There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Food, Hobby, Music-Genre, Pet, Sport.
The attributes have the following possible values:
- Food: avocado, cabbage, garlic, pineapple, radish
- Hobby: collecting, gardening, photography, reading, singing
- Music-Genre: blues, country, funk, pop, rock
- Pet: bird, fish, goldfish, hedgehog, lizard
- Sport: climbing, parkour, sailing, soccer, surfing

Given the following premises about the line of people:
- Pet:fish is on the left or right of Pet:hedgehog
- Hobby:reading is on the right of Food:pineapple
- Pet:lizard is on the right of Music-Genre:blues
- Sport:soccer is on the left of Food:radish
- Food:garlic is on the left of Sport:sailing
- Music-Genre:country is somewhere to the right of Food:avocado
- Music-Genre:rock is on the right of Pet:goldfish
- Hobby:collecting is between Sport:surfing and Pet:fish
- Food:cabbage is on the left or right of Hobby:photography
- Sport:parkour is on the far right
- Music-Genre:blues is on the left or right of Music-Genre:pop
- Pet:lizard == Food:cabbage
- Music-Genre:funk is somewhere to the right of Music-Genre:country
- Hobby:gardening is somewhere to the right of Music-Genre:rock

Answer the following question:

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

Answer: radish


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, Movie-Genre, Pet, Sport.
The attributes have the following possible values:
- Job: entrepreneur, mechanic, nurse, paramedic, software-developer
- Movie-Genre: animation, comedy, crime, disaster, epic
- Pet: cat, ferret, fish, frog, snake
- Sport: badminton, basketball, cricket, rowing, rugby

Given the following premises about the line of people:
- Movie-Genre:disaster and Job:mechanic have different parity positions
- Sport:basketball and Job:entrepreneur have the same parity positions
- Job:mechanic is not to the right of Job:nurse
- Movie-Genre:disaster is not to the left of Job:entrepreneur
- Sport:rowing and Pet:ferret have the same parity positions
- Movie-Genre:epic is somewhere between Movie-Genre:comedy and Job:nurse
- Job:software-developer is somewhere between Sport:cricket and Movie-Genre:crime
- Movie-Genre:epic is not to the right of Movie-Genre:comedy
- Movie-Genre:animation is not to the left of Job:nurse
- Sport:rowing is not to the right of Sport:badminton
- Sport:rugby == Pet:frog or Sport:rugby == Movie-Genre:animation or both
- Sport:basketball == Pet:cat or Pet:cat == Job:mechanic or both
- Job:paramedic and Sport:badminton have the same parity positions
- Sport:basketball and Movie-Genre:crime have the same parity positions
- Movie-Genre:disaster is not to the left of Pet:fish

Answer the following question:

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

Answer: mechanic


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, Beverage, Music-Genre, Nationality, Pet, Transport.
The attributes have the following possible values:
- Job: accountant, chef, journalist, manager
- Beverage: almond-milk, juice, mirinda, water
- Music-Genre: d&b, folk, house, jazz
- Nationality: american, brazilian, japanese, nigerian
- Pet: bird, goat, hedgehog, rabbit
- Transport: roller, subway, trike, van

Given the following premises about the line of people:
- Beverage:almond-milk is not to the right of Music-Genre:jazz
- Beverage:juice is not to the right of Music-Genre:folk
- Transport:subway and Pet:rabbit have different parity positions
- Pet:goat is somewhere to the left of Job:chef
- Pet:rabbit is not to the left of Job:journalist
- Beverage:mirinda is on the far left or far right
- Transport:roller is somewhere between Job:manager and Transport:van
- Music-Genre:d&b and Transport:van have different parity positions
- Pet:hedgehog is somewhere between Beverage:juice and Music-Genre:folk
- Nationality:brazilian is on the left or right of Transport:roller
- Transport:van == Pet:bird or Transport:van == Job:journalist, but not both
- Transport:subway is between Transport:trike and Job:accountant
- Beverage:almond-milk == Nationality:japanese or Pet:bird == Beverage:almond-milk or both
- Nationality:brazilian is somewhere to the left of Music-Genre:house
- Pet:hedgehog is not to the left of Nationality:american

Answer the following question:

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

Answer: japanese


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, Transport.
The attributes have the following possible values:
- Job: chef, doctor, scientist, social-worker
- Hobby: board-games, reading, singing, sudoku
- Movie-Genre: horror, satire, thriller, time-travel
- Music-Genre: blues, country, jazz, punk
- Pet: chinchilla, frog, mouse, snake
- Transport: airplane, motorbike, subway, van

Given the following premises about the line of people:
- Job:social-worker is on the right of Movie-Genre:horror
- Movie-Genre:time-travel is not to the left of Hobby:sudoku
- Hobby:reading is somewhere to the left of Transport:van
- Hobby:reading == Pet:frog
- Transport:motorbike == Pet:frog or Transport:motorbike == Music-Genre:blues, but not both
- Music-Genre:country is in an odd position
- Music-Genre:country and Movie-Genre:satire have different parity positions
- Job:scientist == Transport:van or Job:scientist == Pet:mouse, but not both
- Pet:frog is somewhere to the left of Hobby:sudoku
- Music-Genre:punk is not to the right of Transport:motorbike
- Hobby:board-games is not to the right of Pet:mouse
- Pet:chinchilla == Transport:airplane
- Transport:subway is on the far left or far right
- Music-Genre:punk is somewhere to the right of Music-Genre:blues
- Music-Genre:blues is somewhere to the right of Job:chef

Answer the following question:

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

Answer: satire


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Beverage, Food, Hobby, Music-Genre, Sport, Transport.
The attributes have the following possible values:
- Beverage: 7up, hot-chocolate, lemonade, milk
- Food: cherry, eggplant, spinach, strawberry
- Hobby: camping, collecting, magic-tricks, traveling
- Music-Genre: electronic, funk, gospel, hip-hop
- Sport: biathlon, golf, lacrosse, skiing
- Transport: bike, bus, helicopter, train

Given the following premises about the line of people:
- Music-Genre:gospel and Transport:bus have different parity positions
- Hobby:collecting is not to the left of Sport:biathlon
- Music-Genre:gospel == Beverage:7up or Beverage:7up == Food:cherry or both
- Sport:lacrosse is not to the right of Beverage:hot-chocolate
- Transport:bike is not to the right of Music-Genre:funk
- Music-Genre:gospel == Food:spinach or Music-Genre:gospel == Sport:golf or both
- Transport:bus and Hobby:magic-tricks have the same parity positions
- Food:strawberry is not to the right of Transport:bus
- Beverage:7up is somewhere between Sport:biathlon and Beverage:milk
- Hobby:collecting and Music-Genre:electronic have the same parity positions
- Sport:skiing and Hobby:camping have different parity positions
- Music-Genre:hip-hop is somewhere between Music-Genre:electronic and Food:strawberry
- Sport:biathlon is not to the right of Transport:bike
- Music-Genre:electronic is somewhere between Sport:skiing and Food:eggplant
- Transport:bike and Music-Genre:funk have the same parity positions
- Transport:train is not to the left of Hobby:magic-tricks

Answer the following question:

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

Answer: 7up


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, Food, Hobby, Beverage, Movie-Genre, Sport.
The attributes have the following possible values:
- Job: architect, doctor, journalist, pilot, police-officer
- Food: cauliflower, corn, kiwi, lettuce, potato
- Hobby: camping, collecting, drawing, hiking, puzzles
- Beverage: 7up, almond-milk, iced-tea, milk, water
- Movie-Genre: epic, fantasy, romance, satire, zombie
- Sport: badminton, climbing, lacrosse, rugby, surfing

Given the following premises about the line of people:
- Hobby:drawing and Food:cauliflower have different parity positions
- Movie-Genre:zombie and Beverage:almond-milk have different parity positions
- Job:police-officer is on the left or right of Job:doctor
- Movie-Genre:zombie is somewhere to the right of Beverage:almond-milk
- Movie-Genre:zombie is somewhere to the left of Hobby:hiking
- Hobby:drawing == Job:journalist or Job:journalist == Sport:rugby, but not both
- Sport:surfing is not to the left of Food:cauliflower
- Food:lettuce is somewhere between Food:potato and Sport:climbing
- Job:pilot is not to the right of Movie-Genre:romance
- Sport:surfing == Job:journalist or Sport:surfing == Food:cauliflower or both
- Movie-Genre:fantasy is not to the right of Beverage:7up
- Sport:rugby and Hobby:puzzles have different parity positions
- Hobby:hiking is on the left or right of Beverage:milk
- Job:journalist and Movie-Genre:fantasy have different parity positions
- Sport:badminton is somewhere to the left of Movie-Genre:epic
- Food:potato is somewhere to the right of Sport:climbing
- Food:corn and Hobby:puzzles have the same parity positions
- Movie-Genre:romance == Hobby:camping or Movie-Genre:romance == Beverage:water or both
- Sport:climbing is in an even position
- Hobby:collecting is somewhere between Job:police-officer and Job:journalist
- Hobby:camping is on the left or right of Beverage:water
- Food:cauliflower != Movie-Genre:satire or Hobby:puzzles != Food:cauliflower or both
- Movie-Genre:epic is on the left or right of Movie-Genre:fantasy
- Job:doctor is somewhere to the left of Sport:badminton
- Beverage:7up is not to the right of Sport:lacrosse

Answer the following question:

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

Answer: romance


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, Food, Pet, Transport.
The attributes have the following possible values:
- Nationality: egyptian, indian, indonesian, spanish
- Food: artichoke, avocado, orange, zucchini
- Pet: ferret, fish, goldfish, rabbit
- Transport: car, roller, tram, van

Given the following premises about the line of people:
- Transport:van is on the left or right of Transport:tram
- Pet:goldfish and Food:orange have the same parity positions
- Transport:car is not to the right of Transport:van
- Nationality:egyptian != Pet:ferret
- Pet:ferret is not to the left of Food:zucchini
- Food:artichoke is somewhere to the left of Pet:goldfish
- Transport:van is on the left or right of Transport:car
- Food:artichoke and Transport:tram have different parity positions
- Nationality:spanish is on the far right
- Food:avocado is on the far left or far right
- Pet:rabbit is on the far right
- Pet:fish and Food:zucchini have different parity positions
- Nationality:indonesian is somewhere to the left of Pet:ferret

Answer the following question:

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

Answer: indian


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, Beverage, Music-Genre, Nationality, Pet.
The attributes have the following possible values:
- Job: librarian, nurse, paramedic, pilot, teacher
- Hobby: camping, dancing, gardening, singing, writing
- Beverage: 7up, hot-chocolate, juice, mirinda, water
- Music-Genre: disco, metal, rock, soul, techno
- Nationality: american, brazilian, german, indian, indonesian
- Pet: goat, goldfish, hedgehog, lizard, pony

Given the following premises about the line of people:
- Beverage:7up is on the far right
- Hobby:singing is somewhere to the right of Beverage:mirinda
- Hobby:singing == Beverage:mirinda or Beverage:mirinda == Pet:goat or both
- Music-Genre:soul is somewhere to the left of Nationality:indian
- Hobby:writing is not to the left of Music-Genre:techno
- Music-Genre:disco is not to the left of Pet:lizard
- Pet:pony and Job:librarian have the same parity positions
- Pet:goat is on the left or right of Hobby:writing
- Pet:lizard != Job:teacher
- Job:paramedic is not to the right of Nationality:brazilian
- Nationality:indonesian == Beverage:7up or Beverage:7up == Job:teacher or both
- Hobby:singing is in an odd position
- Nationality:brazilian is on the left or right of Beverage:hot-chocolate
- Hobby:singing is on the left or right of Pet:pony
- Pet:goldfish is somewhere to the right of Job:librarian
- Beverage:mirinda is not to the left of Music-Genre:metal
- Beverage:7up == Hobby:camping or Hobby:camping == Music-Genre:disco, but not both
- Nationality:american and Music-Genre:soul have the same parity positions
- Nationality:german is somewhere to the left of Job:paramedic
- Music-Genre:techno is on the left or right of Nationality:indian
- Pet:goat is somewhere between Beverage:water and Hobby:gardening
- Job:nurse is on the right of Job:pilot
- Music-Genre:soul is not to the left of Job:librarian

Answer the following question:

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

Answer: german


There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes:  Nationality, Movie-Genre, Beverage, Pet, Sport.
The attributes have the following possible values:
- Nationality: canadian, colombian, german, indian, italian, malaysian
- Movie-Genre: comedy, family, martial-arts, mystery, sports, thriller
- Beverage: 7up, coffee, cola, lemonade, sprite, tea
- Pet: bird, frog, goat, hamster, hedgehog, rabbit
- Sport: cycling, sailing, skiing, soccer, surfing, swimming

Given the following premises about the line of people:
- Sport:surfing == Beverage:coffee
- Nationality:italian is somewhere to the right of Beverage:tea
- Movie-Genre:family is on the left or right of Nationality:malaysian
- Pet:rabbit is somewhere between Pet:hedgehog and Nationality:malaysian
- Nationality:italian == Beverage:lemonade
- Beverage:cola is on the left or right of Movie-Genre:sports
- Pet:hamster and Movie-Genre:comedy have the same parity positions
- Beverage:7up is somewhere to the right of Nationality:indian
- Nationality:german is somewhere to the right of Nationality:indian
- Beverage:coffee is somewhere to the left of Movie-Genre:martial-arts
- Movie-Genre:martial-arts is on the left or right of Nationality:italian
- Beverage:tea is not to the left of Nationality:colombian
- Movie-Genre:family is not to the left of Sport:skiing
- Pet:frog is somewhere to the right of Sport:soccer
- Nationality:colombian is on the left of Beverage:lemonade
- Movie-Genre:martial-arts is not to the right of Nationality:indian
- Sport:sailing and Nationality:indian have the same parity positions
- Beverage:cola is on the far left
- Sport:cycling is somewhere between Pet:bird and Nationality:colombian
- Pet:hedgehog is on the left or right of Sport:cycling
- Sport:sailing is somewhere to the left of Pet:goat
- Sport:cycling is somewhere to the left of Movie-Genre:thriller

Answer the following question:

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

Answer: hamster


There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes:  Food, Movie-Genre, Music-Genre, Transport.
The attributes have the following possible values:
- Food: blueberry, grapefruit, lettuce, pear, pepper, pomegranate
- Movie-Genre: disaster, mystery, scientific, spy, thriller, time-travel
- Music-Genre: blues, dubstep, house, pop, reggae, trance
- Transport: airplane, boat, skateboard, taxi, trike, van

Given the following premises about the line of people:
- Movie-Genre:thriller is somewhere between Music-Genre:reggae and Music-Genre:trance
- Movie-Genre:scientific is not to the left of Food:lettuce
- Food:pepper is not to the left of Music-Genre:reggae
- Music-Genre:dubstep != Food:pear
- Food:pear is somewhere to the left of Food:lettuce
- Music-Genre:reggae is not to the right of Transport:skateboard
- Transport:trike == Movie-Genre:thriller or Transport:trike == Food:pomegranate or both
- Movie-Genre:mystery is somewhere to the left of Music-Genre:blues
- Movie-Genre:spy and Transport:skateboard have different parity positions
- Music-Genre:pop is not to the left of Movie-Genre:scientific
- Food:lettuce is somewhere between Food:blueberry and Music-Genre:dubstep
- Food:pear is somewhere to the right of Transport:airplane
- Food:pear is not to the left of Transport:skateboard
- Movie-Genre:scientific == Music-Genre:trance or Music-Genre:trance == Food:lettuce, but not both
- Food:pear == Transport:boat or Transport:boat == Music-Genre:house or both
- Food:grapefruit and Music-Genre:dubstep have different parity positions
- Transport:trike is not to the left of Movie-Genre:disaster
- Transport:van == Music-Genre:trance or Movie-Genre:spy == Transport:van, but not both
- Music-Genre:blues is somewhere to the left of Music-Genre:reggae

Answer the following question:

Question: Question: What is the Movie-Genre of the person who has/is pop??
 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:  Job, Beverage, Movie-Genre, Sport, Transport.
The attributes have the following possible values:
- Job: analyst, coach, journalist, lawyer, videographer
- Beverage: 7up, cola, hot-chocolate, soy-milk, water
- Movie-Genre: action, animation, fantasy, musical, sports
- Sport: baseball, lacrosse, parkour, sailing, snowboarding
- Transport: bike, bus, helicopter, ship, tram

Given the following premises about the line of people:
- Beverage:cola == Job:analyst or Beverage:cola == Transport:bike or both
- Beverage:hot-chocolate is not to the left of Sport:sailing
- Movie-Genre:sports is somewhere between Movie-Genre:animation and Movie-Genre:action
- Job:analyst and Sport:sailing have the same parity positions
- Beverage:soy-milk and Job:coach have the same parity positions
- Movie-Genre:action != Sport:baseball or Movie-Genre:action != Transport:bus or both
- Beverage:soy-milk and Transport:bus have different parity positions
- Beverage:hot-chocolate == Transport:bus or Transport:bus == Movie-Genre:musical or both
- Sport:parkour != Transport:bike or Transport:bike != Movie-Genre:sports or both
- Movie-Genre:sports == Transport:helicopter or Transport:helicopter == Beverage:cola or both
- Job:lawyer != Sport:baseball or Movie-Genre:animation != Job:lawyer or both
- Transport:helicopter == Beverage:water or Beverage:water == Sport:snowboarding, but not both
- Movie-Genre:sports and Transport:helicopter have the same parity positions
- Transport:ship is not to the right of Job:videographer
- Sport:baseball != Movie-Genre:animation or Sport:baseball != Beverage:hot-chocolate or both
- Transport:helicopter == Beverage:hot-chocolate or Transport:helicopter == Sport:baseball or both
- Job:lawyer is not to the right of Sport:sailing
- Sport:lacrosse == Job:coach or Job:coach == Beverage:hot-chocolate, but not both
- Movie-Genre:animation is somewhere between Movie-Genre:fantasy and Movie-Genre:action
- Job:lawyer is somewhere between Job:coach and Movie-Genre:fantasy
- Transport:bus is not to the right of Transport:bike
- Transport:bus == Beverage:water or Beverage:water == Job:journalist or both
- Job:videographer is not to the left of Job:journalist
- Sport:lacrosse and Beverage:hot-chocolate have the same parity positions
- Transport:bike and Movie-Genre:sports have different parity positions
- Job:journalist and Transport:tram have the same parity positions
- Transport:ship == Beverage:7up or Transport:ship == Job:videographer, but not both
- Transport:ship is not to the right of Transport:tram
- Sport:lacrosse is somewhere between Transport:bus and Sport:baseball
- Beverage:soy-milk and Movie-Genre:animation have different parity positions
- Beverage:soy-milk == Sport:lacrosse or Beverage:soy-milk == Movie-Genre:fantasy or both
- Movie-Genre:sports and Transport:ship have the same parity positions
- Sport:parkour == Transport:bike or Sport:parkour == Movie-Genre:musical, but not both
- Sport:lacrosse and Movie-Genre:animation have different parity positions

Answer the following question:

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

Answer: sailing


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, Music-Genre, Transport.
The attributes have the following possible values:
- Job: dancer, lawyer, librarian, nurse, security-guard, videographer
- Hobby: board-games, chess, collecting, puzzles, woodworking, writing
- Music-Genre: folk, gospel, hip-hop, pop, punk, salsa
- Transport: bus, helicopter, jet-ski, roller, scooter, taxi

Given the following premises about the line of people:
- Hobby:board-games is not to the right of Transport:jet-ski
- Hobby:writing and Music-Genre:pop have different parity positions
- Hobby:puzzles and Music-Genre:punk have different parity positions
- Music-Genre:hip-hop is not to the right of Job:lawyer
- Music-Genre:pop is somewhere to the left of Transport:scooter
- Transport:jet-ski == Music-Genre:salsa or Transport:jet-ski == Job:lawyer, but not both
- Transport:roller and Job:lawyer have the same parity positions
- Hobby:board-games is somewhere to the left of Hobby:collecting
- Music-Genre:salsa and Hobby:collecting have different parity positions
- Hobby:chess is somewhere to the left of Job:security-guard
- Music-Genre:salsa != Transport:roller
- Hobby:board-games is somewhere to the right of Job:nurse
- Hobby:writing and Job:security-guard have different parity positions
- Job:videographer is not to the right of Music-Genre:pop
- Job:dancer is not to the right of Job:nurse
- Job:librarian is not to the right of Music-Genre:hip-hop
- Hobby:puzzles and Transport:helicopter have different parity positions
- Hobby:puzzles is not to the right of Job:dancer
- Job:lawyer and Hobby:puzzles have different parity positions
- Music-Genre:punk is somewhere to the right of Music-Genre:gospel
- Music-Genre:folk != Transport:taxi
- Music-Genre:punk is somewhere to the left of Transport:taxi
- Hobby:collecting is not to the left of Hobby:writing
- Music-Genre:salsa and Transport:bus have different parity positions
- Job:librarian is not to the left of Transport:scooter

Answer the following question:

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

Answer: librarian


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, Beverage, Music-Genre, Pet, Transport.
The attributes have the following possible values:
- Job: chef, mechanic, police-officer, project-manager, scientist, software-developer
- Hobby: chess, dancing, drawing, filmmaking, skydiving, video-games
- Beverage: almond-milk, cola, lemonade, mirinda, soy-milk, sprite
- Music-Genre: classical, indie, pop, r&b, salsa, trance
- Pet: chinchilla, fish, horse, mouse, rat, turtle
- Transport: quad-bike, scooter, snowmobile, taxi, trike, van

Given the following premises about the line of people:
- Pet:turtle is not to the right of Music-Genre:classical
- Pet:horse != Hobby:chess
- Beverage:sprite is not to the right of Pet:horse
- Hobby:filmmaking != Pet:turtle
- Hobby:drawing != Pet:turtle
- Transport:trike == Beverage:cola or Transport:trike == Job:police-officer or both
- Transport:quad-bike != Hobby:skydiving
- Job:chef != Beverage:mirinda or Hobby:filmmaking != Job:chef or both
- Job:mechanic == Beverage:almond-milk or Music-Genre:indie == Job:mechanic or both
- Beverage:almond-milk and Job:software-developer have different parity positions
- Pet:rat != Transport:van or Transport:van != Hobby:filmmaking or both
- Music-Genre:indie is not to the right of Music-Genre:classical
- Beverage:almond-milk is not to the left of Pet:chinchilla
- Hobby:filmmaking is not to the left of Hobby:drawing
- Job:software-developer != Pet:horse or Job:software-developer != Transport:quad-bike or both
- Hobby:skydiving is not to the left of Transport:trike
- Transport:trike is not to the right of Beverage:soy-milk
- Job:scientist is not to the right of Pet:horse
- Job:project-manager is not to the right of Job:scientist
- Transport:trike != Hobby:drawing
- Music-Genre:pop is not to the left of Job:chef
- Music-Genre:indie is somewhere between Job:police-officer and Transport:van
- Job:mechanic == Transport:trike or Beverage:mirinda == Job:mechanic or both
- Job:police-officer != Hobby:drawing or Hobby:drawing != Beverage:cola or both
- Transport:scooter and Hobby:dancing have the same parity positions
- Pet:horse is not to the left of Beverage:cola
- Transport:snowmobile is not to the right of Music-Genre:pop
- Beverage:lemonade and Job:police-officer have the same parity positions
- Job:mechanic and Music-Genre:indie have the same parity positions
- Job:project-manager == Pet:turtle or Pet:turtle == Music-Genre:indie or both
- Pet:turtle != Job:mechanic or Job:mechanic != Beverage:lemonade or both
- Beverage:sprite != Music-Genre:indie
- Hobby:drawing is not to the right of Beverage:soy-milk
- Job:scientist is not to the right of Job:software-developer
- Beverage:cola != Transport:van
- Pet:chinchilla != Beverage:sprite or Beverage:sprite != Hobby:filmmaking or both
- Pet:turtle and Transport:quad-bike have different parity positions
- Transport:quad-bike != Hobby:video-games or Transport:quad-bike != Music-Genre:trance or both
- Transport:scooter != Beverage:lemonade or Transport:scooter != Pet:horse or both
- Transport:van != Hobby:drawing
- Music-Genre:salsa != Beverage:soy-milk or Beverage:soy-milk != Job:chef or both
- Job:police-officer is not to the right of Pet:fish
- Pet:turtle and Beverage:almond-milk have different parity positions
- Job:project-manager and Music-Genre:classical have the same parity positions
- Music-Genre:trance == Pet:turtle or Transport:trike == Music-Genre:trance or both
- Beverage:soy-milk and Transport:taxi have the same parity positions
- Pet:horse and Hobby:filmmaking have different parity positions
- Beverage:soy-milk == Job:chef or Beverage:soy-milk == Transport:taxi, but not both
- Job:software-developer is not to the left of Transport:quad-bike
- Beverage:soy-milk is not to the left of Beverage:almond-milk
- Beverage:mirinda is not to the right of Hobby:filmmaking
- Music-Genre:classical is not to the right of Music-Genre:pop
- Beverage:lemonade == Transport:snowmobile or Transport:snowmobile == Pet:turtle, but not both
- Hobby:skydiving is not to the right of Music-Genre:salsa
- Beverage:soy-milk is not to the right of Job:scientist
- Beverage:almond-milk == Transport:snowmobile or Beverage:almond-milk == Music-Genre:classical, but not both
- Transport:van != Pet:fish
- Transport:scooter and Hobby:dancing have the same parity positions
- Pet:mouse == Beverage:lemonade or Pet:mouse == Hobby:video-games or both
- Hobby:skydiving is somewhere between Music-Genre:trance and Beverage:sprite
- Music-Genre:indie and Beverage:cola have different parity positions
- Beverage:soy-milk != Transport:snowmobile
- Job:project-manager and Hobby:skydiving have different parity positions
- Beverage:mirinda and Pet:horse have the same parity positions
- Job:police-officer == Hobby:filmmaking or Transport:trike == Job:police-officer, but not both
- Job:scientist is not to the left of Pet:turtle
- Transport:taxi is not to the right of Beverage:cola
- Music-Genre:salsa is somewhere between Music-Genre:pop and Pet:chinchilla
- Job:software-developer == Transport:trike or Transport:trike == Hobby:video-games or both
- Transport:van == Music-Genre:classical or Transport:van == Beverage:almond-milk or both
- Transport:snowmobile and Hobby:dancing have the same parity positions
- Pet:rat != Music-Genre:pop
- Pet:fish is not to the right of Transport:taxi
- Pet:turtle is not to the left of Beverage:lemonade
- Pet:mouse is not to the left of Hobby:video-games
- Job:mechanic is not to the right of Job:project-manager
- Pet:horse == Transport:quad-bike or Transport:quad-bike == Hobby:filmmaking, but not both
- Beverage:mirinda is not to the left of Hobby:video-games
- Music-Genre:r&b is somewhere between Hobby:skydiving and Beverage:cola
- Pet:chinchilla == Job:project-manager or Music-Genre:indie == Pet:chinchilla, but not both
- Music-Genre:salsa is not to the right of Job:chef

Answer the following question:

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

Answer: scientist


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Food, Hobby, Movie-Genre, Sport.
The attributes have the following possible values:
- Food: kiwi, peas, pineapple, spinach, tomato
- Hobby: baking, chess, cooking, dancing, woodworking
- Movie-Genre: animation, horror, martial-arts, superhero, thriller
- Sport: climbing, cricket, skiing, snowboarding, swimming

Given the following premises about the line of people:
- Food:tomato is on the left or right of Food:pineapple
- Hobby:dancing == Sport:skiing
- Hobby:cooking is somewhere to the left of Food:spinach
- Hobby:woodworking is in an even position
- Movie-Genre:martial-arts is somewhere to the right of Hobby:woodworking
- Movie-Genre:horror == Sport:snowboarding
- Hobby:baking is somewhere to the left of Food:kiwi
- Movie-Genre:animation is somewhere to the right of Sport:skiing
- Food:pineapple is somewhere to the right of Food:peas
- Sport:cricket is in an even position
- Hobby:cooking is in an even position
- Food:kiwi is on the right of Sport:climbing
- Movie-Genre:thriller is somewhere between Food:peas and Food:pineapple
- Movie-Genre:superhero is on the left or right of Movie-Genre:animation

Answer the following question:

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

Answer: martial-arts


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, Hobby, Food, Pet, Sport, Transport.
The attributes have the following possible values:
- Nationality: german, mexican, pakistani, russian
- Hobby: filmmaking, hiking, reading, video-games
- Food: grapes, lime, peach, pumpkin
- Pet: bird, fish, hedgehog, horse
- Sport: badminton, handball, surfing, volleyball
- Transport: car, roller, subway, tram

Given the following premises about the line of people:
- Nationality:mexican is somewhere between Pet:fish and Nationality:german
- Food:lime == Sport:handball or Transport:car == Food:lime, but not both
- Nationality:pakistani == Food:peach or Transport:tram == Nationality:pakistani or both
- Transport:car != Food:peach
- Hobby:filmmaking is not to the right of Sport:handball
- Hobby:reading != Nationality:russian
- Food:lime is not to the left of Food:grapes
- Hobby:hiking is not to the right of Transport:roller
- Sport:handball == Nationality:mexican or Sport:handball == Transport:subway or both
- Hobby:hiking != Food:peach
- Nationality:russian is not to the right of Sport:surfing
- Transport:car and Nationality:russian have different parity positions
- Nationality:mexican == Food:grapes or Nationality:mexican == Hobby:filmmaking or both
- Transport:tram != Hobby:reading or Sport:badminton != Transport:tram or both
- Nationality:mexican != Pet:bird or Nationality:mexican != Food:lime or both
- Sport:badminton is not to the left of Pet:horse
- Food:pumpkin and Sport:surfing have the same parity positions
- Sport:volleyball != Nationality:german or Transport:tram != Sport:volleyball or both
- Food:peach is not to the right of Hobby:video-games
- Nationality:russian is not to the left of Nationality:mexican
- Pet:fish and Nationality:pakistani have different parity positions
- Nationality:russian is somewhere between Pet:hedgehog and Hobby:filmmaking

Answer the following question:

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

Answer: surfing


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, Beverage, Movie-Genre, Nationality, Pet.
The attributes have the following possible values:
- Job: coach, librarian, manager, nurse, security-guard, software-developer
- Beverage: fanta, hot-chocolate, juice, soy-milk, tea, water
- Movie-Genre: animation, comedy, family, martial-arts, musical, scientific
- Nationality: argentine, brazilian, chinese, egyptian, malaysian, pakistani
- Pet: bird, hamster, lizard, pony, rabbit, snake

Given the following premises about the line of people:
- Movie-Genre:scientific != Nationality:malaysian
- Job:coach is not to the right of Job:coach
- Beverage:fanta is somewhere between Pet:snake and Pet:lizard
- Movie-Genre:family == Nationality:brazilian or Nationality:brazilian == Job:librarian or both
- Job:manager != Pet:snake or Pet:snake != Beverage:hot-chocolate or both
- Movie-Genre:animation is not to the left of Pet:snake
- Nationality:argentine and Job:security-guard have different parity positions
- Job:coach is not to the right of Job:software-developer
- Pet:hamster and Job:security-guard have different parity positions
- Job:software-developer is not to the left of Job:software-developer
- Movie-Genre:comedy == Beverage:fanta or Job:security-guard == Movie-Genre:comedy or both
- Job:software-developer is somewhere to the right of Job:coach
- Movie-Genre:family is not to the right of Nationality:malaysian
- Nationality:brazilian != Pet:pony
- Movie-Genre:martial-arts is somewhere to the right of Movie-Genre:musical
- Job:security-guard and Pet:bird have different parity positions
- Movie-Genre:musical != Nationality:brazilian
- Nationality:brazilian is somewhere to the right of Job:manager
- Beverage:soy-milk is not to the left of Nationality:argentine
- Movie-Genre:scientific and Pet:bird have the same parity positions
- Nationality:argentine is somewhere to the right of Pet:rabbit
- Pet:rabbit is somewhere to the right of Pet:bird
- Job:librarian != Nationality:egyptian
- Movie-Genre:animation is not to the left of Beverage:soy-milk
- Job:security-guard != Pet:snake or Job:security-guard != Movie-Genre:comedy or both
- Beverage:tea and Pet:snake have the same parity positions
- Nationality:brazilian != Pet:bird or Job:librarian != Nationality:brazilian or both
- Job:coach == Nationality:egyptian or Job:coach == Pet:hamster, but not both
- Pet:lizard != Job:software-developer
- Movie-Genre:martial-arts is somewhere to the right of Movie-Genre:family
- Beverage:water is not to the right of Pet:hamster
- Beverage:water == Movie-Genre:family or Beverage:water == Job:nurse, but not both
- Movie-Genre:comedy is not to the left of Nationality:argentine
- Job:manager is not to the right of Pet:snake
- Nationality:malaysian is not to the right of Movie-Genre:martial-arts
- Job:coach is not to the left of Job:coach
- Movie-Genre:family == Nationality:malaysian or Beverage:hot-chocolate == Movie-Genre:family, but not both
- Job:coach is not to the left of Beverage:juice
- Nationality:pakistani is somewhere to the right of Movie-Genre:scientific
- Movie-Genre:family is somewhere between Beverage:juice and Job:coach
- Nationality:chinese and Job:librarian have the same parity positions
- Beverage:juice is somewhere between Beverage:water and Nationality:egyptian
- Pet:rabbit is not to the right of Movie-Genre:musical
- Nationality:pakistani is somewhere to the left of Beverage:hot-chocolate
- Movie-Genre:scientific != Job:librarian or Movie-Genre:scientific != Beverage:juice or both

Answer the following question:

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

Answer: hot-chocolate


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, Music-Genre, Beverage, Transport.
The attributes have the following possible values:
- Nationality: dutch, french, indian, pakistani
- Movie-Genre: adventure, drama, family, time-travel
- Music-Genre: country, hip-hop, rock, trance
- Beverage: 7up, hot-chocolate, milk, sprite
- Transport: boat, car, motorbike, skateboard

Given the following premises about the line of people:
- Transport:boat is not to the right of Music-Genre:country
- Beverage:7up is somewhere between Beverage:sprite and Nationality:dutch
- Beverage:sprite is not to the left of Transport:motorbike
- Nationality:french is not to the left of Movie-Genre:time-travel
- Movie-Genre:family == Nationality:indian or Movie-Genre:family == Transport:car or both
- Transport:car and Movie-Genre:family have different parity positions
- Beverage:hot-chocolate and Movie-Genre:adventure have the same parity positions
- Nationality:french and Music-Genre:hip-hop have different parity positions
- Movie-Genre:adventure and Nationality:french have the same parity positions
- Beverage:hot-chocolate is not to the left of Music-Genre:rock
- Nationality:pakistani is somewhere between Nationality:french and Nationality:indian
- Transport:motorbike is not to the left of Movie-Genre:family
- Nationality:french is not to the right of Nationality:indian
- Music-Genre:rock is not to the left of Music-Genre:country

Answer the following question:

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

Answer: hot-chocolate


Let \( f(n) \) be the last non-zero digit in the decimal representation of \( n! \). Show that for distinct integers \( a_i \geq 0 \), \( f(5^{a_1} + 5^{a_2} + \ldots + 5^{a_r}) \) depends only on \( a_1 + \ldots + a_r = a \). Write the value as \( g(a) \). Find the smallest period for \( g \), or show that it is not periodic.

Answer: 4


Three cones are standing on their bases on a table, touching each other. The radii of their bases are 32, 48, and 48, and the apex angles are $\frac{\pi}{3}$, $\frac{2 \pi}{3}$, and $\frac{2 \pi}{3}$ respectively (the apex angle of a cone is the angle between its generatrices in an axial section). A sphere is suspended above the table, touching all the cones. It turns out that the center of the sphere is equidistant from the centers of the bases of all the cones. Find the radius of the sphere.

Answer: 13(\sqrt{3}+1)


For a positive integer \( n \), let \( \mathrm{I}_{\mathrm{n}}=\{1,2, \ldots, \mathrm{n}\} \) and \( \mathrm{P}_{\mathrm{n}}=\left\{\left.\frac{\mathrm{m}}{\sqrt{k}} \right| \mathrm{m} \in \mathrm{I}_{\mathrm{n}}, \mathrm{k} \in \mathrm{I}_{\mathrm{n}}\right\} \). A subset \( \mathrm{A} \) of \( \mathrm{P}_{\mathrm{n}} \) is called a "sparse set" if the sum of any two elements in \( \mathrm{A} \) is not a perfect square. Determine the maximum value of \( n \) such that \( \mathrm{P}_{\mathrm{n}} \) can be partitioned into the union of two disjoint sparse sets.

Answer: 14


In preparation for a game of Fish, Carl must deal 48 cards to 6 players. For each card that he deals, he runs through the entirety of the following process:
1. He gives a card to a random player.
2. A player \( \mathrm{Z} \) is randomly chosen from the set of players who have at least as many cards as every other player (i.e., \( \mathrm{Z} \) has the most cards or is tied for having the most cards).
3. A player \( \mathrm{D} \) is randomly chosen from the set of players other than \( \mathrm{Z} \) who have at most as many cards as every other player (i.e., \( \mathrm{D} \) has the fewest cards or is tied for having the fewest cards).
4. \( \mathrm{Z} \) gives one card to \( \mathrm{D} \).

He repeats steps 1-4 for each card dealt, including the last card. After all the cards have been dealt, what is the probability that each player has exactly 8 cards?

Answer: \frac{5}{6}


In the cells of a $10 \times 10$ table, the natural numbers 1, 2, ..., 99, 100 are placed. We call a figure obtained by removing one cell from a $2 \times 2$ square a "corner." We call a corner good if the number in its cell, bordering on two other cells by sides, is greater than the numbers in these two other cells. What is the maximum possible number of good corners? (Each corner is considered independently of its orientation relative to others; different corners can partially overlap.)

Answer: 162


The product of the digits of any multi-digit number is always less than this number. If we calculate the product of the digits of a given multi-digit number, then the product of the digits of this product, and so on, we will necessarily reach a single-digit number after some number of steps. This number of steps is called the persistence of the number. For example, the number 723 has a persistence of 2 because $7 \cdot 2 \cdot 3 = 42$ (1st step) and $4 \cdot 2 = 8$ (2nd step).

1. Find the largest odd number with distinct digits that has a persistence of 1.
2. Find the largest even number with distinct nonzero digits that has a persistence of 1.
3. Find the smallest natural number that has a persistence of 3.

Answer: 39


In the trapezoid \(ABCD\), point \(K\) is the midpoint of base \(AB\), and \(M\) is the midpoint of base \(CD\). Find the area of the trapezoid, given that \(DK\) is the angle bisector of \(\angle D\), \(BM\) is the angle bisector of \(\angle B\), the largest angle at the lower base is \(60^\circ\), and the perimeter is 30.

Answer: 15\sqrt{3}


A spy-plane circles point A at a distance of 10 km with a speed of 1000 km/h. A missile is fired towards the plane from A at the same speed and moves so that it is always on the line between A and the plane. How long does it take to hit?

Answer: 18\pi\,


Let \(ABCD\) be a convex quadrilateral inscribed in a circle with center \(O\). Let \(P\) be the intersection point of the diagonals and \(Q\) the second intersection point of the circumcircles of triangles \(APD\) and \(BPC\).

Show that \(\widehat{OQP} = 90^\circ\).

Answer: 90


A reservoir has two inlet pipes, Pipe A and Pipe C, and two outlet pipes, Pipe B and Pipe D. To fill the reservoir, it takes 3 hours using only Pipe A and 5 hours using only Pipe C. To empty a full reservoir, it takes 4 hours using only Pipe B and 6 hours using only Pipe D. Currently, the reservoir is $\frac{1}{6}$ full. If the pipes are operated in the following sequence: A, B, C, D, with each pipe running for one hour each day in a cycle, after how much time will the reservoir start overflowing?

Answer: 20\frac{3}{4}


Edward starts at his house located at \((0,0)\) and needs to reach the point \((6,4)\), which is the coordinate for his school. However, there is a park shaped as a square with coordinates \((2,1), (2,3), (4,1), (4,3)\). Edward cannot walk inside the park, but there is a road along the perimeter of the square. Determine how many different shortest routes exist from Edward's house to his school.

Answer: 110


Kelvin the Frog lives in the 2-D plane. Each day, he picks a uniformly random direction (i.e., a uniformly random bearing $\theta \in [0, 2\pi]$) and jumps a mile in that direction. Let $D$ be the number of miles Kelvin is away from his starting point after ten days. Determine the expected value of $D^4$.

Answer: 190


Three congruent circles of radius $2$ are drawn in the plane such that each circle passes through the centers of the other two circles. The region common to all three circles has a boundary consisting of three congruent circular arcs. Let $K$ be the area of the triangle whose vertices are the midpoints of those arcs. If $K = \sqrt{a} - b$ for positive integers $a, b$, find $100a+b$.

Answer: 4806


A positive integer $n$ is said to be $m$-free if $n \leq m!$ and $\gcd(i,n)=1$ for each $i=1,2,...,m$. Let $\mathcal{S}_k$ denote the sum of the squares of all the $k$-free integers. Find the remainder when $\mathcal{S}_7-\mathcal{S}_6$ is divided by $1000$.

Answer: 80


Given an isosceles triangle $\triangle ABC$ where $AB = BC$. A point $K$ is inside $\triangle ABC$ such that $CK = AB = BC$ and $\angle KAC = 30^\circ$. Find $\angle AKB$.

Answer: 150


The following diagram shows an eight-sided polygon $ABCDEFGH$ with side lengths $8, 15, 8, 8, 8, 6, 8,$ and $29$ as shown. All of its angles are right angles. Transform this eight-sided polygon into a six-sided polygon by connecting $B$ to $D$ with an edge and $E$ to $G$ with an edge to form polygon $ABDEGH$. Find the perimeter of $ABDEGH$.

Answer: 80


Let $P$ be the set of positive integers that are prime numbers. Find the number of subsets of $P$ that have the property that the sum of their elements is $34$, such as $\{3, 31\}$.

Answer: 9


Jerry is mowing a rectangular lawn which measures $77$ feet from north to south and $83$ feet from east to west. His lawn mower cuts a path that is $18$ inches wide. Jerry starts mowing by cutting a path from west to east across the north side of the lawn, then makes a right turn to cut a path along the east side of the lawn. He continues mowing by making right turns to mow a path along the next side each time he completes mowing a side.

How many right turns will he make?

Answer: 102


The distance between cities $A$ and $B$ is $30$ km. A bus departs from $A$ and makes a stop every $5$ km for $2$ minutes. The bus travels between stops at a speed of $80$ km/h. Simultaneously with the departure of the bus from $A$, a cyclist leaves $B$ to meet the bus, traveling at a speed of $27$ km/h. How far from $A$ will the cyclist meet the bus?

Answer: 20


In the following diagram, a semicircle is folded along a chord $AN$ and intersects its diameter $MN$ at $B$. Given that $MB : BN = 2 : 3$ and $MN = 10$. If $AN = x$, find $x^2$.

Answer: 80


Jar #1 contains five red marbles, three blue marbles, and one green marble. Jar #2 contains five blue marbles, three green marbles, and one red marble. Jar #3 contains five green marbles, three red marbles, and one blue marble.

You randomly select one marble from each jar. Given that you select one marble of each color, the probability that the red marble came from jar #1, the blue marble came from jar #2, and the green marble came from jar #3 can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 323


A candy store has $100$ pieces of candy to give away. When you get to the store, there are five people in front of you, numbered from $1$ to $5$. The $i$th person in line considers the set of positive integers congruent to $i$ modulo $5$ which are at most the number of pieces of candy remaining. If this set is empty, then they take no candy. Otherwise, they pick an element of this set and take that many pieces of candy. For example, the first person in line will pick an integer from the set $\{1,6,\dots,96\}$ and take that many pieces of candy. How many ways can the first five people take their share of candy so that after they are done there are at least $35$ pieces of candy remaining?

Answer: 3003


Let $\triangle ABC$ have $AB=6$, $BC=7$, and $CA=8$, and denote by $\omega$ its circumcircle. Let $N$ be a point on $\omega$ such that $AN$ is a diameter of $\omega$. Furthermore, let the tangent to $\omega$ at $A$ intersect $BC$ at $T$, and let the second intersection point of $NT$ with $\omega$ be $X$. The length of $\overline{AX}$ can be written in the form $\frac{m}{\sqrt{n}}$ for positive integers $m$ and $n$, where $n$ is not divisible by the square of any prime. Find $100m+n$.

Answer: 9799


Example 1 Given 20 numbers $1,2, \cdots, 20$, two people, A and B, take turns placing a “+” or “-” sign in front of these numbers (the order of placement is not limited). After 20 steps, the absolute value $S$ of the resulting algebraic sum is calculated. A wants to make $S$ as small as possible, while B wants to make $S$ as large as possible. What is the largest $S$ that B can guarantee?

Answer: 30


13. Construct a surface of rotation having as its generators three given lines passing through one point and not lying in the same plane. How many solutions does the problem have? Do the resulting conical surfaces have common generators?

Answer: 4


3. At the hitmen convention, 1000 participants gathered, each receiving a registration number from 1 to 1000. By the end of the convention, it turned out that all hitmen, except number 1, were killed. It is known that each hitman could only kill hitmen with higher numbers, and the number of his victims could not exceed his number. What is the minimum number of hitmen who could have participated in the killings at the convention?

Answer: 10


Problem 9.4. At a round table, 40 knights and 10 samurais are sitting. Exactly 7 knights have a samurai as their right neighbor. What is the maximum number of knights that could be sitting next to two knights?

Answer: 32


11. The common divisors of two numbers that are not both 0 are called their common divisors. Find all the common divisors of 26019 and 354.

Answer: 1,3,59,177


Example 4 In the border desert area, patrol vehicles travel 200 kilometers per day, and each vehicle can carry enough gasoline to travel for 14 days. There are 5 patrol vehicles that set out from base $A$ simultaneously, complete their tasks, and then return along the same route to the base. To allow 3 of them to patrol as far as possible and then return together, vehicles Jia and Yi travel to point $B$ along the way, leave behind the gasoline necessary for their own return to the base, and give the remaining gasoline to the other 3 vehicles to use. How far can these 3 vehicles travel in kilometers?
(1995, Hebei Province Junior High School Mathematics Competition)

Answer: 1800


59. From the integers 1 to 10, at least $\qquad$ numbers must be taken to ensure that there are two numbers among the taken numbers whose sum is 10.

Answer: 7


8. Given $a_{k}$ as the number of integer terms in $\log _{2} k, \log _{3} k, \cdots, \log _{2018} k$. Then $\sum_{k=1}^{2018} a_{k}=$ $\qquad$ .

Answer: 4102


14. Fill in the blanks with 1 and 10, using each number only once. It is required that the number in $\square$ equals the difference (larger minus smaller) of the two numbers above it to the left and right, and the number in $\mathrm{O}$ equals the sum of the two numbers above it to the left and right. Find: the value of $A$.

Fill in the blanks with 1 and 10, using each number only once. It is required that the number in $\square$ equals the difference (larger minus smaller) of the two numbers above it to the left and right, and the number in $\mathrm{O}$ equals the sum of the two numbers above it to the left and right. Find: the value of $A$.

Answer: 9


Answer: 7!\cdot6!


Example 12. Find the inverse function of $y=\sqrt{x+5}(x \geqslant-5)$.

Answer: y=x^{2}-5


15. (3 points) There are 300 chess pieces in black and white. The black crow divides the black and white pieces into 100 piles, with 3 pieces in each pile. Among them, there are 27 piles with only $l$ white piece, 42 piles with 2 or 3 black pieces, and the number of piles with 3 white pieces is equal to the number of piles with 3 black pieces. Therefore, the number of white pieces among these chess pieces is $\qquad$.

Answer: 158


6. A right triangle has a hypotenuse of length 2, and one of its legs has length 1. The altitude to its hypotenuse is drawn. What is the area of the rectangle whose diagonal is this altitude?

Answer: \dfrac{3\sqrt{3}}{16}


Let $\mathbf{N}$ be the set of natural numbers, $k \in \mathbf{N}$. Suppose there is a strictly increasing function $f: \mathbf{N} \rightarrow \mathbf{N}$ such that for every $n \in \mathbf{N}$, $f[f(n)] = k n$. Prove that for every $n \in \mathbf{N}$, $\frac{2k}{k+1} \leqslant f(n) \leqslant \frac{k+1}{2} n$.

Answer: \frac{2k}{k+1} n \leqslant f(n) \leqslant \frac{k+1}{2} n


5. Petya marks four points on a plane so that all of them cannot be crossed out by two parallel lines. From the lines passing through pairs of points, Vasya chooses two, measures the angle between them, and pays Petya an amount equal to the degree measure of the angle. What is the largest sum Petya can guarantee for himself?

Answer: 30


## 22. General Cleaning

A solid joint-stock company occupies three floors in a tower, which has the shape of a parallelepiped and is located in Paris in the La Défense district: the 13th and 14th floors, where the company's offices are located, and the 25th floor, where its board of directors is situated.

To clean these premises, a cleaning crew is called in, who work for four hours in the offices. Then the crew splits into two parts: half goes to clean the 25th floor, while the other half continues to clean the 13th and 14th floors. After another four hours, the crew stops working: the two lower floors are already completely cleaned, while the 25th floor is scheduled to receive another cleaner the next day, who will complete the work in 8 hours.

Assuming that the time required for cleaning is proportional only to the area of the cleaned premises, how many cleaners were in the crew called by the company?

Answer: 8


6. How many solutions does the rebus $\frac{B+O+C+b+M+O+\breve{U}}{K+J+A+C+C}=\frac{22}{29}$ have, where different letters represent different digits, and the same letters represent the same digits? It is known that the digit 0 is not used.

Answer: 0


Problem 8.4. Find any solution to the puzzle

$$
\overline{A B C A}=182 \cdot \overline{C D}
$$

where $A, B, C, D$ are four different non-zero digits (the notation $\overline{X Y \ldots Z}$ means the decimal representation of the number).

As an answer, write the four-digit number $\overline{A B C D}$.

Answer: 2916


2. Let $S$ be a subset of $\{1,2, \ldots, 9\}$ such that the sums of any two elements of $S$ are distinct. For example, the set $\{1,2,3,5\}$ has this property, but the set $\{1,2,3,4,5\}$ does not, because $\{2,3\}$ and $\{1,4\}$ both sum to 5. What is the maximum number of elements that $S$ can contain?

Answer: 5


From point \(A\) to point \(B\), the distance between which is 10 km, a car left at 7:00 AM. After traveling \(\frac{2}{3}\) of the way, the car passed point \(C\), from which a cyclist departed at that moment towards point \(A\). As soon as the car arrived at \(B\), a bus immediately left \(B\) in the opposite direction and arrived at \(A\) at 9:00 AM. How many kilometers from \(B\) did the bus catch up with the cyclist, if the cyclist arrived at point \(A\) at 10:00 AM and the speed of each vehicle is constant?

Answer: 6


10. (10 points) The area of rectangle $A B C D$ is 70 square centimeters. The vertex $F$ of trapezoid $A F G E$ is on $B C$, and $D$ is the midpoint of the non-parallel side $E G$. Find the area of trapezoid $A F G E$.

Answer: 70


There are given 100 distinct positive integers. We call a pair of integers among them good if the ratio of its elements is either 2 or 3. What is the maximum number $g$ of good pairs that these 100 numbers can form? (A same number can be used in several pairs.)

Answer: 180


Let $t$ be TNYWR.

Each of the three lines having equations $x+t y+8=0,5 x-t y+4=0$, and $3 x-k y+1=0$ passes through the same point. What is the value of $k$ ?

##

Answer: \dfrac{5}{6}t


Let $\frac{P_{n}}{Q_{n}}=[\underbrace{1 ; 1, \ldots, 1}_{n}]$. What are $P_{n}$ and $Q_{n}$?

Answer: F_{n+1}


The side of a triangle is divided into five equal parts; lines are drawn through the division points, parallel to the base.

Find the segments of these lines enclosed between the sides of the triangle, if the base is 20.

Answer: 4


Example 1 Figure 1 is a shape composed of 4 unit squares, with each vertex of the unit squares called a grid point. By taking any three grid points as vertices, how many different isosceles right triangles can be formed?

Regarding this problem, the answer given in [1] is 24, [2] supplements it to 30, and [3] provides the correct answer as 32. This shows that although the problem seems simple, a slight oversight can lead to errors in solving it.

This article introduces an effective method for solving such counting problems—the labeling method. It can effectively avoid the aforementioned mistakes.

The basic approach is to label each grid point with a number, and the number labeled at grid point $A$ represents the number of counting objects (such as isosceles right triangles) with $A$ as the special element.

For Example 1, since each isosceles right triangle has a unique right-angle vertex, we can consider the number of isosceles right triangles with each grid point as the right-angle vertex.

Of course, once the right-angle vertex is determined, the isosceles right triangles can be divided into two major categories:
(1) The legs are parallel to the grid lines (called upright);
(2) The legs are not parallel to the grid lines (called slanted).
For each major category, they can be further classified by different directions. Once the direction is determined, they can be classified by the length of the legs.

Answer: 32


A certain natural number is appended with two two-digit numbers on the right. The resulting number equals the cube of the sum of the three original numbers. Find all possible sets of the original three numbers.

Answer: (9, 11, 25)


$33 \quad m$ and $n$ are two positive integers satisfying $1 \leq m \leq n \leq 40$. Find the number of pairs of $(m, n)$ such that their product $m n$ is divisible by 33 .

Answer: 64


8. (10 points) In the expression $(x+y+z)^{2020}+(x-y-z)^{2020}$, the brackets were expanded and like terms were combined. How many monomials $x^{a} y^{b} z^{c}$ with a non-zero coefficient were obtained?

Answer: 1022121


Three. (50 points) Given a square table with four legs $L_{1}, L_{2}, L_{3}, L_{4}$, each of length $n$, where $n$ is a positive integer. How many ordered quadruples $\left(k_{1}, k_{2}, k_{3}, k_{4}\right)$ exist such that after sawing off a segment of length $k_{i}$ from the end of leg $L_{i}$ $(i=1,2,3,4)$, the table remains stable? Here, $k_{1}, k_{2}, k_{3}, k_{4}$ are non-negative integers (the table is considered stable if all four legs touch the ground, and it is allowed to saw off a length of 0).

Answer: \dfrac{(n + 1)(2n^2 + 4n + 3)}{3}


34. From the 20 natural numbers 1 to 20, the Pokémon chooses several numbers (at least two) such that the last digit of their product is 3. The total number of different ways to do this is $\qquad$.

Answer: 62


Along the path between Neznayka's and Sineglazka's houses, there were flowers planted in a row: 15 peonies and 15 tulips mixed together. While visiting Neznayka, Sineglazka watered all the flowers she encountered. After watering the 10th tulip, she ran out of water, leaving 10 flowers unwatered. The next day, while visiting Sineglazka, Neznayka picked all the flowers he encountered. After picking the 6th tulip, he decided he had enough for a bouquet. How many flowers were left growing along the path?

Answer: 19


Problem 7.6. Points $D$ and $E$ are marked on sides $A C$ and $B C$ of triangle $A B C$ respectively. It is known that $A B=B D, \angle A B D=46^{\circ}, \angle D E C=90^{\circ}$. Find $\angle B D E$, if it is known that $2 D E=A D$.

Answer: 67^\circ


5. Xiao Ming and Xiao Lin are playing the water-carrying game. The two use the same water samples to carry water and pour it into the same empty basin. The game is scored as follows: if the basin overflows when Xiao Ming pours water, 10 points are recorded; if the basin overflows when Xiao Lin pours water, 9 points are recorded; if the basin is not full after all the water is poured in one round, 3 points are recorded.
In the first round, Xiao Ming pours 5 buckets of water first, and when Xiao Lin pours 5 buckets of water, the basin overflows:
In the second round, Xiao Ming pours 2 buckets of water first, and Xiao Lin pours 7 buckets of water to end the game, but the basin is still not full:
In the third round, Xiao Ming pours 13 buckets of water alone to end the game.
After the three rounds of the game, the total score for these three rounds is $\qquad$ points.

Answer: 22


8. The equation $(a-1)(\sin 2 x+\cos x)+(a+1) \cdot(\sin x-\cos 2 x)=0$ (parameter $a<0$) has $\qquad$ solutions in the interval $(-\pi, \pi)$.

Answer: 4


Let $a$ be a positive real number, $n$ a positive integer, and define the [i]power tower[/i] $a\uparrow n$ recursively with $a\uparrow 1=a$, and $a\uparrow(i+1)=a^{a\uparrow i}$ for $i=1,2,3,\ldots$.  For example, we have $4\uparrow 3=4^{(4^4)}=4^{256}$, a number which has $155$ digits.  For each positive integer $k$, let $x_k$ denote the unique positive real number solution of the equation $x\uparrow k=10\uparrow (k+1)$.  Which is larger: $x_{42}$ or $x_{43}$?

Answer: x_{42}


A trinitro compound, 1,3,5-tris-(4-nitrophenyl)benzene, on complete reaction with an excess of $\mathrm{Sn} / \mathrm{HCl}$ gives a major product, which on treatment with an excess of $\mathrm{NaNO}_{2} / \mathrm{HCl}$ at $0{ }^{\circ} \mathrm{C}$ provides $\mathbf{P}$ as the product. $\mathbf{P}$, upon treatment with excess of $\mathrm{H}_{2} \mathrm{O}$ at room temperature, gives the product $\mathbf{Q}$. Bromination of $\mathbf{Q}$ in aqueous medium furnishes the product $\mathbf{R}$. The compound $\mathbf{P}$ upon treatment with an excess of phenol under basic conditions gives the product $\mathbf{S}$.

The molar mass difference between compounds $\mathbf{Q}$ and $\mathbf{R}$ is $474 \mathrm{~g} \mathrm{~mol}^{-1}$ and between compounds $\mathbf{P}$ and $\mathbf{S}$ is $172.5 \mathrm{~g} \mathrm{~mol}^{-1}$. What is the number of heteroatoms present in one molecule of $\mathbf{R}$?

[Use: Molar mass (in g mol${ }^{-1}$ ): $\mathrm{H}=1, \mathrm{C}=12, \mathrm{~N}=14, \mathrm{O}=16, \mathrm{Br}=80, \mathrm{Cl}=35.5$

Atoms other than $\mathrm{C}$ and $\mathrm{H}$ are considered as heteroatoms]
Please give the answer directly without any intermediate steps.

Answer: 9


22. Find the number of $n$-permutations with repetition, $N_{n}$, formed by $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ such that $a_{1}$ and $a_{3}$ both appear an even number of times.

Answer: \dfrac{6^n + 2 \cdot 4^n + 2^n}{4}


4. If three dice of different sizes are rolled randomly, the probability that the sum of the numbers shown on two or three of the dice is 7 is $\qquad$ .

Answer: \dfrac{35}{72}


7・15 Given a regular $(2n+1)$-sided polygon, three different vertices are randomly selected from its vertices. If all such selections are equally likely, find the probability that the center of the regular polygon lies inside the triangle formed by the randomly chosen three points.

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


In triangle \(ABC\), with an area of \(S\), the angle bisector \(CE\) and the median \(BD\) intersect at point \(O\). Find the area of quadrilateral \(ADOE\), given that \(BC = a\) and \(AC = b\).

Answer: \dfrac{b(3a + b)}{2(2a + b)(a + b)} S


10. $P$ is any point on the ellipse $\frac{(x+4)^{2}}{9}+\frac{y^{2}}{16}=1$, and $PA, PB$ are two tangent lines to the parabola $y^{2}=4x$, with $A, B$ being the points of tangency. Find the maximum area of $\triangle PAB$, and determine the coordinates of point $P$ when the area of $\triangle PAB$ is maximized.

Answer: \dfrac{137\sqrt{137}}{16}


A rectangular parallelepiped has edges meeting at a vertex in the ratio $1: 2: 3$. How do the lateral surface areas of the right circular cylinders circumscribed around the parallelepiped compare?

Answer: \sqrt{13} : 2\sqrt{10} : 3\sqrt{5}


Problem 8.4. Points $D$ and $E$ are marked on sides $A C$ and $B C$ of triangle $A B C$ respectively, such that $A D=E C$. It turns out that $B D=E D, \angle B D C=\angle D E B$. Find the length of segment $A C$, given that $A B=7$ and $B E=2$.

Answer: 12


1. Determine the number of all four-digit natural numbers that are divisible by six and in whose notation exactly two ones appear.

Answer: 41


When admitting students to the university, each applicant is assigned a cover code consisting of five digits. The thorough, yet superstitious associate professor decided to eliminate from all possible codes (i.e., 00000 to 99999) those that contained the number 13, meaning the digit 3 immediately following the digit 1. How many codes did the professor have to eliminate?

Answer: 3970


【Example 1】Four people, A, B, C, and D, are practicing table tennis doubles. Two people form a team. How many different ways are there?

Answer: 3


78. A has several dogs, and B has several sheep. If B exchanges all the sheep for the same number of dogs, then B needs to pay A an additional 320 yuan; if B does not do so, B will get 5 fewer dogs. It is known that the price of 1 dog is 32 yuan less than the price of 2 sheep. B originally had $\qquad$ sheep.

Answer: 20


3rd Centromerican 2001 Problem A2 C and D are points on the circle diameter AB such that ∠AQB = 2 ∠COD. The tangents at C and D meet at P. The circle has radius 1. Find the distance of P from its center.

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


5. In the border desert area, patrol vehicles travel 200 kilometers per day, and each vehicle can carry enough gasoline to travel for 14 days. There are 5 patrol vehicles that set out from base $A$ simultaneously, complete their tasks, and then return along the same route to the base. To allow three of the vehicles to patrol as far as possible (and then return together), vehicles 甲 and 乙 travel to a point $B$ along the way, leaving only enough gasoline for their return to the base, and giving the remaining gasoline to the other three vehicles. How far can the other three vehicles travel in kilometers?

Answer: 1800


For a single-story apartment, we know the following: a) there is at most 1 door between any two rooms; b) from any room, there is at most 1 door leading outside the apartment; c) there are 12 doors in the apartment. What is the minimum number of rooms in the apartment?

Answer: 5


Given that there are only three positive integers between the fractions $\frac{112}{19}$ and $\frac{112+x}{19+x}$. Find the sum of all possible integer values of $x$.

Answer: 2310


A triangle $OAB$ with $\angle A=90^{\circ}$ lies inside another triangle with vertex $O$. The altitude of $OAB$ from $A$ until it meets the side of angle $O$ at $M$. The distances from $M$ and $B$ to the second side of angle $O$ are $2$ and $1$ respectively. Find the length of $OA$.

Answer: \sqrt{2}


3. Color each cell of a $12 \times 12$ grid either black or white, such that any $3 \times 4$ or $4 \times 3$ subgrid contains at least one black cell. Find the minimum number of black cells.
（Liang Yingde, problem contributor）

Answer: 12


Answer: 45


Consider the parabola $ C: y \equal{} x^2$ in the $ x \minus{} y$ plane. Let $ l_1$ be the tangent line of $ C$ with slope $ \sqrt {3}$ and $ l_2$ be the tangent line of $ C$ with negative slope which makes angle $ 45^\circ$ with $ l_1$. Find the area bounded by $ C,\ l_1$ and $ l_2$.

Answer: \dfrac{5 + 3\sqrt{3}}{6}


Two workers, $A$ and $B$, completed a task assigned to them as follows. First, only $A$ worked for $\frac{2}{3}$ of the time it would take $B$ to complete the entire task alone. Then, $B$ took over from $A$ and finished the work. In this way, the work took 2 hours longer than if they had started working together and completed it collaboratively. If they had worked together in the latter manner, $A$ would have done half as much work as $B$ actually left for $B$ to complete. How many hours would it take for $A$ and $B$ to complete the work individually?

Answer: 6


1. Given various real numbers $x$ and $y$ such that $x / y + x = y / x + y$. What can $1 / x + 1 / y$ be equal to? Find all possible values and prove that there are no others.

Answer: -1


10. Let the line $y=\sqrt{3} x+b$ intersect the parabola $y^{2}=2 p x(p>0)$ at points $A, B$. The circle passing through $A, B$ intersects the parabola $y^{2}=2 p x(p>0)$ at two other distinct points $C, D$. Find the angle of intersection between the lines $A B$ and $C D$.

Answer: 60^\circ


6. Let $A=\{1,3,5,7,9\}, B=\{2,4,6,8\}$. Now, by tossing a coin, numbers are drawn without replacement from $A$ and $B$ to form a nine-digit number. The rule is: if the coin shows heads, the smallest number from set $A$ is taken; if the coin shows tails, the smallest number from set $B$ is taken. When the numbers from one set are exhausted and the other set still has numbers, the remaining numbers from the other set are added in ascending order. According to this rule, the probability that the drawn numbers are exactly 123456789 is . $\qquad$

Answer: \dfrac{1}{256}


Three. (Full marks 25 points) On the blackboard, all natural numbers from 1 to 1997 are written. Students $A$ and $B$ take turns to perform the following operations: Student $A$ subtracts the same natural number from each number on the blackboard (the number subtracted can be different in different operations); Student $B$ erases two numbers from the blackboard and writes down their sum. Student $A$ goes first, and the operations continue until only one number remains on the blackboard. If this number is non-negative, find this number.

Answer: 1


A sphere is inscribed in a truncated triangular pyramid, touching the bases at points \( T_1 \) and \( T_2 \). Let \( h \) be the height of the pyramid, \( R_1 \) and \( R_2 \) be the radii of the circles described around its bases, and \( O_1 \) and \( O_2 \) be the centers of these circles. Prove that:

\[
R_1R_2h^2 = (R_1^2 - O_1T_1^2)(R_2^2 - O_2T_2^2).
\]

Answer: R_1R_2h^2 = (R_1^2 - O_1T_1^2)(R_2^2 - O_2T_2^2)


2. Determine the maximum possible value of the expression

$$
\frac{a_{1} a_{2} \ldots a_{n}}{\left(1+a_{1}\right)\left(a_{1}+a_{2}\right) \ldots\left(a_{n-1}+a_{n}\right)\left(a_{n}+2^{n+1}\right)}
$$

where $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ are positive real numbers.

Answer: \dfrac{1}{3^{n+1}}


[ Trigonometric equations ] [ Divisibility of numbers. General properties ]

Authors: Begun $\underline{\text { A.V. }}, \underline{\text { Goryashin D.V. }}$

What is the maximum number of factors of the form $\sin \frac{n \pi}{x}$ that can be crossed out in the left-hand side of the equation

$\sin \frac{\pi}{x} \sin \frac{2 \pi}{x} \sin \frac{3 \pi}{x} \ldots \sin \frac{2015 \pi}{x}=0$ so that the number of its natural roots does not change?

Answer: 1007


6. Let $P$ be a fixed point on the circle $x^{2}+y^{2}=1$, and points $Q$ and $R$ are inside or on the circumference of the circle, and $\triangle P Q R$ is an equilateral triangle with side length $\frac{2}{\sqrt{3}}$. Then the minimum value of $|O Q|^{2}+|O R|^{2}$ is

Answer: \dfrac{2}{3}


B4. We write the numbers from 1 to 30000 in a row so that a long sequence of digits is formed:

$$
123456789101112 \ldots 30000
$$

How many times does 2013 appear in this sequence?

Answer: 25


8.1. Parallelogram $A B C D$ was folded along the diagonal $B D$ so that vertex $C$ remained in place, and vertex $A$ took the position $A^{\prime}$. Segments $B C$ and $A^{\prime} D$ intersected at point $K$, and $B K: K C=3: 2$. Find the area of triangle $A^{\prime} K C$, if the area of parallelogram $A B C D$ is 27.

Answer: \dfrac{18}{5}


Example 3. In an equilateral $\triangle ABC$, take a point $D$ inside such that $DA = DB$; also take a point $E$ outside $\triangle ABC$ such that $\angle DBE = \angle DBC$, and $BE = BA$. Find $\angle BED$. (1992, Sichuan Province Junior High School Mathematics League)

Answer: 30^\circ


Example 9. Factorize $a b(a-b)+b c(b-c)+c a(c-a)$.

Answer: -(a - b)(b - c)(c - a)


9.4. To the number $A$, consisting of eight non-zero digits, a seven-digit number, consisting of identical digits, was added, and the eight-digit number $B$ was obtained. It turned out that the number $B$ can be obtained from the number $A$ by rearranging some of the digits. What digit can the number $A$ start with if the last digit of the number $B$ is 5?

Answer: 5


The 58th question: Given that $p$ is an odd prime, if a natural number $x$ satisfies that for any natural number $t, p \nmid\left(x-t^{2}\right)$, then $x$ is called a quadratic non-residue modulo $p$. Let the set $A=\{a \mid a \in Z, 1 \leq a<p, a$ and $4-$ $\mathrm{a}$ are both quadratic non-residues modulo $\mathrm{p}\}$. Find the remainder of the product of all elements in $\mathrm{A}$ divided by $\mathrm{p}$.

Answer: 2


18. (3 points) Li Shuang rides a bike at a speed of 320 meters per minute from location $A$ to location $B$. On the way, due to a bicycle malfunction, he pushes the bike and walks for 5 minutes to a place 1800 meters from $B$ to repair the bike. After 15 minutes, he continues towards $B$ at 1.5 times his original riding speed, and arrives at $B$ 17 minutes later than the expected time. What is Li Shuang's walking speed in meters per minute?

Answer: 72


A natural number \( a \) has 107 distinct divisors, including 1 and \( a \). Find the sum and the product of these divisors.

Answer: \frac{p^{107} - 1}{p - 1}


1. Let $p$ be a prime number. Find all natural numbers $x$ and $y$ that satisfy the equation $p \cdot(x-5)=x \cdot y$.

Answer: (x, y) = (5p, p - 1)


A metro network has at least 4 stations on each line, with no more than three interchange stations. No interchange station intersects more than two lines. What is the maximum number of lines that such a network can have if you can travel from any station to any other by making no more than two interchanges?

Answer: 10


## Task 1 - 270811

Steffen presents the following task to the members of the AG Mathematics group:

"Each of you think of a number, multiply this number by 2, add 30 to the product, divide the sum by 2, and subtract the initially thought number from the obtained result!

Write down the result!"

It turned out that all students in the working group had the same result. Must Steffen's classmates have necessarily thought of the same number?

Answer: 15


Consider a round-robin tournament with $2 n+1$ teams, where each team plays each other team exactly once. We say that three teams $X, Y$ and $Z$, form a cycle triplet if $X$ beats $Y, Y$ beats $Z$, and $Z$ beats $X$. There are no ties.

(a) Determine the minimum number of cycle triplets possible.

(b) Determine the maximum number of cycle triplets possible.

Answer: 0


\section*{Exercise 3 - V11033}

Fritz obtained the result of a division problem as 57 remainder 52. When he checked his work, he got 17380. This is incorrect because he had written the numbers unclearly and read a 6 as 0 in the tens place of the divisor during the check.

What is the correct problem? How did you find the result?

Answer: 20800


5. A five-digit number with all distinct digits, if the sum of any two of its digits is not 7, is called a “hidden seven number”. There are $\qquad$ such “hidden seven numbers”.

Answer: 7008


Consider those real triangles whose vertices have coordinates from the numbers $-1,0,1$ in a plane Cartesian coordinate system. How far can the centroid of these triangles be from the origin?

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


3. Let $A, B \in M_{2}(\mathbb{Q})$ such that $A B = B A$, $\operatorname{det}(A) = -3$, and $\operatorname{det}(A + \sqrt{3} B) = 0$. Calculate $\operatorname{det}\left(A^{2} + B^{2} - A B\right)$.

Local Olympiad, Constanța 2014

Answer: 13


36 children are standing in a circle, each wearing either a red or blue sweater. It is known that next to each boy stands a girl, and next to each girl stands a person in a blue sweater. Find the maximum possible number of girls wearing red sweaters.

Answer: 24


40. Solve the equation $x\left(y^{2}-1\right) d x+y\left(x^{2}+1\right) d y=0$.

Answer: (x^2 + 1)(y^2 - 1) = C


6. On the sides $A B$ and $A C$ of an acute-angled triangle $A B C$, two equal rectangles $A M N B$ and $A P Q C$ are constructed outwardly. Find the distance between the vertices $N$ and $Q$ of the rectangles, if the lengths of the sides $A B$ and $A C$ are $2 \sqrt{2}$ and 1, respectively, and the angle at vertex $A$ of the triangle is $45^{\circ}$.

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


# 8. Problem 8.10

Pentagon $A B C D E$ is circumscribed around a circle.

Angles $\angle A B C, \angle B A E, \angle C D E$ are each $104^{\circ}$. Find $\angle A D B$. Write the answer in degrees (the number only, without specifying the unit of measurement).

#

Answer: 38


A $100 \times 100$ square is divided into $2 \times 2$ squares. Then it is divided into dominos (rectangles $1 \times 2$ and $2 \times 1$). What is the smallest number of dominos that could have been inside the divided squares?

Answer: 100


S is a collection of subsets of {1, 2, ... , n} of size 3. Any two distinct elements of S have at most one common element. Show that S cannot have more than n(n-1)/6 elements. Find a set S with n(n-4)/6 elements.

Answer: \frac{n(n-1)}{6}


a) What is the maximum number of bishops that can be placed on a standard chessboard (comprising 64 squares) such that no two bishops threaten each other? Solve the same problem for a chessboard consisting of \( n^2 \) squares.
b) What is the minimum number of bishops that can be placed on a standard chessboard (comprising 64 squares) such that they threaten all the squares on the board? Solve the same problem for a chessboard consisting of \( n^2 \) squares.

Answer: 8


We are interested in the set of words of length 6 that can be written with the letters \( x, y, \) and \( t \). We allow the following transformations:

(i) \( xy \mapsto yyx \) and \( yyx \mapsto xy \)

(ii) \( xt \mapsto ttx \) and \( ttx \mapsto xt \)

(iii) \( yt \mapsto ty \) and \( ty \mapsto yt \)

The first condition means, for example, that if we have a word in which the letters \( x \) and \( y \) appear next to each other, we are allowed to replace these two letters with the three letters \( y, y, \) and \( x \). We are also allowed to reverse this, which is the second part of condition (i).

Determine whether the following words are equivalent:

(i) \( xx yy \) and \( x yyyy x \);

(ii) \( xytx \) and \( txyt \);

(iii) \( xy \) and \( xt \).

Answer: Not\ Equivalent


A number is written on the board. In one move, you can either increase or decrease any of its digits by three (if the resulting digit is valid) or swap two adjacent digits. Show how to obtain the number 654321 from the number 123456 in 11 moves.

Answer: 654321


Let \( f(x, y) \) be a two-variable polynomial that satisfies the following conditions:
(1) \( f(1,2) = 2 \);
(2) \( y f(x, f(x, y)) = x f(f(x, y), y) = (f(x, y))^2 \).

Determine all possible such polynomials \( f(x, y) \).

Answer: f(x, y) = x y


A six-digit (decimal) number has six different digits, none of them 0, and is divisible by 37. Show that you can obtain at least 23 other numbers which are divisible by 37 by permuting the digits.

Answer: 23


Is it possible to use 500 small $1 \times 2$ rectangles as shown in the figure to form a large $5 \times 200$ rectangle, such that each row and each column of the $5 \times 200$ rectangle contains an even number of stars? Please provide a reason.

Answer: \text{Yes}


Let \( c \in \mathbb{R} \) be a non-zero constant such that \( f(x+1) = f(x) + c \) and \( f\left(x^2\right) = f(x)^2 \) for all real \( x \). Show that \( c = 1 \) and \( f \) is the identity function.

Answer: f(x) = x


Given a pyramid \( S A B C \) with a triangular base \( A B C \) that is equilateral, and all the plane angles at the vertex \( S \) are equal to \( \alpha \). What is the smallest \( \alpha \) for which we can claim that this pyramid is regular?

Answer: 60^\circ


Merlin has two $100 \times 100$ grids. One grid is blank and hangs on a cliff at the entrance of the cave. The other grid has a number in each cell and hangs on the wall inside the cave. You can specify a square subgrid of any size (from $1 \times 1$ to $100 \times 100$) in the blank grid and pay Merlin a shilling to find out the sum of all the numbers in the corresponding subgrid in the second grid. What is the minimum number of shillings needed to find the sum of all the numbers along the diagonal of the second grid?

Answer: 100


An equilateral triangle with a side length of 3 is divided into nine triangular cells, as shown in the figure. Initially, zeros are written in these cells. In one move, you can choose two numbers located in adjacent cells and either add one to both or subtract one from both. Petya wants to make several moves such that, after these moves, the cells contain consecutive natural numbers \( n, n+1, \ldots, n+8 \) in some order. For which \( n \) can he achieve this?

Answer: 2


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

## Clues:
1. The person who enjoys mountain retreats is Alice.
2. The Norwegian is somewhere to the left of the person whose birthday is in September.
3. The person who prefers city breaks is the British person.
4. Alice and Eric are next to each other.
5. The person who likes going on cruises is not in the fourth house.
6. Alice is somewhere to the left of the person whose birthday is in January.
7. The person who enjoys mountain retreats is the Norwegian.
8. Alice is directly left of the Swedish person.
9. Eric is in the third house.
10. The person whose birthday is in April is Peter.
11. The Dane is the person whose birthday is in September.
 What is the value of attribute House for the person whose attribute Nationality is brit? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Eric`, `Arnold`, `Alice`
 - The people keep unique animals: `dog`, `fish`, `bird`, `horse`, `cat`
 - The people are of nationalities: `dane`, `swede`, `brit`, `german`, `norwegian`
 - Each mother is accompanied by their child: `Meredith`, `Bella`, `Timothy`, `Samantha`, `Fred`
 - Each person has a unique type of pet: `bird`, `fish`, `hamster`, `cat`, `dog`

## Clues:
1. The dog owner is not in the third house.
2. The person who keeps a pet bird is in the fourth house.
3. The German is Eric.
4. Bob is somewhere to the left of the person's child is named Fred.
5. The person with a pet hamster is Arnold.
6. The British person is the dog owner.
7. The person's child is named Meredith is not in the third house.
8. The fish enthusiast is the person's child is named Samantha.
9. The person who is the mother of Timothy is the German.
10. The person who has a cat is in the fifth house.
11. Peter is not in the fourth house.
12. The Swedish person and the person's child is named Samantha are next to each other.
13. The Swedish person is Peter.
14. The Dane is in the second house.
15. Eric and the cat lover are next to each other.
16. The German is in the first house.
17. The person with an aquarium of fish is somewhere to the right of the bird keeper.
 What is the value of attribute Animal for the person whose attribute Pet is fish? Please reason step by step, and put your final answer within \boxed{}

Answer: horse


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

## Clues:
1. The Dane is the person who loves a carnations arrangement.
2. The Swedish person is Arnold.
3. The Chinese is the person who loves the rose bouquet.
4. The Norwegian and the person who loves the rose bouquet are next to each other.
5. The person who loves the vase of tulips is Arnold.
6. There are two houses between Bob and the person who loves the boquet of iris.
7. The person who loves a carnations arrangement is directly left of the person who loves a bouquet of daffodils.
8. The Dane is in the second house.
9. The person who loves a carnations arrangement is directly left of Alice.
10. The person who loves a bouquet of daffodils and the Swedish person are next to each other.
11. The German is somewhere to the right of the person who loves a carnations arrangement.
12. The Norwegian is somewhere to the right of Peter.
13. The Norwegian is Eric.
 What is the value of attribute Name for the person whose attribute Flower is lilies? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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

## Clues:
1. The person whose mother's name is Penny is in the fourth house.
2. Eric is The person whose mother's name is Penny.
3. The person who keeps horses is the person who loves swimming.
4. The person who loves baseball is the person's child is named Meredith.
5. The dog owner is directly left of the person who loves basketball.
6. The person who loves soccer is the bird keeper.
7. The person who loves swimming is directly left of the person's child is named Fred.
8. The person who loves tennis is The person whose mother's name is Kailyn.
9. The person who is the mother of Timothy is The person whose mother's name is Aniya.
10. Eric is somewhere to the right of the person's child is named Samantha.
11. The person who loves basketball is Bob.
12. The person who is the mother of Timothy is the fish enthusiast.
13. The person's child is named Meredith is somewhere to the right of Peter.
14. The person who loves basketball is somewhere to the left of the person who keeps horses.
15. The bird keeper is Alice.
16. The person whose mother's name is Holly is the bird keeper.
17. Bob is not in the third house.
 What is the value of attribute Children for the person whose attribute Animal is cat? Please reason step by step, and put your final answer within \boxed{}

Answer: Meredith


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

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

Answer: white


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`, `Eric`, `Bob`, `Alice`, `Peter`
 - Each person has a unique birthday month: `feb`, `mar`, `sept`, `jan`, `may`, `april`
 - Everyone has something unique for lunch: `stew`, `soup`, `grilled cheese`, `stir fry`, `spaghetti`, `pizza`
 - People have unique heights: `very short`, `average`, `super tall`, `short`, `very tall`, `tall`
 - People own unique car models: `chevrolet silverado`, `ford f150`, `bmw 3 series`, `tesla model 3`, `toyota camry`, `honda civic`

## Clues:
1. The person who owns a Honda Civic is the person who is short.
2. The person who owns a Ford F-150 is in the fifth house.
3. The person who loves stir fry is somewhere to the left of Eric.
4. The person whose birthday is in May is somewhere to the left of Carol.
5. The person who is very short is somewhere to the left of the person whose birthday is in April.
6. The person who owns a BMW 3 Series is not in the third house.
7. There are two houses between the person who loves stir fry and the person who is a pizza lover.
8. The person who loves the soup is directly left of Eric.
9. The person who loves the spaghetti eater and the person whose birthday is in May are next to each other.
10. Alice is directly left of the person who owns a BMW 3 Series.
11. The person who owns a Tesla Model 3 is somewhere to the left of the person who is tall.
12. The person who is very tall is the person who owns a Toyota Camry.
13. Peter is directly left of the person who is a pizza lover.
14. The person who loves the stew is not in the third house.
15. There is one house between the person whose birthday is in September and the person who is very short.
16. There is one house between the person whose birthday is in March and the person who is super tall.
17. The person who is tall is Bob.
18. The person whose birthday is in May is somewhere to the right of Alice.
19. The person who is very short is in the fourth house.
20. The person whose birthday is in March is the person who is short.
21. Carol is the person who owns a Tesla Model 3.
22. Eric is the person whose birthday is in January.
 What is the value of attribute House for the person whose attribute CarModel is chevrolet silverado? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


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

Answer: 3


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

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

Answer: 1


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

## Clues:
1. The Dunhill smoker is not in the second house.
2. The person who loves rock music is directly left of the person who loves classical music.
3. Peter is the person who loves jazz music.
4. Arnold is directly left of the Dane.
5. The root beer lover is the German.
6. The tea drinker is the Swedish person.
7. The Swedish person is the person who loves classical music.
8. The Dunhill smoker is not in the fourth house.
9. The person who loves jazz music is the Prince smoker.
10. Eric and the person who loves hip-hop music are next to each other.
11. There are two houses between the person partial to Pall Mall and the Swedish person.
12. Arnold is somewhere to the right of Eric.
13. The coffee drinker is somewhere to the right of the Dunhill smoker.
14. The Norwegian is somewhere to the right of the person who smokes Blue Master.
15. There is one house between the one who only drinks water and the person who smokes many unique blends.
16. Alice is the tea drinker.
 What is the value of attribute House for the person whose attribute Drink is water? Please reason step by step, and put your final answer within \boxed{}

Answer: 2