There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Eric`, `Bob`, `Peter`
 - Each person lives in a unique style of house: `victorian`, `colonial`, `ranch`, `modern`, `craftsman`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `oneplus 9`, `samsung galaxy s21`, `huawei p50`
 - Everyone has a unique favorite cigar: `blue master`, `dunhill`, `pall mall`, `blends`, `prince`
 - People have unique favorite book genres: `mystery`, `science fiction`, `biography`, `romance`, `fantasy`

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

Answer: iphone 13


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Bob`, `Peter`, `Arnold`, `Carol`
 - People have unique heights: `very tall`, `tall`, `super tall`, `average`, `very short`, `short`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `samsung galaxy s21`, `iphone 13`, `huawei p50`, `xiaomi mi 11`

## Clues:
1. Bob is directly left of the person who is tall.
2. Peter is somewhere to the left of the person who uses an iPhone 13.
3. The person who is very short is somewhere to the right of the person who uses a Google Pixel 6.
4. Carol is the person who is very tall.
5. There is one house between the person who uses a Google Pixel 6 and the person who is short.
6. The person who uses a Samsung Galaxy S21 is not in the first house.
7. The person who uses a OnePlus 9 is directly left of the person who is short.
8. The person who is tall is Arnold.
9. The person who is super tall is in the first house.
10. The person who uses a Xiaomi Mi 11 is Carol.
11. The person who uses a Google Pixel 6 is somewhere to the right of Eric.
12. The person who is short is in the sixth house.
 What is the value of attribute Height for the person whose attribute PhoneModel is google pixel 6? Please reason step by step, and put your final answer within \boxed{}

Answer: tall


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

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

Answer: 3


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Bob`, `Alice`, `Carol`, `Peter`
 - The mothers' names in different houses are unique: `Sarah`, `Holly`, `Janelle`, `Aniya`, `Penny`, `Kailyn`
 - Each person has a unique type of pet: `hamster`, `dog`, `bird`, `cat`, `fish`, `rabbit`

## Clues:
1. Bob is not in the second house.
2. There are two houses between the person who has a cat and the person who owns a rabbit.
3. The person who has a cat is directly left of The person whose mother's name is Holly.
4. The person with a pet hamster is directly left of the person who owns a rabbit.
5. The person who owns a rabbit is Eric.
6. There is one house between the person who owns a dog and the person who has a cat.
7. The person who has a cat is The person whose mother's name is Janelle.
8. Alice is directly left of Carol.
9. Carol is The person whose mother's name is Aniya.
10. Arnold is the person who has a cat.
11. The person whose mother's name is Kailyn is the person who owns a rabbit.
12. The person with an aquarium of fish is The person whose mother's name is Sarah.
 What is the value of attribute Mother for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: Penny


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Carol`, `Bob`, `Alice`, `Eric`, `Peter`
 - People have unique favorite sports: `swimming`, `basketball`, `volleyball`, `soccer`, `tennis`, `baseball`
 - People have unique favorite music genres: `rock`, `pop`, `jazz`, `country`, `hip hop`, `classical`
 - Each person has a unique level of education: `doctorate`, `trade school`, `bachelor`, `high school`, `associate`, `master`

## Clues:
1. Alice is in the fourth house.
2. Alice is somewhere to the right of the person with a doctorate.
3. The person with a high school diploma is somewhere to the right of the person who loves basketball.
4. The person who loves volleyball is the person with a bachelor's degree.
5. Alice is directly left of the person who loves jazz music.
6. The person with a master's degree is the person who loves swimming.
7. There is one house between the person who loves rock music and the person with a bachelor's degree.
8. Carol is in the second house.
9. The person who loves rock music is somewhere to the left of the person who loves pop music.
10. The person who loves swimming is somewhere to the left of Carol.
11. Eric is directly left of the person who loves hip-hop music.
12. Alice is the person with an associate's degree.
13. Alice is the person who loves classical music.
14. Arnold is the person who loves soccer.
15. The person who loves basketball and the person who loves volleyball are next to each other.
16. Peter is the person who loves tennis.
17. The person with a doctorate is Peter.
 What is the value of attribute FavoriteSport for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: swimming


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`
 - Each person has a unique hobby: `cooking`, `photography`, `gardening`
 - The people are of nationalities: `dane`, `brit`, `swede`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`
 - Everyone has a favorite smoothie: `cherry`, `desert`, `watermelon`

## Clues:
1. The Desert smoothie lover is the person who enjoys gardening.
2. The photography enthusiast is the Swedish person.
3. Peter is somewhere to the left of the Desert smoothie lover.
4. The person who is a pizza lover is in the third house.
5. The person who enjoys gardening is the Dane.
6. There is one house between the person who loves eating grilled cheese and Arnold.
7. Arnold is the Swedish person.
8. The Watermelon smoothie lover is somewhere to the left of the person who likes Cherry smoothies.
 What is the value of attribute Hobby for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: gardening


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 Food for the person whose attribute Cigar is pall mall? Please reason step by step, and put your final answer within \boxed{}

Answer: stir fry


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Eric`, `Arnold`, `Alice`
 - Each person has a favorite color: `blue`, `red`, `green`, `yellow`, `white`
 - Everyone has a unique favorite cigar: `dunhill`, `prince`, `pall mall`, `blends`, `blue master`
 - People have unique hair colors: `blonde`, `brown`, `red`, `gray`, `black`
 - Each person has a unique favorite drink: `tea`, `milk`, `coffee`, `water`, `root beer`
 - The people are of nationalities: `dane`, `swede`, `german`, `brit`, `norwegian`

## Clues:
1. The Norwegian is directly left of the Prince smoker.
2. The person who loves yellow is somewhere to the right of the tea drinker.
3. The person who has blonde hair is the Norwegian.
4. The root beer lover is somewhere to the left of the Dane.
5. There is one house between the person whose favorite color is red and the person who has black hair.
6. The German is in the third house.
7. The person who loves blue is not in the fourth house.
8. The person whose favorite color is red and the Dunhill smoker are next to each other.
9. The root beer lover is directly left of the German.
10. The Norwegian is Eric.
11. The British person is the person who likes milk.
12. Peter is the person who smokes many unique blends.
13. The coffee drinker is directly left of the Swedish person.
14. The person who smokes Blue Master is somewhere to the left of the person who smokes many unique blends.
15. The person who has gray hair is directly left of the person who has red hair.
16. The person who loves blue is somewhere to the right of the one who only drinks water.
17. Arnold is the person who has brown hair.
18. The person who has gray hair is not in the third house.
19. The coffee drinker is the person whose favorite color is green.
20. The Dunhill smoker is Alice.
 What is the value of attribute Drink for the person whose attribute Cigar is blends? Please reason step by step, and put your final answer within \boxed{}

Answer: water


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`, `Alice`
 - People have unique favorite sports: `basketball`, `soccer`, `swimming`, `tennis`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `samsung galaxy s21`, `iphone 13`
 - People own unique car models: `toyota camry`, `ford f150`, `tesla model 3`, `honda civic`
 - They all have a unique favorite flower: `roses`, `daffodils`, `lilies`, `carnations`

## Clues:
1. The person who owns a Honda Civic is the person who uses an iPhone 13.
2. The person who loves tennis is the person who uses an iPhone 13.
3. The person who uses a Google Pixel 6 is somewhere to the right of the person who loves basketball.
4. The person who uses an iPhone 13 is directly left of Eric.
5. Alice is the person who owns a Ford F-150.
6. Arnold is the person who loves a bouquet of daffodils.
7. The person who loves the boquet of lilies is in the third house.
8. The person who loves soccer is Peter.
9. The person who uses a OnePlus 9 is the person who loves the rose bouquet.
10. The person who loves a carnations arrangement is not in the fourth house.
11. Peter is the person who loves the boquet of lilies.
12. The person who owns a Toyota Camry is directly left of the person who owns a Tesla Model 3.
 What is the value of attribute PhoneModel for the person whose attribute CarModel is tesla model 3? Please reason step by step, and put your final answer within \boxed{}

Answer: google pixel 6


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - People own unique car models: `toyota camry`, `ford f150`, `tesla model 3`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `victorian`
 - Each person has a unique type of pet: `cat`, `dog`, `fish`
 - Each person has an occupation: `engineer`, `doctor`, `teacher`
 - Each person prefers a unique type of vacation: `city`, `mountain`, `beach`

## Clues:
1. The person with an aquarium of fish is in the first house.
2. The person who owns a Toyota Camry is in the second house.
3. The person who enjoys mountain retreats is not in the second house.
4. The person who prefers city breaks is not in the second house.
5. The person in a ranch-style home is somewhere to the left of Peter.
6. The person who owns a Toyota Camry is directly left of the person living in a colonial-style house.
7. Arnold is the person who has a cat.
8. Eric is somewhere to the left of the person who enjoys mountain retreats.
9. The person who is an engineer is not in the third house.
10. The person who owns a Tesla Model 3 is somewhere to the left of the person who is a teacher.
11. The person who owns a dog is the person who is an engineer.
 What is the value of attribute House for the person whose attribute Vacation is city? 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: `Bob`, `Arnold`, `Peter`, `Carol`, `Alice`, `Eric`
 - Everyone has something unique for lunch: `stir fry`, `stew`, `soup`, `pizza`, `grilled cheese`, `spaghetti`
 - Each person has an occupation: `doctor`, `artist`, `engineer`, `teacher`, `lawyer`, `nurse`
 - Each person has a unique type of pet: `dog`, `bird`, `hamster`, `cat`, `fish`, `rabbit`
 - People have unique favorite music genres: `pop`, `country`, `jazz`, `classical`, `rock`, `hip hop`
 - Everyone has a favorite smoothie: `blueberry`, `desert`, `watermelon`, `cherry`, `lime`, `dragonfruit`

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

Answer: fish


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Bob`, `Eric`
 - Each person has a favorite color: `white`, `blue`, `yellow`, `red`, `green`
 - Everyone has a unique favorite cigar: `blue master`, `dunhill`, `prince`, `pall mall`, `blends`
 - People have unique favorite book genres: `biography`, `mystery`, `romance`, `fantasy`, `science fiction`

## Clues:
1. There are two houses between the Prince smoker and the person who loves yellow.
2. The person who loves white is Bob.
3. The person who smokes Blue Master is directly left of Bob.
4. Alice is somewhere to the left of the person who loves science fiction books.
5. The person who loves biography books is not in the first house.
6. Arnold is somewhere to the left of the person who smokes Blue Master.
7. The person who loves mystery books and the person who loves biography books are next to each other.
8. The person who loves fantasy books is Bob.
9. The person whose favorite color is green is the person who smokes Blue Master.
10. The person partial to Pall Mall is in the fourth house.
11. The person who loves science fiction books is the Dunhill smoker.
12. The person whose favorite color is red is not in the fifth house.
13. Peter is directly left of the Dunhill smoker.
 What is the value of attribute Color for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: yellow


5. Jure drew a regular 9-sided polygon. He wanted to arrange the numbers from 1 to 9 at its vertices such that the sum of the numbers in any three consecutive vertices would not exceed a natural number $n$. For what smallest number $n$ would he have succeeded?

## 53rd Mathematical Competition for High School Students in Slovenia

Selection Competition, April 1, 2009

## Solutions to the Problems

Each problem is worth 7 points. All mathematically and logically correct solutions are equally valid. When evaluating each problem, reasonably consider the provided scoring guide. A contestant should not receive more than 3 points for any problem if the partial solution does not clearly lead to the final solution of the problem.

Answer: 17


Example 4 Let $A=\left\{x \mid x^{2}+(p+2) x+1=0, x \in \mathbf{R}\right\}$, if $A \varsubsetneqq \mathbf{R}^{-}$, find the range of real number $p$.

Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.

Answer: (-4, +\infty)


Three, (50 points) A positive integer is called "simple" if it does not have any square factors greater than 1. Determine how many numbers in $1,2, \cdots, 2010$ are simple.

Answer: 1221


OKOSBANK's safe has several different locks. In the bank, $n$ tellers work, each having keys to some of the locks. (A teller can have multiple keys, and multiple tellers can have keys to the same lock.) We know that any $k$ tellers together can always open the safe, but $k-1$ tellers never can. What is the minimum number of locks on the safe?

Answer: \dbinom{n}{k-1}


A test consisted of 4 questions, each with 3 predetermined possible answers. What is the maximum number of people who could have participated in the test, given that for any 3 test-takers, there was a question where all three marked different answers as correct?

Answer: 9


There are six piles of apples, and the number of apples in each pile forms an arithmetic sequence. Junjun selects one of the piles and takes out 150 apples, then distributes these to the remaining 5 piles with 10, 20, 30, 40, and 50 apples respectively. After the distribution, Junjun magically finds that the number of apples in these 5 piles correspond to the 2, 3, 4, 5, and 6 times the number of apples in the pile he selected. How many apples are there in total in the six piles?

Answer: 735


15. Given the plane point set $A=\{(x, y) \mid x=2 \sin \alpha+2 \sin \beta, y=2 \cos \alpha+2 \cos \beta\}$
$$
B=\{(x, y) \mid \sin (x+y) \cos (x+y) \geqslant 0\}
$$

Find the area of the region represented by $A \cap B$.

Answer: 8\pi


When a Brilliant Idea comes to the Scattered Scholar, he writes it down on a piece of paper, but immediately realizes that the idea is not brilliant, crumples the paper, and throws it under the table, where there are two wastebaskets. The Scholar misses the first wastebasket with a probability of $p>0.5$, and with the same probability, he misses the second one. In the morning, the Scholar threw five crumpled brilliant ideas under the table. Find the probability that each wastebasket ended up with at least one of the morning ideas.

Answer: 1 - 2p^5 + (2p - 1)^5


$[$ [extension of a tetrahedron to a parallelepiped]

Segment $A B(A B=1)$, being a chord of a sphere with radius 1, is positioned at an angle of $60^{\circ}$ to the diameter $C D$ of this sphere. The distance from the end $C$ of the diameter to the nearest end $A$ of the chord $A B$ is $\sqrt{2}$. Find $B D$.

#

Answer: 1


Problem 8.2. In a volleyball tournament for the Euro- African cup the European teams are 9 more than the African teams. Every two teams met exactly once and the European teams gained 9 times more points than the African teams (the winner takes 1 point and the loser takes 0 point). What are the maximum possible points gained by an African team?

Ivan Tonov

Answer: 11


Let $C$ be the set of all 100-digit numbers consisting of only the digits $1$ and $2$. Given a number in $C$, we may transform the number by considering any $10$ consecutive digits $x_0x_1x_2 \dots x_9$ and transform it into $x_5x_6\dots x_9x_0x_1\dots x_4$. We say that two numbers in $C$ are similar if one of them can be reached from the other by performing finitely many such transformations. Let $D$ be a subset of $C$ such that any two numbers in $D$ are not similar. Determine the maximum possible size of $D$.

Answer: 21^5


Let \( m \) and \( n \) be nonnegative integers. Prove that \( m!n!(m+n)! \) divides \( (2m)!(2n)! \).

Answer: m!n!(m+n)! \text{ divides } (2m)!(2n)!


3. In a cube $A B C D-$ $A_{1} B_{1} C_{1} D_{1}$ with edge length $a$, $X$ and $Y$ are the centers of the squares $A A_{1} B B_{1}$ and $B B_{1} C_{1} C$, respectively, and $Z$ is on the diagonal $B D$ such that $D Z=3 Z B$. Then the area of the section through $X$, $Y$, and $Z$ is $\qquad$

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


Example 5 (1990 Canadian Mathematical Olympiad) Arrange $\frac{n(n+1)}{2}$ different numbers randomly into a triangular array as shown in Figure $12-1$. Let $M_{k}$ be the maximum number in the $k$-th row (from top to bottom). Try to find the probability that $M_{1}<M_{2}<\cdots<M_{n}$.

Answer: \dfrac{2^n}{(n+1)!}


3. On a plane, a certain number of triangles are drawn, the lengths of whose sides are ten-digit natural numbers, containing only threes and eights in their decimal representation. No segment belongs to two triangles, and the sides of all triangles are distinct. What is the maximum number of triangles that can be drawn?

Answer: 341


6. (10 points) For a certain project, if working alone, Team A needs 24 days, Team B needs 36 days, and Team C needs 60 days; it is known that all three teams worked for an integer number of days, and the task was completed within 18 days (including 18 days). How many days did Team A work at least? $\qquad$ days.

Answer: 6


We are given one red and $k>1$ blue cells, and a pack of $2n$ cards, enumerated by the numbers from $1$ to $2n$.  Initially, the pack is situated on the red cell and arranged in an arbitrary order.  In each move, we are allowed to take the top card from one of the cells and place it either onto the top of another cell on which the number on the top card is greater by $1$, or onto an empty cell.  Given $k$, what is the maximal $n$ for which it is always possible to move all the cards onto a blue cell?

Answer: k-1


Problem 8. Given an isosceles triangle $K L M(K L=L M)$ with the angle at the vertex equal to $114^{\circ}$. Point $O$ is located inside triangle $K L M$ such that $\angle O M K=30^{\circ}$, and $\angle O K M=27^{\circ}$. Find the measure of angle $\angle L O M$.

Answer: 150^\circ


7. On the side $AC$ of triangle $ABC$, a circle is constructed with $AC$ as its diameter, which intersects sides $AB$ and $BC$ at points $D$ and $E$ respectively. The angle $EDC$ is $30^{\circ}$, $EC=1$, and the area of triangle $DBE$ is to the area of triangle $ABC$ as $1:2$. Find the length of segment $BO$, where $O$ is the intersection point of segments $AE$ and $CD$.

Answer: 2


Let \( S \) be the set of lattice points inside the circle \( x^{2}+y^{2}=11 \). Let \( M \) be the greatest area of any triangle with vertices in \( S \). How many triangles with vertices in \( S \) have area \( M \)?

Answer: 16


Inside a large rectangle of size $A \times B$ there is a smaller rectangle of size $a \times b$ (see the figure).

Find the difference between the total area of the yellow quadrilaterals and the total area of the green quadrilaterals, given that $A=20$, $B=30$, $a=4$, $b=7$.

Answer: 20


The perimeter of the parallelogram \(ABCD\) is 20. The bisector of angle \(B\) intersects the lines \(AD\) and \(CD\) at points \(K\) and \(L\) respectively. Find \(CL\), given that \(DK = 4\).

Answer: 7


As shown in the image, 7 small squares form a large square. If the side lengths of these 7 small squares, in increasing order, are $1, 1, 2, 3, 5, 8, 13$, what is the perimeter of this large square?

Answer: 68


In the figure, the triangle $\triangle ABC$ is isosceles, with $\angle BAC = 20^\circ$.

Given that $BC = BD = BE$, determine the measure of the angle $\angle BDE$.

Answer: 60


The country of HMMTLand has 8 cities. The government decides to construct several two-way roads between pairs of distinct cities. After construction, it turns out that each city can reach exactly 3 other cities via a single road, and for any pair of distinct cities, either exactly 0 or 2 other cities can be reached from both cities by a single road. Compute the number of ways HMMTLand could have constructed the roads.

Answer: 875


Let \( ABCD \) be a rectangle and \( E \) be a point on segment \( AD \). We are given that quadrilateral \( BCDE \) has an inscribed circle \( \omega_{1} \) that is tangent to \( BE \) at \( T \). If the incircle \( \omega_{2} \) of \( ABE \) is also tangent to \( BE \) at \( T \), then find the ratio of the radius of \( \omega_{1} \) to the radius of \( \omega_{2} \).

Answer: \frac{3+\sqrt{5}}{2}


As shown in Figure 1, in the right trapezoid \(ABCD\), \(AD \parallel BC\), \(\angle BAD = \frac{\pi}{2}\), \(AB = BC = 1\), \(AD = 2\). Point \(E\) is the midpoint of \(AD\), and point \(O\) is the intersection of \(AC\) and \(BE\). Triangle \(\triangle ABE\) is folded along \(BE\) to the position of triangle \(\triangle A_1BE\) as shown in Figure 2.
(1) Prove that \(CD \perp\) plane \(A_1OC\).
(2) If plane \(A_1BE \perp\) plane \(BCDE\), find the sine value of the dihedral angle \(B-A_1C-D\).

Answer: \frac{\sqrt{3}}{3}


Regular hexagon \( P_{1} P_{2} P_{3} P_{4} P_{5} P_{6} \) has side length 2. For \( 1 \leq i \leq 6 \), let \( C_{i} \) be a unit circle centered at \( P_{i} \) and \( \ell_{i} \) be one of the internal common tangents of \( C_{i} \) and \( C_{i+2} \), where \( C_{7} = C_{1} \) and \( C_{8} = C_{2} \). Assume that the lines \( \{ \ell_{1}, \ell_{2}, \ell_{3}, \ell_{4}, \ell_{5}, \ell_{6} \} \) bound a regular hexagon. The area of this hexagon can be expressed as \( \sqrt{\frac{a}{b}} \), where \( a \) and \( b \) are relatively prime positive integers. Compute \( 100a + b \).

Answer: 1603


Tanya sequentially wrote down numbers of the form \( n^{7} - 1 \) for natural numbers \( n = 2, 3, \ldots \) and noticed that for \( n = 8 \), the resulting number is divisible by 337. For what smallest \( n > 1 \) will she get a number divisible by 2022?

Answer: 79


In the figure, the center of square \(ABCD\) is \(O\), and its area is \(1989 \text{ cm}^2\). Point \(P\) is a point inside the square such that \(\angle OPB = 45^\circ\) and \(PA : PB = 5 : 14\). Find \(PB\) in centimeters.

Answer: 42


In math class, $2 / 3$ of the students had a problem set, and $4 / 5$ of them brought a calculator. Among those who brought a calculator, the same proportion did not have a problem set as those who did not bring a calculator. What fraction of the students had both a problem set and a calculator?

Answer: \dfrac{8}{15}


Task B-1.5. An isosceles trapezoid, which has no right angles, is divided by a diagonal into two isosceles triangles. Determine the angles of the trapezoid.

Answer: 72^\circ


5. The museum has 16 halls, arranged as shown in the diagram. In half of them, paintings are exhibited, and in the other half, sculptures. From any hall, you can go to any adjacent one (sharing a common wall). During any tour of the museum, the halls alternate: a hall with paintings - a hall with sculptures - a hall with paintings, and so on. The tour starts in hall A, where paintings are displayed, and ends in hall B.

a) Mark with crosses all the halls where paintings are exhibited. Solution. See the diagram.

b) A tourist wants to visit as many halls as possible (travel from hall A to hall B), but visit each hall no more than once. What is the maximum number of halls he can visit? Draw one of his longest routes and prove that he could not have visited more halls.

Answer: 15


10. For any positive integer $n$, let $N_{n}$ be the set of integers from 1 to $n$, i.e., $N_{n}=$ $\{1,2,3, \cdots, n\}$. Now assume that $n \geq 10$. Determine the maximum value of $n$ such that the following inequality
$$
\mathrm{i}_{\substack{a, b \in A \\ a \neq b}}|a-b| \leq 10
$$
holds for each $A \subseteq N_{n}$ with $|A| \geq 10$.

Answer: 99


3. Determine the largest prime number whose all digits are distinct such that any permutation of its digits yields another prime number.

Answer: 97


Each of the spots in a $8\times 8$ chessboard is occupied by either a black or white “horse”. At most how many black horses can be on the chessboard so that none of the horses attack more than one black horse?

[b]Remark:[/b] A black horse could attack another black horse.

Answer: 16


10. On the side $AC$ of $\triangle ABC$, take a point $D$ such that $\angle BDC = \angle ABC$. If $BC = 1$, what is the minimum possible distance between the circumcenter of $\triangle ABC$ and the circumcenter of $\triangle ABD$?

Answer: \dfrac{1}{2}


[ Pythagorean Theorem (direct and inverse) ]  [ Isosceles, inscribed, and circumscribed trapezoids ]

In an isosceles trapezoid \(ABCD\), the base \(AD\) is 4, and the base \(BC\) is 3. Points \(M\) and \(N\) lie on the diagonal \(BD\), with point \(M\) located between points \(B\) and \(N\), and segments \(AM\) and \(CN\) are perpendicular to the diagonal \(BD\). Find \(CN\) if \(BM: DN = 2: 3\).

Answer: \dfrac{\sqrt{15}}{2}


## 3. Math Puzzle $8 / 65$

A crane is being transported to a construction site. To measure the length of the crane, Peter walks from the tip of the crane to the end and takes 15 steps (step length 80 $\mathrm{cm}$).

To walk back from the end to the tip of the crane while the train is moving at a constant speed, he needs 75 steps at the same walking speed.

How long is the crane?

Answer: 20


Example 8. In isosceles $\triangle A B C$, the vertex angle $B=20^{\circ}$. Points $D, E$ are taken on sides $B C, A B$ respectively, such that $\angle D A C=$ $60^{\circ}, \angle A C E=50^{\circ}$. Find $\angle A D E$.

Answer: 30^\circ


4. How to form a rectangle, with all sides greater than 1, from 13 rectangles of sizes $1 \times 1, 2 \times 1, 3 \times 1, \ldots, 13 \times 1$?

Answer: 7 \times 13


Two equilateral triangles \(ABC\) and \(CDE\) are located on the same side of the line \(AE\) and share a single point \(C\). Let \(M, N,\) and \(K\) be the midpoints of segments \(BD, AC,\) and \(CE\) respectively. Prove that the triangle \(MNK\) is equilateral.

Answer: \triangle MNK \text{ is equilateral}


## Task 3 - 200923

Given two circles $k_{1}$ and $k_{2}$ with radii $r_{1}$ and $r_{2}$ respectively, where $r_{1}>r_{2}$.

Furthermore, it is assumed that both circles touch each other externally, meaning they have exactly one common internal tangent. This internal tangent intersects one of the common external tangents of both circles at $P$ and the other common tangent at $Q$.

Determine, under these conditions, the length $P Q$ from $r_{1}$ and $r_{2}$!

Answer: 2\sqrt{r_1 r_2}


9.1. Find the smallest four-digit number that is not a multiple of 10 and has the following property: if the digits are reversed, the resulting number is a divisor of the original number, and the quotient is different from one.

Answer. 8712.

Answer: 8712


3. A circle $\omega$ is circumscribed around triangle $ABC$. A line tangent to $\omega$ at point $C$ intersects the ray $BA$ at point $P$. On the ray $PC$ beyond point $C$, a point $Q$ is marked such that $PC = QC$. The segment $BQ$ intersects the circle $\omega$ again at point $K$. On the smaller arc $BK$ of the circle $\omega$, a point $L$ is marked such that $\angle LAK = \angle CQB$. Find the angle $\angle PCA$, given that $\angle ALQ = 60^{\circ}$.

Answer: 30^{\circ}


Example 4. As shown in the figure, points $A, B, C, D$ lie on the same circle, and $BC=DC=4, AE=6$. The lengths of segments $BE$ and $DE$ are both positive integers. What is the length of $BD$? (1988
National Junior High School Mathematics Competition)

Answer: 7


6. Choose two different non-empty subsets $A$ and $B$ of the set $S=\{1,2, \ldots, 10\}$. The probability that the smallest number in $B$ is greater than the largest number in $A$ is $\qquad$

Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.

Answer: \dfrac{4097}{1045506}


Problem 4. Given is $\triangle A B C$ with the angle bisector of the angle at vertex $A$, $A L(L \in B C)$. The point $M \in A C$ is such that $M L \| A B$. Let $\beta^{\prime}$ be the exterior angle at vertex $B$ of $\triangle A B C$ and $\measuredangle A C B: \beta^{\prime}=5: 11$.

a) Determine the ratio $\measuredangle C A B$ : $\measuredangle A C B$.

b) If $\measuredangle A L B=80^{\circ}$, determine $\measuredangle A L M$.

Answer: 30


9. Buratino the Statistician (from 7th grade, 2 points). Every month, Buratino plays in the "6 out of 45" lottery organized by Karabas-Barabas. In the lottery, there are 45 numbered balls, and in each draw, 6 random winning balls are drawn.

Buratino noticed that in each subsequent draw, there are no balls that appeared in the previous draw: in the second draw, there were no balls from the first, in the third draw, there were no balls from the second, and so on.

Buratino does not believe in events with a probability of less than 0.01. If such an event occurs, Buratino begins to suspect something is amiss. After which draw will Buratino start to suspect that Karabas-Barabas is cheating?

Answer: 7


78. A person is walking along the road, and a car comes from the opposite direction. He asks the driver: "Was there a bicycle behind you?" The driver answers: "I passed a bicycle 10 minutes ago." The person continues walking for 10 minutes and meets the bicycle. If the bicycle's speed is 3 times the walking speed, then the car's speed is $\qquad$ times the walking speed.

Answer: 7


The set $M= \{1;2;3;\ldots ; 29;30\}$ is  divided in $k$
subsets such that if $a+b=n^2, (a,b \in M, a\neq b, n$ is an
integer number $)$, then $a$ and $b$ belong different subsets.
Determine the minimum value of $k$.

Answer: 3


Problem 3. Find the smallest positive integer $n$, for which there exist $n$ different positive integers $a_{1}, a_{2}, \ldots, a_{n}$ satisfying the conditions:

a) the smallest common multiple of $a_{1}, a_{2}, \ldots, a_{n}$ is 1995 ;

b) for each $i, j \in\{1,2, \ldots, n\}$ the numbers $a_{i}$ and $a_{j}$ have a common divisor $\neq 1$;

c) the product $a_{1} a_{2} \ldots a_{n}$ is a perfect square and is divisible by 243 .

Find all $n$-ples $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$, satisfying a), b) and c).

Answer: 7


8. In rectangle $A B C D$, $A D=a, A B=b, b>a$, fold the rectangle so that point $A$ coincides with
point $C$, obtaining the crease $M N, M, V$ are on $A B, C D$ respectively, then unfold it, making the half-plane $M N D^{\prime} A^{\prime}$ form a $18^{\circ}$ dihedral angle with the half-plane $M N C B$, then the angle formed by line $A^{\prime} C$ and line $M N$ is $\qquad$.

Answer: 90^\circ


In cyclic quadrilateral $ABCD$, $\angle DBC = 90^\circ$ and $\angle CAB = 30^\circ$. The diagonals of $ABCD$ meet at $E$. If $\frac{BE}{ED} = 2$ and $CD = 60$, compute $AD$. (Note: a cyclic quadrilateral is a quadrilateral that can be inscribed in a circle.)

Answer: \dfrac{30\sqrt{7}}{7}


5. In the diagram, point $P$ is inside quadrilateral $A B C D$. Also, $D A=D P=D C$ and $A P=A B$. If $\angle A D P=\angle C D P=2 x^{\circ}, \angle B A P=(x+5)^{\circ}$, and $\angle B P C=(10 x-5)^{\circ}$, what is the value of $x$ ?

Answer: 13


17. Streak of Luck (from 9th grade, 4 points). A coin is tossed $n$ times. A streak of luck of length $k$ is a sequence of $k$ consecutive heads. Find the probability that the longest streak of luck in 10 coin tosses will be of length 4.

Answer: \dfrac{139}{1024}


12.222. The base of the pyramid is a rhombus, one of the angles of which is $\alpha$. The lateral faces are equally inclined to the plane of the base. A plane is drawn through the midpoints of two adjacent sides of the base and the vertex of the pyramid, forming an angle $\beta$ with the plane of the base. The area of the resulting section is $S$. Find the side of the rhombus.

Answer: \sqrt{\dfrac{8 S \cos \beta}{\sin \alpha}}


A plane figure's orthogonal projection on two perpendicular planes is a 2 unit side square on each. What is the perimeter of the plane figure if one of its sides is $\sqrt{5}$ units?

Answer: 2\sqrt{5} + 2\sqrt{7}


a) The image on the left shows six points that lie three on each of four lines. Prove that there are 24 different ways to map this set of six points onto itself such that each group of three points lying on one line is mapped into another group of three points lying on a single line.

b) The image on the right shows nine points lying three on each of nine lines, and through each point, three such lines pass. These nine points and nine lines form the famous Pascal configuration. How many ways can we map our set of nine points onto itself so that each group of three points lying on one of our nine lines is mapped onto a group of three points that also lies on one of the lines in our configuration?

c) The same question for the Desargues configuration (consisting of ten points and ten lines) shown in the lower image.

Answer: 24


Quadrangle $A B C D$ is inscribed in a circle with radius 1 in such a way that one diagonal, $A C$, is a diameter of the circle, while the other diagonal, $B D$, is as long as $A B$. The diagonals intersect in $P$. It is known that the length of $P C$ is $\frac{2}{5}$. How long is the side $C D$ ?

![](https://cdn.mathpix.com/cropped/2024_04_17_31b1db04c15387bffd81g-4.jpg?height=271&width=320&top_left_y=207&top_left_x=594)

Figure 1

![](https://cdn.mathpix.com/cropped/2024_04_17_31b1db04c15387bffd81g-4.jpg?height=252&width=323&top_left_y=228&top_left_x=912)

Figure 2

![](https://cdn.mathpix.com/cropped/2024_04_17_31b1db04c15387bffd81g-4.jpg?height=271&width=234&top_left_y=224&top_left_x=1299)

Figure 3

Answer: \dfrac{2}{3}


A collective farm consists of four villages, located at the vertices of a square with a side length of 10 km. It has the means to lay 28 km of roads. Can the collective farm build a road system such that it is possible to travel from any village to any other village?

Answer: Yes


8. Given that the vertex of a parabola is on the $x$-axis, the directrix is the $y$-axis, and the shortest distance from point $A(4,0)$ to a point on the parabola is 2. Then, the number of such parabolas is $\qquad$.

Answer: 3


Answer: 1990


Example 2-44 How many regions does $n$ lines divide the plane into? Assume no $=$ lines intersect at a common point, and each pair of lines intersects.

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


Question 11 Given $\alpha+\beta+\gamma=180^{\circ}(\alpha, \beta, \gamma \geqslant$ $0)$. Find the maximum value of $3 \cos \alpha+4 \cos \beta+5 \cos \gamma$.

Answer: \dfrac{769}{120}


A 9-digit number contains exactly the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 each exactly once, and every pair of adjacent digits form a two-digit number that can be expressed as the product of two single-digit numbers. What is the 9-digit number?

Answer: 728163549


A graph has $30$ points, and each point has $6$ edges. Find the total number of triples such that each pair of points is either joined or not joined.

Answer: 4060


Two types of pieces, bishops and rooks, are to be placed on a $10 \times 10$ chessboard (without necessarily filling it) such that each piece occupies exactly one square of the board. A bishop $B$ is said to attack a piece $P$ if $B$ and $P$ are on the same diagonal and there are no pieces between $B$ and $P$ on that diagonal; a rook $R$ is said to attack a piece $P$ if $R$ and $P$ are on the same row or column and there are no pieces between $R$ and $P$ on that row or column. A piece $P$ is chocolate if no other piece $Q$ attacks $P$. What is the maximum number of chocolate pieces there may be, after placing some pieces on the chessboard?

Answer: 50


On each non-boundary unit segment of an $8 \times 8$ chessboard, we write the number of dissections of the board into dominoes in which this segment lies on the border of a domino. What is the last digit of the sum of all the written numbers?

Answer: 0


Kelvin the Frog was bored in math class one day, so he wrote all ordered triples $(a, b, c)$ of positive integers such that $abc=2310$ on a sheet of paper. Find the sum of all the integers he wrote down. In other words, compute $$\sum_{\substack{abc=2310 \\ a,b,c\in \mathbb{N}}} (a+b+c),$$ where $\mathbb{N}$ denotes the positive integers.

Answer: 49140


You have a $3 \times 2021$ chessboard from which one corner square has been removed. You also have a set of $3031$ identical dominoes, each of which can cover two adjacent chessboard squares. Let $m$ be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps. What is the remainder when $m$ is divided by $19$?

Answer: 1


Archit and Ayush are walking around on the set of points $(x,y)$ for all integers $-1 \leq x,y \leq 1$. Archit starts at $(1,1)$ and Ayush starts at $(1,0)$. Each second, they move to another point in the set chosen uniformly at random among the points with distance $1$ away from them. If the probability that Archit goes to the point $(0,0)$ strictly before Ayush does can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m,n$, compute $m+n$.

Answer: 7


An equilateral triangle is given. A point lies on the incircle of this triangle. If the smallest two distances from the point to the sides of the triangle are $1$ and $4$, the side length of this equilateral triangle can be expressed as $\frac{a\sqrt{b}}{c}$, where $(a, c) = 1$ and $b$ is not divisible by the square of an integer greater than $1$. Find $a + b + c$.

Answer: 34


Suppose that $a_1 = 1$, and for all $n \ge 2$, \[ a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1. \] Furthermore, suppose that \[ b_n = a_1 + a_2 + \ldots + a_n \] for all $n$. If \[ b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k \] for some $k$, find $k$.

Answer: 2022


An infinite sequence of integers, $a_0, a_1, a_2, \dots$, with $a_0 > 0$, has the property that for $n \ge 0$, $a_{n+1} = a_n - b_n$, where $b_n$ is the number having the same sign as $a_n$, but having the digits written in reverse order. For example, if $a_0 = 1210$, $a_1 = 1089$, and $a_2 = -8712$, etc. Find the smallest value of $a_0$ such that $a_n \neq 0$ for all $n \ge 1$.

Answer: 1012


Consider $m$ segments on the real line. Each segment has its two endpoints in the set of integers $\{1, 2, \ldots, 2024\}$, and no two segments have the same length. No segment is entirely contained in another segment, but two segments may partially overlap each other.

What is the maximum value of $m$?

Answer: 1012


Let  $ABCD$  be a rhombus with angle  $\angle A = 60^\circ$ . Let  $E$  be a point, different from  $D$ , on the line  $AD$ . The lines  $CE$  and  $AB$  intersect at  $F$ . The lines  $DF$  and  $BE$  intersect at  $M$ . Determine the angle  $\angle BMD$  as a function of the position of  $E$  on  $AD.$

Answer: 120


On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (Thus the front of a car traveling 52 kilometers per hour will be four car lengths behind the back of the car in front of it.) A photoelectric eye by the side of the road counts the number of cars that pass in one hour. Assuming that each car is 4 meters long and that the cars can travel at any speed, let  $ M$  be the maximum whole number of cars that can pass the photoelectric eye in one hour. Find the quotient when  $ M$  is divided by 10.

Answer: 375


Let  $P$  be the set of all  $2012$  tuples  $(x_1, x_2, \dots, x_{2012})$ , where  $x_i \in \{1,2,\dots 20\}$  for each  $1\leq i \leq 2012$ . The set  $A \subset P$  is said to be decreasing if for each  $(x_1,x_2,\dots ,x_{2012} ) \in A$  any  $(y_1,y_2,\dots, y_{2012})$  satisfying  $y_i \leq x_i (1\leq i \leq 2012)$  also belongs to  $A$ . The set  $B \subset P$  is said to be increasing if for each  $(x_1,x_2,\dots ,x_{2012} ) \in B$  any  $(y_1,y_2,\dots, y_{2012})$  satisfying  $y_i \geq x_i (1\leq i \leq 2012)$  also belongs to  $B$ . Find the maximum possible value of  $f(A,B)= \dfrac {|A\cap B|}{|A|\cdot |B|}$ , where  $A$  and  $B$  are nonempty decreasing and increasing sets ( $\mid \cdot \mid$  denotes the number of elements of the set).

Answer: \frac{1}{20^{2012}}


Let  $A = (a_1, a_2, \ldots, a_{2001})$  be a sequence of positive integers. Let  $m$  be the number of 3-element subsequences  $(a_i,a_j,a_k)$  with  $1 \leq i < j < k \leq 2001$ , such that  $a_j = a_i + 1$  and  $a_k = a_j + 1$ .  Considering all such sequences  $A$ , find the greatest value of  $m$ .

Answer: 667^3


In a space, let  $B$  be a sphere (including the inside) with radius of 1. Line  $l$  intersects with  $B$ , the length of the common part is the line segment with the length of  $\sqrt{3}$ .

(1) Find the distance between the center of  $B$  and  $l$ .

(2) Find the volume of the solid generated by rotating  $B$  about  $l$ .

Answer: \pi^2


We have  $2021$  colors and  $2021$  chips of each color. We place the  $2021^2$  chips in a row. We say that a chip  $F$  is *bad* if there is an odd number of chips that have a different color to  $F$  both to the left and to the right of  $F$ .

(a) Determine the minimum possible number of bad chips.

(b) If we impose the additional condition that each chip must have at least one adjacent chip of the same color, determine the minimum possible number of bad chips.

Answer: 1010


While out for a stroll, you encounter a vicious velociraptor. You start running away to the northeast at  $ 10 \text{m/s}$ , and you manage a three-second head start over the raptor. If the raptor runs at  $ 15\sqrt{2} \text{m/s}$ , but only runs either north or east at any given time, how many seconds do you have until it devours you?

Answer: 6


Answer: 508032


A *strip* is the region between two parallel lines.  Let  $A$  and  $B$  be two strips in a plane.  The intersection of strips  $A$  and  $B$  is a parallelogram  $P$ .  Let  $A'$  be a rotation of  $A$  in the plane by  $60^\circ$ .  The intersection of strips  $A'$  and  $B$  is a parallelogram with the same area as  $P$ .  Let  $x^\circ$  be the measure (in degrees) of one interior angle of  $P$ .  What is the greatest possible value of the number  $x$ ?

Answer: 150


Let  $M_1, M_2, . . ., M_{11}$  be  $5-$ element sets such that  $M_i \cap M_j \neq {\O}$  for all  $i, j \in \{1, . . ., 11\}$ . Determine the minimum possible value of the greatest number of the given sets that have nonempty intersection.

Answer: 4


Richard likes to solve problems from the IMO Shortlist. In 2013, Richard solves  $5$  problems each Saturday and  $7$  problems each Sunday. He has school on weekdays, so he ``only'' solves  $2$ ,  $1$ ,  $2$ ,  $1$ ,  $2$  problems on each Monday, Tuesday, Wednesday, Thursday, and Friday, respectively -- with the exception of December 3, 2013, where he solved  $60$  problems out of boredom. Altogether, how many problems does Richard solve in 2013?

*Proposed by Evan Chen*

Answer: 1099


Alex, a rookie Python developer, is trying to find out how many different regex patterns he can write in a week to match specific text sequences. Each day, he spends 3 hours learning about regex and 2 hours practicing by writing patterns. After 5 days, he has written a total of 15 unique regex patterns. On the weekend, he dedicates 6 hours each day solely to writing new regex patterns. If he writes regex patterns at the same rate on the weekend as he did during his practice sessions on weekdays, how many unique regex patterns can Alex write by the end of the weekend?

Answer: 33


Alex is a young entrepreneur who is excited about the potential of Industry 4.0, particularly in improving efficiency through automation and data analysis. Alex decides to invest in three smart machines for their small manufacturing startup.

The first machine costs $2,500 and can produce 100 units per day. The second machine costs $3,200 and can produce 150 units per day. The third machine costs $4,500 and can produce 200 units per day. Alex estimates that with these machines, they can increase production by 30% due to optimizations in the workflow.

If Alex's startup currently produces 500 units per day, how much will Alex's total daily production be after purchasing and utilizing all three machines with the additional 30% increase in efficiency?

Answer: 1235


A game designer is creating a new board game and has asked their partner, who has a great strategic mindset, to help develop the game's scoring system. In the game, players earn points by completing different tasks.

The designer and their partner decide on the following scoring system:
- Each task completed gives 5 points.
- If a player completes 3 tasks in a row, they earn a bonus of 10 points.
- If a player completes 5 tasks in a single turn, they earn an extra bonus of 20 points.

Their partner tests the game and completes 12 tasks in 3 turns. During the first turn, they complete 3 tasks, during the second turn, they complete 5 tasks, and during the third turn, they complete 4 tasks.

How many total points did their partner earn by the end of the third turn?

Answer: 110


Jamie is teaching their child, Alex, how to track animals in the forest. They find a trail that a deer has been using. Jamie explains that the deer visits a nearby river to drink water twice a day, once in the morning and once in the evening. Each trip to the river takes the deer 20 minutes, and it spends 10 minutes drinking water before returning. If Alex and Jamie spend 3 hours tracking the deer in the forest and observe it making 4 round trips to the river in that time, how much time in total does the deer spend at the river, including the drinking time, during their tracking session?

Answer: 40


The local organic farmer, who supplies fresh produce for the spa's wellness cafe, grows three types of vegetables: carrots, cucumbers, and kale. Each week, the wellness cafe needs 30 pounds of carrots, 25 pounds of cucumbers, and 20 pounds of kale. The farmer's garden produces 5 pounds of carrots, 8 pounds of cucumbers, and 4 pounds of kale each day. If the farmer wants to ensure they have enough produce to supply the cafe for the entire week, how many total pounds of vegetables must the garden produce in one week?

Answer: 119


Dr. Emma, a marine biologist from New Zealand, is studying the population of shell creatures in a coastal area. She discovers that there are 120 hermit crabs, 75 sea snails, and 45 scallops living in the region. During her study, she finds that every week, the population of hermit crabs increases by 10%, while the sea snails decrease by 5%, and the scallops remain constant. After one week, how many shell creatures are there in total in the area?

Answer: 248.25


A film director is working on a new movie project and needs to schedule some meetings with Brian Helgeland to discuss the script revisions. They have already had 3 meetings this month, each lasting 2 hours. They plan to have twice as many meetings next month, with each meeting lasting 30 minutes longer than the previous ones. How many total hours will the film director and Brian Helgeland spend in meetings next month?

Answer: 15


A former diplomat is attending a conference to discuss diplomatic conflicts. There are 15 countries participating in the conference. Each country brings 3 delegates, including the former diplomat himself, who represents his own country. During the conference, the former diplomat holds a private meeting with 2 delegates from each of the other countries to gain insider perspectives on various conflicts. If each meeting takes 15 minutes, how many minutes in total does the former diplomat spend in private meetings with the delegates?

Answer: 210


Professor Smith, an esteemed professor of military history, is conducting a demonstration of a famous historical battle. He sets up a battlefield on a large table with miniature soldiers to recreate the scene. In his demonstration, he uses 75 red soldiers to represent one army and 120 blue soldiers to represent the opposing army. For a special effect, he adds 3 cannons for every 25 soldiers.

How many cannons are used for both armies combined?

Answer: 21


A music composer named Alex has written 12 original songs, and each song is protected by copyright for 70 years. Alex decides to release these songs gradually over 6 years, releasing an equal number of songs each year. Additionally, Alex earns $50 in royalties per song each year after a song is released.

1. How many songs does Alex release each year?
2. How much in royalties does Alex earn from all the released songs in the first year?
3. How much total royalty income will Alex earn from all the songs by the end of the 6th year, assuming each song continues to earn $50 per year once released?

Answer: 2100


A historian is studying the development of the automobile industry in the United States. She is particularly interested in how the number of cars produced has changed over time. In the year 1900, the U.S. produced approximately 4,000 cars. By 1929, this number had increased to about 4.8 million cars. If the historian wants to examine the average annual increase in car production between 1900 and 1929, how many more cars were produced on average each year during this period?

Answer: 165379


A novelist is developing a story with a character who has two conflicting beliefs: faith and doubt. To illustrate this complexity, the novelist decides to represent faith with the number 7 and doubt with the number 3. In the story, the character's journey is divided into 5 chapters. In each chapter, the strength of the character's faith increases by 2 due to various experiences, while the strength of their doubt decreases by 1. At the end of each chapter, the overall strength of the character's beliefs is calculated by multiplying the strength of faith and doubt together. What is the total strength of the character's beliefs at the end of all 5 chapters?

Answer: -20


Jamie is a data scientist who loves sharing research papers about data-driven approaches in psychology. Each month, Jamie writes 4 new research papers. To share these papers with the community, Jamie uses a special platform that allows each paper to be shared 3 times with different psychology groups. Additionally, Jamie receives feedback from each group, which takes him 2 days per paper to analyze. If Jamie continues this process for 5 months, how many days will Jamie spend analyzing feedback in total?

Answer: 120


Detective Sam is investigating a case and has been given a list of TV shows that might contain clues. However, Sam despises one particular TV show and is skeptical about using it as a source of clues. Instead, Sam decides to focus on finding clues from other sources and comes up with a meticulous plan.

Sam spends 2 hours each day at the library researching old case files. On Mondays, Sam spends an additional 1 hour at a local café interviewing a potential witness. On Wednesdays, Sam dedicates 3 hours to cross-referencing information from online databases, and on Fridays, Sam spends 2 hours analyzing CCTV footage from the crime scene.

If Sam continues this routine for two weeks, how many total hours will Sam spend on the investigation, excluding any time related to the TV show?

Answer: 40


Let $S = \{1, 2, ..., n\}$, where $n \ge 1$. Each of the $2^n$ subsets of $S$ is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set $T \subseteq S$, we then write $f(T)$ for the number of subsets of T that are blue.
Determine the number of colorings that satisfy the following condition: for any subsets $T_1$ and $T_2$ of $S$, \[f(T_1)f(T_2) = f(T_1 \cup T_2)f(T_1 \cap T_2).\]

Solution
Define function: $C(T)=1$ if the set T is colored blue, and $C(T)=0$ if $T$ is colored red.
Define the $\text{Core} =\text{intersection of all } T \text{ where } C(T)=1$.
The empty set is denoted as $\varnothing$, $\cap$ denotes intersection, and $\cup$ denotes union. Let $S_n=\{n\}$ are one-element subsets.
Let $m_{c_k}=\dbinom{m}{k} = \frac{m!}{k!(m-k)!}$ denote m choose k.

(Case I) $f(\varnothing)=1$. Then for distinct m and k, $f(S_m \cup S_k)=f(S_m)f(S_k)$, meaning only if $S_m$ and $S_k$ are both blue is their union blue. Namely $C(S_m \cup S_k)=C(S_m)C(S_k).$
Similarly, for distinct $m,n,k$, $f(S_m \cup S_k \cup Sn)=f(S_m \cup S_k)f(S_n)$, $C(S_m \cup S_k \cup S_n)=C(S_m)C(S_k)C(S_n)$. This procedure of determination continues to $S$. Therefore, if $T=\{a_1,a_2, \cdots a_k\}$, then $C(T)=C(S_{a_1})C(S_{a_2}) \cdots C(S_{a_k})$. All colorings thus determined by the free colors chosen for subsets of one single elements satisfy the condition.  There are $2^n$ colorings in this case.
(Case II.)  $f(\varnothing)=0$.
(Case II.1)  $\text{Core}=\varnothing$. Then either (II.1.1) there exist two nonintersecting subsets A and B, $C(A)=C(B)=1$, but f$(A)f(B)=0$, a contradiction, or (II.1.2) all subsets has $C(T)=0$, which is easily confirmed to satisfy the condition $f(T_1)f(T_2)=f(T_1 \cap T_2)f(T_1 \cup T_2)$.  There is one coloring in this case.
(Case II.2) Core = a subset of 1 element. WLOG, $C(S_1)=1$. Then $f(S_1)=1$, and subsets containing element 1 may be colored blue. $f(S_1 \cup S_m)f(S_1\cup S_n)=f(S_1 \cup S_m \cup S_n)$, namely $C(S_1 \cup S_m \cup S_n)=C(S_m \cup S_1)C(S_n \cup S_1)$. Now S_1 functions as the $\varnothing$ in case I, with $n-1$ elements to combine into a base of $n-1$ two-element sets, and all the other subsets are determined. There are $2^{n-1}$ colorings for each choice of core. However, there are nC1 = n such cores. Hence altogether there are $n2^{n-1}$ colorings in this case.
(Case II.3) Core = a subset of 2 elements. WLOG, let $C(S_1 \cup S_2)=1$. Only subsets containing the core may be colored blue. With the same reasoning as in the preceding case, there are $(nC2)2^{n-2}$ colorings.
$\dots$
(Case II.n+1) Core = S. Then $C(S)=1$, with all other subsets $C(T)=0$, there is $1=\dbinom{n}{n}2^0$
Combining all the cases, we have $1+\left[2^n+\dbinom{n}{1}2^{n-1}+\dbinom{n}{2}2^{n-2}+ \cdots + \dbinom{n}{n}2^0\right]=\boxed{1+3^n}$ colorings. sponsored by ALLEN

Answer: 1 + 3^n


[Inscribed Quadrilaterals (Miscellaneous)]

Quadrilateral $ABCD$ is inscribed in a circle. The bisectors of angles $B$ and $C$ intersect at a point lying on the segment $AD$.

Find $AD$, if $AB=5, CD=3$.

Answer: 8


2. On the board, there are several (not necessarily distinct) prime numbers such that their product is $2020$ times greater than their sum. Determine their smallest possible number. (Patrik Bak)

Answer: 7


Let \( n \geq 4 \) be an integer. Consider strictly positive integers \( a_{1}, \ldots, a_{n} \) placed on a circle. Suppose that each term \( a_{i}(1 \leq i \leq n) \) divides the sum of its two neighbors, i.e., there exists an integer \( k_{i} \) such that

\[ \frac{a_{i-1}+a_{i+1}}{a_{i}} = k_{i} \]

with the convention \( a_{0} = a_{n} \) and \( a_{n+1} = a_{1} \). Show that

\[ 2n \leq k_{1} + k_{2} + \cdots + k_{n} < 3n \]

Answer: 2n \leq k_{1} + k_{2} + \cdots + k_{n} < 3n


Example 4.3.3 $n$ points, no three of which are collinear, are connected by line segments, and these segments are colored with two colors, red and blue. If for any coloring, there must exist 12 monochromatic triangles, find the minimum value of $n$.

Answer: 9


7. As shown in Figure 2, in plane $\alpha$, $\triangle A B C$ and $\triangle A_{1} B_{1} C_{1}$ are on opposite sides of line $l$, with no common points with $l$, and are symmetric about line $l$. Now, if plane $\alpha$ is folded along line $l$ to form a right dihedral angle, then the six points $A, B, C, A_{1}, B_{1}, C_{1}$ can determine $\qquad$ planes (answer with a number).

Answer: 11


9.2. The bisectors of the external angles $B$ and $C$ of triangle $ABC$ intersect at point $M$. a) Can angle $BMC$ be obtuse? b) Find angle $BAC$ if it is known that $\angle BMC = \frac{\angle BAM}{2}$.

Answer: \text{No}


10. (10 points) Teacher Guo has a cake to share with 4 or 5 children, so Teacher Guo cuts the cake into several pieces, which may not be of the same size; in this way, whether 4 or 5 children come, they can take some pieces so that each person gets the same amount. Therefore, Teacher Guo must cut the cake into at least $\qquad$ pieces.

Answer: 8


10. The sequence $a_{0}, a_{1}, a_{2}, \cdots, a_{n}$ satisfies $a_{0}=\sqrt{3}, a_{n+1}=\left[a_{n}\right]+\frac{1}{\left\{a_{n}\right\}}\left(\left[a_{n}\right],\left\{a_{n}\right\}\right.$ represent the integer part and fractional part of $a_{n}$, respectively), then $a_{2016}=$ $\qquad$ .

Answer: 3024 + \sqrt{3}


We say that a group of $k$ boys is $n-acceptable$ if removing any boy from the group one can always find, in the other $k-1$ group,  a group of $n$ boys such that everyone knows each other. For each $n$, find the biggest $k$ such that in any group of $k$ boys that is $n-acceptable$ we must always have a group of $n+1$ boys such that everyone knows each other.

Answer: 2n-1


Let $n \in \mathbf{N}_{+}$. Find the positive real number $T(n)$ such that the set of real numbers $x$ satisfying the inequality $\sum_{k=1}^{n} \frac{k}{x-k} \geqslant T(n)$ is the union of disjoint intervals, and the total length of these intervals is 2009.

Answer: \dfrac{n(n+1)}{4018}


4. On the bank of the river stand 10 sheikhs, each with a harem of 100 wives. Also at the bank stands an $n$-person yacht. By law, a woman should not be on the same bank, on the yacht, or even at a transfer point with a man if her husband is not present. What is the smallest $n$ for which all the sheikhs and their wives can cross to the other bank without breaking the law?

Answer: 10


18. (6 points) There are 300 balls in total, including white balls and red balls, and 100 boxes. Each box contains 3 balls, with 27 boxes containing 1 white ball, 42 boxes containing 2 or 3 red balls, and the number of boxes containing 3 white balls is the same as the number of boxes containing 3 red balls. Therefore, the total number of white balls is.

balls.

Answer: 158


$ABCD$ is a cyclic quadrilateral such that $AB = BC = CA$. Diagonals $AC$ and $BD$ intersect at $E$. Given that $BE = 19$ and $ED = 6$, find the possible values of $AD$.

Answer: 10


3. Given is a rectangle $A B C D$ whose side lengths differ by $4 \mathrm{~cm}$, and its perimeter is 16 $\mathrm{dm}$, with $|A B|>|B C|$. Point $M$ is the midpoint of side $\overline{B C}$ of the rectangle. Point $N$ lies on side $\overline{C D}$ and it is given that $|N C|=3|D N|$. Segments $\overline{A M}$ and $\overline{B N}$ intersect at point $P$. Is the area of triangle $A B P$ greater or the area of quadrilateral $P M C N$? By how much?

Answer: \dfrac{399}{2}


Exercise 5. Let ABCD be a convex quadrilateral, with $\widehat{A B C}=90^{\circ}, \widehat{B A D}=\widehat{A D C}=80^{\circ}$. Let $M$ and $N$ be points on $[A D]$ and $[B C]$ such that $\widehat{C D N}=\widehat{A B M}=20^{\circ}$. Suppose finally $M D=A B$. What is $\widehat{M N B}$?

Answer: 70^{\circ}


Determine the number of ten-digit positive integers with the following properties:
$\bullet$ Each of the digits $0, 1, 2, . . . , 8$ and $9$ is contained exactly once.
$\bullet$ Each digit, except $9$, has a neighbouring digit that is larger than it.
(Note. For example, in the number $1230$, the digits $1$ and $3$ are the neighbouring digits of $2$  while $2$ and $0$ are the neighbouring digits of $3$. The digits $1$ and $0$ have only one neighbouring digit.)

[i](Karl Czakler)[/i]

Answer: 256


$7 \cdot 82$ Given 155 birds are perched on a circle $C$. If $\overparen{P_{i} P_{j}} \leqslant 10^{\circ}$, then the birds $P_{i}$ and $P_{j}$ are said to be mutually visible. Find the minimum number of pairs of mutually visible birds (you can assume that multiple birds can occupy the same position).

Answer: 270


In the diagram, $A B C D H E F G$ is a truncated square-based pyramid. (Note that $D$ is hidden.) In particular, $A B C D$ is a square with side length $7, E F G H$ is a square with side length 1, and $A E=B F=C G=D H$. Each of the four trapezoidal faces is partially shaded. On each face, only the region that is below both diagonals is unshaded. If the height of the truncated pyramid is 4 , what is the total

![](https://cdn.mathpix.com/cropped/2024_04_17_8918b015730641e4f574g-1.jpg?height=317&width=596&top_left_y=1912&top_left_x=1358)
shaded area?

Answer: \dfrac{75}{4}


[ The ratio of the areas of triangles with a common base or a common height ] Complex [ The area of a figure is equal to the sum of the areas of the figures into which it is divided ] Class

On side $A B$ of triangle $A B C$, point $E$ is taken, and on side $B C$, point $D$ is taken, such that $A E=2$, and $C D=11$. Lines $A D$ and $C E$ intersect at point $O$. Find the area of quadrilateral $B D O E$, if $A B=B C=8$, and $A C=6$.

#

Answer: \dfrac{189\sqrt{55}}{88}


Given the set $P$ consisting of positive integers not exceeding 2012, explicitly $P = \{1,2,3, \cdots, 2012\}$. The set $A$ is a subset of $P$, where $|A|$ denotes the number of elements in set $A$, and $S(A)$ denotes the sum of all elements in set $A$.

1. If the difference between any two numbers in set $A$ is not a multiple of 101, find the maximum value of $|A|$.
2. If the difference between any two numbers in set $A$ is not a multiple of 101, and the sum of any two numbers is also not a multiple of 101, find the maximum value of $|A|$.
3. If the difference between any two numbers in set $A$ is not a multiple of 101, the sum of any two numbers is also not a multiple of 101, and $S(A) = 2012$, find the maximum value of $|A|$.

Answer: 101


8. As shown in Figure 3, in quadrilateral $A B C D$, $A D / / B C, A B=A C$, a perpendicular line is drawn from point $D$ to $B C$, with the foot of the perpendicular being $F, D F$ intersects $A C$ at point $E$. If $B E=2 D F$, and $B E$ bisects $\angle A B C$, then $\angle B A D=$ $\qquad$ . (China)

Answer: 144^\circ


There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes:  Job, Hobby, Nationality, Pet, Transport.
The attributes have the following possible values:
- Job: designer, entrepreneur, freelancer, manager, musician, videographer
- Hobby: camping, photography, puzzles, reading, sudoku, traveling
- Nationality: american, argentine, egyptian, french, japanese, nigerian
- Pet: fish, frog, goat, horse, mouse, turtle
- Transport: bike, helicopter, motorbike, roller, ship, tram

Given the following premises about the line of people:
- Hobby:photography == Pet:turtle or Hobby:photography == Nationality:argentine or both
- Transport:roller is on the left or right of Transport:motorbike
- Nationality:japanese is somewhere to the right of Nationality:american
- Job:freelancer != Nationality:argentine
- Transport:motorbike is on the left of Transport:tram
- Nationality:nigerian is on the right of Nationality:french
- Nationality:japanese is somewhere to the right of Transport:motorbike
- Job:entrepreneur == Nationality:japanese or Job:entrepreneur == Transport:helicopter, but not both
- Nationality:french and Transport:roller have the same parity positions
- Pet:horse is on the left of Transport:motorbike
- Hobby:sudoku and Transport:ship have the same parity positions
- Hobby:photography is somewhere between Transport:roller and Pet:fish
- Hobby:puzzles and Transport:bike have the same parity positions
- Pet:frog is somewhere to the left of Hobby:sudoku
- Transport:motorbike and Hobby:camping have different parity positions
- Job:manager and Hobby:camping have different parity positions
- Transport:ship is on the right of Hobby:reading
- Nationality:japanese == Transport:roller or Nationality:japanese == Pet:mouse or both
- Job:videographer == Transport:helicopter or Nationality:egyptian == Job:videographer or both
- Job:musician is between Pet:goat and Pet:horse
- Pet:goat != Hobby:camping
- Hobby:traveling is somewhere between Pet:turtle and Pet:horse
- Hobby:puzzles != Transport:tram
- Job:entrepreneur is on the right of Job:freelancer
- Nationality:argentine is on the right of Pet:goat

Answer the following question:

Question: Question: What is the Pet of the person who has/is traveling??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: goat


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Hobby, Beverage, Music-Genre, Nationality, Pet.
The attributes have the following possible values:
- Job: analyst, engineer, lawyer, librarian
- Hobby: camping, collecting, gardening, writing
- Beverage: hot-chocolate, milk, tea, water
- Music-Genre: country, folk, rock, trance
- Nationality: canadian, indian, indonesian, japanese
- Pet: frog, goat, goldfish, horse

Given the following premises about the line of people:
- Nationality:indonesian is on the right of Hobby:gardening
- Beverage:milk is between Hobby:gardening and Hobby:collecting
- Music-Genre:trance is on the left or right of Pet:goat
- Beverage:water is somewhere to the right of Hobby:writing
- Job:analyst is on the right of Job:librarian
- Music-Genre:trance is somewhere to the left of Beverage:tea
- Nationality:canadian != Job:analyst
- Pet:frog is on the right of Pet:goldfish
- Beverage:water is somewhere to the left of Nationality:japanese
- Job:lawyer is somewhere to the right of Music-Genre:country
- Job:lawyer is in an odd position
- Music-Genre:country is somewhere to the left of Pet:goat
- Nationality:indonesian == Music-Genre:folk
- Beverage:water is on the right of Beverage:hot-chocolate

Answer the following question:

Question: Question: What is the Music-Genre of the person who is analyst??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: trance


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Hobby, Food, Pet, Sport, Transport.
The attributes have the following possible values:
- Job: accountant, coach, fisherman, security-guard
- Hobby: camping, fishing, rock-climbing, traveling
- Food: blueberry, cranberry, papaya, strawberry
- Pet: ferret, goldfish, hamster, horse
- Sport: cricket, skiing, volleyball, weightlifting
- Transport: bike, bus, motorbike, trike

Given the following premises about the line of people:
- Hobby:traveling and Sport:cricket have the same parity positions
- Hobby:camping and Transport:trike have different parity positions
- Job:fisherman is not to the left of Pet:hamster
- Sport:cricket is not to the right of Transport:bus
- Pet:ferret and Job:accountant have different parity positions
- Food:cranberry and Transport:bus have the same parity positions
- Sport:weightlifting != Hobby:rock-climbing
- Job:security-guard and Transport:bus have the same parity positions
- Transport:motorbike is not to the right of Pet:hamster
- Food:blueberry is not to the right of Transport:bike
- Sport:volleyball and Job:security-guard have the same parity positions
- Sport:cricket != Job:accountant
- Food:strawberry is not to the right of Food:papaya
- Food:strawberry == Hobby:fishing or Hobby:fishing == Job:accountant or both
- Pet:goldfish is not to the left of Pet:ferret
- Food:blueberry != Transport:motorbike or Food:blueberry != Hobby:camping or both
- Transport:bus and Pet:goldfish have the same parity positions
- Job:coach is somewhere between Hobby:camping and Job:accountant
- Pet:horse is not to the left of Hobby:camping
- Pet:hamster and Food:strawberry have the same parity positions
- Job:security-guard is somewhere between Sport:skiing and Transport:trike
- Transport:bus == Sport:cricket or Food:blueberry == Transport:bus or both

Answer the following question:

Question: Question: What is the Hobby of the person who has/is weightlifting??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: fishing


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Nationality, Movie-Genre, Beverage, Pet.
The attributes have the following possible values:
- Nationality: american, german, italian, polish
- Movie-Genre: comedy, scientific, time-travel, zombie
- Beverage: almond-milk, coffee, fanta, milk
- Pet: chinchilla, hamster, mouse, snake

Given the following premises about the line of people:
- Beverage:coffee is somewhere to the right of Movie-Genre:comedy
- Movie-Genre:time-travel is somewhere between Pet:chinchilla and Beverage:fanta
- Nationality:american is between Pet:hamster and Pet:mouse
- Nationality:american is somewhere to the right of Pet:mouse
- Nationality:german and Movie-Genre:time-travel have the same parity positions
- Nationality:polish is not to the right of Movie-Genre:comedy
- Nationality:german is on the left of Movie-Genre:scientific
- Beverage:almond-milk is somewhere to the right of Pet:hamster
- Pet:snake is not to the left of Beverage:coffee

Answer the following question:

Question: Question: What is the Nationality of the person who has/is comedy??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: polish


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Hobby, Movie-Genre, Nationality, Sport, Transport.
The attributes have the following possible values:
- Job: accountant, firefighter, manager, scientist
- Hobby: baking, collecting, cooking, singing
- Movie-Genre: documentary, family, romance, zombie
- Nationality: brazilian, chinese, malaysian, turkish
- Sport: climbing, lacrosse, skateboarding, volleyball
- Transport: snowmobile, subway, train, trike

Given the following premises about the line of people:
- Job:firefighter is somewhere between Job:manager and Transport:trike
- Job:accountant is somewhere to the right of Nationality:turkish
- Nationality:chinese is somewhere between Sport:climbing and Nationality:turkish
- Movie-Genre:documentary is on the far left or far right
- Sport:volleyball is somewhere to the right of Sport:lacrosse
- Nationality:turkish is on the left or right of Hobby:baking
- Movie-Genre:zombie is on the left or right of Hobby:collecting
- Hobby:cooking == Transport:trike or Hobby:cooking == Job:firefighter or both
- Job:firefighter is not to the right of Nationality:turkish
- Nationality:turkish is somewhere to the left of Nationality:malaysian
- Movie-Genre:family and Sport:skateboarding have the same parity positions
- Hobby:singing is not to the left of Nationality:turkish
- Sport:lacrosse is not to the left of Nationality:turkish
- Transport:snowmobile is between Movie-Genre:zombie and Movie-Genre:romance
- Hobby:singing == Nationality:malaysian or Movie-Genre:zombie == Hobby:singing, but not both
- Hobby:baking == Transport:train or Hobby:baking == Nationality:malaysian or both
- Transport:subway is on the far left or far right

Answer the following question:

Question: Question: What is the Movie-Genre of the person who is firefighter??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: family


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Hobby, Music-Genre, Pet, Sport.
The attributes have the following possible values:
- Hobby: board-games, card-games, collecting, dancing, skydiving
- Music-Genre: classical, disco, jazz, reggae, techno
- Pet: ferret, hamster, rabbit, rat, snake
- Sport: baseball, cricket, sailing, swimming, weightlifting

Given the following premises about the line of people:
- Hobby:board-games is in the middle
- Music-Genre:techno is on the left or right of Music-Genre:classical
- Hobby:card-games is somewhere to the right of Music-Genre:reggae
- Pet:rat is somewhere to the right of Sport:swimming
- Hobby:board-games == Music-Genre:disco or Pet:rabbit == Hobby:board-games or both
- Pet:hamster is somewhere to the right of Hobby:card-games
- Music-Genre:classical is somewhere between Hobby:card-games and Sport:baseball
- Music-Genre:disco is somewhere between Pet:rat and Music-Genre:techno
- Sport:baseball is somewhere to the left of Sport:sailing
- Hobby:collecting is not to the left of Music-Genre:jazz
- Hobby:skydiving == Sport:cricket or Hobby:skydiving == Pet:snake, but not both
- Hobby:card-games == Sport:cricket or Hobby:card-games == Music-Genre:disco or both
- Pet:snake != Hobby:skydiving or Pet:snake != Sport:swimming or both

Answer the following question:

Question: Question: What is the Pet of the person who has/is skydiving??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: snake


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Job, Beverage, Movie-Genre, Music-Genre, Pet, Sport.
The attributes have the following possible values:
- Job: chef, doctor, lawyer, security-guard, writer
- Beverage: almond-milk, fanta, hot-chocolate, milk, sprite
- Movie-Genre: comedy, mystery, scientific, sports, time-travel
- Music-Genre: blues, disco, folk, rock, trance
- Pet: cat, frog, goat, guinea-pig, hedgehog
- Sport: basketball, cricket, golf, handball, skiing

Given the following premises about the line of people:
- Sport:golf is on the left or right of Music-Genre:blues
- Movie-Genre:scientific is on the far left
- Beverage:almond-milk is on the left or right of Movie-Genre:mystery
- Job:doctor is on the left of Movie-Genre:mystery
- Sport:skiing is somewhere to the left of Beverage:milk
- Pet:goat is on the far left or far right
- Sport:handball == Pet:hedgehog
- Music-Genre:trance is between Music-Genre:disco and Beverage:sprite
- Beverage:fanta is on the left or right of Sport:basketball
- Job:writer == Pet:frog
- Sport:basketball is on the left or right of Music-Genre:rock
- Beverage:sprite is somewhere to the left of Pet:frog
- Pet:frog is on the right of Music-Genre:rock
- Sport:skiing is on the right of Job:security-guard
- Movie-Genre:time-travel is in an even position
- Music-Genre:blues is somewhere to the right of Beverage:almond-milk
- Job:chef is on the far left or far right
- Pet:guinea-pig is in an even position
- Music-Genre:disco is somewhere to the right of Movie-Genre:sports

Answer the following question:

Question: Question: What is the Music-Genre of the person who has/is time-travel??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: trance


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Job, Hobby, Music-Genre, Nationality, Pet.
The attributes have the following possible values:
- Job: freelancer, lawyer, musician, project-manager, teacher
- Hobby: board-games, hiking, magic-tricks, sudoku, woodworking
- Music-Genre: disco, r&b, salsa, soul, techno
- Nationality: argentine, german, indonesian, malaysian, nigerian
- Pet: dog, fish, frog, goat, mouse

Given the following premises about the line of people:
- Job:freelancer is between Nationality:malaysian and Pet:goat
- Music-Genre:salsa is on the left or right of Nationality:nigerian
- Hobby:magic-tricks is not to the left of Music-Genre:r&b
- Music-Genre:soul is somewhere to the left of Hobby:hiking
- Hobby:board-games is in the middle
- Music-Genre:disco is not to the right of Hobby:sudoku
- Pet:fish != Hobby:sudoku or Pet:fish != Nationality:malaysian or both
- Nationality:argentine is on the left or right of Pet:fish
- Hobby:woodworking and Pet:mouse have different parity positions
- Job:musician == Music-Genre:disco or Nationality:nigerian == Job:musician or both
- Music-Genre:r&b is on the left or right of Pet:goat
- Music-Genre:soul is on the left or right of Hobby:hiking
- Pet:dog is somewhere to the right of Job:lawyer
- Nationality:nigerian is on the left or right of Job:teacher
- Nationality:nigerian is not to the right of Hobby:woodworking
- Music-Genre:r&b is on the far left or far right
- Nationality:nigerian and Music-Genre:techno have different parity positions
- Job:lawyer == Music-Genre:disco or Job:lawyer == Hobby:sudoku or both
- Nationality:german is on the far left or far right

Answer the following question:

Question: Question: What is the Nationality of the person who has/is board-games??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: indonesian


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Job, Hobby, Movie-Genre, Music-Genre, Sport, Transport.
The attributes have the following possible values:
- Job: analyst, designer, dressmaker, engineer, pilot
- Hobby: hiking, photography, rock-climbing, skydiving, writing
- Movie-Genre: adventure, comedy, mystery, romance, spy
- Music-Genre: folk, gospel, pop, reggae, salsa
- Sport: baseball, ice-hockey, lacrosse, rugby, soccer
- Transport: bike, jet-ski, taxi, train, trike

Given the following premises about the line of people:
- Music-Genre:folk is on the far left or far right
- Transport:trike is in an even position
- Job:analyst is on the far right
- Sport:ice-hockey is on the left or right of Movie-Genre:romance
- Hobby:hiking is between Music-Genre:pop and Transport:jet-ski
- Movie-Genre:spy == Music-Genre:gospel
- Transport:train is somewhere to the right of Transport:taxi
- Job:dressmaker == Sport:soccer
- Sport:rugby is somewhere between Movie-Genre:romance and Job:designer
- Hobby:rock-climbing is on the left or right of Movie-Genre:adventure
- Hobby:writing is somewhere to the right of Hobby:hiking
- Hobby:hiking is on the left of Movie-Genre:comedy
- Job:dressmaker is on the left or right of Movie-Genre:comedy
- Transport:bike is on the right of Hobby:rock-climbing
- Hobby:photography is somewhere between Transport:taxi and Transport:train
- Movie-Genre:adventure is on the left of Movie-Genre:spy
- Music-Genre:gospel == Sport:baseball
- Hobby:rock-climbing == Job:pilot
- Sport:lacrosse is somewhere to the right of Music-Genre:reggae

Answer the following question:

Question: Question: What is the Sport of the person who has/is salsa??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: lacrosse


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Nationality, Hobby, Pet, Transport.
The attributes have the following possible values:
- Nationality: french, german, indonesian, polish, turkish
- Hobby: board-games, hiking, magic-tricks, puzzles, woodworking
- Pet: cat, chinchilla, goat, goldfish, rat
- Transport: bike, helicopter, skateboard, train, tram

Given the following premises about the line of people:
- Hobby:magic-tricks and Nationality:turkish have different parity positions
- Nationality:french is somewhere to the right of Nationality:indonesian
- Transport:bike is somewhere to the right of Transport:tram
- Hobby:magic-tricks is somewhere between Transport:train and Nationality:german
- Transport:bike and Hobby:magic-tricks have the same parity positions
- Transport:train is not to the right of Nationality:polish
- Nationality:german == Transport:train or Nationality:german == Pet:rat, but not both
- Pet:goat is somewhere to the right of Pet:cat
- Hobby:magic-tricks is somewhere to the right of Hobby:woodworking
- Pet:cat and Transport:train have different parity positions
- Nationality:turkish != Pet:chinchilla
- Pet:goldfish and Nationality:german have different parity positions
- Nationality:polish is somewhere to the left of Nationality:indonesian
- Transport:train is not to the right of Nationality:turkish
- Transport:helicopter is somewhere to the left of Hobby:woodworking
- Hobby:puzzles is not to the left of Hobby:woodworking
- Hobby:board-games is not to the left of Hobby:hiking

Answer the following question:

Question: Question: What is the Nationality of the person who has/is goldfish??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: french


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Hobby, Movie-Genre, Music-Genre, Pet.
The attributes have the following possible values:
- Job: bartender, electrician, manager, musician
- Hobby: cooking, fishing, magic-tricks, puzzles
- Movie-Genre: action, comedy, martial-arts, scientific
- Music-Genre: ambient, electronic, hip-hop, metal
- Pet: dog, lizard, pony, snake

Given the following premises about the line of people:
- Hobby:magic-tricks is not to the left of Pet:pony
- Movie-Genre:action and Job:electrician have different parity positions
- Pet:snake is on the left of Pet:lizard
- Movie-Genre:martial-arts is on the left or right of Job:manager
- Job:electrician is somewhere between Music-Genre:metal and Movie-Genre:comedy
- Hobby:magic-tricks is not to the right of Music-Genre:electronic
- Pet:pony != Music-Genre:ambient
- Hobby:magic-tricks is on the right of Movie-Genre:scientific
- Hobby:cooking is not to the right of Music-Genre:hip-hop
- Movie-Genre:comedy is somewhere to the right of Pet:pony
- Job:musician is not to the left of Hobby:fishing
- Music-Genre:metal is not to the left of Hobby:magic-tricks
- Music-Genre:hip-hop is on the right of Job:bartender

Answer the following question:

Question: Question: What is the Movie-Genre of the person who has/is ambient??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: scientific


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Nationality, Food, Hobby, Beverage, Sport, Transport.
The attributes have the following possible values:
- Nationality: brazilian, chinese, french, malaysian, polish
- Food: cucumber, kiwi, lime, pumpkin, spinach
- Hobby: chess, photography, rock-climbing, singing, sudoku
- Beverage: almond-milk, coffee, lemonade, sprite, water
- Sport: cycling, lacrosse, parkour, snowboarding, swimming
- Transport: boat, scooter, snowmobile, train, van

Given the following premises about the line of people:
- Nationality:chinese == Hobby:sudoku
- Sport:snowboarding == Beverage:coffee
- Transport:train is somewhere to the right of Food:pumpkin
- Sport:lacrosse == Food:kiwi
- Beverage:sprite is on the right of Nationality:french
- Nationality:polish == Sport:cycling
- Nationality:polish is somewhere to the right of Beverage:water
- Transport:boat is on the left of Transport:snowmobile
- Sport:swimming is somewhere to the right of Food:cucumber
- Sport:lacrosse is on the left or right of Transport:scooter
- Hobby:rock-climbing is somewhere to the left of Hobby:photography
- Sport:cycling is on the left or right of Hobby:rock-climbing
- Nationality:polish is on the left of Sport:parkour
- Nationality:brazilian == Food:cucumber
- Transport:van is somewhere to the right of Beverage:lemonade
- Food:lime is somewhere to the left of Hobby:chess
- Beverage:almond-milk is on the left or right of Nationality:french

Answer the following question:

Question: Question: What is the Hobby of the person who has/is cycling??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: photography


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Job, Pet, Sport, Transport.
The attributes have the following possible values:
- Job: doctor, entrepreneur, lawyer, project-manager, social-worker
- Pet: cat, chinchilla, frog, hamster, horse
- Sport: biathlon, skateboarding, swimming, tennis, weightlifting
- Transport: bike, boat, helicopter, train, tram

Given the following premises about the line of people:
- Job:lawyer is not to the right of Sport:swimming
- Job:doctor == Pet:hamster or Job:doctor == Transport:boat or both
- Pet:chinchilla and Job:lawyer have the same parity positions
- Sport:biathlon is on the left of Pet:cat
- Job:social-worker == Pet:chinchilla or Job:social-worker == Sport:weightlifting or both
- Transport:boat is on the left or right of Sport:biathlon
- Transport:bike is somewhere between Sport:biathlon and Sport:tennis
- Job:project-manager and Transport:boat have the same parity positions
- Pet:frog == Sport:tennis or Sport:tennis == Transport:train or both
- Transport:helicopter is between Sport:skateboarding and Sport:tennis
- Pet:horse is on the far right
- Job:entrepreneur is on the left or right of Transport:train

Answer the following question:

Question: Question: What is the Sport of the person who is social-worker??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: weightlifting


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Hobby, Nationality, Pet.
The attributes have the following possible values:
- Job: freelancer, librarian, photographer, security-guard
- Hobby: card-games, hiking, photography, writing
- Nationality: british, french, german, turkish
- Pet: chinchilla, frog, lizard, rabbit

Given the following premises about the line of people:
- Job:librarian is on the left or right of Pet:chinchilla
- Nationality:british is not to the left of Hobby:card-games
- Job:photographer is not to the right of Hobby:card-games
- Job:photographer is on the right of Nationality:turkish
- Job:security-guard and Pet:rabbit have different parity positions
- Pet:chinchilla is somewhere to the left of Hobby:writing
- Hobby:photography is on the right of Hobby:hiking
- Hobby:writing and Pet:lizard have different parity positions
- Nationality:british is somewhere between Pet:frog and Pet:lizard
- Nationality:german is between Job:librarian and Nationality:turkish

Answer the following question:

Question: Question: What is the Job of the person who has/is photography??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: photographer


There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes:  Job, Food, Beverage, Movie-Genre, Music-Genre, Nationality.
The attributes have the following possible values:
- Job: designer, entrepreneur, freelancer, musician, paramedic, scientist
- Food: apple, blueberry, cauliflower, garlic, grapefruit, lettuce
- Beverage: 7up, coffee, juice, lemonade, milk, sprite
- Movie-Genre: action, documentary, family, fantasy, martial-arts, scientific
- Music-Genre: funk, pop, r&b, rock, techno, trance
- Nationality: brazilian, canadian, indian, japanese, mexican, spanish

Given the following premises about the line of people:
- Job:musician is somewhere to the right of Food:blueberry
- Food:lettuce is somewhere between Job:designer and Food:blueberry
- Job:paramedic is somewhere between Movie-Genre:scientific and Beverage:juice
- Nationality:canadian is somewhere to the left of Beverage:coffee
- Nationality:japanese == Beverage:sprite or Nationality:japanese == Movie-Genre:action, but not both
- Beverage:7up is somewhere to the right of Music-Genre:r&b
- Job:musician is somewhere between Nationality:indian and Nationality:spanish
- Nationality:canadian == Movie-Genre:family or Nationality:canadian == Beverage:coffee or both
- Food:lettuce is somewhere to the right of Job:designer
- Nationality:indian is somewhere to the left of Nationality:mexican
- Beverage:juice == Movie-Genre:action or Beverage:juice == Music-Genre:techno, but not both
- Movie-Genre:documentary is somewhere to the right of Movie-Genre:family
- Beverage:sprite == Nationality:indian
- Beverage:lemonade and Music-Genre:techno have different parity positions
- Nationality:brazilian == Music-Genre:pop
- Music-Genre:pop is on the far left
- Food:cauliflower is somewhere to the right of Food:grapefruit
- Job:entrepreneur == Nationality:mexican or Beverage:lemonade == Job:entrepreneur or both
- Food:garlic == Beverage:coffee or Movie-Genre:martial-arts == Food:garlic or both
- Music-Genre:techno is somewhere between Job:designer and Music-Genre:r&b
- Music-Genre:funk is somewhere to the left of Music-Genre:techno
- Music-Genre:trance is on the right of Beverage:7up
- Music-Genre:rock == Job:musician or Job:musician == Food:cauliflower, but not both
- Nationality:indian is on the left of Food:apple
- Job:scientist == Nationality:spanish or Movie-Genre:family == Job:scientist, but not both
- Job:musician is not to the right of Movie-Genre:family

Answer the following question:

Question: Question: What is the Beverage of the person who has/is r&b??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: lemonade