3. Find all positive integers $n$, such that the numbers in the set $S=$ $\{1,2, \cdots, n\}$ are colored red or blue, and satisfy the following condition: the set $S \times S \times S$ contains exactly 2007 ordered triples $(x, y, z)$, such that
(1) $x, y, z$ are of the same color;
(2) $x+y+z$ is divisible by $n$.

Answer: 69


Given a parallelogram $ABCD$. Let $BP$ and $CQ$ be perpendiculars dropped from vertices $B$ and $C$ to diagonals $AC$ and $BD$ respectively (point $P$ lies on segment $AC$, and point $Q$ lies on segment $BD$). Find the ratio $\frac{10BD}{AC}$ if $\frac{AP}{AC} = \frac{4}{9}$ and $\frac{DQ}{DB} = \frac{28}{81}$.

Answer: 6


## Task A-4.5.

In a circle, a finite number of real numbers are arranged. Each number is colored red, white, or blue. Each red number is twice as small as the sum of its two neighboring numbers, each white number is equal to the sum of its two neighboring numbers, and each blue number is twice as large as the sum of its two neighboring numbers. Let $b$ be the sum of all white numbers, and $p$ be the sum of all blue numbers, where both sums are different from 0.

Determine the ratio $\frac{b}{p}$.

Answer: -\dfrac{3}{2}


7. A rectangle $A B C D$ with a diagonal of length $20 \text{ cm}$ is circumscribed by a circle. The side $\overline{C D}$ of the rectangle $A B C D$ is the base of an isosceles triangle whose third vertex $E$ is on the shorter arc determined by the chord $\overline{C D}$ of the circle circumscribed around the rectangle. What is the length of the side $\overline{A D}$ of the rectangle if the area of the rectangle $A B C D$ is equal to the area of triangle DCE?

SCHOOL/CITY COMPETITION IN MATHEMATICS21st January 2016.8th grade-elementary school

Problems worth 6 points:

Answer: 4


8,9

In a parallelogram, the ratio of the sides and the ratio of the diagonals are the same and equal to $\frac{1}{2}$. From the vertex of the obtuse angle A, a height $A E$ is dropped to the larger side $C D$. Find the ratio $\frac{D E}{C E}$.

Answer: \dfrac{3}{5}


04.1. 27 balls, labelled by numbers from 1 to 27, are in a red, blue or yellow bowl. Find the possible numbers of balls in the red bowl, if the averages of the labels in the red, blue, and yellow bowl are 15, 3 ja 18, respectively.

Answer: 11


14th Swedish 1974 Problem 3 Let a 1 = 1, a 2 = 2 a 1 , a 3 = 3 a 2 , a 4 = 4 a 3 , ... , a 9 = 9 a 8 . Find the last two digits of a 9 .

Answer: 21


5.69 11 theater troupes participate in a performance, with some of the troupes scheduled to perform each day, while the rest join the general audience. By the end of the performance, each troupe, apart from their own performance days, must have watched at least one performance by each of the other troupes. How many days at minimum must the performance be scheduled for?

Answer: 6


4. All possible 6-digit numbers, in each of which the digits occur in non-increasing order (from left to right, e.g., 877550) are written as a sequence in increasing order. Find the 2005 -th number in this sequence.

Answer: 864110


Let $A_1A_2 \dots A_{4000}$ be a regular $4000$-gon. Let $X$ be the foot of the altitude from $A_{1986}$ onto diagonal $A_{1000}A_{3000}$, and let $Y$ be the foot of the altitude from $A_{2014}$ onto $A_{2000}A_{4000}$. If $XY = 1$, what is the area of square $A_{500}A_{1500}A_{2500}A_{3500}$?

[i]Proposed by Evan Chen[/i]

Answer: 2


In a regular quadrilateral pyramid, two identical spheres of radius $r$ are placed, with their centers located on the axis of symmetry of the pyramid. One of the spheres touches all the lateral faces of the pyramid, while the second touches the base of the pyramid and the first sphere. Find the height of the pyramid for which the volume of the pyramid is the smallest.

Answer: 2r(3 + \sqrt{3})


Given \( n \geq 2, x_{1}, x_{2}, \cdots, x_{n} \) are all real numbers, and \( \sum_{i=1}^{n} x_{i}^{2} + \sum_{i=1}^{n-1} x_{i} x_{i+1} = 1 \). For each fixed \( k \) (\( k \in \mathbf{Z}, 1 \leq k \leq n \)), find the maximum value of \( \left|x_{k}\right| \).

Answer: \sqrt{ \dfrac{2k(n - k + 1)}{n + 1} }


2. It is known that $\frac{\cos x-\sin x}{\cos y}=\frac{\sqrt{2}}{3} \operatorname{tg} \frac{x+y}{2}$ and $\frac{\sin x+\cos x}{\sin y}=3 \sqrt{2} \operatorname{ctg} \frac{x+y}{2}$. Find all possible values of the expression $\operatorname{tg}(x+y)$, given that there are at least three.

Answer: 1


Wu starts out with exactly one coin. Wu flips every coin he has at once after each year. For each heads he flips, Wu receives a coin, and for every tails he flips, Wu loses a coin. He will keep repeating this process each year until he has $0$ coins, at which point he will stop. The probability that Wu will stop after exactly five years can be expressed as $\frac{a}{2^b}$, where $a, b$ are positive integers such that $a$ is odd. Find $a+b$.

Answer: 71622400


Aida made three cubes with positive integer side lengths $a$, $b$, and $c$. She decided to dismantle these cubes into unit cubes and tried to construct a larger cube with side length $a+b+c$. Unfortunately, she was 648 blocks short. How many possible ordered triples $(a, b, c)$ exist?

Answer: 7


I have 6 friends and during a vacation, I met them during several dinners. I found that:

- I dined with all 6 friends exactly on 1 day.
- I dined with every 5 of them on 2 days.
- I dined with every 4 of them on 3 days.
- I dined with every 3 of them on 4 days.
- I dined with every 2 of them on 5 days.

Additionally, every friend was present at 7 dinners and absent at 7 dinners. How many dinners did I have alone?

Answer: 1


Answer: 181


Among the coordinates $(x,y)$ where $1 \leq x,y \leq 101$, choose some points such that there do not exist $4$ points which form an isosceles trapezium with its base parallel to either the $x$ or $y$ axis (including rectangles). Find the maximum number of coordinate points that can be chosen.

Answer: 201


Let \( \{x\} \) denote the fractional part of \( x \). For example, \( \{5.5\} = 0.5 \). Find the smallest prime \( p \) such that the inequality \[ \sum_{n=1}^{p^2}\left\{\frac{n^p}{p^2}\right\} > 2016 \] holds.

Answer: 67


Define the minimum real $C$ such that for any real numbers $0 = a_0 < a_1 < \dots < a_{1000}$, the inequality $$\min_{0 \le k \le 1000} (a_k^2 + (1000-k)^2) \le C(a_1 + \dots + a_{1000})$$ holds. Find $\lfloor 100C \rfloor$.

Answer: 127


A regular polygon with $20$ vertices is given. Alice colors each vertex in one of two colors. Bob then draws a diagonal connecting two opposite vertices. Now Bob draws perpendicular segments to this diagonal, each segment having vertices of the same color as endpoints. He gets a fish from Alice for each such segment he draws. How many fish can Bob guarantee getting, no matter Alice's goodwill?

Answer: 4


How many six-letter words formed from the letters of "AMC" do not contain the substring "AMC"? For example, "AMAMMC" has this property, but "AAMCCC" does not.

Answer: 622


For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First, she plants six tree saplings at the end of Day $0$. On each day afterwards, Emilia attempts to use her magic to turn each sapling into a perfectly-shaped apple tree. For each sapling, she succeeds in turning it into a perfectly-shaped apple tree that day with a probability of $\frac{1}{2}$. Once a sapling is turned into a perfectly-shaped apple tree, it will remain so.

The expected number of days it will take Emilia to obtain six perfectly-shaped apple trees is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.

Answer: 789953


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`, `Peter`, `Arnold`
 - They all have a unique favorite flower: `lilies`, `carnations`, `roses`, `daffodils`
 - Each person has a favorite color: `white`, `green`, `yellow`, `red`
 - Each person has a unique level of education: `bachelor`, `associate`, `master`, `high school`
 - Each person has a unique hobby: `painting`, `photography`, `gardening`, `cooking`
 - Everyone has a unique favorite cigar: `dunhill`, `blue master`, `prince`, `pall mall`

## Clues:
1. The person partial to Pall Mall is the person with an associate's degree.
2. The person who loves white is the Prince smoker.
3. The person with a high school diploma is not in the fourth house.
4. The person with a master's degree is not in the second house.
5. Arnold is not in the fourth house.
6. The person whose favorite color is red is not in the third house.
7. The person whose favorite color is red is somewhere to the left of Eric.
8. The person who loves cooking is somewhere to the right of Peter.
9. The person who loves a bouquet of daffodils is somewhere to the left of the person who paints as a hobby.
10. The person who loves white is the person who loves the boquet of lilies.
11. Alice is the photography enthusiast.
12. Arnold is somewhere to the right of the person partial to Pall Mall.
13. The person who loves white is in the second house.
14. The person whose favorite color is green is directly left of the person who enjoys gardening.
15. The Dunhill smoker is the person with a high school diploma.
16. The person who loves the rose bouquet is somewhere to the right of the person whose favorite color is green.
 What is the value of attribute House for the person whose attribute Cigar is dunhill? 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`, `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 Birthday for the person whose attribute Children is Bella? Please reason step by step, and put your final answer within \boxed{}

Answer: jan


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

## Clues:
1. The Watermelon smoothie lover is the person who is very short.
2. The person who likes Cherry smoothies is the person who loves romance books.
3. The Desert smoothie lover is the person who enjoys gardening.
4. The Watermelon smoothie lover is somewhere to the right of the person with a high school diploma.
5. The person with an associate's degree is directly left of the person with a high school diploma.
6. The photography enthusiast is the person with an associate's degree.
7. The person who likes Cherry smoothies is Arnold.
8. The person who loves science fiction books is somewhere to the right of the person who enjoys gardening.
9. The person who is short is not in the first house.
10. The Watermelon smoothie lover is somewhere to the right of Eric.
 What is the value of attribute Name for the person whose attribute Smoothie is watermelon? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Bob`, `Alice`, `Arnold`, `Carol`, `Eric`
 - The people are of nationalities: `swede`, `chinese`, `norwegian`, `german`, `brit`, `dane`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`, `chevrolet silverado`, `honda civic`, `bmw 3 series`
 - Each person has a unique favorite drink: `root beer`, `milk`, `tea`, `water`, `coffee`, `boba tea`

## Clues:
1. The boba tea drinker is the person who owns a Ford F-150.
2. The British person is directly left of Eric.
3. Alice is the person who owns a Honda Civic.
4. The tea drinker is not in the first house.
5. Eric is the Chinese.
6. The tea drinker is the person who owns a BMW 3 Series.
7. The German is in the second house.
8. The person who owns a Chevrolet Silverado is somewhere to the left of the person who likes milk.
9. Bob is in the fifth house.
10. The person who owns a Toyota Camry is directly left of the one who only drinks water.
11. The root beer lover is directly left of the Norwegian.
12. The Dane is somewhere to the right of the person who owns a BMW 3 Series.
13. Arnold is in the second house.
14. The person who owns a Honda Civic is directly left of the person who owns a Ford F-150.
15. Peter is not in the third house.
16. The person who owns a Toyota Camry is in the third house.
 What is the value of attribute CarModel for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: tesla model 3


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Arnold`, `Eric`
 - People own unique car models: `toyota camry`, `tesla model 3`, `honda civic`, `ford f150`
 - People have unique heights: `average`, `very short`, `short`, `tall`
 - The mothers' names in different houses are unique: `Aniya`, `Kailyn`, `Holly`, `Janelle`
 - Each person has a unique level of education: `bachelor`, `high school`, `master`, `associate`
 - The people keep unique animals: `cat`, `fish`, `horse`, `bird`

## Clues:
1. The bird keeper is The person whose mother's name is Kailyn.
2. The person who owns a Ford F-150 is the person who is tall.
3. The person whose mother's name is Kailyn is not in the third house.
4. The person with a master's degree is the fish enthusiast.
5. The person who is very short is somewhere to the left of the cat lover.
6. The fish enthusiast is the person who is short.
7. The person whose mother's name is Janelle is directly left of the person who owns a Toyota Camry.
8. The person whose mother's name is Kailyn is not in the first house.
9. The person who is short is somewhere to the right of Alice.
10. The person with a high school diploma is the person who has an average height.
11. There is one house between the person who is very short and The person whose mother's name is Holly.
12. The person with a master's degree is somewhere to the right of the person with a bachelor's degree.
13. The person with a master's degree and Peter are next to each other.
14. The person who owns a Tesla Model 3 is Eric.
15. The person who has an average height is somewhere to the right of Arnold.
16. The person who keeps horses is directly left of the person with a master's degree.
 What is the value of attribute Height for the person whose attribute CarModel is toyota camry? Please reason step by step, and put your final answer within \boxed{}

Answer: short


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

## Clues:
1. Arnold is the person who loves baseball.
2. The Swedish person is somewhere to the left of Arnold.
3. There are two houses between the person who loves basketball and the person who has black hair.
4. The Norwegian is in the fourth house.
5. Peter is the person who has black hair.
6. The person who has red hair is in the second house.
7. Bob is the German.
8. The person who has blonde hair is not in the third house.
9. The person who loves swimming is the person who has red hair.
10. The German is not in the first house.
11. The person who loves basketball is directly left of the Dane.
12. The person who loves soccer is directly left of Arnold.
13. Alice is the person who has gray hair.
 What is the value of attribute FavoriteSport for the person whose attribute HairColor is red? Please reason step by step, and put your final answer within \boxed{}

Answer: swimming


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`, `Bob`, `Carol`, `Alice`
 - Each person prefers a unique type of vacation: `cruise`, `beach`, `cultural`, `mountain`, `city`, `camping`
 - People have unique hair colors: `gray`, `black`, `brown`, `blonde`, `auburn`, `red`
 - People use unique phone models: `oneplus 9`, `huawei p50`, `xiaomi mi 11`, `iphone 13`, `google pixel 6`, `samsung galaxy s21`
 - Each person has a unique favorite drink: `root beer`, `boba tea`, `milk`, `coffee`, `tea`, `water`

## Clues:
1. There is one house between the boba tea drinker and Bob.
2. There are two houses between the person who uses a Google Pixel 6 and the person who enjoys camping trips.
3. The person who goes on cultural tours is Carol.
4. The person who enjoys camping trips is Eric.
5. The person who uses a Google Pixel 6 is not in the fifth house.
6. The person who uses a OnePlus 9 and the person who goes on cultural tours are next to each other.
7. The person who has gray hair and the person who goes on cultural tours are next to each other.
8. The person who uses a Samsung Galaxy S21 is somewhere to the right of the person who has red hair.
9. The person who enjoys mountain retreats is somewhere to the right of the person who uses a Xiaomi Mi 11.
10. The person who uses a OnePlus 9 is Peter.
11. The person who likes going on cruises is the person who has brown hair.
12. The coffee drinker is the person who goes on cultural tours.
13. The person who likes milk is directly left of the person who has black hair.
14. The person who uses a Xiaomi Mi 11 is directly left of the person who uses a Samsung Galaxy S21.
15. The person who prefers city breaks is not in the sixth house.
16. The person who has blonde hair is Arnold.
17. The person who goes on cultural tours is not in the sixth house.
18. The one who only drinks water and the person who goes on cultural tours are next to each other.
19. The tea drinker is somewhere to the left of the person who enjoys mountain retreats.
20. The person who likes milk is the person who uses a Huawei P50.
21. The person who has blonde hair is directly left of the person who has gray hair.
22. Bob is not in the third house.
23. Peter is the person who prefers city breaks.
 What is the value of attribute Drink for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: tea


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Eric`, `Bob`, `Arnold`, `Carol`
 - Everyone has a unique favorite cigar: `pall mall`, `yellow monster`, `dunhill`, `blue master`, `prince`, `blends`
 - People have unique favorite music genres: `hip hop`, `jazz`, `country`, `pop`, `classical`, `rock`
 - Each person has a unique favorite drink: `water`, `milk`, `boba tea`, `tea`, `root beer`, `coffee`
 - The mothers' names in different houses are unique: `Kailyn`, `Penny`, `Janelle`, `Holly`, `Sarah`, `Aniya`
 - Everyone has something unique for lunch: `soup`, `pizza`, `spaghetti`, `stir fry`, `stew`, `grilled cheese`

## Clues:
1. Carol is directly left of the person who loves eating grilled cheese.
2. Eric is not in the second house.
3. The person whose mother's name is Holly is somewhere to the right of Carol.
4. The person who loves eating grilled cheese is somewhere to the right of the person who loves rock music.
5. Eric is directly left of Carol.
6. The person who loves pop music is not in the third house.
7. Eric is the person who loves country music.
8. The person who loves classical music is in the sixth house.
9. The coffee drinker is Bob.
10. The person who smokes many unique blends is Peter.
11. The person who loves the stew is not in the fifth house.
12. The root beer lover is directly left of The person whose mother's name is Janelle.
13. There are two houses between The person whose mother's name is Sarah and the person who smokes Yellow Monster.
14. Eric is the tea drinker.
15. The person partial to Pall Mall is somewhere to the right of the person who loves stir fry.
16. The person who loves the soup is Bob.
17. The person who loves hip-hop music is directly left of The person whose mother's name is Kailyn.
18. Arnold is somewhere to the right of The person whose mother's name is Kailyn.
19. The one who only drinks water is directly left of the person who smokes Blue Master.
20. The person who loves the spaghetti eater is somewhere to the left of the person who smokes many unique blends.
21. The person whose mother's name is Sarah is directly left of the person who loves jazz music.
22. The person who loves hip-hop music is directly left of the root beer lover.
23. The one who only drinks water is the person who loves the stew.
24. The Dunhill smoker is not in the second house.
25. The person who likes milk is The person whose mother's name is Janelle.
26. Eric is The person whose mother's name is Aniya.
 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: Janelle


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Alice`, `Peter`, `Eric`
 - People have unique favorite sports: `baseball`, `swimming`, `tennis`, `soccer`, `basketball`
 - People have unique heights: `tall`, `short`, `average`, `very tall`, `very short`
 - Each person prefers a unique type of vacation: `city`, `beach`, `mountain`, `cruise`, `camping`
 - Each person lives in a unique style of house: `modern`, `victorian`, `craftsman`, `colonial`, `ranch`

## Clues:
1. Arnold is the person who has an average height.
2. Arnold is not in the first house.
3. The person who loves soccer is the person who likes going on cruises.
4. The person who is very tall is the person who loves basketball.
5. The person who is very short is in the third house.
6. The person in a modern-style house is somewhere to the right of the person who loves soccer.
7. The person who enjoys camping trips is in the fourth house.
8. The person who is short is not in the fourth house.
9. The person living in a colonial-style house is the person who prefers city breaks.
10. The person who loves tennis is directly left of Eric.
11. Alice and the person in a ranch-style home are next to each other.
12. The person in a modern-style house is somewhere to the left of the person in a ranch-style home.
13. The person who enjoys mountain retreats and the person in a Craftsman-style house are next to each other.
14. The person in a Craftsman-style house is somewhere to the left of Peter.
15. The person who loves basketball is the person living in a colonial-style house.
16. The person who loves swimming is the person who enjoys mountain retreats.
17. The person in a Craftsman-style house is the person who loves tennis.
18. Eric is not in the third house.
 What is the value of attribute House for the person whose attribute Height is tall? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


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

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

Answer: victorian


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Bob`, `Alice`, `Eric`, `Carol`
 - The mothers' names in different houses are unique: `Holly`, `Aniya`, `Kailyn`, `Janelle`, `Penny`, `Sarah`
 - Everyone has something unique for lunch: `stir fry`, `stew`, `spaghetti`, `soup`, `pizza`, `grilled cheese`
 - The people keep unique animals: `dog`, `horse`, `cat`, `fish`, `rabbit`, `bird`
 - Everyone has a favorite smoothie: `watermelon`, `desert`, `lime`, `cherry`, `blueberry`, `dragonfruit`
 - Each person lives in a unique style of house: `mediterranean`, `craftsman`, `victorian`, `colonial`, `modern`, `ranch`

## Clues:
1. The person whose mother's name is Penny is the person residing in a Victorian house.
2. The person in a modern-style house is The person whose mother's name is Kailyn.
3. The person in a modern-style house is somewhere to the right of the person residing in a Victorian house.
4. The person who drinks Blueberry smoothies is not in the sixth house.
5. The person whose mother's name is Sarah is the person who drinks Blueberry smoothies.
6. The person who likes Cherry smoothies is not in the third house.
7. The person who likes Cherry smoothies is Eric.
8. The person whose mother's name is Janelle is not in the third house.
9. The person who loves the spaghetti eater is in the third house.
10. The fish enthusiast is The person whose mother's name is Holly.
11. The person who keeps horses is in the first house.
12. The person in a Mediterranean-style villa is the person who loves the stew.
13. The bird keeper is in the third house.
14. The person living in a colonial-style house is the person who drinks Blueberry smoothies.
15. Carol is the person who keeps horses.
16. There are two houses between the Desert smoothie lover and Peter.
17. The person who loves eating grilled cheese is the Dragonfruit smoothie lover.
18. The person who is a pizza lover is somewhere to the right of the Dragonfruit smoothie lover.
19. The person whose mother's name is Janelle is somewhere to the left of Arnold.
20. Carol is directly left of The person whose mother's name is Aniya.
21. The person in a ranch-style home is The person whose mother's name is Aniya.
22. Arnold is somewhere to the right of the person in a Craftsman-style house.
23. The cat lover is Bob.
24. There is one house between the Watermelon smoothie lover and the person who is a pizza lover.
25. There are two houses between the cat lover and the person who loves stir fry.
26. Arnold is the rabbit owner.
27. Peter is The person whose mother's name is Janelle.
 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: 5


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Peter`, `Bob`
 - Each person has a unique type of pet: `fish`, `bird`, `cat`, `dog`, `hamster`
 - People use unique phone models: `samsung galaxy s21`, `iphone 13`, `google pixel 6`, `huawei p50`, `oneplus 9`
 - People have unique heights: `short`, `very tall`, `very short`, `tall`, `average`
 - Each mother is accompanied by their child: `Fred`, `Meredith`, `Samantha`, `Timothy`, `Bella`

## Clues:
1. The person with an aquarium of fish is Arnold.
2. The person who uses a Huawei P50 is not in the fifth house.
3. The person with an aquarium of fish is in the fourth house.
4. The person who uses a Samsung Galaxy S21 is in the second house.
5. The person's child is named Bella is the person who owns a dog.
6. Arnold is somewhere to the left of the person with a pet hamster.
7. The person who uses a Google Pixel 6 is directly left of the person's child is named Meredith.
8. The person's child is named Bella is directly left of the person who is short.
9. The person who keeps a pet bird is somewhere to the left of the person's child is named Bella.
10. The person who has an average height and the person who uses a OnePlus 9 are next to each other.
11. Alice is the person who has a cat.
12. Bob is somewhere to the right of the person who is very short.
13. The person who uses a Google Pixel 6 is not in the fourth house.
14. Eric is the person who uses an iPhone 13.
15. The person's child is named Fred is the person who keeps a pet bird.
16. The person who is very tall is in the first house.
17. There is one house between the person who uses an iPhone 13 and the person who is the mother of Timothy.
 What is the value of attribute House for the person whose attribute Height is average? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


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

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

Answer: very short


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Alice`, `Eric`, `Arnold`
 - The mothers' names in different houses are unique: `Penny`, `Kailyn`, `Aniya`, `Janelle`, `Holly`
 - Everyone has a unique favorite cigar: `blue master`, `pall mall`, `prince`, `dunhill`, `blends`
 - The people keep unique animals: `bird`, `cat`, `horse`, `fish`, `dog`
 - Everyone has something unique for lunch: `grilled cheese`, `stir fry`, `pizza`, `spaghetti`, `stew`

## Clues:
1. The person who loves the stew is somewhere to the left of the fish enthusiast.
2. The person who is a pizza lover is somewhere to the right of the Dunhill smoker.
3. Peter is the cat lover.
4. The dog owner is directly left of the person who loves the spaghetti eater.
5. The person who loves stir fry is The person whose mother's name is Aniya.
6. There are two houses between the fish enthusiast and Eric.
7. The person who smokes many unique blends is The person whose mother's name is Kailyn.
8. The fish enthusiast is the person who loves the spaghetti eater.
9. The fish enthusiast is in the second house.
10. The person whose mother's name is Kailyn is the person who loves the stew.
11. The person whose mother's name is Janelle is not in the fifth house.
12. The person whose mother's name is Holly is the person who smokes Blue Master.
13. Arnold is The person whose mother's name is Aniya.
14. The person who loves the spaghetti eater and the person partial to Pall Mall are next to each other.
15. Peter is not in the third house.
16. Arnold is the person who keeps horses.
17. The person whose mother's name is Holly is Bob.
 What is the value of attribute House for the person whose attribute Animal is horse? Please reason step by step, and put your final answer within \boxed{}

Answer: 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`, `Peter`, `Eric`, `Bob`, `Alice`
 - People use unique phone models: `google pixel 6`, `samsung galaxy s21`, `huawei p50`, `oneplus 9`, `iphone 13`
 - People have unique favorite book genres: `science fiction`, `romance`, `biography`, `mystery`, `fantasy`

## Clues:
1. Peter is the person who uses a Huawei P50.
2. The person who loves fantasy books is directly left of Alice.
3. Bob is in the second house.
4. There are two houses between the person who uses a OnePlus 9 and the person who loves biography books.
5. Arnold is the person who loves romance books.
6. The person who loves fantasy books is the person who uses a OnePlus 9.
7. The person who uses a Samsung Galaxy S21 is not in the fifth house.
8. The person who uses an iPhone 13 is the person who loves science fiction books.
9. Peter is directly left of the person who uses a OnePlus 9.
 What is the value of attribute House for the person whose attribute PhoneModel is google pixel 6? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - Each person has an occupation: `engineer`, `teacher`, `doctor`
 - Each person has a unique favorite drink: `water`, `milk`, `tea`
 - People use unique phone models: `google pixel 6`, `samsung galaxy s21`, `iphone 13`

## Clues:
1. The person who uses a Samsung Galaxy S21 is directly left of Arnold.
2. The person who uses a Google Pixel 6 is the one who only drinks water.
3. The person who likes milk is the person who is an engineer.
4. Eric is the person who likes milk.
5. The person who is a teacher is directly left of the tea drinker.
 What is the value of attribute House for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


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

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

Answer: grilled cheese


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Carol`, `Alice`, `Eric`, `Arnold`, `Peter`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `soup`, `grilled cheese`, `stew`, `stir fry`
 - Each person has a unique level of education: `associate`, `doctorate`, `trade school`, `high school`, `master`, `bachelor`
 - The people keep unique animals: `horse`, `fish`, `rabbit`, `cat`, `bird`, `dog`
 - Each person lives in a unique style of house: `ranch`, `victorian`, `modern`, `craftsman`, `mediterranean`, `colonial`

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

Answer: high school


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`, `Bob`, `Eric`, `Alice`
 - People have unique hair colors: `gray`, `brown`, `black`, `red`, `blonde`
 - Each person prefers a unique type of vacation: `city`, `mountain`, `camping`, `cruise`, `beach`
 - They all have a unique favorite flower: `roses`, `tulips`, `carnations`, `lilies`, `daffodils`
 - Everyone has something unique for lunch: `stew`, `stir fry`, `spaghetti`, `pizza`, `grilled cheese`
 - Each person has a unique hobby: `painting`, `photography`, `gardening`, `knitting`, `cooking`

## Clues:
1. The person who loves the stew is directly left of the person who loves the vase of tulips.
2. The person who enjoys camping trips is the person who loves eating grilled cheese.
3. The person who has black hair is somewhere to the left of Bob.
4. The photography enthusiast is the person who has gray hair.
5. The person who is a pizza lover is the person who likes going on cruises.
6. Alice is the person who prefers city breaks.
7. The person who loves beach vacations is directly left of the person who paints as a hobby.
8. The person who loves the spaghetti eater is Arnold.
9. Bob is in the fifth house.
10. The person who loves the boquet of lilies is in the third house.
11. The person who enjoys knitting is in the fourth house.
12. The person who has brown hair is directly left of the person who paints as a hobby.
13. There are two houses between the person who likes going on cruises and the person who enjoys knitting.
14. The person who has red hair is the person who loves the spaghetti eater.
15. The person who loves a carnations arrangement is not in the second house.
16. Eric is somewhere to the left of the person who enjoys gardening.
17. The photography enthusiast is the person who loves the stew.
18. Eric is the person who loves a bouquet of daffodils.
 What is the value of attribute Food for the person whose attribute Hobby is cooking? Please reason step by step, and put your final answer within \boxed{}

Answer: pizza


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

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

Answer: red


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: `Arnold`, `Eric`
 - Each person has a unique birthday month: `april`, `sept`
 - Each person has a favorite color: `red`, `yellow`
 - The people are of nationalities: `dane`, `brit`
 - People have unique hair colors: `black`, `brown`
 - Each mother is accompanied by their child: `Fred`, `Bella`

## Clues:
1. The person who loves yellow is the British person.
2. The person who has black hair is the person's child is named Fred.
3. The person who loves yellow is the person's child is named Bella.
4. The British person is in the second house.
5. Eric is the person who loves yellow.
6. The person whose birthday is in April is Eric.
 What is the value of attribute Children for the person whose attribute HairColor is black? Please reason step by step, and put your final answer within \boxed{}

Answer: Fred


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`, `Alice`, `Arnold`, `Eric`, `Carol`, `Bob`
 - Each person has a unique favorite drink: `water`, `milk`, `root beer`, `tea`, `coffee`, `boba tea`
 - Each person has an occupation: `artist`, `doctor`, `teacher`, `engineer`, `nurse`, `lawyer`

## Clues:
1. The person who is a lawyer is somewhere to the left of the one who only drinks water.
2. The boba tea drinker is Alice.
3. Arnold is directly left of the coffee drinker.
4. Peter and the person who is an artist are next to each other.
5. Arnold is the person who is a nurse.
6. Carol is not in the first house.
7. The person who is a teacher is somewhere to the right of the person who is a doctor.
8. The root beer lover is somewhere to the left of the tea drinker.
9. The person who is a lawyer is Carol.
10. Bob is the one who only drinks water.
11. Eric is not in the fourth house.
12. The tea drinker is not in the sixth house.
13. The person who is a teacher is the root beer lover.
14. The one who only drinks water is somewhere to the left of Alice.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Peter`, `Carol`, `Bob`, `Arnold`
 - Everyone has a favorite smoothie: `watermelon`, `blueberry`, `desert`, `cherry`, `dragonfruit`, `lime`
 - Everyone has something unique for lunch: `stew`, `pizza`, `grilled cheese`, `stir fry`, `soup`, `spaghetti`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `xiaomi mi 11`, `huawei p50`, `samsung galaxy s21`, `oneplus 9`
 - People own unique car models: `tesla model 3`, `honda civic`, `toyota camry`, `ford f150`, `chevrolet silverado`, `bmw 3 series`
 - Each person lives in a unique style of house: `craftsman`, `ranch`, `modern`, `victorian`, `mediterranean`, `colonial`

## Clues:
1. Bob is the person who uses a Xiaomi Mi 11.
2. The person who loves the soup is in the fourth house.
3. The Dragonfruit smoothie lover is somewhere to the left of the person in a ranch-style home.
4. There is one house between the person who owns a Chevrolet Silverado and the person residing in a Victorian house.
5. The person in a Mediterranean-style villa is the person who drinks Lime smoothies.
6. Eric is in the sixth house.
7. The Desert smoothie lover is the person who is a pizza lover.
8. The person living in a colonial-style house is the person who drinks Blueberry smoothies.
9. The Dragonfruit smoothie lover and the person who uses a Google Pixel 6 are next to each other.
10. The person who loves the soup is Peter.
11. Alice is somewhere to the right of the person who owns a BMW 3 Series.
12. The person who loves stir fry is the person in a ranch-style home.
13. The person who owns a Ford F-150 is the person living in a colonial-style house.
14. The person in a Craftsman-style house is somewhere to the right of the person in a modern-style house.
15. The person who loves the stew is directly left of the person in a ranch-style home.
16. The person who owns a Tesla Model 3 is directly left of the person who loves stir fry.
17. The person who loves eating grilled cheese is the person who owns a Honda Civic.
18. The person in a Mediterranean-style villa is the person who uses a Google Pixel 6.
19. The person in a Craftsman-style house is the Watermelon smoothie lover.
20. The person in a modern-style house is Carol.
21. The person who uses an iPhone 13 is in the first house.
22. The person who owns a Honda Civic is somewhere to the left of the person who owns a BMW 3 Series.
23. The person who uses a Xiaomi Mi 11 is the person who is a pizza lover.
24. The person who uses a Samsung Galaxy S21 is the person who loves stir fry.
25. The person residing in a Victorian house is directly left of the person who uses a OnePlus 9.
 What is the value of attribute PhoneModel for the person whose attribute HouseStyle is mediterranean? Please reason step by step, and put your final answer within \boxed{}

Answer: google pixel 6


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`
 - Everyone has a unique favorite cigar: `prince`, `pall mall`, `blue master`, `dunhill`
 - Each person has a unique favorite drink: `milk`, `coffee`, `water`, `tea`
 - People have unique hair colors: `blonde`, `black`, `red`, `brown`

## Clues:
1. The person who has brown hair is not in the fourth house.
2. The person who likes milk is the person who has brown hair.
3. The person partial to Pall Mall is somewhere to the right of the person who has black hair.
4. Peter is somewhere to the left of the Dunhill smoker.
5. The person who smokes Blue Master is directly left of Alice.
6. The tea drinker is Peter.
7. Eric is the person who has brown hair.
8. Eric is not in the third house.
9. Alice is the Dunhill smoker.
10. The person who has blonde hair is the Prince smoker.
11. The one who only drinks water is not in the fourth house.
 What is the value of attribute House for the person whose attribute Cigar is pall mall? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


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

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

Answer: rock


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 Food is pizza? 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: `Eric`, `Arnold`, `Peter`, `Alice`
 - People have unique favorite sports: `basketball`, `swimming`, `tennis`, `soccer`
 - People own unique car models: `tesla model 3`, `ford f150`, `honda civic`, `toyota camry`
 - Everyone has a unique favorite cigar: `blue master`, `pall mall`, `prince`, `dunhill`
 - Each person has an occupation: `engineer`, `doctor`, `artist`, `teacher`
 - Each person has a unique type of pet: `bird`, `dog`, `cat`, `fish`

## Clues:
1. The Dunhill smoker is in the second house.
2. The person who keeps a pet bird is the person who loves swimming.
3. The person who is an engineer is Arnold.
4. There are two houses between the person who is a teacher and the person partial to Pall Mall.
5. The person who is an artist is somewhere to the right of the Prince smoker.
6. The person who owns a Tesla Model 3 is not in the fourth house.
7. Eric is somewhere to the right of the person who owns a dog.
8. The person who loves tennis and the person who is an engineer are next to each other.
9. The person with an aquarium of fish is Alice.
10. Eric is the person who owns a Tesla Model 3.
11. The person who owns a Honda Civic is not in the first house.
12. Peter is the person who loves soccer.
13. The person who owns a Toyota Camry is the person who is a doctor.
14. The person who smokes Blue Master is the person who loves swimming.
15. Peter is the person who is a doctor.
 What is the value of attribute CarModel for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: toyota camry


13. A student participates in military training and must shoot 10 times. In the 6th, 7th, 8th, and 9th shots, he scored 9.0, 8.4, 8.1, and 9.3 points, respectively. The average score of his first 9 shots is higher than the average score of his first 5 shots. If he wants the average score of 10 shots to exceed 8.8 points, how many points does he need to score at least in the 10th shot? (The points scored in each shot are accurate to 0.1 points)

Answer: 9.9


Problem 6.2. A square with a side of 1 m is cut into three rectangles with equal perimeters. What can these perimeters be? List all possible options and explain why there are no others.

Answer: \dfrac{5}{2}


One, (20 points) In a certain competition, each player plays exactly one game against every other player. The winner of each game gets 1 point, the loser gets 0 points, and in the case of a draw, both get 0.5 points. After the competition, it is found that each player's score is exactly half from games played against the 10 lowest-scoring players (the 10 lowest-scoring players each have exactly half of their points from games played against each other). Find the number of participants in the competition.

Answer: 25


3. Let $M=\{1,2, \cdots, 2009\}$. If $n \in M$, such that $S_{n}=\frac{1}{n}\left(1^{3}+2^{3}+\cdots+n^{3}\right)$ is a perfect square, then the number of such $n$ is $\qquad$.

Answer: 44


3A. For the angles $\alpha, \beta$ and $\gamma$ in triangle $ABC$, the following equality holds:

$$
\frac{\sin ^{2} \beta+\sin ^{2} \gamma-\sin ^{2} \alpha}{\sin \beta \sin \gamma}=1
$$

Calculate the angle $\alpha$.

Answer: \dfrac{\pi}{3}


12. Given the parabola $y^{2}=2 p x(p>0)$, with its focus at $F$, a line passing through $F$ with an inclination angle of $\theta$ intersects the parabola at points $A$ and $B$. The maximum area of $\triangle A B O$ is $\qquad$ (where $O$ is the origin).

Answer: \dfrac{p^2}{2}


6. For the complex number $x$, the minimum value of the expression $u=|x|+|x-i|+|x-\sqrt{3}|$ is $\qquad$ .

Answer: \sqrt{7}


Alphonse and Beryl play a game involving $n$ safes. Each safe can be opened by a unique key and each key opens a unique safe. Beryl randomly shuffles the $n$ keys, and after placing one key inside each safe, she locks all of the safes with her master key. Alphonse then selects $m$ of the safes (where $m < n$), and Beryl uses her master key to open just the safes that Alphonse selected. Alphonse collects all of the keys inside these $m$ safes and tries to use these keys to open up the other $n - m$ safes. If he can open a safe with one of the $m$ keys, he can then use the key in that safe to try to open any of the remaining safes, repeating the process until Alphonse successfully opens all of the safes, or cannot open any more. Let $P_m(n)$ be the probability that Alphonse can eventually open all $n$ safes starting from his initial selection of $m$ keys.

(a) Show that $P_2(3) = \frac23$.

(b) Prove that $P_1(n) = \frac1n$.

(c) For all integers $n \geq 2$, prove that $$P_2(n) = \frac2n \cdot P_1(n-1) + \frac{n-2}{n} \cdot P_2(n-1).$$

(d) Determine a formula for $P_2 (n)$.

Answer: \dfrac{2}{3}


1. Mrs. Pekmezac loves fruit. In her garden, she harvested $5 \frac{3}{4} \mathrm{~kg}$ of blueberries, $4 \frac{1}{2} \mathrm{~kg}$ of cherries, $2 \frac{19}{20} \mathrm{~kg}$ of raspberries, and $1 \frac{4}{5} \mathrm{~kg}$ of blackberries. She decided to store the fruit in the refrigerator, in bags of 75 dag. Fruit of different types can be stored in the same bag. How many bags does she need to prepare at a minimum?

Answer: 20


3 [ Completing the square. Sums of squares ]

 Find all real solutions of the equation with 4 unknowns: $x^{2}+y^{2}+z^{2}+t^{2}=x(y+z+t)$.

Answer: (0, 0, 0, 0)


Given a triangle $A B C$ such that the foot $P$ of the perpendicular from point $C$ to line $A B$ lies inside segment $A B$. From point $P$, perpendiculars $p, q$ are drawn to lines $A C$ and $B C$ (in that order). Let $S$ be the intersection of line $B C$ and line $q$, and $T$ be the intersection of line $A C$ and line $p$. Calculate the measure of angle $A C B$, given that $|\nless A P T|+|\Varangle B P S|=20^{\circ}$.

Answer: 20^\circ


Let \(\Omega_{1}\) be a circle tangent internally to another circle \(\Omega_{2}\) at point \(A\). Let \(P\) be a point on \(\Omega_{2}\). The first tangent to \(\Omega_{1}\) passing through \(P\) intersects \(\Omega_{1}\) at \(X\) and \(\Omega_{2}\) at \(Q\). The second tangent intersects \(\Omega_{1}\) at \(Y\) and \(\Omega_{2}\) at \(R\). Show that \(\widehat{Q A R} = 2 \widehat{X A Y}\).

Answer: \widehat{QAR} = 2 \widehat{XAY}


Problem 4. Each cell of a $3 \times 3$ table is painted in one of three colors such that cells sharing a side have different colors, and not all three colors need to be used. How many such colorings exist?

Answer: 246


Example 10 We notice that $6!=8 \cdot 9 \cdot 10$. Try to find the largest positive integer $n$ such that $n!$ can be expressed as the product of $n-3$ consecutive natural numbers.

Answer: 23


2. Given $I$ as the incenter of $\triangle A B C$, $M$ as the midpoint of side $A C$, $A I \perp I W$, and $C I$ intersects the circumcircle of $\triangle A B C$ at a second point $W$. Find $\frac{C I}{I W}$.

Answer: 2


4. On a magnetic board, there are the first 16 natural numbers not divisible by 4. Elena intends to fill a rectangle with 3 rows and 5 columns with distinct numbers from the board such that the 3 row sums are equal and the 5 column sums are equal. She starts with the largest number on the board and succeeds in her goal.

a) Which number on the board was not used?

b) Provide an example of the arrangement of numbers that satisfies the conditions in the statement!

Gabriel Daniilescu, Brăila

Answer: 6


Example 6 Arrange the terms of the arithmetic sequence $2,6,10,14, \cdots, 2006$ tightly together to form a “large number”: $A=261014 \cdots 2006$. Find the remainder when $A$ is divided by 9.

Answer: 8


41st Putnam 1980 Problem B3 Define a n by a 0 = α, a n+1 = 2a n - n 2 . For which α are all a n positive? Solution

Answer: [3, \infty)


Exercise 2. Let $S$ be the set of two-digit numbers that do not contain the digit 0. Two numbers in $S$ are called friends if their largest digits are equal, and if the difference between their smallest digits is 1. For example, 68 and 85 are friends, 78 and 88 are friends, but 58 and 75 are not friends.

Determine the largest integer $m$ such that there exists a subset $T$ of $S$ with $m$ elements, such that any two elements of $T$ are not friends.

Answer: 45


Let $MATH$ be a square with $MA = 1$. Point $B$ lies on $AT$ such that $\angle MBT = 3.5 \angle BMT$. What is the area of $\vartriangle BMT$?

Answer: \dfrac{\sqrt{3} - 1}{2}


In a society of 30 people, any two people are either friends or enemies, and everyone has exactly six enemies. How many ways are there to choose three people from the society so that any two of them are either all friends or all enemies?

Answer: 1990


8,9

All edges of a regular triangular prism are equal to $a$. Find the area of the section of the prism by a plane passing through an edge of the base and the midpoint of the non-parallel edge of the other base.

Answer: \dfrac{3\sqrt{19}}{16}a^2


2. A box contains 9 good items and 3 defective items. Each time an item is taken, it is not replaced. What is the probability that 3 defective items have been taken before 2 good items are taken? $\qquad$

Answer: \dfrac{1}{55}


Given a square $ABCD$ with side length $6$. We draw line segments from the midpoints of each side to the vertices on the opposite side. For example, we draw line segments from the midpoint of side $AB$ to vertices $C$ and $D$. The eight resulting line segments together bound an octagon inside the square. What is the area of this octagon?

Answer: 6


5. [5 points] A magician has a set of $20^{2}$ different cards. Each card has one side red and the other side blue; on each card, there is a natural number from 1 to 20 written on both sides. We will call a card a duplicate if the numbers on both sides of the card are the same. The magician wants to draw two cards such that at least one of them is a duplicate, and no number appears on both drawn cards simultaneously. In how many ways can he do this?

Answer: 7030


8. Given positive integers $a, b, c, x, y, z$ satisfying
$$
a \geqslant b \geqslant c \geqslant 1, x \geqslant y \geqslant z \geqslant 1 \text {, }
$$

and $\left\{\begin{array}{l}2 a+b+4 c=4 x y z, \\ 2 x+y+4 z=4 a b c .\end{array}\right.$
then the number of six-tuples $(a, b, c, x, y, z)$ that satisfy the conditions is $\qquad$.

Answer: 0


Let $x$ be the answer to this problem. For what real number $a$ is the answer to this problem also $a-x$?

Answer: 0


Triangle $ABC$ has side lengths $AB = 14$, $BC = 15$, and $CA = 13$. We draw a circle with diameter $AB$ such that it intersects $BC$ again at $D$ and intersects $CA$ again at $E$. If the circumradius of $\triangle CDE$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are coprime positive integers, determine $100m+n$.

Answer: 3308


Antoine, Benoît, Claude, Didier, Étienne, and Françoise go to the cinema together to see a movie. The six of them want to sit in a single row of six seats. However, Antoine, Benoît, and Claude are mortal enemies and refuse to sit next to either of the other two. How many different seating arrangements are possible?

Answer: 144


A semicircle has diameter $\overline{AD}$ with $AD = 30$. Points $B$ and $C$ lie on $\overline{AD}$, and points $E$ and $F$ lie on the arc of the semicircle. The two right triangles $\vartriangle BCF$ and $\vartriangle CDE$ are congruent. The area of $\vartriangle BCF$ is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m + n$.

Answer: 52


Solomiya wrote the numbers $1, 2, \ldots, 2024$ on the board. In one move, she can erase any two numbers $a, b$ from the board and write the sum $a+b$ instead of each of them. After some time, all the numbers on the board became equal. What is the minimum number of moves Solomiya could make to achieve this?

Answer: 1012


A set of five-digit numbers $\{N_1, ..., N_k\}$ is such that any five-digit number, all of whose digits are in non-decreasing order, coincides in at least one digit with at least one of the numbers $N_1, ..., N_k$. Find the smallest possible value of $k$.

Answer: 9


A positive integer $n$ is called *nice* if it has at least 3 proper divisors and it is equal to the sum of its three largest proper divisors. For example, $6$ is *nice* because its largest three proper divisors are $3, 2, 1$ and $6 = 3 + 2 + 1$. Find the number of *nice* integers not greater than $3000$.

Answer: 200


GM Bisain's IQ is so high that he can move around in $10$-dimensional space. He starts at the origin and moves in a straight line away from the origin, stopping after $3$ units. How many lattice points can he land on? A lattice point is one with all integer coordinates.

Answer: 88660


Determine the largest two-digit number $d$ with the following property: for any six-digit number $\overline{aabbcc}$, the number $d$ is a divisor of $\overline{aabbcc}$ if and only if $d$ is a divisor of the corresponding three-digit number $\overline{abc}$.

Note: The numbers $a \ne 0$, $b$, and $c$ need not be different.

Answer: 99


Given that there are $24$ primes between $3$ and $100$, inclusive, determine the number of ordered pairs $(p, a)$ where $p$ is a prime number such that $3 \le p < 100$, and $1 \le a < p$. The condition is that the sum:
\[
a + a^2 + a^3 + \cdots + a^{(p-2)!}
\]
is not divisible by $p$.

Answer: 27


There are 16 cities participating in a football match. Each city has two teams: Team A and Team B. According to the rules, each team can compete with any other team from different cities at most once. After a few days, it is observed that the number of matches played by each team, except for Team A from City 1, is unique. Determine the number of matches that Team A from City 1 has played.

Answer: 15


If $x$ is a real number, let $\lfloor x \rfloor$ be the greatest integer that is less than or equal to $x$. If $n$ is a positive integer, let $S(n)$ be defined by
\[
  S(n)
    = \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
        + 10 \left( n - 10^{\lfloor \log n \rfloor}
               \cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
             \right) \, .
\]
(All the logarithms are base 10.) How many integers $n$ from 1 to 2011 (inclusive) satisfy $S(S(n)) = n$?

Answer: 108


Let $s_1, s_2, s_3$ be the three roots of the polynomial $x^3 + x^2 + \frac{9}{2}x + 9$.

Find the value of $a + b + c$ if the expression $\prod_{i=1}^{3}(4s^4_i + 81)$ can be written as $2^a3^b5^c$.

Answer: 16


For a positive integer $n$, an $n$-branch $B$ is an ordered tuple $(S_1, S_2, \dots, S_m)$ of nonempty sets (where $m$ is any positive integer) satisfying $S_1 \subset S_2 \subset \dots \subset S_m \subseteq \{1,2,\dots,n\}$. An integer $x$ is said to appear in $B$ if it is an element of the last set $S_m$. Define an $n$-plant to be an (unordered) set of $n$-branches $\{ B_1, B_2, \dots, B_k\}$, and call it perfect if each of $1$, $2$, \dots, $n$ appears in exactly one of its branches.

Let $T_n$ be the number of distinct perfect $n$-plants (where $T_0=1$), and suppose that for some positive real number $x$ we have the convergence \[ \ln \left( \sum_{n \ge 0} T_n \cdot \frac{\left( \ln x \right)^n}{n!} \right) = \frac{6}{29}. \] If $x = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.

Answer: 76


Let $\mathbf{u}$ and $\mathbf{v}$ be unit vectors, and let $\mathbf{w}$ be a vector such that
\[\mathbf{w} + \mathbf{w} \times \mathbf{u} = \mathbf{v}.\]Find the largest possible value of $(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 3


In triangle $ABC$, $AB = 5$, $BC = 4$, and $CA = 3$.

[asy]
defaultpen(1);

pair C=(0,0), A = (0,3), B = (4,0);
draw(A--B--C--cycle);

label("\(A\)",A,N);
label("\(B\)",B,E);
label("\(C\)",C,SW);
[/asy]

Point $P$ is randomly selected inside triangle $ABC$.  What is the probability that $P$ is closer to $C$ than it is to either $A$ or $B$?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 3


Calculate
\[\prod_{n = 1}^{13} \frac{n(n + 2)}{(n + 4)^2}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 161843


In the figure below, parallelograms $ABCD$ and $BFEC$ have areas $1234\, \text{cm}^2$ and $2804\, \text{cm}^2$, respectively. Points $M$ and $N$ are chosen on sides $AD$ and $FE$, respectively, so that segment $MN$ passes through $B$. Find the area of $\triangle MNC$.

Answer: 2019


Let $AB$ be a chord of a circle with center $O$. Let $C$ be a point on the circle such that $\angle ABC = 30^\circ$ and $O$ lies inside the triangle $ABC$. Let $D$ be a point on $AB$ such that $\angle DCO = \angle OCB = 20^\circ$. Find the measure of $\angle CDO$ in degrees.

Answer: 80


Suppose there are 36 objects placed at equal distances along a circle. Determine the number of ways to choose 3 objects such that no two chosen objects are adjacent or diametrically opposite.

Answer: 5412


Kelvin the frog is hopping on the coordinate plane $\mathbb{R}^{2}$. He starts at the origin, and every second, he hops one unit to the right, left, up, or down, such that he always remains in the first quadrant $\{(x, y): x \geq 0, y \geq 0\}$. In how many ways can Kelvin make his first 14 jumps such that his 14th jump lands at the origin?

Answer: 613470


Line $\ell$ passes through point $A$ and into the interior of the equilateral triangle $ABC$. Points $D$ and $E$ are the orthogonal projections of points $B$ and $C$ onto line $\ell$, respectively. Given that $DE = 1$ and $2BD = CE$, determine the area of triangle $ABC$ expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find the value of $m+n$.

Answer: 10


Given a triangle $ABC$ with an area of $1$. Let $A'$, $B'$, and $C'$ be the midpoints of the sides $[BC]$, $[CA]$, and $[AB]$ respectively. What is the minimal possible area of the common part of the two triangles $A'B'C'$ and $KLM$, if the points $K$, $L$, and $M$ are lying on the segments $[AB']$, $[CA']$, and $[BC']$ respectively?

Answer: 0


Given a natural number $n$, suppose that $N$ is the maximum number of elephants (bishops) that can be placed on a chessboard measuring $2 \times n$ such that no two elephants are mutually under attack. Determine the number of ways to place $N$ elephants on a $2 \times n$ chessboard so that no two elephants attack each other.

Alternative Formulation: Determine the number of ways to place $2015$ elephants on a chessboard measuring $2 \times 2015$ so that no two elephants attack each other.

Answer: 2


For any positive integer $k$, define $f_1(k)$ as the square of the digital sum of $k$ in the decimal system. For $n > 1$, define $f_{n}(k)=f_1(f_{n-1}(k))$. Compute $f_{1992}(2^{1991})$.

Answer: 256


Alice, Bob, Charlie, Diana, Emma, and Fred sit in a circle, in that order, and each roll a six-sided die. Each person looks at their own roll and also looks at the roll of either the person to the right or to the left, deciding at random. Then, at the same time, Alice, Bob, Charlie, Diana, Emma, and Fred each state the expected sum of the dice rolls based on the information they have. All six people say different numbers; in particular, Alice, Bob, Charlie, and Diana say $19$, $22$, $21$, and $23$, respectively. Compute the product of the dice rolls.

Answer: 1440


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

Answer: 17


Identify a four-digit number $m$ which is the product of three distinct prime factors $x$, $y$, and $10x+y$. Moreover, introduce another prime factor $z$, such that $10z + x$ must also be a prime. Here $x$, $y$, and $z$ are distinct primes each less than 10. What is the maximum value of $m$ under these conditions?

Answer: 1533


Given that $ABCDEH$ is a rectangle, where $AD = 4$ and $AB = 2$. Inside rectangle $ABCDEH$, a regular hexagon $FGHIJK$ is inscribed such that its vertices touch the midpoint of each side of the rectangle. Find the total area of the hexagon.

Answer: 6\sqrt{3}


Consider two solid spherical balls where the first ball is centered at $\left( 0, 0, 10 \right)$ with radius 5, and the second ball is centered at $(0, 0, 2)$ with radius 4. Calculate the volume of the intersection of these two spheres.

Answer: \frac{205\pi}{96}


Modern widescreen television screens are rectangles and are typically measured by the length of their diagonals. The ratio of the horizontal length to the height in these widescreen televisions is $16:9$. Calculate the horizontal length (in inches) of a ``50-inch'' widescreen television screen.

Answer: 43.6


A parabola with the equation $y=ax^2+bx+c$ is graphed below:

[asy]
xaxis(-3,7);
yaxis(-5,32);
real g(real x)
{
return 3(x-3)^2-9;
}
draw(graph(g,-1,5));
dot((3,-9));
label("Vertex: $(3,-9)$", (3,-9), SE);
dot((5,6));
label("$(5,6)$", (5,6), E);
[/asy]

The zeros of the quadratic $ax^2 + bx + c$ are at $x=p$ and $x=q$, where $p>q$. What is $p-q$?

Answer: \frac{4\sqrt{15}}{5}


The ratio of irises to roses in Nikki's garden is 3:7. Initially, she has 35 roses but gives away 15 roses. After this, she plans to add 25 more roses and wants to plant enough new irises to maintain the same ratio. How many irises will she need to have in total after these adjustments?

Answer: 20


In a new sequence, the first term is $b_1 = 3010$ and the second term is $b_2 = 3011$. Additionally, the terms are designed such that $b_n + b_{n+1} + b_{n+2} = n + 4$ for all $n \ge 1$. Determine $b_{1000}$.

Answer: 3343


A frustum of a cone has a small radius of 2 inches, a large radius of 3 inches, and a height of 5 inches. What is the radius of a sphere which has the same volume as this frustum?

Answer: \sqrt[3]{\frac{95}{4}}


In a literature class, students are tasked with reading a 980-page book. Two friends, Daniel and Emma, decide to split the reading. Daniel reads a page in 50 seconds, and Emma reads a page in 40 seconds. They plan to divide the book such that each spends an equal amount of time reading. Determine the last page that Daniel should read to ensure both spend the same amount of time reading.

Answer: 436


Math City now has ten streets, each straight and no two streets are parallel. Additionally, two involvements are such that these streets cross each other precisely at an angle of 180 degrees, effectively not creating an intersection. How many police officers are needed to manage each intersection, assuming each intersection is manned by one officer?

Answer: 43


Football tickets now cost $\$15$ each. Jane has $\$180$. If Jane buys more than 10 tickets, she receives a discount of $\$2$ per ticket for every ticket. What is the maximum number of tickets Jane can buy?

Answer: 13


Let \( z = re^{i\theta} \) be a complex number such that \( |z| = r \neq 1 \) and \( r > 0 \). Find the real part of \( \frac{1}{1 - z^2} \).

Answer: \frac{1 - r^2 \cos(2\theta)}{1 - 2r^2 \cos(2\theta) + r^4}


A $\frac{1}{p}$-array is structured where each row forms a sequence such that the first entry of each row is $\frac{1}{2p}$ times the first entry of the previous row, and each succeeding term in a row is $\frac{1}{p^2}$ times the previous term. Construct and find the sum of all terms in a $\frac{1}{2008}$-array, written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Then, provide the remainder when $m+n$ is divided by 2008.

Answer: 1


A circular spinner for a game has a radius of 15 cm. The probability of winning on one spin of this spinner has two possibilities with each winning sector having an equal chance, and each winning probability is $\frac{1}{6}$. Calculate the total area, in sq cm, of the WIN sectors combined. Express your answer in terms of $\pi$.

Answer: 75\pi


The number of students in a school competition is more than 100 but fewer than 200. The number of students is 2 less than a multiple of 4, 3 less than a multiple of 5, and 4 less than a multiple of 6. Find how many students were in the competition.

Answer: 182


Suppose the function $g(x)$ is defined on domain $\{x_1,x_2,x_3\}$ such that the graph of $y=g(x)$ consists of just three points, forming a triangle of area $48$. Consider the graph of $y = 3g(\frac{x}{4})$. What is the area of the triangle formed by these three points?

Answer: 576 \text{ unit}^2


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - People use unique phone models: `iphone 13`, `samsung galaxy s21`, `google pixel 6`
 - People have unique heights: `very short`, `average`, `short`
 - Each person lives in a unique style of house: `colonial`, `ranch`, `victorian`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`

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

Answer: toyota camry


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Peter`, `Alice`, `Eric`
 - Each person has a unique favorite drink: `milk`, `root beer`, `coffee`, `water`, `tea`
 - Each person has a unique type of pet: `dog`, `bird`, `hamster`, `cat`, `fish`
 - Each mother is accompanied by their child: `Timothy`, `Meredith`, `Fred`, `Bella`, `Samantha`
 - The mothers' names in different houses are unique: `Kailyn`, `Penny`, `Aniya`, `Janelle`, `Holly`
 - People have unique favorite sports: `tennis`, `soccer`, `swimming`, `baseball`, `basketball`

## Clues:
1. The tea drinker is the person who loves tennis.
2. The person whose mother's name is Aniya is Arnold.
3. The person whose mother's name is Janelle is somewhere to the right of the person who loves swimming.
4. The person whose mother's name is Janelle is the person's child is named Bella.
5. The person's child is named Samantha is the person who likes milk.
6. The coffee drinker is not in the fifth house.
7. Bob is the person's child is named Bella.
8. The person who loves swimming and the person who loves basketball are next to each other.
9. The person who is the mother of Timothy is somewhere to the left of the person who owns a dog.
10. The person whose mother's name is Kailyn is in the fifth house.
11. The root beer lover is Alice.
12. The person whose mother's name is Holly is directly left of the root beer lover.
13. The person who likes milk is not in the second house.
14. There is one house between the person who has a cat and the person who loves baseball.
15. The person whose mother's name is Penny is somewhere to the right of the person who loves tennis.
16. The person's child is named Fred is the person who keeps a pet bird.
17. The person who likes milk is directly left of the person with a pet hamster.
18. The person who keeps a pet bird is Alice.
19. There are two houses between Eric and The person whose mother's name is Aniya.
 What is the value of attribute House for the person whose attribute Drink is water? 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: `Alice`, `Arnold`, `Eric`, `Peter`
 - Each person has a unique favorite drink: `water`, `tea`, `coffee`, `milk`
 - Each person has a unique level of education: `master`, `associate`, `high school`, `bachelor`
 - The mothers' names in different houses are unique: `Kailyn`, `Holly`, `Aniya`, `Janelle`
 - Each person has a favorite color: `yellow`, `white`, `green`, `red`

## Clues:
1. Eric is the person with an associate's degree.
2. The person whose mother's name is Kailyn is the person who loves yellow.
3. The person who likes milk is in the third house.
4. The person who likes milk is somewhere to the right of the person who loves white.
5. The person with a bachelor's degree is in the first house.
6. The person whose mother's name is Aniya is not in the second house.
7. The one who only drinks water is Arnold.
8. The person whose favorite color is green is Peter.
9. The person whose favorite color is red is somewhere to the left of the person who loves white.
10. The person with a high school diploma is directly left of The person whose mother's name is Janelle.
11. The person with a high school diploma is the tea drinker.
 What is the value of attribute House for the person whose attribute Mother is Janelle? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Peter`
 - People have unique favorite sports: `swimming`, `soccer`, `tennis`, `basketball`
 - Each person has a unique type of pet: `bird`, `fish`, `cat`, `dog`
 - Everyone has a favorite smoothie: `dragonfruit`, `desert`, `watermelon`, `cherry`
 - People have unique hair colors: `brown`, `black`, `blonde`, `red`
 - Each person has a unique birthday month: `jan`, `feb`, `sept`, `april`

## Clues:
1. The person whose birthday is in September is the person who likes Cherry smoothies.
2. The person whose birthday is in February is not in the first house.
3. The Desert smoothie lover is the person who loves basketball.
4. Peter is the person who has black hair.
5. The person who loves tennis is somewhere to the right of the person who has a cat.
6. The person with an aquarium of fish is not in the fourth house.
7. Arnold is directly left of the person who has blonde hair.
8. There is one house between the person who has red hair and the person whose birthday is in January.
9. The person whose birthday is in September is directly left of the person who loves soccer.
10. Peter is directly left of the person who loves basketball.
11. The person who has blonde hair is the person whose birthday is in September.
12. Arnold is the person who keeps a pet bird.
13. The person whose birthday is in January is the Watermelon smoothie lover.
14. The person who has blonde hair is Eric.
 What is the value of attribute HairColor for the person whose attribute Birthday is feb? Please reason step by step, and put your final answer within \boxed{}

Answer: red


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

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

Answer: lime


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

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

Answer: very tall