`, `dane`, `german`, `norwegian`
 - Each person has a unique type of pet: `bird`, `rabbit`, `fish`, `cat`, `hamster`, `dog`

## Clues:
1. The person with an aquarium of fish is in the first house.
2. The person who owns a dog is not in the third house.
3. The British person is the person who loves historical fiction books.
4. Bob is the person who owns a dog.
5. The person with a pet hamster is the British person.
6. Eric is in the sixth house.
7. There is one house between the person who smokes Yellow Monster and the person who owns a dog.
8. Alice is directly left of the person who has a cat.
9. The British person is directly left of the person who smokes Blue Master.
10. Carol is in the second house.
11. The person who loves fantasy books is the Swedish person.
12. The person who loves romance books is directly left of the Dunhill smoker.
13. The person partial to Pall Mall is the Swedish person.
14. The Prince smoker is the Chinese.
15. The person who loves biography books is somewhere to the left of the British person.
16. The person who loves science fiction books is in the fourth house.
17. The British person is Alice.
18. The Norwegian is the person who loves biography books.
19. The Dane and the Dunhill smoker are next to each other.
20. Peter is not in the third house.
21. The person who owns a rabbit is the person who loves romance books.
 What is the value of attribute Pet for the person whose attribute Nationality is brit? Please reason step by step, and put your final answer within \boxed{}

Answer: hamster


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Eric`, `Arnold`
 - The people keep unique animals: `horse`, `cat`, `fish`, `bird`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `iphone 13`, `samsung galaxy s21`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`
 - Each person lives in a unique style of house: `colonial`, `victorian`, `ranch`, `craftsman`

## Clues:
1. The fish enthusiast is the person who uses an iPhone 13.
2. Alice is in the fourth house.
3. Peter is in the first house.
4. The person residing in a Victorian house is the person whose favorite color is red.
5. The person who loves white is the person in a Craftsman-style house.
6. The person whose favorite color is green is in the fourth house.
7. The person living in a colonial-style house is Eric.
8. The bird keeper is Peter.
9. The person who uses a Google Pixel 6 is in the fourth house.
10. The person who loves white is the person who uses a Samsung Galaxy S21.
11. The person living in a colonial-style house is the cat lover.
12. Eric is not in the third house.
 What is the value of attribute House for the person whose attribute HouseStyle is victorian? Please reason step by step, and put your final answer within \boxed{}

Answer: 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: `Eric`, `Peter`, `Arnold`, `Alice`
 - People have unique favorite music genres: `rock`, `jazz`, `pop`, `classical`
 - Everyone has a favorite smoothie: `dragonfruit`, `watermelon`, `desert`, `cherry`
 - People have unique hair colors: `red`, `black`, `brown`, `blonde`
 - People have unique favorite book genres: `science fiction`, `fantasy`, `romance`, `mystery`

## Clues:
1. The person who has red hair is not in the third house.
2. There is one house between Eric and the person who likes Cherry smoothies.
3. The person who loves science fiction books is Peter.
4. The person who loves fantasy books is Alice.
5. The Dragonfruit smoothie lover is the person who loves pop music.
6. The Dragonfruit smoothie lover is the person who has red hair.
7. The Watermelon smoothie lover is somewhere to the right of the person who loves classical music.
8. The person who has black hair is the person who likes Cherry smoothies.
9. There is one house between the person who loves pop music and the person who loves fantasy books.
10. The person who loves romance books is somewhere to the right of the person who loves jazz music.
11. The person who has red hair is directly left of the person who has black hair.
12. The person who has brown hair and the Dragonfruit smoothie lover are next to each other.
13. The person who loves science fiction books is the Dragonfruit smoothie lover.
 What is the value of attribute BookGenre for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: science fiction


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Alice`, `Eric`
 - They all have a unique favorite flower: `roses`, `daffodils`, `carnations`, `lilies`
 - Each person has a unique hobby: `photography`, `painting`, `cooking`, `gardening`
 - Each person has a unique type of pet: `dog`, `fish`, `bird`, `cat`
 - Each person has a favorite color: `red`, `yellow`, `green`, `white`
 - Each person lives in a unique style of house: `craftsman`, `colonial`, `ranch`, `victorian`

## Clues:
1. The person in a Craftsman-style house is Arnold.
2. The person who loves the rose bouquet is somewhere to the right of Peter.
3. The photography enthusiast is the person who owns a dog.
4. The person who loves a bouquet of daffodils is not in the fourth house.
5. The person who loves the rose bouquet is the person whose favorite color is red.
6. The person in a Craftsman-style house is in the second house.
7. Eric is the person residing in a Victorian house.
8. The person with an aquarium of fish is the person who loves white.
9. The person who loves cooking is somewhere to the right of the person whose favorite color is red.
10. The person who loves white is the person who loves a carnations arrangement.
11. The person who loves white is somewhere to the right of the person who enjoys gardening.
12. The person who loves a bouquet of daffodils is the person who loves yellow.
13. The person living in a colonial-style house is the person whose favorite color is red.
14. The person who has a cat is Eric.
 What is the value of attribute Color for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: red


43rd IMO 2002 shortlist Problem C3 A sequence of n positive integers is full if for eack k > 1, k only occurs if k-1 occurs before the last occurrence of k. How many full sequences are there for each n?

Answer: n!


1. If a positive integer cannot be expressed as the difference of squares of two positive integers, then this positive integer is called a "non-wise number". If these non-wise numbers are arranged in ascending order, then the 2009th non-wise number is $\qquad$

Answer: 8026


Three, (Full marks 30 points) In $\triangle A B C$, $A B=A C$, the altitude $A D=5$ on $B C$, $M$ is a point on $A D$, $M D=1$, and $\angle B M C=3 \angle B A C$. Try to find the perimeter of $\triangle A B C$.

Answer: \dfrac{10(2\sqrt{14} + \sqrt{7})}{7}


10. (14 points) As shown in Figure 1, $A$ is the right vertex of the hyperbola $\frac{x^{2}}{4}-y^{2}=1$. Two perpendicular lines passing through $A$ intersect the right branch of the hyperbola at points $M$ and $N$, respectively. Is the line $MN$ guaranteed to pass through a fixed point on the $x$-axis? If such a fixed point does not exist, please explain the reason; if there is such a fixed point $P$, try to find the coordinates of this fixed point $P$.

Answer: \left( \dfrac{10}{3}, 0 \right)


14, 43 students, each carrying a different amount of money ranging from 8 cents to 5 yuan. Each student spent all their money on picture cards. There are only two types of picture cards, 3 cents each and 5 cents each, and each student tried to buy as many 5-cent cards as possible. How many 3-cent cards did they buy in total?

Answer: 84


On a table, there is a certain number $N$ of candies. Aline and Bruna agree that, alternately, each one must eat at least one, but no more than half of the existing quantity. The one who eats the last candy wins the game. Ana always starts the game.

a) For $N=5$, which of the two has the winning position? (A winning position is one in which the player can win regardless of the sequence of moves made by their opponent.)

b) For $N=20$, which of the two has the winning position?

c) What are the values of $N$, where $100<N<200$, that give Bruna the winning position?

Answer: 191


Three cockroaches run along a circle in the same direction. They start simultaneously from a point $S$. Cockroach $A$ runs twice as slow than $B$, and thee times as slow than $C$. Points $X, Y$ on segment $SC$ are such that $SX = XY =YC$. The lines $AX$ and $BY$ meet at point $Z$. Find the locus of centroids of triangles $ZAB$.

Answer: (0,0)


4. Let the sequence $\left\{a_{n}\right\}$ satisfy $a_{1}=1, a_{2}=3, a_{n}=3 a_{n-1}-a_{n-2}, n \in \mathbf{N}^{*}, n \geqslant 3$. Does there exist a positive integer $n$ such that $2^{2016} \| a_{n}$ (i.e., $2^{2016}$ divides $a_{n}$, but $2^{2017}$ does not divide $a_{n}$)? If it exists, find the smallest positive integer $n$; if not, explain why.

Answer: 3 \times 2^{2013}


11.1. The ore contains $21 \%$ copper, enriched - $45 \%$ copper. It is known that during the enrichment process, $60 \%$ of the mined ore goes to waste. Determine the percentage content of ore in the waste.

Answer: 5


## Problem 1.

Consider the set $A$ of four-digit numbers that are at most equal to 2014. Determine the maximum number of elements of a subset of $A$ that contains only perfect squares, any two of which are coprime.

Answer: 6


4. Through a random point inside a given triangle, three lines parallel to the sides of the triangle pass. These lines divide the area of the triangle into six parts, three of which are triangles with areas $s_{1}, s_{2}, s_{3}$. Determine the area of the given triangle.

Answer: (\sqrt{s_1} + \sqrt{s_2} + \sqrt{s_3})^2


7. A wire of length $150 \mathrm{~cm}$ is to be cut into $n(n>2)$ segments, each of which has an integer length of no less than $1(\mathrm{~cm})$. If no three segments can form a triangle, then the maximum value of $n$ is $\qquad$ .

Answer: 10


Given the vectors
$$
\begin{array}{l}
\boldsymbol{m}=(\sqrt{3} \sin \omega x, \cos \omega x), \\
\boldsymbol{n}=(\cos \omega x,-\cos \omega x),
\end{array}
$$
where $\omega>0$. The function $f(x)=\boldsymbol{m} \cdot \boldsymbol{n}$ has a minimum positive period of $\frac{\pi}{2}$.

(1) Find the value of $\omega$;

(2) Given a triangle $\triangle ABC$ with sides $a, b, c$ satisfying $b^2 = ac$, and the angle opposite side $b$ is $x$, if the equation $f(x) = k$ has two distinct real solutions, determine the range of the real number $k$.

Answer: 2


4. Let $A L, B M, C N$ be the medians of triangle $A B C$, intersecting at point $K$. It is known that a circle can be circumscribed around quadrilateral $C L K M$, and $A B=2$. Find the length of the median $C N$.

Answer: \sqrt{3}


Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$, and let $O$ be an arbitrary point in the interior of $\vartriangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell_2$, and $\ell_3$ and the sides of $\vartriangle ABC$ form a hexagon whose area is $A$. Compute the minimum value of $A$.

Answer: 280


$\left[\begin{array}{ll}\text { Ratios of areas } \\ \text { P } & \text { Product of lengths of chord segments and lengths of secant segments }]\end{array}\right]$ [ Ratio of areas of triangles with a common angle ]

In triangle $A B C$, points $B_{1}$ and $C_{1}$ are chosen on sides $A B$ and $A C$ respectively, such that $A B_{1}: A B=1: 3$ and $A C_{1}: A C=1: 2$. A circle is drawn through points $A, B_{1}$, and $C_{1}$. A line is drawn through point $B_{1}$, intersecting segment $A C_{1}$ at point $D$, and the circle at point $E$. Find the area of triangle $B_{1} C_{1} E$, if $A C_{1}=4, A D=1$, $D E=2$, and the area of triangle $A B C$ is 12.

Answer: \dfrac{7}{2}


Example 8 A math game is played between two very smart students, A and B. The referee first writes the integers $2,3, \cdots, 2006$ on the blackboard, then randomly erases one number. Next, A and B take turns erasing one number (i.e., B erases one of the numbers first, then A erases one number, and so on). If the last two remaining numbers are coprime, A wins; otherwise, B wins. According to this game rule, find the probability of A winning (answer with a specific number).

Answer: \dfrac{1003}{2005}


\( p(x, y) \) is a symmetric polynomial with the factor \( (x - y) \). Show that \( (x - y)^2 \) is a factor.

Answer: (x - y)^2 \text{ is a factor of } p(x, y)


Let $[x]$ denote the largest integer less than or equal to the real number $x$. Define
$$
S_{n} = \left[1 + \frac{1}{2} + \cdots + \frac{1}{n}\right] \quad (n = 1, 2, \cdots).
$$

Prove that there exist infinitely many $n$ such that $\mathrm{C}_{n}^{S_{n}}$ is even.

Answer: \text{There exist infinitely many } n \text{ such that } \mathrm{C}_{n}^{S_{n}} \text{ is even.}


Inside a semicircle with a radius of 12, there is a circle with a radius of 6 and a small semicircle, all pairwise tangent to each other, as shown in the diagram. Find the radius of the small semicircle.

Answer: 4


Rax $* \star$ Given that the wooden tetrahedron has three pairs of opposite edges all equal, each being $13 \mathrm{~cm}$, $14 \mathrm{~cm}$, and $15 \mathrm{~cm}$. If it is to be carved into a sphere, what is the maximum radius of the sphere?

Answer: \dfrac{3\sqrt{55}}{8}


## [ [Mean proportionals in a right triangle]

[ Median to the hypotenuse ]

In a right triangle $A B C$, angle $A C B$ is a right angle. Let $E$ be the point of intersection of the angle bisector of angle $A B C$ with side $A C$. Point $D$ is the midpoint of side $A B$, and $O$ is the point of intersection of segments $B E$ and $C D$. A perpendicular is drawn from point $O$ to $B O$ until it intersects side $B C$ at point $F$. It is known that

$F C=b, O C=3 b / 2$. Find the area of triangle $A B C$.

#

Answer: \dfrac{8\sqrt{11}}{5}b^2


2. Katya decided to calculate the sum of the cubes of all natural divisors of some natural number, and she got the result $M A T H$. But then she discovered that she had forgotten one of the divisors. Adding its cube, she got the correct result - MASS. Find the smallest possible value of the number $M A T H$. (MATH and $M A S S$ are four-digit numbers, where each digit is replaced by a letter, with the same digits replaced by the same letters, and different digits by different letters.)

Answer: 2017


31. The oldest teacher at the Ocean School is the Turtle Teacher, who says: “Our class, including Nemo, has a total of 10 little fish, and each little fish is the same age as Nemo. Nemo's father is exactly 5 times as old as Nemo. If we add up the ages of all the students in our class and then add Nemo's father's age, it equals my age.” The Turtle Teacher's age is \qquad times Nemo's father's age.

Answer: 3


We have two perpendicular lines - like the axes of an ellipse - and a tangent $(t)$ with its point of tangency $(T)$. Construct the endpoints of the ellipse's axes $(A, B, C, D)$! How many solutions are there?

Answer: 1


6.3. Six pennies lie on the table, forming a closed chain (i.e., the first penny touches the second, the second touches the third, and so on, the sixth touches the first). A seventh penny, also lying on the table, rolls without slipping along the outer side of the chain, touching each of the six pennies in the chain in turn. How many revolutions will this penny make before returning to its original position?

## QUADRILATERAL AND FOUR CIRCLES

Answer: 4


9. Let the complex number $z_{1}=(6-a)+(4-b) \mathrm{i}$,
$$
\begin{array}{l}
z_{2}=(3+2 a)+(2+3 b) \mathrm{i}, \\
z_{3}=(3-a)+(3-2 b) \mathrm{i},
\end{array}
$$

where, $a, b \in \mathbf{R}$. When $\left|z_{1}\right|+\left|z_{2}\right|+\left|z_{3}\right|$ achieves its minimum value, $3 a+4 b=$ $\qquad$

Answer: 12


7. An Unusual Client. A certain man brought 1000 dollars in one-dollar bills and 10 empty bags to the bank and, addressing the clerk, said:

- I would appreciate it if you could distribute this money into the bags in such a way that any amount of money I need can always be given out in one or several bags without opening any of them.

How should the money be distributed? The bank should be able to provide any required amount only once, and the amount is limited only by the size of the deposit. In other words, the depositor has the right to request any amount from 1 to 1000 dollars (the number of dollars must be a whole number).

Answer: [1, 2, 4, 8, 16, 32, 64, 128, 256, 489]


A circular track with diameter $500$ is externally tangent at a point A to a second circular track with diameter $1700.$  Two runners start at point A at the same time and run at the same speed.  The first runner runs clockwise along the smaller track while the second runner runs clockwise along the larger track.  There is a first time after they begin running when their two positions are collinear with the point A.  At that time each runner will have run a distance of $\frac{m\pi}{n}$ where  $m$  and  $n$ are relatively prime positive integers.  Find  $m + n. $

Answer: 2128


Each of the distinct letters in the following addition problem represents a different digit.  Find the number represented by the word MEET.

$ \begin{array}{cccccc}P&U&R&P&L&E\\&C&O&M&E&T\\&&M&E&E&T\\ \hline Z&Z&Z&Z&Z&Z\end{array} $

Answer: 9221


The diagram below contains eight line segments, all the same length.  Each of the angles formed by the intersections of two segments is either a right angle or a $45$ degree angle.  If the outside square has area $1000$, find the largest integer less than or equal to the area of the inside square.

Answer: 171


$24! = 620,448,401,733,239,439,360,000$ ends in four zeros, and $25!=15,511,210,043,330,985,984,000,000$ ends in six zeros. Thus, there is no integer $n$ such that  $n!$ ends in exactly five zeros. Let $S$ be the set of all $k$ such that for no integer n does $n!$ end in exactly $k$ zeros. If the numbers in $S$ are listed in increasing order, 5 will be the first number. Find the 100th number in that list.

Answer: 504


The four points $A(-1,2), B(3,-4), C(5,-6),$ and $D(-2,8)$ lie in the coordinate plane. Compute the minimum possible value of $PA + PB + PC + PD$ over all points P .

Answer: 23


A right circular cone pointing downward forms an angle of $60^\circ$ at its vertex. Sphere $S$ with radius $1$ is set into the cone so that it is tangent to the side of the cone. Three congruent spheres are placed in the cone on top of S so that they are all tangent to each other, to sphere $S$, and to the side of the cone. The radius of these congruent spheres can be written as $\tfrac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime. Find $a + b + c$.

Answer: 364


The [i]Purple Comet! Math Meet[/i] runs from April 27 through May 3, so the sum of the calendar dates for these seven days is $27 + 28 + 29 + 30 + 1 + 2 + 3 = 120.$ What is the largest sum of the calendar dates for seven consecutive Fridays occurring at any time in any year?

Answer: 142


Towers grow at points along a line. All towers start with height 0 and grow at the rate of one meter per second. As soon as any two adjacent towers are each at least 1 meter tall, a new tower begins to grow at a point along the line exactly half way between those two adjacent towers. Before time 0 there are no towers, but at time 0 the first two towers begin to grow at two points along the line. Find the total of all the heights of all towers at time 10 seconds.

Answer: 1033


Five men and seven women stand in a line in random order. Let m and n be relatively prime positive integers so that $\tfrac{m}{n}$ is the probability that each man stands next to at least one woman. Find $m + n.$

Answer: 287


Alan, Barb, Cory, and Doug are on the golf team, Doug, Emma, Fran, and Greg are on the swim team, and Greg, Hope, Inga, and Alan are on the tennis team. These nine people sit in a circle in random order. The probability that no two people from the same team sit next to each other is $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Answer: 2521


There are two rows of seats with three side-by-side seats in each row. Two little boys, two little girls, and two adults sit in the six seats so that neither little boy sits to the side of either little girl. In how many different ways can these six people be seated?

Answer: 176


Grisha: Masha, Sasha, and Natasha told the truth.

How many children actually told the truth?

Answer: 1


There are \(10!\) permutations \( s_{0}, s_{1}, \ldots, s_{9} \) of the numbers \(0, 1, \ldots, 9\). How many of these permutations satisfy \( s_{k} \geq k - 2 \) for \( k = 0, 1, \ldots, 9 \)?

Answer: 13122


On the sides \(AB\) and \(AD\) of rectangle \(ABCD\), points \(M\) and \(N\) are marked, respectively. It is known that \(AN=7\), \(NC=39\), \(AM=12\), and \(MB=3\).

(a) Find the area of the rectangle \(ABCD\).

(b) Find the area of triangle \(MNC\).

Answer: 286.5


Let \(\pi\) be a uniformly random permutation of the set \(\{1,2, \ldots, 100\}\). The probability that \(\pi^{20}(20) = 20\) and \(\pi^{21}(21) = 21\) can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are relatively prime positive integers. Compute \(100a + b\). (Here, \(\pi^{k}\) means \(\pi\) iterated \(k\) times.)

Answer: 1025


In a convex quadrilateral \(ABCD\), angles \(B\) and \(D\) are equal, \(CD = 4BC\), and the angle bisector of angle \(A\) passes through the midpoint of side \(CD\).

What can be the ratio \(AD: AB\)?

Answer: \frac{2}{3}


The points \( S, T, U \) lie on the sides of the triangle \( PQR \) such that \( QS = QU \) and \( RS = RT \). Given that \(\angle TSU = 40^\circ\), what is \(\angle TPU\)?

A) \(60^\circ\)
B) \(70^\circ\)
C) \(80^\circ\)
D) \(90^\circ\)
E) \(100^\circ\)

Answer: 100


A randomly selected phone number consists of 5 digits. What is the probability that: 1) all digits are different; 2) all digits are odd?

Answer: 0.03125


Suppose that point \( D \) lies on side \( BC \) of triangle \( ABC \) such that \( AD \) bisects \( \angle BAC \), and let \(\ell\) denote the line through \( A \) perpendicular to \( AD \). If the distances from \( B \) and \( C \) to \(\ell\) are 5 and 6, respectively, compute \( AD \).

Answer: \frac{60}{11}


In a $3 \times 4$ table, 12 numbers are arranged such that all seven sums of these numbers in the rows and columns of the table are distinct. What is the maximum number of numbers in this table that can be equal to zero?

Answer: 8


Vasya had 100 segments, and no three of them could form a triangle. He made himself one more segment and now can form triangles in several ways. What is the maximum number of ways he can form a triangle?

Answer: 100


For each positive integer \( n \), define \( s(n) \) to be the sum of the digits of \( n \). For example, \( s(2014) = 2+0+1+4 = 7 \). Determine all positive integers \( n \) with \( 1000 \leq n \leq 9999 \) for which \( \frac{n}{s(n)} = 112 \).

Answer: 1008,1344,1680,2688


Let \( P(x) = x^3 + a x^2 + b x + 2015 \) be a polynomial all of whose roots are integers. Given that \( P(x) \geq 0 \) for all \( x \geq 0 \), find the sum of all possible values of \( P(-1) \).

Answer: 9496


Space is partitioned into three disjoint sets. Show that for each \( d > 0 \), we can find two points a distance \( d \) apart in one of the sets.

Answer: 3


Let $\left\{b_{n}\right\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $b_{0}=1$, $b_{n}=2+\sqrt{b_{n-1}}-2 \sqrt{1+\sqrt{b_{n-1}}}$. Calculate the sum of $b_{n} 2^{n}$ from $n=1$ to $\infty$. The original answer includes a constant term and a logarithmic term. Please provide the integer part of the answer.

Answer: 2


Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_{i}=i^{k}+i$ for $i=0,1, \\ldots, p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_{2}$ upon division by $p$? Please provide the sum of the possible remainders.

Answer: 5


We say that a real number $V$ is good if there exist two closed convex subsets $X, Y$ of the unit cube in $\mathbb{R}^{3}$, with volume $V$ each, such that for each of the three coordinate planes (that is, the planes spanned by any two of the three coordinate axes), the projections of $X$ and $Y$ onto that plane are disjoint. Find $\sup \{V \mid V$ is good $\}$. The original answer is in \frac{m}{n} format, please give the value of m + n.

Answer: 5


Evaluate the expression $\tan \left(\arctan 2+\arctan \frac{2}{2^{2}}+\cdots+\arctan \frac{2}{12^{2}}\right)$. The original answer is in the form of $-\frac{m}{n}$. Please provide the value of m + n.

Answer: 31


Evaluate$$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\left(\left[\frac{2n}{k}\right]-2\left[\frac{n}{k}\right]-\ln4\right)$$
and express your answer in the form $\log a-b$, with $a$ and $b$ positive integers.
Here $[x]$ is defined to be the integer such that $[x]\leqq x<[x]+1$ and $\log x$ is the logarithm of $x$ to base $e$.

Answer: -1


Suppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite "word" $A^{n_1}B^{n_2}A^{n_3}\cdots B^{n_k}$, where $n_1,\ldots,n_k$ are any integers, and $A^0=B^0=1$ as usual. Also, suppose that $A^{4}=B^{7}=ABA^{-1}B=1,A^{2}\neq1$, and $B\neq1$.
 How many elements of $G$ are of the form $C^{2}$ with $C$ in $G$?

Answer: 8


Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length 3 units and the remaining four sides of length 2 units. Give the answer in the form $r+s\sqrt{t}$ with $r$, $s$, and $t$ positive integers，writing r.

Answer: 13


For each positive integer $n$, let $a(n)$ be the number of zeros in the base $3$ representation of $n$. For which positive real numbers $x$ does the series $$ \sum_{n=1}^{\infty} \frac{x^{a(n)}}{n^{3}} $$ converge? Please provide the sum of the bounds of the interval where x satisfies the condition.

Answer: 25


Find the equations of the two straight lines each of which cuts all the four straight lines
$x=1, y=0 ; \quad y=1, z=0 ; \quad z=1, x=0 ; \quad x=y=-6 z$.
If \(L_{1}\) is given by
$$y = a_{1}x + b_{1} = c_{1}z + d_{1}$$
and \(L_{2}\) is given by
$$y = a_{2}x + b_{2} = c_{2}z + d_{2}$$
Find the expression of \(a_{1} + b_{1} + c_{1}z + d_{1} + a_{2} + b_{2} + c_{2}z + d_{2}\).

Answer: 1


Find the equation of the parabola which touches the $x$-axis at the point $(1,0)$ and the $y$-axis at the point $(0,2)$. Find the coordinates of its vertex.If the coordinates of the vertex are $(\frac{a}{b},\frac{c}{d})$, find the value of $a + b + c + d$.

Answer: 68


For which real numbers $c$ is there a straight line that intersects the curve
$$
y=x^{4}+9 x^{3}+c x^{2}+9 x+4
$$
in four distinct points? The original answer is in the format $c < \frac{m}{n}$. Please provide the value of $m + n$.

Answer: 251


Let $C$ be the unit circle $x^{2}+y^{2}=1$. A point $p$ is chosen randomly on the circumference $C$ and another point $q$ is chosen randomly from the interior of $C$ (these points are chosen independently and uniformly over their domains). Let $R$ be the rectangle with sides parallel to the $x$-and $y$-axes with diagonal $p q$. What is the probability that no point of $R$ lies outside of $C$? The original answer is in $\frac{n}{\pi^m}$format, please give the value of $m + n$.

Answer: 6


Let \( D \) be a point inside the acute triangle \( \triangle ABC \), such that \( \angle ADB = \angle ACB + 90^\circ \) and \( AC \cdot BD = AD \cdot BC \). Calculate the ratio \( \frac{AB \cdot CD}{AC \cdot BD} \).

Answer: \sqrt{2}


Let \( A_{1}A_{2}A_{3} \) be a triangle and \( P_{0} \) a point in the plane. We define \( A_{s} = A_{s-3} \) for \( s \geq 4 \). We construct a sequence \( P_{1}, P_{2}, P_{3}, \ldots \) such that \( P_{k} \) is the image of \( P_{k-1} \) by the rotation with center \( A_{k+1} \) and angle \( \frac{2 \pi}{3} \). Show that if \( P_{2007} = P_{0} \), then the triangle \( A_{1}A_{2}A_{3} \) is equilateral.

Answer: \text{The triangle } A_1A_2A_3 \text{ is equilateral.}


2. Given: $b_{1}, b_{2}, b_{3}, b_{4}$ are positive integers, the polynomial $g(z)=(1-z)^{b_{1}}\left(1-z^{2}\right)^{b_{2}}\left(1-z^{3}\right)^{b_{3}}\left(1-z^{4}\right)^{b_{4}}$ when expanded and terms higher than 4th degree are omitted, becomes $1-2 z$. Also, $\alpha$ is the largest root of the polynomial $f(x)=x^{3}-b_{4} x^{2}+b_{2}$. Try to find the remainder when $\left[\alpha^{1995}\right]$ is divided by 9. Here $[x]$ denotes the greatest integer less than or equal to $x$.

Answer: 5


6. A circle passes through vertices $A$ and $B$ of triangle $ABC$ and intersects its sides $AC$ and $BC$ at points $Q$ and $N$ respectively, such that $AQ: QC = 5: 2$ and $CN: NB = 5: 2$. Find $AB$, if $QN = 5 \sqrt{2}$.

Answer: 7\sqrt{5}


10. Question: Among $1,2,3, \cdots, 1999,2000,2001$, what is the maximum number of numbers that can be chosen such that the sum of any three chosen numbers is divisible by 21?

Answer: 95


12. For any set $S$, use $|S|$ to denote the number of elements in set $S$, and use $n(S)$ to denote the number of subsets of set $S$. If $A, B, C$ are three finite sets, and satisfy
(1) $|A|=|B|=2016$;
(2) $n(A)+n(B)+n(C)=n(A \cup B \cup C)$.
Then the maximum value of $|A \cap B \cap C|$ is $\qquad$

Answer: 2015


3. In the regular triangular frustum $A^{\prime} B^{\prime} C^{\prime}-A B C$, $A^{\prime} B^{\prime} : A B = 5 : 7$, the sections $A B C^{\prime}$ and $A^{\prime} B C^{\prime}$ divide the frustum into three pyramids $C^{\prime}-A B C$, $B-A^{\prime} B^{\prime} C^{\prime}$, and $C^{\prime}-A A B A^{\prime}$. The volume ratio of $V_{1}, V_{2}, V_{3}$ is $\qquad$

Answer: 49:25:35


One, (25 points) In trapezoid $ABCD$, $AB \parallel CD, AB=125, CD=$ $DA=80$. Can diagonal $BD$ divide the trapezoid into two similar triangles? If not, provide a proof; if so, find the lengths of $BC, BD$.

Answer: 64


7,8,9 |
| :---: | :---: | :---: |
|  | [ Pairing and grouping; bijections |  |
|  | [ Pigeonhole Principle (other). |  |
|  | Estimation + example |  |

![](https://cdn.mathpix.com/cropped/2024_05_06_ed1a9e9dd82795d4db2eg-26.jpg?height=55&width=297&top_left_y=1779&top_left_x=0)

At a joint conference of the party of liars and the party of truth-tellers, 32 people were elected to the presidium and seated in four rows of eight. During a break, each member of the presidium claimed that among their neighbors there were representatives of both parties. It is known that liars always lie, and truth-tellers always tell the truth. What is the minimum number of liars in the presidium for the described situation to be possible? (Two members of the presidium are neighbors if one of them sits to the left, right, in front of, or behind the other.)

Answer: 8


8. Two Go teams, Team A and Team B, each have 5 players who compete in a relay match according to a predetermined order. The No. 1 players of both teams start the match, and the loser is eliminated; then the No. 2 player of the losing team competes against the winner, and the loser is eliminated. This continues until all players of one team are eliminated, at which point the other team is declared the winner. Assuming each player has equal strength, the expected number of Team A's players who have not played when the match ends, $E(X)=$ $\qquad$

---

I hope this translation meets your needs. If you have any further requests or need additional assistance, feel free to let me know!

Answer: \dfrac{187}{256}


12.059. Find the acute angle of the rhombus, knowing that the volumes of the bodies obtained by rotating the rhombus around its larger diagonal and around its side are in the ratio of $1: 2 \sqrt{5}$.

Answer: \arccos\left(\frac{1}{9}\right)


I1.1 Let $P$ be the unit digit of $3^{2003} \times 5^{2002} \times 7^{2001}$. Find the value of $P$.
I1.2 If the equation $\left(x^{2}-x-1\right)^{x+P-1}=1$ has $Q$ integral solutions, find the value of $Q$.
I1.3 Let $x, y$ be real numbers and $x y=1$.
If the minimum value of $\frac{1}{x^{4}}+\frac{1}{Q y^{4}}$ is $R$, find the value of $R$.
11.4 Let $x_{R}, x_{R+1}, \ldots, x_{K}(K>R)$ be $K-R+1$ distinct positive integers and $x_{R}+x_{R+1}+\ldots+x_{K}=2003$ If $S$ is the maximum possible value of $K$, find the value of $S$. (Reference: 2004 HI4)

Answer: 5


Let \( M \) be an arbitrary natural number greater than 3, and \( S \) be the sum of all natural numbers \( x \) not exceeding \( M \) such that \( x^{2} - x + 1 \) is divisible by \( M \). Prove that \( S \) is divisible by \( M + 1 \). (In the case where there are no such numbers \( x \), assume \( S \) is equal to zero).

Answer: S \text{ is divisible by } M + 1


5. A circle is circumscribed around a right triangle $\mathrm{ABC}$ with hypotenuse $\mathrm{AB}$. On the larger leg $\mathrm{AC}$, a point $\mathrm{P}$ is marked such that $\mathrm{AP}=\mathrm{BC}$. On the arc $\mathrm{ACB}$, its midpoint $\mathrm{M}$ is marked. What can the angle $\mathrm{PMC}$ be equal to?

Answer: 90^\circ


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
 - Each person prefers a unique type of vacation: `camping`, `city`, `beach`, `cruise`, `mountain`
 - People have unique heights: `very tall`, `average`, `very short`, `short`, `tall`
 - They all have a unique favorite flower: `tulips`, `roses`, `carnations`, `lilies`, `daffodils`
 - Each person has a unique level of education: `bachelor`, `master`, `associate`, `high school`, `doctorate`
 - People own unique car models: `bmw 3 series`, `tesla model 3`, `toyota camry`, `ford f150`, `honda civic`

## Clues:
1. Eric is the person who prefers city breaks.
2. Arnold is the person who is tall.
3. The person who loves beach vacations is the person who loves the boquet of lilies.
4. The person who owns a BMW 3 Series is the person who is tall.
5. The person with an associate's degree is the person who loves the boquet of lilies.
6. The person who has an average height is Eric.
7. Peter is the person who owns a Toyota Camry.
8. The person who has an average height is somewhere to the right of the person who loves a bouquet of daffodils.
9. The person who owns a Honda Civic is somewhere to the left of the person who loves the boquet of lilies.
10. Eric is the person who owns a Ford F-150.
11. The person who loves the vase of tulips is in the fourth house.
12. The person who is very short is not in the fourth house.
13. Eric is the person with a high school diploma.
14. The person who enjoys mountain retreats is somewhere to the right of the person with a master's degree.
15. The person who likes going on cruises is not in the third house.
16. Bob is the person who is short.
17. The person with an associate's degree is directly left of Arnold.
18. The person with a bachelor's degree is the person who is very tall.
19. The person who loves a carnations arrangement is directly left of the person with a doctorate.
 What is the value of attribute Flower for the person whose attribute Education is doctorate? Please reason step by step, and put your final answer within \boxed{}

Answer: tulips


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Eric`, `Arnold`
 - Each person has a unique favorite drink: `coffee`, `tea`, `milk`, `water`
 - People have unique favorite book genres: `fantasy`, `science fiction`, `mystery`, `romance`
 - People have unique hair colors: `red`, `black`, `blonde`, `brown`

## Clues:
1. The coffee drinker is the person who has blonde hair.
2. The person who has brown hair is the person who likes milk.
3. Arnold is the person who loves fantasy books.
4. The person who has red hair is in the fourth house.
5. The person who has red hair is the person who loves science fiction books.
6. The person who loves romance books is the tea drinker.
7. The person who has blonde hair is not in the third house.
8. The tea drinker is directly left of the person who loves mystery books.
9. The person who has black hair is not in the second house.
10. The person who loves romance books is somewhere to the left of Eric.
11. The person who loves fantasy books is somewhere to the left of Peter.
 What is the value of attribute House for the person whose attribute BookGenre is romance? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


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`, `Eric`, `Alice`
 - The mothers' names in different houses are unique: `Holly`, `Kailyn`, `Aniya`, `Janelle`
 - People have unique favorite book genres: `science fiction`, `mystery`, `fantasy`, `romance`
 - Each person has a favorite color: `green`, `red`, `white`, `yellow`
 - Each person prefers a unique type of vacation: `mountain`, `cruise`, `beach`, `city`

## Clues:
1. The person who loves romance books and the person who likes going on cruises are next to each other.
2. Arnold is The person whose mother's name is Janelle.
3. The person who prefers city breaks is the person who loves yellow.
4. The person who loves science fiction books is directly left of Peter.
5. The person who loves science fiction books is somewhere to the left of The person whose mother's name is Kailyn.
6. The person who loves beach vacations is not in the first house.
7. There is one house between Alice and the person who likes going on cruises.
8. The person who loves mystery books is the person who loves white.
9. Peter is the person who prefers city breaks.
10. The person whose favorite color is green is in the third house.
11. The person whose mother's name is Aniya is Peter.
12. The person whose favorite color is green is The person whose mother's name is Janelle.
 What is the value of attribute Mother for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: Janelle


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Alice`, `Arnold`, `Carol`, `Bob`
 - People have unique heights: `very tall`, `very short`, `tall`, `super tall`, `short`, `average`
 - Each person prefers a unique type of vacation: `cruise`, `mountain`, `camping`, `beach`, `city`, `cultural`
 - People have unique hair colors: `blonde`, `red`, `gray`, `brown`, `auburn`, `black`
 - Each person has a unique favorite drink: `tea`, `boba tea`, `milk`, `water`, `coffee`, `root beer`
 - They all have a unique favorite flower: `lilies`, `carnations`, `iris`, `tulips`, `roses`, `daffodils`

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

Answer: short


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`, `Bob`, `Arnold`, `Peter`, `Carol`
 - People have unique favorite music genres: `classical`, `rock`, `jazz`, `pop`, `country`, `hip hop`
 - Each person has an occupation: `teacher`, `artist`, `nurse`, `engineer`, `lawyer`, `doctor`
 - Each person prefers a unique type of vacation: `cruise`, `camping`, `city`, `cultural`, `mountain`, `beach`
 - Each person has a favorite color: `blue`, `white`, `red`, `yellow`, `purple`, `green`

## Clues:
1. The person who loves beach vacations is Arnold.
2. The person who is an engineer is the person who enjoys mountain retreats.
3. Peter is not in the second house.
4. The person whose favorite color is red is in the third house.
5. The person who loves beach vacations and the person who is a teacher are next to each other.
6. The person who is a nurse is Carol.
7. The person who goes on cultural tours is the person who loves pop music.
8. The person who loves rock music and the person who loves country music are next to each other.
9. The person whose favorite color is green is in the second house.
10. The person whose favorite color is red is the person who enjoys camping trips.
11. Peter is the person who loves pop music.
12. The person who loves classical music is the person who loves purple.
13. The person who loves blue is directly left of Alice.
14. The person who loves rock music is in the fourth house.
15. The person who loves white is the person who is a doctor.
16. The person who loves jazz music is the person whose favorite color is red.
17. Carol is somewhere to the right of the person who prefers city breaks.
18. The person who loves rock music is the person who is an artist.
19. The person who enjoys mountain retreats is in the first house.
20. Bob is the person who loves classical music.
 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: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Carol`, `Eric`, `Peter`, `Arnold`
 - Each person has a unique hobby: `knitting`, `woodworking`, `photography`, `cooking`, `gardening`, `painting`
 - Each person lives in a unique style of house: `ranch`, `modern`, `mediterranean`, `colonial`, `craftsman`, `victorian`
 - People have unique favorite music genres: `classical`, `jazz`, `rock`, `hip hop`, `pop`, `country`
 - People have unique hair colors: `blonde`, `red`, `gray`, `black`, `brown`, `auburn`

## Clues:
1. The woodworking hobbyist is somewhere to the right of the person living in a colonial-style house.
2. The person who loves jazz music is in the sixth house.
3. The person who has black hair is directly left of Bob.
4. The person who has blonde hair is the person living in a colonial-style house.
5. Arnold is directly left of the person in a ranch-style home.
6. The person who paints as a hobby is somewhere to the left of the photography enthusiast.
7. Peter is the person who has black hair.
8. Eric is not in the first house.
9. The person who enjoys gardening is the person who has brown hair.
10. The person who has red hair is the person who loves pop music.
11. The person in a Craftsman-style house is in the sixth house.
12. Carol is in the sixth house.
13. The person who has blonde hair is directly left of the person residing in a Victorian house.
14. The person who enjoys knitting is Bob.
15. The person who paints as a hobby is the person who has blonde hair.
16. The person who enjoys knitting is the person who has auburn hair.
17. The photography enthusiast is somewhere to the left of the person in a Craftsman-style house.
18. There is one house between the person who loves hip-hop music and the person who loves country music.
19. The person who has black hair is somewhere to the right of the person who loves hip-hop music.
20. The person who enjoys gardening is the person residing in a Victorian house.
21. The person who loves pop music is somewhere to the left of the person in a Mediterranean-style villa.
22. The person who loves rock music is in the second house.
 What is the value of attribute HairColor for the person whose attribute House is 6? Please reason step by step, and put your final answer within \boxed{}

Answer: gray


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`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`
 - Each person has a unique type of pet: `cat`, `fish`, `dog`
 - People have unique favorite book genres: `science fiction`, `mystery`, `romance`
 - Everyone has something unique for lunch: `spaghetti`, `pizza`, `grilled cheese`

## Clues:
1. The person who owns a Ford F-150 is not in the first house.
2. The person who owns a Ford F-150 is not in the third house.
3. The person who loves science fiction books is somewhere to the right of the person who is a pizza lover.
4. The person with an aquarium of fish is the person who is a pizza lover.
5. The person who is a pizza lover is somewhere to the left of Peter.
6. Peter is the person who owns a Tesla Model 3.
7. The person who loves science fiction books is directly left of the person who loves mystery books.
8. Arnold is the person who loves the spaghetti eater.
9. The person who has a cat is the person who loves the spaghetti eater.
 What is the value of attribute BookGenre for the person whose attribute CarModel is toyota camry? Please reason step by step, and put your final answer within \boxed{}

Answer: romance


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

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

Answer: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Alice`, `Carol`, `Eric`, `Arnold`
 - People have unique favorite sports: `swimming`, `tennis`, `baseball`, `soccer`, `volleyball`, `basketball`
 - People have unique heights: `super tall`, `average`, `very short`, `very tall`, `short`, `tall`
 - People own unique car models: `chevrolet silverado`, `tesla model 3`, `toyota camry`, `ford f150`, `bmw 3 series`, `honda civic`

## Clues:
1. The person who is very short is the person who loves swimming.
2. The person who loves baseball is the person who is short.
3. Alice is the person who owns a Honda Civic.
4. The person who loves baseball is somewhere to the right of Bob.
5. The person who loves soccer is somewhere to the right of Carol.
6. There are two houses between the person who loves volleyball and the person who has an average height.
7. Arnold is in the third house.
8. The person who loves tennis is not in the fourth house.
9. The person who owns a Chevrolet Silverado and the person who owns a Toyota Camry are next to each other.
10. Alice is directly left of the person who is super tall.
11. The person who owns a Ford F-150 is the person who loves swimming.
12. Eric is in the fourth house.
13. The person who is short is the person who owns a Chevrolet Silverado.
14. The person who loves baseball is in the fifth house.
15. Eric is the person who owns a BMW 3 Series.
16. The person who loves swimming is somewhere to the left of the person who is very tall.
17. The person who has an average height is the person who owns a BMW 3 Series.
 What is the value of attribute Height for the person whose attribute FavoriteSport is tennis? Please reason step by step, and put your final answer within \boxed{}

Answer: super tall


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

## Clues:
1. The person whose birthday is in January is Peter.
2. The person who loves eating grilled cheese is the person whose birthday is in May.
3. The person whose birthday is in April is the person who loves stir fry.
4. The person who loves the stew is the one who only drinks water.
5. The person whose birthday is in September is not in the fourth house.
6. The person who likes milk is somewhere to the left of the person who loves stir fry.
7. The coffee drinker is in the fifth house.
8. The person who loves eating grilled cheese is directly left of the one who only drinks water.
9. The person whose birthday is in May is directly left of Arnold.
10. The root beer lover is somewhere to the left of Alice.
11. There are two houses between Eric and the person who is a pizza lover.
12. The person who loves the spaghetti eater is the person who is very short.
13. The person who is super tall is the tea drinker.
14. The person who is tall is not in the third house.
15. Peter is directly left of the person who is a pizza lover.
16. The person who likes milk is somewhere to the right of the one who only drinks water.
17. The person who is very short is the coffee drinker.
18. The person who is super tall is Bob.
19. The person who is super tall and the person whose birthday is in March are next to each other.
20. Carol is somewhere to the left of the person who has an average height.
21. The person who is very tall is the person who loves the stew.
22. There is one house between the person who loves eating grilled cheese and Peter.
 What is the value of attribute Food for the person whose attribute Birthday is mar? Please reason step by step, and put your final answer within \boxed{}

Answer: spaghetti


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

## Clues:
1. The person who prefers city breaks is the Swedish person.
2. The person who likes going on cruises is directly left of the person who loves beach vacations.
3. Arnold is in the second house.
4. Arnold is the person who likes milk.
5. The person who prefers city breaks is not in the fourth house.
6. The coffee drinker is the Norwegian.
7. The person who loves beach vacations and Peter are next to each other.
8. Eric is the one who only drinks water.
9. The person who loves beach vacations is directly left of the Dane.
 What is the value of attribute House for the person whose attribute Nationality is brit? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


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

## Clues:
1. Alice is the person with a master's degree.
2. The person who uses a OnePlus 9 is the person with a doctorate.
3. The person who loves blue is somewhere to the left of Bob.
4. The person who prefers city breaks is Peter.
5. The person who loves beach vacations is the person whose favorite color is green.
6. The person who loves white is the person who uses an iPhone 13.
7. The person who uses a OnePlus 9 is the person who enjoys mountain retreats.
8. The person who uses a Huawei P50 is somewhere to the left of the person whose favorite color is red.
9. The person who loves beach vacations is not in the first house.
10. Alice is in the fourth house.
11. Eric and Peter are next to each other.
12. Alice is the person whose favorite color is red.
13. There is one house between Peter and the person who uses a OnePlus 9.
14. The person with a bachelor's degree is the person who uses a Samsung Galaxy S21.
15. The person who enjoys camping trips is the person who loves blue.
16. The person who loves beach vacations is the person with a high school diploma.
 What is the value of attribute PhoneModel for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: samsung galaxy s21


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Eric`, `Peter`, `Bob`, `Carol`
 - They all have a unique favorite flower: `daffodils`, `roses`, `tulips`, `iris`, `carnations`, `lilies`
 - People have unique favorite sports: `basketball`, `swimming`, `soccer`, `volleyball`, `tennis`, `baseball`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `lime`, `blueberry`, `dragonfruit`, `desert`
 - Each person has an occupation: `lawyer`, `doctor`, `teacher`, `artist`, `nurse`, `engineer`

## Clues:
1. The person who loves a bouquet of daffodils is the person who is an artist.
2. The person who loves tennis is not in the second house.
3. The person who is an artist is somewhere to the right of the person who likes Cherry smoothies.
4. The person who loves soccer is somewhere to the right of the person who is a teacher.
5. The person who loves basketball is directly left of the person who loves the vase of tulips.
6. The person who is a nurse is in the sixth house.
7. The person who loves a bouquet of daffodils and the Dragonfruit smoothie lover are next to each other.
8. The person who loves tennis is not in the third house.
9. The person who loves the boquet of iris is not in the second house.
10. The person who is a teacher is not in the second house.
11. The person who loves a carnations arrangement is the Watermelon smoothie lover.
12. There are two houses between the person who loves a bouquet of daffodils and the person who is a nurse.
13. The person who loves volleyball is the Dragonfruit smoothie lover.
14. Peter is the person who is a nurse.
15. There are two houses between the person who is a doctor and the person who is a lawyer.
16. The person who loves baseball is somewhere to the left of the person who drinks Blueberry smoothies.
17. The person who drinks Lime smoothies is somewhere to the left of Carol.
18. The person who loves the vase of tulips is the Desert smoothie lover.
19. The person who drinks Blueberry smoothies is Alice.
20. There is one house between the person who is an artist and Arnold.
21. Eric is directly left of Arnold.
22. There is one house between the person who is a doctor and Alice.
23. The person who loves the boquet of lilies is in the first house.
24. The person who is a doctor is not in the fifth house.
 What is the value of attribute FavoriteSport for the person whose attribute Flower is daffodils? Please reason step by step, and put your final answer within \boxed{}

Answer: swimming


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

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

Answer: victorian


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Alice`, `Peter`
 - People have unique favorite music genres: `pop`, `rock`, `jazz`, `classical`
 - Everyone has something unique for lunch: `pizza`, `stew`, `grilled cheese`, `spaghetti`
 - Each person has a unique level of education: `master`, `associate`, `bachelor`, `high school`
 - Each mother is accompanied by their child: `Samantha`, `Bella`, `Meredith`, `Fred`

## Clues:
1. The person's child is named Samantha is in the second house.
2. Arnold is the person's child is named Samantha.
3. The person who loves the spaghetti eater is directly left of the person with a master's degree.
4. The person who loves eating grilled cheese is the person who loves classical music.
5. The person who loves the stew is the person's child is named Bella.
6. The person who loves pop music is Alice.
7. The person who loves jazz music is directly left of the person's child is named Samantha.
8. The person with a high school diploma is Arnold.
9. The person with a bachelor's degree is Peter.
10. The person who loves the spaghetti eater is the person with an associate's degree.
11. The person who loves classical music is the person's child is named Fred.
 What is the value of attribute Food for the person whose attribute Education is high school? Please reason step by step, and put your final answer within \boxed{}

Answer: pizza


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: `average`, `tall`, `very short`, `short`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`, `dragonfruit`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`

## Clues:
1. Eric is the person who loves the rose bouquet.
2. The Dragonfruit smoothie lover is somewhere to the left of the person who loves the rose bouquet.
3. Arnold is the person who loves a carnations arrangement.
4. The Desert smoothie lover is in the third house.
5. The person who is very short is the Dragonfruit smoothie lover.
6. Alice is the person who likes Cherry smoothies.
7. The person who loves a bouquet of daffodils is the person who likes Cherry smoothies.
8. The person who loves a carnations arrangement is in the fourth house.
9. The person who is very short is directly left of the person who is tall.
10. The person who is short is the Desert smoothie lover.
 What is the value of attribute Height for the person whose attribute Smoothie is desert? Please reason step by step, and put your final answer within \boxed{}

Answer: short


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Arnold`, `Eric`, `Alice`
 - Everyone has something unique for lunch: `stew`, `stir fry`, `spaghetti`, `grilled cheese`, `pizza`
 - Everyone has a favorite smoothie: `watermelon`, `lime`, `desert`, `dragonfruit`, `cherry`
 - People have unique favorite book genres: `fantasy`, `biography`, `mystery`, `science fiction`, `romance`
 - Each person has a unique hobby: `photography`, `knitting`, `cooking`, `gardening`, `painting`

## Clues:
1. The person who loves stir fry is the Desert smoothie lover.
2. The person who loves the spaghetti eater is the person who enjoys gardening.
3. Bob is somewhere to the left of the person who loves mystery books.
4. The person who paints as a hobby is directly left of Alice.
5. The Desert smoothie lover is the person who loves romance books.
6. The photography enthusiast and the person who loves eating grilled cheese are next to each other.
7. The person who loves cooking is somewhere to the left of the Desert smoothie lover.
8. The person who likes Cherry smoothies is Alice.
9. There is one house between the person who drinks Lime smoothies and the person who enjoys gardening.
10. The Watermelon smoothie lover is Eric.
11. The person who loves eating grilled cheese is directly left of the Dragonfruit smoothie lover.
12. The person who is a pizza lover is Alice.
13. The person who loves romance books is Peter.
14. The person who loves fantasy books is the Watermelon smoothie lover.
15. Alice is the person who loves biography books.
16. Eric is not in the third house.
 What is the value of attribute House for the person whose attribute Smoothie is cherry? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Carol`, `Peter`, `Bob`, `Arnold`, `Alice`
 - They all have a unique favorite flower: `tulips`, `lilies`, `iris`, `roses`, `carnations`, `daffodils`

## Clues:
1. The person who loves the vase of tulips is directly left of the person who loves a bouquet of daffodils.
2. Alice is somewhere to the right of Peter.
3. There are two houses between Carol and Peter.
4. The person who loves the boquet of lilies is Bob.
5. The person who loves the boquet of iris is not in the second house.
6. The person who loves the boquet of iris and the person who loves a carnations arrangement are next to each other.
7. Arnold is not in the second house.
8. The person who loves a bouquet of daffodils is Peter.
9. Bob is in the fifth house.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 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`, `Alice`, `Carol`, `Peter`
 - The people are of nationalities: `german`, `brit`, `dane`, `chinese`, `swede`, `norwegian`
 - People have unique favorite music genres: `pop`, `jazz`, `hip hop`, `classical`, `country`, `rock`
 - Each person has a unique level of education: `doctorate`, `associate`, `master`, `bachelor`, `high school`, `trade school`

## Clues:
1. The Chinese is the person who loves pop music.
2. The person with a bachelor's degree is directly left of Arnold.
3. The person with a master's degree is in the first house.
4. Carol is somewhere to the left of Peter.
5. The Dane is the person with a doctorate.
6. The person who loves classical music is the person with a master's degree.
7. The person with a bachelor's degree is in the fifth house.
8. Eric is somewhere to the right of Bob.
9. Carol is the British person.
10. The person with a doctorate is somewhere to the right of the Norwegian.
11. The person with a doctorate is the person who loves rock music.
12. The Norwegian is somewhere to the right of the German.
13. The person who attended trade school is somewhere to the left of the person with a high school diploma.
14. The person who attended trade school and the Chinese are next to each other.
15. The person who loves jazz music is Arnold.
16. The person who loves hip-hop music is directly left of Alice.
 What is the value of attribute MusicGenre for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: rock


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`
 - Everyone has a unique favorite cigar: `blue master`, `prince`, `pall mall`
 - Each person has a unique hobby: `photography`, `gardening`, `cooking`
 - Each person has a unique level of education: `high school`, `associate`, `bachelor`
 - Each person has a unique favorite drink: `tea`, `milk`, `water`

## Clues:
1. The person partial to Pall Mall is Peter.
2. The person who likes milk is directly left of the person with a high school diploma.
3. Eric is the tea drinker.
4. Arnold and the Prince smoker are next to each other.
5. The person who enjoys gardening is somewhere to the left of the Prince smoker.
6. The person who likes milk is the person with an associate's degree.
7. The person with a bachelor's degree is directly left of the photography enthusiast.
 What is the value of attribute Name for the person whose attribute Hobby is gardening? 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: `Alice`, `Arnold`, `Eric`, `Peter`, `Bob`
 - People have unique hair colors: `black`, `red`, `gray`, `blonde`, `brown`
 - The people are of nationalities: `dane`, `norwegian`, `swede`, `brit`, `german`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `modern`, `craftsman`, `victorian`
 - Each person has a unique birthday month: `sept`, `jan`, `mar`, `april`, `feb`
 - Each person has an occupation: `engineer`, `doctor`, `lawyer`, `teacher`, `artist`

## Clues:
1. The person who has brown hair is the person whose birthday is in March.
2. The person who is a doctor is the Dane.
3. Eric is the person whose birthday is in March.
4. The person whose birthday is in February is directly left of the person who is an engineer.
5. There is one house between the person in a ranch-style home and the German.
6. Bob is the person whose birthday is in January.
7. The person who is an artist is the Swedish person.
8. The Swedish person is in the second house.
9. Peter is not in the fourth house.
10. The person who is an artist is somewhere to the left of the person who is a teacher.
11. The person who is an artist is directly left of the person who has gray hair.
12. Alice is the person who has red hair.
13. The person whose birthday is in September is not in the third house.
14. Arnold is somewhere to the left of the person living in a colonial-style house.
15. The person in a modern-style house is in the fifth house.
16. The person who has black hair is the Norwegian.
17. There are two houses between the person whose birthday is in January and the British person.
18. The person who has red hair is the person in a Craftsman-style house.
19. The person who is an engineer and the British person are next to each other.
20. There is one house between Arnold and the person who has red hair.
 What is the value of attribute HairColor for the person whose attribute Nationality is brit? 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`, `Carol`, `Bob`, `Peter`, `Eric`, `Arnold`
 - They all have a unique favorite flower: `lilies`, `tulips`, `roses`, `daffodils`, `iris`, `carnations`
 - The people are of nationalities: `norwegian`, `dane`, `swede`, `german`, `brit`, `chinese`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `desert`, `dragonfruit`, `blueberry`, `lime`
 - Each person has a unique hobby: `photography`, `cooking`, `knitting`, `woodworking`, `painting`, `gardening`
 - Each person has a unique type of pet: `rabbit`, `hamster`, `fish`, `cat`, `bird`, `dog`

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

Answer: photography


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

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

Answer: mountain


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Bob`, `Peter`, `Alice`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `lime`, `cherry`, `dragonfruit`
 - The people are of nationalities: `german`, `swede`, `norwegian`, `dane`, `brit`

## Clues:
1. The Dragonfruit smoothie lover is somewhere to the left of Eric.
2. The Dragonfruit smoothie lover is in the second house.
3. Peter is not in the first house.
4. The Dane and the British person are next to each other.
5. The Desert smoothie lover is not in the fifth house.
6. The Swedish person is somewhere to the left of the Dragonfruit smoothie lover.
7. There are two houses between the person who drinks Lime smoothies and the Dane.
8. Bob is the Dane.
9. Alice is the Norwegian.
10. Alice is in the third house.
11. The Watermelon smoothie lover is in the third house.
 What is the value of attribute Nationality for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: german


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Peter`, `Eric`
 - They all have a unique favorite flower: `roses`, `carnations`, `daffodils`, `lilies`
 - Each person lives in a unique style of house: `victorian`, `craftsman`, `colonial`, `ranch`
 - Everyone has a unique favorite cigar: `prince`, `dunhill`, `blue master`, `pall mall`

## Clues:
1. The person in a ranch-style home is in the third house.
2. The person who smokes Blue Master is somewhere to the left of the person residing in a Victorian house.
3. There are two houses between Eric and the person who loves the rose bouquet.
4. The Prince smoker is Peter.
5. The person in a Craftsman-style house is the Dunhill smoker.
6. The Dunhill smoker is the person who loves a carnations arrangement.
7. Arnold is in the fourth house.
8. The Prince smoker is in the second house.
9. The person who loves the boquet of lilies is the person who smokes Blue Master.
 What is the value of attribute Cigar for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: pall mall