There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`, `Carol`
 - People have unique favorite sports: `volleyball`, `soccer`, `basketball`, `baseball`, `swimming`, `tennis`
 - Each person has a unique hobby: `gardening`, `cooking`, `painting`, `woodworking`, `knitting`, `photography`
 - Each mother is accompanied by their child: `Bella`, `Meredith`, `Alice`, `Samantha`, `Fred`, `Timothy`
 - Each person has a unique birthday month: `feb`, `jan`, `mar`, `may`, `sept`, `april`
 - Everyone has something unique for lunch: `stir fry`, `grilled cheese`, `pizza`, `spaghetti`, `stew`, `soup`

## Clues:
1. The person who loves soccer is somewhere to the right of Eric.
2. The person who paints as a hobby is somewhere to the left of the woodworking hobbyist.
3. The person whose birthday is in February is not in the third house.
4. The person who is the mother of Timothy is directly left of Arnold.
5. The person who loves cooking is somewhere to the left of the person's child is named Fred.
6. Alice is the person who paints as a hobby.
7. The person's child is named Alice is the person who loves cooking.
8. The person whose birthday is in March is the person's child is named Samantha.
9. The person who loves the soup is the person whose birthday is in May.
10. Carol is somewhere to the right of the person whose birthday is in May.
11. The person whose birthday is in March and the person who is a pizza lover are next to each other.
12. The person whose birthday is in February is not in the fourth house.
13. There are two houses between the person who loves baseball and the person who loves the stew.
14. The person whose birthday is in May is somewhere to the left of the person who loves volleyball.
15. The person whose birthday is in January is somewhere to the right of Arnold.
16. The person who loves cooking is not in the third house.
17. The person who enjoys knitting is the person whose birthday is in September.
18. The person who enjoys gardening is the person's child is named Meredith.
19. The person who is a pizza lover is the person who loves swimming.
20. There is one house between the person who loves basketball and the person whose birthday is in May.
21. The person who loves the soup is directly left of the person who loves the spaghetti eater.
22. The person who loves the stew is the person's child is named Bella.
23. The person who enjoys knitting is somewhere to the left of the person who loves the soup.
24. Bob is somewhere to the left of the person who loves the stew.
25. Arnold is the person who loves stir fry.
26. The woodworking hobbyist is directly left of the person who loves the soup.
 What is the value of attribute House for the person whose attribute Food is stir fry? 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`, `Carol`, `Peter`, `Eric`, `Bob`, `Alice`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `modern`, `craftsman`, `mediterranean`, `victorian`
 - Everyone has something unique for lunch: `pizza`, `stew`, `spaghetti`, `grilled cheese`, `stir fry`, `soup`
 - Each person prefers a unique type of vacation: `cultural`, `cruise`, `mountain`, `camping`, `city`, `beach`
 - People have unique heights: `average`, `very tall`, `very short`, `short`, `tall`, `super tall`
 - Everyone has a unique favorite cigar: `yellow monster`, `prince`, `dunhill`, `pall mall`, `blue master`, `blends`

## Clues:
1. Alice is in the fifth house.
2. The person who loves stir fry is the person living in a colonial-style house.
3. Alice is the person who loves the spaghetti eater.
4. Arnold is the person who loves the stew.
5. There is one house between the person who has an average height and Peter.
6. The person in a Craftsman-style house is not in the third house.
7. The person who has an average height is the person who loves stir fry.
8. The person who loves beach vacations is the person in a ranch-style home.
9. Eric is in the fourth house.
10. There is one house between the person living in a colonial-style house and the person who enjoys camping trips.
11. The person who enjoys mountain retreats is the person who smokes Yellow Monster.
12. The person who enjoys mountain retreats is the person who is very tall.
13. The person who enjoys mountain retreats and the Dunhill smoker are next to each other.
14. The person who loves the spaghetti eater is the person residing in a Victorian house.
15. The person who is tall is the person who loves beach vacations.
16. The person who is tall is somewhere to the left of the person residing in a Victorian house.
17. The person who loves stir fry is directly left of Bob.
18. The person in a modern-style house is somewhere to the left of Alice.
19. The person in a Craftsman-style house is somewhere to the left of the person who is short.
20. The person who loves stir fry is somewhere to the left of the Prince smoker.
21. There are two houses between the person who loves eating grilled cheese and the person who is super tall.
22. The person in a ranch-style home is the person who smokes Blue Master.
23. The person who smokes many unique blends is directly left of the person who smokes Blue Master.
24. The person who goes on cultural tours is the person who is a pizza lover.
25. The person who is a pizza lover is somewhere to the left of the person who likes going on cruises.
 What is the value of attribute HouseStyle for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: mediterranean


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`
 - Each person has a unique birthday month: `jan`, `april`, `sept`
 - Each person has a unique type of pet: `cat`, `dog`, `fish`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `city`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`

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

Answer: 3


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

## Clues:
1. Arnold is in the fifth house.
2. The person who is short is the person who owns a Tesla Model 3.
3. The person who has an average height is somewhere to the right of Alice.
4. The person who owns a Honda Civic is not in the third house.
5. The person who owns a BMW 3 Series is directly left of Arnold.
6. The person who uses a OnePlus 9 is directly left of the person who owns a Ford F-150.
7. Alice is directly left of the person who is very short.
8. The person who uses an iPhone 13 is not in the second house.
9. The person who uses a Google Pixel 6 is Arnold.
10. The person who uses a Google Pixel 6 is the person who is tall.
11. Peter is the person who uses a Huawei P50.
12. Peter is directly left of the person who uses a OnePlus 9.
13. Bob is the person who owns a Tesla Model 3.
 What is the value of attribute House for the person whose attribute CarModel is ford f150? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


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 own unique car models: `honda civic`, `tesla model 3`, `toyota camry`, `ford f150`
 - They all have a unique favorite flower: `roses`, `daffodils`, `lilies`, `carnations`
 - Each person has a unique favorite drink: `milk`, `water`, `tea`, `coffee`
 - People have unique heights: `very short`, `average`, `tall`, `short`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`

## Clues:
1. The person who owns a Toyota Camry is somewhere to the left of the person who owns a Ford F-150.
2. The coffee drinker is the person who owns a Ford F-150.
3. The person who has an average height is in the third house.
4. The person whose favorite color is green and the person who owns a Tesla Model 3 are next to each other.
5. There are two houses between the person who loves the rose bouquet and the person who owns a Tesla Model 3.
6. The person who is very short is the tea drinker.
7. Alice is the person who loves white.
8. The person who loves the boquet of lilies is the person who owns a Tesla Model 3.
9. Peter is the person whose favorite color is red.
10. The person who owns a Toyota Camry is Eric.
11. The person who is short is somewhere to the left of the person who loves the boquet of lilies.
12. The person who loves a bouquet of daffodils is the person who loves white.
13. The coffee drinker is in the second house.
14. The person who owns a Tesla Model 3 is the person who likes milk.
 What is the value of attribute House for the person whose attribute Height is tall? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - The people keep unique animals: `horse`, `bird`, `cat`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `desert`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`
 - People have unique favorite music genres: `pop`, `classical`, `rock`
 - Each person has a favorite color: `yellow`, `white`, `red`

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

Answer: Holly


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Arnold`, `Peter`, `Eric`
 - Each person has a unique level of education: `doctorate`, `bachelor`, `associate`, `master`, `high school`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `google pixel 6`, `iphone 13`, `huawei p50`
 - People have unique favorite music genres: `rock`, `classical`, `jazz`, `pop`, `hip hop`
 - The people keep unique animals: `bird`, `dog`, `horse`, `cat`, `fish`
 - Each person has a unique hobby: `painting`, `cooking`, `knitting`, `gardening`, `photography`

## Clues:
1. The person who loves hip-hop music is in the second house.
2. The person who loves classical music is not in the fifth house.
3. The person who loves classical music is the fish enthusiast.
4. The person who uses an iPhone 13 is the person with a doctorate.
5. The person with a master's degree is not in the second house.
6. The person who enjoys knitting is not in the first house.
7. The person with a high school diploma is somewhere to the left of the photography enthusiast.
8. The person who paints as a hobby is somewhere to the right of the person who loves rock music.
9. Arnold is the bird keeper.
10. The person with a bachelor's degree is the person who uses a Google Pixel 6.
11. The person with a bachelor's degree is somewhere to the right of the person who paints as a hobby.
12. Bob is not in the fourth house.
13. The person who loves cooking is the person who loves classical music.
14. The person who loves cooking is the person who uses a Huawei P50.
15. The person who keeps horses is somewhere to the right of Arnold.
16. The person who paints as a hobby is the person who uses a OnePlus 9.
17. The person who uses a OnePlus 9 is somewhere to the left of the person with an associate's degree.
18. The person who uses an iPhone 13 is the dog owner.
19. The person with a high school diploma is Alice.
20. Peter is the person who enjoys gardening.
21. The person who paints as a hobby is the person who loves jazz music.
 What is the value of attribute Education for the person whose attribute MusicGenre is hip hop? Please reason step by step, and put your final answer within \boxed{}

Answer: high school


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - People have unique favorite music genres: `classical`, `rock`, `pop`
 - Each person has a unique type of pet: `cat`, `fish`, `dog`
 - The mothers' names in different houses are unique: `Janelle`, `Aniya`, `Holly`
 - The people keep unique animals: `cat`, `bird`, `horse`

## Clues:
1. The person whose mother's name is Aniya is not in the first house.
2. The person whose mother's name is Aniya is directly left of The person whose mother's name is Holly.
3. The person whose mother's name is Janelle is Peter.
4. The person who loves pop music is directly left of the person who owns a dog.
5. The person who keeps horses is in the first house.
6. Peter and the cat lover are next to each other.
7. Eric is the person who loves rock music.
8. The person whose mother's name is Janelle and the person who has a cat are next to each other.
 What is the value of attribute Mother for the person whose attribute MusicGenre is rock? Please reason step by step, and put your final answer within \boxed{}

Answer: Holly


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Peter`, `Bob`, `Arnold`
 - People have unique favorite sports: `soccer`, `baseball`, `swimming`, `tennis`, `basketball`
 - Each person lives in a unique style of house: `modern`, `victorian`, `colonial`, `craftsman`, `ranch`
 - Each person prefers a unique type of vacation: `beach`, `city`, `cruise`, `mountain`, `camping`
 - Each person has an occupation: `artist`, `engineer`, `lawyer`, `doctor`, `teacher`
 - The mothers' names in different houses are unique: `Penny`, `Aniya`, `Janelle`, `Holly`, `Kailyn`

## Clues:
1. There is one house between The person whose mother's name is Janelle and Eric.
2. Arnold is not in the fifth house.
3. The person whose mother's name is Janelle is Peter.
4. The person who enjoys mountain retreats is The person whose mother's name is Holly.
5. The person whose mother's name is Aniya is the person in a modern-style house.
6. The person who likes going on cruises is The person whose mother's name is Kailyn.
7. The person who prefers city breaks is the person living in a colonial-style house.
8. The person who loves soccer is The person whose mother's name is Holly.
9. The person who is an artist is Eric.
10. The person who is a doctor is in the second house.
11. The person who is a teacher is somewhere to the left of the person in a modern-style house.
12. The person whose mother's name is Penny is not in the second house.
13. The person in a ranch-style home is in the third house.
14. The person who loves swimming is The person whose mother's name is Kailyn.
15. Arnold is not in the third house.
16. The person who loves swimming is in the fourth house.
17. The person who loves baseball and the person who prefers city breaks are next to each other.
18. The person who loves basketball is somewhere to the left of the person who loves soccer.
19. Bob is somewhere to the left of the person in a modern-style house.
20. The person who enjoys camping trips is in the fifth house.
21. The person in a Craftsman-style house is the person who is an engineer.
22. There are two houses between the person living in a colonial-style house and the person in a modern-style house.
 What is the value of attribute Name for the person whose attribute Mother is Janelle? 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: `Arnold`, `Peter`, `Eric`, `Alice`, `Bob`
 - Each person has a unique hobby: `painting`, `cooking`, `knitting`, `gardening`, `photography`
 - People have unique heights: `very tall`, `tall`, `very short`, `average`, `short`
 - Everyone has something unique for lunch: `stew`, `grilled cheese`, `stir fry`, `spaghetti`, `pizza`

## Clues:
1. Bob is the photography enthusiast.
2. The person who loves eating grilled cheese is the person who is tall.
3. Peter is not in the second house.
4. The person who is tall is directly left of the person who loves stir fry.
5. The person who loves cooking is the person who has an average height.
6. Alice is directly left of the person who is a pizza lover.
7. The person who loves the spaghetti eater is not in the second house.
8. Eric is not in the fifth house.
9. The person who is short is Peter.
10. The person who has an average height and the person who enjoys gardening are next to each other.
11. The person who paints as a hobby is directly left of the person who loves eating grilled cheese.
12. The person who is very short is in the fifth house.
13. The person who is tall is in the third house.
14. Alice is somewhere to the right of the photography enthusiast.
 What is the value of attribute House for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Peter`
 - The people keep unique animals: `cat`, `fish`, `horse`, `bird`
 - Each person has a favorite color: `yellow`, `green`, `white`, `red`
 - Each person prefers a unique type of vacation: `cruise`, `city`, `beach`, `mountain`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `stew`, `grilled cheese`
 - People have unique favorite sports: `soccer`, `swimming`, `basketball`, `tennis`

## Clues:
1. The person whose favorite color is red is the person who loves the spaghetti eater.
2. The person who loves the stew is the cat lover.
3. The person who loves yellow is not in the fourth house.
4. Alice is somewhere to the right of Arnold.
5. The person who loves the spaghetti eater is somewhere to the right of the person who loves soccer.
6. The person who loves white is the person who loves tennis.
7. The fish enthusiast is in the second house.
8. Arnold is the person who loves the stew.
9. The person who prefers city breaks is the person who is a pizza lover.
10. The person who prefers city breaks is directly left of the person who likes going on cruises.
11. The person who loves swimming is the person who loves beach vacations.
12. The person who keeps horses and the person whose favorite color is red are next to each other.
13. The person whose favorite color is green is directly left of Eric.
14. The person who prefers city breaks is somewhere to the left of the cat lover.
 What is the value of attribute Food for the person whose attribute Color is yellow? Please reason step by step, and put your final answer within \boxed{}

Answer: stew


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`
 - Each person has a unique level of education: `associate`, `bachelor`, `high school`
 - Each person has a unique favorite drink: `water`, `tea`, `milk`
 - People have unique favorite book genres: `science fiction`, `mystery`, `romance`
 - Each person has a unique hobby: `photography`, `cooking`, `gardening`
 - People own unique car models: `ford f150`, `tesla model 3`, `toyota camry`

## Clues:
1. The person who likes milk is somewhere to the right of Eric.
2. The person who loves science fiction books is directly left of the person who owns a Toyota Camry.
3. The person who loves cooking is Peter.
4. The person who loves romance books is the person with a bachelor's degree.
5. The person who owns a Tesla Model 3 is Peter.
6. The person with an associate's degree is the person who loves mystery books.
7. The person who loves mystery books is the photography enthusiast.
8. The one who only drinks water is not in the second house.
9. The person who owns a Toyota Camry is not in the third house.
10. The person who owns a Ford F-150 is the person who loves mystery books.
 What is the value of attribute Drink for the person whose attribute CarModel is ford f150? Please reason step by step, and put your final answer within \boxed{}

Answer: milk


4. The continuation of the height $B H$ of triangle $A B C$ intersects the circumscribed circle around it at point $D$ (points $B$ and $D$ lie on opposite sides of line $A C$). The degree measures of arcs $A D$ and $C D$, not containing point $B$, are $120^{\circ}$ and $90^{\circ}$, respectively. Determine in what ratio the segment $B D$ is divided by side $A C$.

Answer: 1:\sqrt{3}


Example 2. Given that $D, F$ are points on the sides $A B$, $A C$ of $\triangle A B C$ respectively, and $A D: D B=C F: F A=2: 3$. Connect $D F$ to intersect the extension of side $B C$ at point $E$. Then, $E F: F D=$ $\qquad$
(3rd Zu Chongzhi Cup Junior High School Mathematics Invitational Competition)

Answer: 2: 1


1. In a computer game, a turtle moves across a grid on the computer screen, which contains 5 columns and 7 rows. Initially, it is located at the bottom-left corner of the screen - on the cell with coordinates $(0,0)$. If the program instructs the turtle to move off the screen, it reappears on the opposite side - for example, taking one step up from the cell $(3,6)$, the turtle will end up in the cell $(3,0)$. Where will the turtle be after executing the following program:
1) 1 step down; 2) 2 steps to the right; 3) 3 steps up; 4) 4 steps to the left; 5) 5 steps down; 6) 6 steps to the right; ... ; 2016) 2016 steps to the left; 2017) 2017 steps down?

Answer: (2,6)


2. In the room, there are knights who always tell the truth, and liars who always lie. 10 of them said: "In this room, there are more knights than liars." 15 said: "In this room, there are more liars than knights." The remaining 25 said: "In this room, there are an equal number of liars and knights." How many liars could there be in the room? If there are multiple possible answers, write them in any order separated by a semicolon.

Answer: 25;35


4. Adam and Bohouš participated in a tournament played in a round-robin system, where each player was supposed to play one match per day. However, Adam and Bohouš were the only ones who did not complete the tournament due to illness. Bohouš withdrew five days earlier than Adam. In total, 350 matches were played. How many matches did Adam play? Did he play against Bohouš?

Answer: 15


10.208. Perpendiculars are drawn from the vertex of the acute angle of a rhombus to the lines containing the sides of the rhombus to which this vertex does not belong. The length of each perpendicular is 3 cm, and the distance between their bases is $3 \sqrt{3}$ cm. Calculate the lengths of the diagonals of the rhombus.

Answer: 6


2. The eight-digit number $\square 2 \square 0 \square 2 \square 2$ (the digits in $\square$ can repeat) is a multiple of 72. There are $\qquad$ such eight-digit numbers.

Answer: 200


2. (12 points) In a family, there are four children of different ages. Their total age is 31 years. Four years ago, the total age of all the children in the family was 16 years, 7 years ago it was 8 years, and 11 years ago it was 1 year. How old are the children at present? (Age is always expressed as a whole number of years.)

Answer: 3


| $A$ | $B$ | $C$ | $D$ | $E$ |
| :--- | :--- | :--- | :--- | :--- |
| $E$ | $D$ | $C$ | $B$ | $A$ |
| $F$ | $F$ | $F$ | $F$ | $F$ |$+$

A fragment of a conversation on the beach:

Feri: Don't rush me now, I need to write digits here instead of the letters (Fig. 1), so that the addition is correct, the same digit for the same letter, and different digits for different letters. - Gyurka (after looking at the problem): What's the fuss about this? I can write down 77 solutions for you. - F: I doubt that. - Gy: I'll buy you as many ice creams as the number of solutions you miss, just come and swim now. - F: I'll go, but is the bet on? - How did the bet turn out?

Answer: 32


8. Find all positive integers $a$ such that for any positive integer $n \geqslant 5$, we have $\left(2^{n}-n^{2}\right) \mid\left(a^{n}-n^{a}\right)$.

Answer: 2


On colors some cells of an $8 \times 8$ chessboard in red. How many cells can be colored at most if we want there to be no red tromino? How many cells can be colored at least if we want every tromino to have at least one red cell?

Answer: 32


Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$.  It is given that there exist points $X$ and $Y$ on the circumference of $\omega$ such that $\angle BXC=\angle BYC=90^\circ$.  Suppose further that $X$, $I$, and $Y$ are collinear.  If $AB=80$ and $AC=97$, compute the length of $BC$.

Answer: 59


Let $n$ and $k$ be two integers which are greater than $1$. Let $a_1,a_2,\ldots,a_n,c_1,c_2,\ldots,c_m$ be non-negative real numbers such that
i) $a_1\ge a_2\ge\ldots\ge a_n$ and $a_1+a_2+\ldots+a_n=1$;
ii) For any integer $m\in\{1,2,\ldots,n\}$, we have that $c_1+c_2+\ldots+c_m\le m^k$.
Find the maximum of $c_1a_1^k+c_2a_2^k+\ldots+c_na_n^k$.

Answer: 1


[ Arithmetic. Mental calculation, etc.]

$[\quad$ Invariants $]$

The rabbits are sawing the log again, but now both ends of the log are secured. Ten middle pieces fell, while the two end pieces remained secured. How many cuts did the rabbits make?

Answer: 11


36. Mathematician Gauss invented the floor function $[x]$ when studying integer problems, denoting by $[x]$ the greatest integer not exceeding $x$.
Question: When the value of the natural number $n$ is taken as $1, 2, 3, \ldots, 2019$, how many different possible values does $\left[\frac{n}{2}\right]+\left[\frac{n}{3}\right]+\left[\frac{n}{6}\right]$ have?

Answer: 1347


8. On the edge $AS$ of the tetrahedron $S-ABC$, mark points $M, N$ such that $AM=MN=NS$. If the areas of $\triangle ABC$, $\triangle MBC$, and $\triangle SBC$ are $1$, $2$, and $\sqrt{37}$, respectively, find the area of $\triangle NBC$.

Answer: 4


[ Concerning the sphere ] $[$ Regular tetrahedron $]$

Four spheres of radius 1 touch each other pairwise. Find the radius of the sphere that touches all four spheres.

Answer: \sqrt{6} - 2


\( ABCD \) is a cyclic quadrilateral. A perpendicular to \( BA \) erected at point \( A \) intersects line \( CD \) at point \( M \); a perpendicular to \( DA \) erected at point \( A \) intersects line \( BC \) at point \( N \). Prove that \( MN \) passes through the center of the circle.

Answer: MN \text{ passes through the center of the circle}


2. In $\square A B C D$, $\angle B<90^{\circ}, A B<B C$. From point $D$ draw tangents to the circumcircle $\Gamma$ of $\triangle A B C$, the points of tangency are $E$ and $F$. It is known that $\angle E D A=\angle F D C$. Find $\angle A B C$

Answer: 60^\circ


Triangle $\triangle ABC$ has circumcenter $O$ and incircle $\gamma$. Suppose that $\angle BAC =60^\circ$ and $O$ lies on $\gamma$. If \[ \tan B \tan C = a + \sqrt{b} \] for positive integers $a$ and $b$, compute $100a+b$.

[i]Proposed by Kaan Dokmeci[/i]

Answer: 408


3. In triangle $A B C$, point $M$ is the midpoint of $A C$, moreover, $B C=2 A C / 3$ and $\angle B M C=2 \angle A B M$. Find the ratio $A M / A B$.

Answer: \dfrac{3\sqrt{5}}{10}


11. Given a regular quadrilateral pyramid $V-A B C D$ with all edges equal to $a$, the midpoints of the lateral edges $V B$ and $V D$ are $H$ and $K$, respectively. If the plane passing through points $A$, $H$, and $K$ intersects the lateral edge $V C$ at $L$, then the area of quadrilateral $A H L K$ is $\qquad$

Answer: \dfrac{a^2 \sqrt{5}}{6}


## Task B-1.1.

Write the expression $\left[27^{-2 m+3} \cdot\left(\frac{1}{9}\right)^{3-2 m}\right]^{-2}: 81^{1+m}-6 \cdot 81^{-3}$, where $m$ is an integer, in the form of a power with a positive exponent.

Answer: \left(\frac{1}{3}\right)^{11}


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Eric`, `Peter`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `lilies`, `roses`
 - Everyone has a unique favorite cigar: `dunhill`, `blue master`, `pall mall`, `prince`
 - Each mother is accompanied by their child: `Meredith`, `Samantha`, `Bella`, `Fred`
 - The mothers' names in different houses are unique: `Janelle`, `Aniya`, `Holly`, `Kailyn`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `oneplus 9`, `samsung galaxy s21`

## Clues:
1. The person whose mother's name is Kailyn is the Prince smoker.
2. The person whose mother's name is Holly is the person who loves a bouquet of daffodils.
3. Arnold is in the fourth house.
4. The person's child is named Bella is somewhere to the left of the person who uses a OnePlus 9.
5. The person's child is named Fred is not in the second house.
6. The person's child is named Meredith is directly left of The person whose mother's name is Janelle.
7. The person who loves a bouquet of daffodils is Eric.
8. The person who loves the boquet of lilies is the person partial to Pall Mall.
9. The person whose mother's name is Janelle is the person who loves a carnations arrangement.
10. Alice is somewhere to the right of the person's child is named Fred.
11. The person who uses an iPhone 13 is directly left of the Dunhill smoker.
12. The person who loves the rose bouquet is somewhere to the right of the Dunhill smoker.
13. The person who loves the rose bouquet is somewhere to the left of the person who uses a Google Pixel 6.
 What is the value of attribute PhoneModel for the person whose attribute Cigar is prince? Please reason step by step, and put your final answer within \boxed{}

Answer: oneplus 9


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Bob`, `Peter`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `dragonfruit`, `lime`, `desert`
 - People have unique favorite music genres: `hip hop`, `classical`, `rock`, `pop`, `jazz`
 - Everyone has something unique for lunch: `grilled cheese`, `pizza`, `spaghetti`, `stir fry`, `stew`
 - Each person has a unique favorite drink: `tea`, `root beer`, `milk`, `water`, `coffee`

## Clues:
1. The coffee drinker is not in the first house.
2. The person who loves jazz music is the Desert smoothie lover.
3. The Watermelon smoothie lover is in the fourth house.
4. The coffee drinker is directly left of Peter.
5. The one who only drinks water is the person who loves hip-hop music.
6. The Dragonfruit smoothie lover is directly left of the one who only drinks water.
7. The Desert smoothie lover is in the fifth house.
8. The person who loves stir fry is somewhere to the right of Bob.
9. The person who is a pizza lover is the person who likes Cherry smoothies.
10. The person who loves classical music is the person who loves the stew.
11. The one who only drinks water is Bob.
12. The root beer lover is the Dragonfruit smoothie lover.
13. There is one house between Arnold and the person who likes Cherry smoothies.
14. The person who loves the spaghetti eater is in the fifth house.
15. The person who likes milk is in the fifth house.
16. The person who likes Cherry smoothies is Bob.
17. The Dragonfruit smoothie lover is the person who loves the stew.
18. Eric and Peter are next to each other.
19. The person who loves rock music is not in the first house.
 What is the value of attribute Food for the person whose attribute Smoothie is lime? Please reason step by step, and put your final answer within \boxed{}

Answer: grilled cheese


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`, `Peter`, `Alice`
 - People have unique heights: `very short`, `short`, `tall`, `average`
 - Each person prefers a unique type of vacation: `cruise`, `beach`, `city`, `mountain`
 - Each person has a favorite color: `green`, `red`, `yellow`, `white`
 - Each mother is accompanied by their child: `Fred`, `Bella`, `Meredith`, `Samantha`

## Clues:
1. The person who loves beach vacations is the person who is very short.
2. The person who is very short is the person who loves white.
3. The person who has an average height is the person's child is named Bella.
4. The person who has an average height is the person whose favorite color is red.
5. Peter is the person's child is named Samantha.
6. The person whose favorite color is green is directly left of the person who loves white.
7. Alice is in the third house.
8. The person who likes going on cruises is the person's child is named Bella.
9. The person who is short is the person who loves yellow.
10. The person who is very short is the person's child is named Fred.
11. Eric is the person who loves yellow.
12. The person who prefers city breaks is Eric.
13. The person who is short is somewhere to the left of the person who is tall.
 What is the value of attribute Height for the person whose attribute Color is green? Please reason step by step, and put your final answer within \boxed{}

Answer: tall


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`, `Alice`, `Bob`
 - Each person has a unique level of education: `associate`, `doctorate`, `high school`, `bachelor`, `master`
 - Each mother is accompanied by their child: `Fred`, `Meredith`, `Bella`, `Timothy`, `Samantha`
 - Everyone has a unique favorite cigar: `prince`, `pall mall`, `blue master`, `dunhill`, `blends`
 - The people are of nationalities: `swede`, `brit`, `dane`, `german`, `norwegian`
 - People have unique favorite book genres: `romance`, `mystery`, `biography`, `science fiction`, `fantasy`

## Clues:
1. The person with a master's degree is the person partial to Pall Mall.
2. The person with a high school diploma is in the fourth house.
3. The person's child is named Samantha is in the first house.
4. The Norwegian and the person with a master's degree are next to each other.
5. The German is the Dunhill smoker.
6. The person who is the mother of Timothy is directly left of the person with a master's degree.
7. Arnold and the Dane are next to each other.
8. The Swedish person is the person who loves mystery books.
9. The Swedish person is Alice.
10. Arnold is the person with an associate's degree.
11. The person with a doctorate is in the second house.
12. Peter is somewhere to the right of the person who loves biography books.
13. The person who smokes many unique blends is Alice.
14. The person who loves fantasy books is in the first house.
15. Peter is the Norwegian.
16. The person with a doctorate is the person's child is named Meredith.
17. The Prince smoker is somewhere to the left of the person who smokes many unique blends.
18. The person partial to Pall Mall is the person's child is named Bella.
19. The person who loves romance books is Bob.
 What is the value of attribute Children for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: Samantha


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - Each person prefers a unique type of vacation: `city`, `beach`, `mountain`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `cherry`
 - People own unique car models: `toyota camry`, `tesla model 3`, `ford f150`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `lilies`

## Clues:
1. The person who likes Cherry smoothies is directly left of the person who loves a bouquet of daffodils.
2. Eric is the person who loves beach vacations.
3. The person who loves a carnations arrangement is somewhere to the left of the person who loves a bouquet of daffodils.
4. The person who owns a Toyota Camry is not in the first house.
5. The Watermelon smoothie lover is directly left of the person who is a pizza lover.
6. Peter is somewhere to the left of Arnold.
7. Arnold is not in the third house.
8. The person who loves beach vacations is the person who owns a Ford F-150.
9. The person who loves the spaghetti eater is the person who prefers city breaks.
10. Peter and the person who loves the spaghetti eater are next to each other.
 What is the value of attribute CarModel for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: tesla model 3


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`, `Alice`, `Bob`, `Eric`, `Peter`
 - Each person has a favorite color: `green`, `red`, `white`, `yellow`, `blue`
 - Everyone has something unique for lunch: `grilled cheese`, `stew`, `pizza`, `spaghetti`, `stir fry`
 - People use unique phone models: `google pixel 6`, `huawei p50`, `oneplus 9`, `samsung galaxy s21`, `iphone 13`
 - Each person has a unique level of education: `master`, `associate`, `doctorate`, `bachelor`, `high school`

## Clues:
1. The person who uses a Samsung Galaxy S21 is somewhere to the left of the person whose favorite color is green.
2. The person whose favorite color is red is Alice.
3. Bob is directly left of the person whose favorite color is red.
4. Alice is the person with an associate's degree.
5. The person with a high school diploma is the person who loves eating grilled cheese.
6. The person who uses a Google Pixel 6 is the person who loves yellow.
7. The person who loves stir fry is the person with a bachelor's degree.
8. The person who is a pizza lover is not in the first house.
9. The person who loves white is the person with a master's degree.
10. The person who loves blue is Eric.
11. The person who uses a Huawei P50 is the person who loves the stew.
12. The person who uses an iPhone 13 is the person with a high school diploma.
13. The person whose favorite color is green is the person who loves stir fry.
14. The person with a doctorate and the person who loves white are next to each other.
15. The person who loves the stew is not in the second house.
16. Arnold is somewhere to the right of the person who loves eating grilled cheese.
17. The person who loves yellow is in the first house.
 What is the value of attribute PhoneModel for the person whose attribute Education is high school? Please reason step by step, and put your final answer within \boxed{}

Answer: iphone 13


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `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 House is 5? Please reason step by step, and put your final answer within \boxed{}

Answer: green


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`
 - Each person has a unique birthday month: `sept`, `april`, `feb`, `jan`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`, `stew`
 - Each person has a unique hobby: `cooking`, `gardening`, `painting`, `photography`
 - Each person has an occupation: `teacher`, `artist`, `engineer`, `doctor`
 - Each mother is accompanied by their child: `Bella`, `Samantha`, `Fred`, `Meredith`

## Clues:
1. The person who is a teacher is the person who loves cooking.
2. The person whose birthday is in January is directly left of Eric.
3. The person who is an artist is not in the fourth house.
4. The person who loves the spaghetti eater is not in the third house.
5. The person's child is named Bella is the person who paints as a hobby.
6. The person whose birthday is in January is the person who is a teacher.
7. The person who paints as a hobby is the person who is a doctor.
8. The person whose birthday is in September is directly left of Arnold.
9. The person who is a pizza lover is Peter.
10. The person who loves the stew is somewhere to the left of the photography enthusiast.
11. Arnold is the person's child is named Meredith.
12. The person whose birthday is in January is directly left of the person whose birthday is in September.
13. The person whose birthday is in April is in the fourth house.
14. The person's child is named Fred is directly left of the person whose birthday is in April.
15. Alice is in the first house.
16. The person whose birthday is in April is the person who enjoys gardening.
 What is the value of attribute House for the person whose attribute Birthday is feb? Please reason step by step, and put your final answer within \boxed{}

Answer: 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: `Alice`, `Arnold`, `Eric`, `Peter`
 - Each person has a unique level of education: `master`, `bachelor`, `associate`, `high school`
 - People have unique favorite sports: `tennis`, `soccer`, `swimming`, `basketball`
 - Each person has an occupation: `teacher`, `engineer`, `doctor`, `artist`
 - They all have a unique favorite flower: `carnations`, `daffodils`, `lilies`, `roses`
 - People have unique heights: `very short`, `average`, `tall`, `short`

## Clues:
1. Eric is the person who loves a carnations arrangement.
2. The person who loves swimming is somewhere to the left of the person who is an artist.
3. Arnold is directly left of the person who is tall.
4. The person who loves swimming and the person who is a teacher are next to each other.
5. The person who loves basketball is the person with a bachelor's degree.
6. The person who is a doctor is the person who loves the rose bouquet.
7. The person who loves the boquet of lilies is the person who loves tennis.
8. The person who loves a carnations arrangement is the person with a master's degree.
9. The person who is a teacher is directly left of Alice.
10. The person who is an artist is the person with an associate's degree.
11. The person who loves a bouquet of daffodils is the person who is an artist.
12. The person who loves soccer is not in the fourth house.
13. The person who loves a bouquet of daffodils is somewhere to the right of the person who is short.
14. The person with a master's degree is the person who is very short.
 What is the value of attribute Flower for the person whose attribute FavoriteSport is tennis? Please reason step by step, and put your final answer within \boxed{}

Answer: lilies


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

Answer: 6


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

## Clues:
1. Alice is the person who loves cooking.
2. Bob is not in the second house.
3. The person who enjoys gardening is the person who smokes Blue Master.
4. The person partial to Pall Mall is the person who enjoys knitting.
5. The person who enjoys gardening is directly left of the Dunhill smoker.
6. The person who smokes many unique blends is directly left of Carol.
7. The person partial to Pall Mall is directly left of Bob.
8. The Dunhill smoker is the person who paints as a hobby.
9. The person who smokes Yellow Monster is in the second house.
10. The woodworking hobbyist is somewhere to the right of the Dunhill smoker.
11. Eric is somewhere to the left of the person who enjoys gardening.
12. There are two houses between the photography enthusiast and Peter.
 What is the value of attribute Hobby for the person whose attribute Cigar is prince? Please reason step by step, and put your final answer within \boxed{}

Answer: woodworking


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Arnold`, `Eric`, `Peter`
 - People have unique favorite book genres: `romance`, `mystery`, `biography`, `fantasy`, `science fiction`
 - Each person has a unique birthday month: `sept`, `feb`, `jan`, `mar`, `april`
 - Each person prefers a unique type of vacation: `mountain`, `beach`, `city`, `camping`, `cruise`
 - The people keep unique animals: `horse`, `bird`, `fish`, `dog`, `cat`

## Clues:
1. The person whose birthday is in April is the person who loves mystery books.
2. The person whose birthday is in March is somewhere to the left of the person who loves mystery books.
3. The person who loves beach vacations is Arnold.
4. The person who loves beach vacations is the person whose birthday is in January.
5. The bird keeper is directly left of Peter.
6. The person who loves science fiction books is somewhere to the left of the person whose birthday is in February.
7. The fish enthusiast is somewhere to the right of Bob.
8. The fish enthusiast is Eric.
9. The fish enthusiast is the person who loves fantasy books.
10. The person who enjoys mountain retreats is directly left of the person whose birthday is in April.
11. The person whose birthday is in April is the person who enjoys camping trips.
12. The person whose birthday is in September is the person who loves science fiction books.
13. The cat lover is the person who prefers city breaks.
14. The person who loves romance books is somewhere to the right of the person who loves science fiction books.
15. The cat lover is the person whose birthday is in February.
16. The person who keeps horses is Peter.
 What is the value of attribute BookGenre for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: mystery


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Bob`, `Arnold`, `Carol`, `Eric`, `Alice`
 - Everyone has something unique for lunch: `spaghetti`, `stir fry`, `stew`, `soup`, `grilled cheese`, `pizza`
 - Everyone has a unique favorite cigar: `blends`, `pall mall`, `dunhill`, `yellow monster`, `prince`, `blue master`
 - Each person has a unique favorite drink: `boba tea`, `water`, `coffee`, `root beer`, `milk`, `tea`
 - Each person has a unique level of education: `associate`, `bachelor`, `trade school`, `doctorate`, `high school`, `master`

## Clues:
1. Bob is not in the second house.
2. The person who smokes many unique blends is somewhere to the left of the person who is a pizza lover.
3. The person partial to Pall Mall is somewhere to the right of the Dunhill smoker.
4. The person who loves the soup is not in the first house.
5. The person who likes milk is directly left of the person who loves the stew.
6. The person who loves the soup is the person with a doctorate.
7. The coffee drinker is the person with a high school diploma.
8. There are two houses between the person who loves the spaghetti eater and the person with an associate's degree.
9. The Prince smoker is not in the third house.
10. The Dunhill smoker and Bob are next to each other.
11. There are two houses between Alice and the boba tea drinker.
12. The person with a high school diploma is Eric.
13. The person who attended trade school is the Prince smoker.
14. There is one house between the Dunhill smoker and the person who loves stir fry.
15. The Dunhill smoker is directly left of the person who is a pizza lover.
16. There is one house between the person partial to Pall Mall and the person who smokes Blue Master.
17. The person who smokes Blue Master is the coffee drinker.
18. The person with a bachelor's degree is the Dunhill smoker.
19. Alice is the tea drinker.
20. Arnold is directly left of the root beer lover.
21. The person with a master's degree is Peter.
22. Arnold is somewhere to the left of the person who likes milk.
 What is the value of attribute Education for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: master


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`, `Peter`, `Bob`, `Alice`
 - Everyone has a favorite smoothie: `watermelon`, `lime`, `cherry`, `dragonfruit`, `desert`
 - The people keep unique animals: `fish`, `bird`, `horse`, `cat`, `dog`
 - Each person prefers a unique type of vacation: `camping`, `beach`, `mountain`, `city`, `cruise`
 - Each person has a unique favorite drink: `root beer`, `water`, `milk`, `tea`, `coffee`

## Clues:
1. The person who keeps horses is the person who drinks Lime smoothies.
2. There is one house between the Watermelon smoothie lover and the person who likes Cherry smoothies.
3. Bob and the coffee drinker are next to each other.
4. The fish enthusiast is the one who only drinks water.
5. The bird keeper is Bob.
6. The person who enjoys mountain retreats is directly left of the root beer lover.
7. The person who loves beach vacations is somewhere to the left of the dog owner.
8. Eric is the person who likes Cherry smoothies.
9. The tea drinker is Alice.
10. The person who prefers city breaks is somewhere to the right of the person who enjoys mountain retreats.
11. Peter is somewhere to the left of the coffee drinker.
12. Eric is the person who loves beach vacations.
13. The person who enjoys mountain retreats is Alice.
14. There are two houses between the Watermelon smoothie lover and Alice.
15. The person who enjoys camping trips is not in the first house.
16. The dog owner is not in the fifth house.
17. The fish enthusiast and the Dragonfruit smoothie lover are next to each other.
 What is the value of attribute House for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - People own unique car models: `ford f150`, `tesla model 3`, `toyota camry`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `city`
 - The people are of nationalities: `dane`, `swede`, `brit`
 - People have unique hair colors: `blonde`, `black`, `brown`
 - Each person lives in a unique style of house: `colonial`, `ranch`, `victorian`

## Clues:
1. The Swedish person is the person who owns a Ford F-150.
2. The person who has blonde hair is not in the second house.
3. The person who owns a Toyota Camry and Arnold are next to each other.
4. The person who has black hair is somewhere to the right of the person in a ranch-style home.
5. The person who has black hair is the Dane.
6. The person residing in a Victorian house is not in the second house.
7. Peter is directly left of the person who loves beach vacations.
8. The person living in a colonial-style house is the person who has blonde hair.
9. The person who has black hair is Arnold.
10. The Dane is the person who loves beach vacations.
11. Peter is the person who prefers city breaks.
 What is the value of attribute Vacation for the person whose attribute HouseStyle is colonial? Please reason step by step, and put your final answer within \boxed{}

Answer: mountain


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

## Clues:
1. The person who is a teacher is the person in a Craftsman-style house.
2. Bob is in the second house.
3. There are two houses between the person residing in a Victorian house and Eric.
4. The cat lover is directly left of the person who keeps horses.
5. The person who is an engineer is somewhere to the left of the person in a modern-style house.
6. The person in a Craftsman-style house is directly left of the person who is an artist.
7. The person in a Mediterranean-style villa is somewhere to the right of the person who is a doctor.
8. The dog owner is not in the first house.
9. The person in a Mediterranean-style villa is somewhere to the left of the cat lover.
10. The person in a ranch-style home is in the sixth house.
11. The person in a Mediterranean-style villa is Alice.
12. The fish enthusiast is the person in a ranch-style home.
13. The bird keeper is the person in a modern-style house.
14. The person living in a colonial-style house is Peter.
15. The person who is a lawyer is Arnold.
 What is the value of attribute Animal for the person whose attribute HouseStyle is ranch? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Peter`
 - The mothers' names in different houses are unique: `Holly`, `Kailyn`, `Aniya`, `Janelle`
 - Each person has a unique birthday month: `april`, `feb`, `sept`, `jan`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `victorian`, `craftsman`

## Clues:
1. Peter is the person whose birthday is in February.
2. The person whose birthday is in February is not in the fourth house.
3. The person whose birthday is in January is somewhere to the right of the person in a ranch-style home.
4. The person whose birthday is in January is directly left of Eric.
5. The person in a ranch-style home is the person whose birthday is in April.
6. The person whose mother's name is Janelle is the person whose birthday is in September.
7. The person in a ranch-style home and The person whose mother's name is Kailyn are next to each other.
8. Arnold is The person whose mother's name is Holly.
9. The person whose mother's name is Kailyn is directly left of the person residing in a Victorian house.
10. The person whose birthday is in January is the person in a Craftsman-style house.
 What is the value of attribute Mother for the person whose attribute Birthday is jan? Please reason step by step, and put your final answer within \boxed{}

Answer: Kailyn


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Bob`, `Alice`, `Carol`
 - The people keep unique animals: `rabbit`, `dog`, `fish`, `bird`, `cat`, `horse`
 - Each person has a unique hobby: `cooking`, `knitting`, `painting`, `woodworking`, `photography`, `gardening`
 - People have unique favorite sports: `basketball`, `soccer`, `swimming`, `tennis`, `baseball`, `volleyball`
 - Each person prefers a unique type of vacation: `cruise`, `cultural`, `city`, `camping`, `mountain`, `beach`
 - Each person has a unique favorite drink: `water`, `root beer`, `boba tea`, `tea`, `milk`, `coffee`

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

Answer: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Peter`, `Bob`, `Eric`, `Arnold`, `Alice`
 - Each person has a unique birthday month: `jan`, `may`, `mar`, `feb`, `april`, `sept`
 - Each person has a unique type of pet: `rabbit`, `bird`, `cat`, `dog`, `fish`, `hamster`
 - People have unique favorite music genres: `hip hop`, `rock`, `pop`, `country`, `classical`, `jazz`
 - People have unique favorite book genres: `fantasy`, `romance`, `historical fiction`, `mystery`, `biography`, `science fiction`

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

Answer: sept


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`
 - Each person has a unique birthday month: `sept`, `feb`, `april`, `jan`
 - The mothers' names in different houses are unique: `Aniya`, `Kailyn`, `Holly`, `Janelle`
 - The people keep unique animals: `horse`, `bird`, `cat`, `fish`
 - Each person has a favorite color: `green`, `white`, `yellow`, `red`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `soccer`

## Clues:
1. The bird keeper is The person whose mother's name is Kailyn.
2. The fish enthusiast is not in the fourth house.
3. The person who loves white is Arnold.
4. The person who keeps horses is Peter.
5. The person who keeps horses is not in the first house.
6. The person whose mother's name is Kailyn is Arnold.
7. The person whose birthday is in January is in the first house.
8. The person whose birthday is in April is somewhere to the left of the person whose birthday is in September.
9. Alice is the person whose birthday is in September.
10. The person whose mother's name is Aniya and the person whose birthday is in September are next to each other.
11. Peter is directly left of the person whose favorite color is red.
12. There is one house between the person whose birthday is in April and the person who loves soccer.
13. The fish enthusiast is the person whose birthday is in September.
14. The person who loves swimming is The person whose mother's name is Holly.
15. The person whose favorite color is green is the person who loves swimming.
16. The person who loves yellow is the person who loves basketball.
 What is the value of attribute Birthday for the person whose attribute Color is red? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`
 - People have unique favorite music genres: `pop`, `rock`, `classical`
 - The people keep unique animals: `cat`, `horse`, `bird`
 - Each mother is accompanied by their child: `Meredith`, `Bella`, `Fred`

## Clues:
1. The bird keeper and the person's child is named Meredith are next to each other.
2. The person who loves classical music is Arnold.
3. Eric is the person's child is named Fred.
4. The person who keeps horses is somewhere to the left of the person's child is named Bella.
5. The person's child is named Meredith is directly left of Arnold.
6. The bird keeper is not in the third house.
7. Arnold is not in the second house.
8. Eric is the person who loves rock music.
 What is the value of attribute House for the person whose attribute Animal is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`
 - Each person prefers a unique type of vacation: `beach`, `city`, `mountain`
 - People have unique favorite sports: `soccer`, `basketball`, `tennis`
 - Each person has a unique birthday month: `april`, `jan`, `sept`

## Clues:
1. The person whose birthday is in April is not in the third house.
2. There is one house between the person who enjoys mountain retreats and the person who loves beach vacations.
3. The person who loves soccer is Peter.
4. The person who loves tennis is the person who prefers city breaks.
5. Eric is directly left of the person who loves tennis.
6. The person who loves beach vacations is the person who loves soccer.
7. The person who prefers city breaks is the person whose birthday is in January.
 What is the value of attribute House for the person whose attribute Birthday is april? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`
 - People have unique favorite book genres: `science fiction`, `mystery`, `romance`
 - Each mother is accompanied by their child: `Bella`, `Fred`, `Meredith`
 - People have unique hair colors: `black`, `blonde`, `brown`
 - Each person has a unique level of education: `high school`, `associate`, `bachelor`
 - They all have a unique favorite flower: `daffodils`, `lilies`, `carnations`

## Clues:
1. The person who loves the boquet of lilies is directly left of the person who loves romance books.
2. Peter is the person with a high school diploma.
3. The person who loves the boquet of lilies is the person who has brown hair.
4. The person's child is named Bella is Arnold.
5. The person who loves science fiction books is the person's child is named Fred.
6. The person's child is named Fred is the person who loves the boquet of lilies.
7. The person who has black hair is the person with an associate's degree.
8. The person's child is named Meredith is not in the first house.
9. The person with a high school diploma is the person who loves a carnations arrangement.
10. Peter is not in the third house.
 What is the value of attribute Name for the person whose attribute Flower is carnations? 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`, `Arnold`, `Eric`
 - Each person has a unique hobby: `cooking`, `gardening`, `painting`, `photography`
 - People own unique car models: `honda civic`, `toyota camry`, `tesla model 3`, `ford f150`
 - Everyone has something unique for lunch: `grilled cheese`, `pizza`, `spaghetti`, `stew`
 - Each person has an occupation: `doctor`, `artist`, `engineer`, `teacher`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`

## Clues:
1. The person who loves the stew is Eric.
2. The person who is a teacher is directly left of Peter.
3. The person who loves a carnations arrangement is the person who is an artist.
4. The person who paints as a hobby is the person who loves the spaghetti eater.
5. There is one house between the person who is an artist and the photography enthusiast.
6. The person who enjoys gardening is in the first house.
7. The person who loves the spaghetti eater is somewhere to the right of the person who owns a Honda Civic.
8. The person who is a pizza lover is directly left of the photography enthusiast.
9. The person who loves a carnations arrangement is Alice.
10. The person who owns a Tesla Model 3 is directly left of Eric.
11. The person who is an engineer is the person who loves a bouquet of daffodils.
12. The person who loves the boquet of lilies is somewhere to the left of the person who loves the stew.
13. There is one house between the person who enjoys gardening and the person who owns a Toyota Camry.
14. The person who is a teacher is in the third house.
 What is the value of attribute Flower for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: roses


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`, `Arnold`, `Eric`, `Alice`, `Carol`
 - People have unique hair colors: `red`, `gray`, `auburn`, `blonde`, `brown`, `black`
 - People have unique favorite book genres: `historical fiction`, `biography`, `romance`, `fantasy`, `mystery`, `science fiction`
 - People have unique heights: `very short`, `very tall`, `tall`, `short`, `super tall`, `average`
 - Each person has a unique level of education: `associate`, `doctorate`, `high school`, `trade school`, `bachelor`, `master`
 - Each person has a unique favorite drink: `boba tea`, `milk`, `water`, `root beer`, `coffee`, `tea`

## Clues:
1. The person with a doctorate and Bob are next to each other.
2. There is one house between Carol and the person who has black hair.
3. Peter is the person with a doctorate.
4. The person who loves science fiction books is Eric.
5. The one who only drinks water is the person who has an average height.
6. Alice is the person who loves mystery books.
7. The person who loves romance books is Bob.
8. The person who has brown hair is in the second house.
9. The person who has an average height is in the fourth house.
10. The boba tea drinker is the person who is very tall.
11. The person with a high school diploma is Bob.
12. The person who is short is somewhere to the left of the person who is super tall.
13. The person who loves historical fiction books and the person with a bachelor's degree are next to each other.
14. The person who attended trade school is the person who has red hair.
15. Carol is the person who has auburn hair.
16. The person who is very short is somewhere to the right of the person who has gray hair.
17. There are two houses between Eric and the person who loves fantasy books.
18. There are two houses between the tea drinker and the person who is very short.
19. Eric is somewhere to the left of the person with a bachelor's degree.
20. The root beer lover is Alice.
21. The person with an associate's degree is Arnold.
22. The person who is tall is not in the second house.
23. The person who loves science fiction books is the boba tea drinker.
24. The person who has black hair is not in the fourth house.
25. The person who is short is the coffee drinker.
26. The coffee drinker and the person with a high school diploma are next to each other.
27. The coffee drinker is not in the second house.
 What is the value of attribute Education for the person whose attribute HairColor is brown? Please reason step by step, and put your final answer within \boxed{}

Answer: high school


There are 2 houses, numbered 1 to 2 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`
 - Each person lives in a unique style of house: `victorian`, `colonial`
 - People have unique heights: `very short`, `short`
 - Each person has a unique level of education: `associate`, `high school`

## Clues:
1. The person who is short is directly left of Eric.
2. The person residing in a Victorian house is in the first house.
3. The person who is short is the person with an associate's degree.
 What is the value of attribute Height for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: very short


Let \( ABC \) be an acute triangle and \( O \) be its circumcenter. Let \( D \) be the midpoint of \( [AB] \). The circumcircle of \( \triangle ADO \) meets \( [AC] \) at \( A \) and \( E \). If \( |AE|=7 \), \( |DE|=8 \), and \( m(\widehat{AOD}) = 45^\circ \), what is the area of \( \triangle ABC \) in the form \( x \sqrt{2} \)? Find the value of \( x \).

Answer: 56


Let $(1+\sqrt{2})^{2012}=a+b\sqrt{2}$, where $a$ and $b$ are integers. Find the greatest common divisor of $b$ and $81$.

Answer: 3


Let \(M\) be the intersection of the diagonals \(AC\) and \(BD\) of cyclic quadrilateral \(ABCD\). If \(|AB|=5\), \(|CD|=3\), and \(m(\widehat{AMB}) = 60^\circ\), find the circumradius of the quadrilateral. If the circumradius is expressed in the form \(\frac{a\sqrt{b}}{c}\), where \(a\), \(b\), and \(c\) are integers, what is the value of \(a + b + c\)?

Answer: 13


Two obvious approximations to the length of the perimeter of the ellipse with semi-axes $a$ and $b$ are $\pi(a+b)$ and $2 \pi(a b)^{1/2}$. Which one comes nearer the truth when the ratio $b / a$ is very close to 1? Write your answer in the form of $\pi(x+y)$, where $x$ and $y$ are integers. Find the value of $x + y$.

Answer: 2


The solution of the equation \(7^{x+7} = 8^x\) can be expressed in the form \(x = \log_b 7^7\). The original answer is in the format \(\frac{k}{m}\). Please find the value of \(k + m\).

Answer: 15


Let \(M\) be the intersection of diagonals of the convex quadrilateral \(ABCD\), where \(m(\widehat{AMB})=60^\circ\). Let the points \(O_1\), \(O_2\), \(O_3\), \(O_4\) be the circumcenters of the triangles \(ABM\), \(BCM\), \(CDM\), \(DAM\), respectively. The original answer is in the form \(\frac{k}{m}\), where k and m are integers. Please give the value of k + m.

Answer: 5


The Hawks scored a certain number of points, and the total points scored by both teams together is 82. If the difference between the points scored by the Eagles and the Hawks is 18, and the spectator claimed that the Hawks scored 40 points, then determine the actual number of points the Hawks scored.

Answer: 32


Consider the set of all four-digit rising numbers using the digits 1 through 7. Find the digit that the 35th number in the list from smallest to largest does not contain.

Answer: 3


In an isosceles triangle, one of the angles measures $60^\circ$. Determine the sum of the three possible values of another angle $y^\circ$ in the triangle.

Answer: 180^\circ


Consider a modified finite sequence of four-digit integers where the tens, hundreds, and units digits of each term are, respectively, the thousands, hundreds, and tens digits of the next term, and the tens, hundreds, and units digits of the last term are, respectively, the thousands, hundreds, and tens digits of the first term. Let S be the sum of all terms in the sequence. Determine the largest prime factor that always divides S.

Answer: 101


Jonas sets his watch correctly at 8:00 AM and notices that his watch reads 9:48 AM at the actual time of 10:00 AM. Assuming his watch loses time at a constant rate, calculate the actual time when his watch will first read 5:00 PM.

Answer: 6:00 PM


Square $ABCD$ has its vertex $A$ at the origin of the coordinate system, and side length $AB = 2$. Vertex $E$ of isosceles triangle $\triangle ABE$, where $AE=BE$, is inside the square. A circle is inscribed in $\triangle ABE$ tangent to side $AB$ at point $G$, exactly at the midpoint of $AB$. Find the area of $\triangle ABF$, where $F$ is the point of intersection of diagonal $BD$ of the square and line segment $AE$.

Answer: 1


Jack drove 150 miles in 2.5 hours. His average speed during the first hour was 50 mph. After a 15-minute stop, he resumed travel for another hour at an average speed of 55 mph. Calculate his average speed, in mph, during the last 30 minutes.

Answer: 90


The interior of a quadrilateral is bounded by the graphs of $(x+by)^2 = 9b^2$ and $(bx-y)^2 = 4b^2$, where $b$ is a positive real number. Determine the area of this region in terms of $b$, valid for all $b > 0$.

Answer: \frac{24b^2}{1 + b^2}


In our number system, the base is ten. If the base were changed to seven, count the twenty-fifth number in the new base.

Answer: 34_7


Twenty-five percent of the audience listened to the entire 90-minute talk, and fifteen percent did not pay attention at all. Of the remainder, 40% caught half of the talk, and the rest heard only one fourth of it. Calculate the average time in minutes the talk was heard by the audience members.

Answer: 41.4


Calculate the sum $E(1)+E(2)+E(3)+\cdots+E(500)$, where $E(n)$ denotes the sum of the even digits of $n$. For example, $E(5681) = 6 + 8 = 14$.
A) 2000
B) 2200
C) 2400
D) 2500
E) 2600

Answer: 2600


Given circle $O$, point $C$ is on the opposite side of diameter $\overline{AB}$ from point $A$, and point $D$ is on the same side as point $A$. Given $\angle AOC = 40^{\circ}$, and $\angle DOB = 60^{\circ}$, calculate the ratio of the area of the smaller sector $COD$ to the area of the circle.

Answer: \frac{4}{9}


7. (4 points) With the number written on the board, one of the following operations is allowed:
1) If there is a digit in the original number that is not equal to 9 and has two neighboring digits greater than 0, you can increase this digit by 1, and decrease the neighboring digits by 1.
2) Subtract 1 from any non-zero digit except the last one, and add 3 to the next digit.
3) Decrease any sufficiently large digit by 7.

If, as a result of any of these operations, there are zeros in the number on one or more of the leading positions, they are automatically discarded.

Initially, the number consisting of one hundred nines was written on the board. In the end, a single-digit number remained. Which one exactly?

Answer: 3


Given the figure, $\odot O$ is the circumcircle of $\triangle A B C$. The circle $\odot J$ is inscribed in $\odot O$ and tangential to $A B$ and $A C$ at points $D$ and $E$, respectively. The line segment $F G$ is tangent to $\odot O$ at point $A$ and satisfies $A F = A G = A D$. The circumcircle of $\triangle A F B$ intersects $\odot J$ at point $S$. Prove that the circumcircle of $\triangle A S G$ is tangent to $\odot J$.

Answer: \text{The circumcircle of } \triangle ASG \text{ is tangent to } \odot J


$\underline{\text { Tolkpy A.K. }}$

A circle is divided into seven arcs such that the sum of any two adjacent arcs does not exceed $103^{\circ}$.

Name the largest number $A$ such that in any such division, each of the seven arcs contains at least $A^{\circ}$.

Answer: 51


As shown in Figure 2, given a square $ABCD$, extend $BC$ and $DC$ to $M$ and $N$ respectively, such that $S_{\triangle QMN} = S_{\text{square } ABCD}$. Determine the degree measure of $\angle MAN$.

Answer: 45


Choose positive integers \( b_{1}, b_{2}, \ldots \) satisfying
\[
1=\frac{b_{1}}{1^{2}}>\frac{b_{2}}{2^{2}}>\frac{b_{3}}{3^{2}}>\frac{b_{4}}{4^{2}}>\cdots
\]
and let \( r \) denote the largest real number satisfying \( \frac{b_{n}}{n^{2}} \geq r \) for all positive integers \( n \). What are the possible values of \( r \) across all possible choices of the sequence \( \left( b_{n} \right) \)?

Answer: [0, \frac{1}{2}]


II. (40 points) Find all positive integers $m, n$ such that $\frac{m^{3}+n^{3}-m^{2} n^{2}}{(m+n)^{2}}$ is a non-negative integer.

Answer: (2, 2)


Kazitsyna T.V.

In triangle $ABC$, points $X$ and $Y$ are taken on sides $AC$ and $BC$ such that $\angle ABX = \angle YAC$, $\angle AYB = \angle BXC$, and $XC = YB$. Find the angles of triangle $ABC$.

Answer: 60^\circ


15. Given the parabola $y=a x^{2}$ passes through the point $P(-1,1)$, a line $l$ with a positive slope is drawn through the point $Q\left(-\frac{1}{2}, 0\right)$ intersecting the parabola at points $M, N$ (point $M$ is between $Q$ and $N$). A line parallel to the $x$-axis is drawn through point $M$, intersecting $O P$ at $A$ and $O N$ at $B$. The areas of $\triangle P M A$ and $\triangle O A B$ are denoted as $S_{1}$ and $S_{2}$, respectively. Compare $S_{1}$ with $3 S_{2}$, and explain the reasoning.

Answer: S_1 > 3S_2


Define $ n!!$ to be $ n(n\minus{}2)(n\minus{}4)\ldots3\cdot1$ for $ n$ odd and $ n(n\minus{}2)(n\minus{}4)\ldots4\cdot2$ for $ n$ even. When $ \displaystyle \sum_{i\equal{}1}^{2009} \frac{(2i\minus{}1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $ 2^ab$ with $ b$ odd. Find $ \displaystyle \frac{ab}{10}$.

Answer: 401


# Task 2. (3 points)

In a certain company, the $20 \%$ most useful employees perform $80 \%$ of the work. What is the smallest percentage of work that the $40 \%$ most useful employees can perform?

We will consider an employee more useful if they perform more work.

Answer: 85


41. $y=\lg (x-2)$.

The above text is translated into English as follows, retaining the original text's line breaks and format:

41. $y=\lg (x-2)$.

Answer: (2, \infty)


5. [5 points] Given the numbers $\log _{\sqrt{29-x}}\left(\frac{x}{7}+7\right), \log _{(x+1)^{2}}(29-x), \log _{\sqrt{\frac{x}{7}+7}}(-x-1)$. For which $x$ are two of these numbers equal, and the third one greater than them by 1?

Answer: -7


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 MusicGenre for the person whose attribute HairColor is red? Please reason step by step, and put your final answer within \boxed{}

Answer: pop


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

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

Answer: mystery


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`, `Alice`, `Eric`, `Bob`, `Peter`
 - Each person lives in a unique style of house: `victorian`, `colonial`, `ranch`, `modern`, `craftsman`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `oneplus 9`, `samsung galaxy s21`, `huawei p50`
 - Everyone has a unique favorite cigar: `blue master`, `dunhill`, `pall mall`, `blends`, `prince`
 - People have unique favorite book genres: `mystery`, `science fiction`, `biography`, `romance`, `fantasy`

## Clues:
1. There is one house between the person residing in a Victorian house and the person who uses a Samsung Galaxy S21.
2. The person who smokes many unique blends is Eric.
3. The person who smokes many unique blends is the person who loves fantasy books.
4. Alice is somewhere to the right of the person who uses a Samsung Galaxy S21.
5. Bob is directly left of Eric.
6. The person who loves mystery books is directly left of the person who loves fantasy books.
7. Bob is somewhere to the left of the person who uses an iPhone 13.
8. The person who uses a Google Pixel 6 is somewhere to the right of Peter.
9. The person who loves romance books is directly left of the person in a ranch-style home.
10. The person living in a colonial-style house is in the fourth house.
11. The person who loves fantasy books is in the fourth house.
12. The person partial to Pall Mall is Peter.
13. The person in a modern-style house and the person who smokes Blue Master are next to each other.
14. The person who loves science fiction books is the Dunhill smoker.
15. Eric is the person who uses a Samsung Galaxy S21.
16. The person who uses a Huawei P50 and the person who loves fantasy books are next to each other.
 What is the value of attribute PhoneModel for the person whose attribute Cigar is prince? Please reason step by step, and put your final answer within \boxed{}

Answer: huawei p50


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - The people keep unique animals: `horse`, `bird`, `cat`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `desert`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`
 - People have unique favorite music genres: `pop`, `classical`, `rock`
 - Each person has a favorite color: `yellow`, `white`, `red`

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

Answer: red