There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Food, Hobby, Movie-Genre, Pet, Sport, Transport.
The attributes have the following possible values:
- Food: apricot, lemon, orange, radish
- Hobby: board-games, collecting, filmmaking, hiking
- Movie-Genre: mystery, superhero, time-travel, western
- Pet: ferret, hamster, mouse, pony
- Sport: basketball, cricket, sailing, snowboarding
- Transport: motorbike, ship, tram, van

Given the following premises about the line of people:
- Food:radish is on the left or right of Sport:basketball
- Pet:pony is somewhere to the right of Hobby:hiking
- Movie-Genre:western is somewhere to the left of Hobby:hiking
- Pet:ferret == Transport:motorbike or Movie-Genre:mystery == Pet:ferret or both
- Transport:ship and Food:lemon have different parity positions
- Hobby:board-games is somewhere between Movie-Genre:western and Food:lemon
- Movie-Genre:superhero and Food:apricot have different parity positions
- Movie-Genre:time-travel is not to the left of Sport:cricket
- Transport:tram and Movie-Genre:time-travel have the same parity positions
- Pet:mouse is not to the left of Hobby:board-games
- Hobby:collecting is not to the left of Movie-Genre:time-travel
- Transport:tram == Movie-Genre:mystery or Transport:tram == Pet:pony or both
- Movie-Genre:superhero is somewhere to the right of Sport:cricket
- Movie-Genre:superhero and Hobby:board-games have different parity positions
- Hobby:board-games is on the left or right of Pet:pony
- Pet:hamster is somewhere to the right of Sport:snowboarding
- Pet:ferret and Food:radish have the same parity positions

Answer the following question:

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

Answer: superhero


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, Music-Genre, Beverage, Sport, Transport.
The attributes have the following possible values:
- Nationality: argentine, canadian, mexican, thai
- Hobby: card-games, cooking, rock-climbing, woodworking
- Music-Genre: house, metal, reggae, salsa
- Beverage: coffee, juice, tea, water
- Sport: golf, sailing, skateboarding, surfing
- Transport: jet-ski, motorbike, train, van

Given the following premises about the line of people:
- Hobby:rock-climbing and Transport:motorbike have the same parity positions
- Music-Genre:house is somewhere to the left of Beverage:coffee
- Music-Genre:reggae and Hobby:woodworking have different parity positions
- Transport:motorbike is somewhere to the right of Hobby:card-games
- Sport:surfing is somewhere to the left of Nationality:canadian
- Hobby:card-games is somewhere to the right of Music-Genre:metal
- Nationality:thai == Hobby:rock-climbing or Transport:jet-ski == Nationality:thai or both
- Nationality:mexican is on the left or right of Sport:sailing
- Music-Genre:metal != Sport:surfing
- Transport:jet-ski is on the left or right of Transport:van
- Nationality:argentine == Beverage:tea or Beverage:tea == Music-Genre:reggae or both
- Sport:golf is somewhere between Hobby:card-games and Music-Genre:salsa
- Hobby:cooking != Music-Genre:metal
- Hobby:rock-climbing and Transport:jet-ski have different parity positions
- Transport:train == Nationality:argentine or Transport:train == Hobby:woodworking or both
- Transport:van is somewhere to the right of Beverage:juice
- Music-Genre:house is not to the right of Nationality:mexican

Answer the following question:

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

Answer: water


There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes:  Beverage, Movie-Genre, Pet, Sport.
The attributes have the following possible values:
- Beverage: cola, iced-tea, juice, milk, mirinda, soy-milk
- Movie-Genre: action, adventure, crime, musical, scientific, thriller
- Pet: ferret, goat, hamster, horse, mouse, pony
- Sport: badminton, biathlon, cricket, rugby, swimming, water-polo

Given the following premises about the line of people:
- Beverage:iced-tea is between Movie-Genre:scientific and Beverage:mirinda
- Movie-Genre:crime is between Movie-Genre:action and Movie-Genre:scientific
- Movie-Genre:adventure and Pet:goat have different parity positions
- Beverage:soy-milk is not to the right of Beverage:mirinda
- Sport:cricket is between Pet:hamster and Movie-Genre:scientific
- Pet:hamster is somewhere between Pet:goat and Pet:pony
- Movie-Genre:scientific is somewhere to the right of Pet:pony
- Sport:water-polo is somewhere to the left of Pet:mouse
- Movie-Genre:crime is somewhere to the left of Movie-Genre:musical
- Pet:horse == Beverage:juice or Pet:horse == Sport:swimming or both
- Beverage:soy-milk is not to the left of Pet:goat
- Movie-Genre:musical is somewhere to the right of Sport:biathlon
- Sport:rugby is somewhere between Beverage:juice and Beverage:cola
- Beverage:milk is somewhere to the left of Pet:pony
- Sport:rugby is somewhere to the right of Sport:badminton

Answer the following question:

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

Answer: hamster


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Job, Hobby, Movie-Genre, Transport.
The attributes have the following possible values:
- Job: accountant, musician, nurse, paramedic, social-worker
- Hobby: camping, dancing, hiking, rock-climbing, traveling
- Movie-Genre: action, documentary, martial-arts, mystery, scientific
- Transport: car, helicopter, motorbike, ship, snowmobile

Given the following premises about the line of people:
- Hobby:hiking == Movie-Genre:documentary or Hobby:hiking == Transport:ship or both
- Job:accountant is not to the right of Movie-Genre:martial-arts
- Hobby:camping is not to the left of Movie-Genre:action
- Hobby:rock-climbing != Job:social-worker or Hobby:rock-climbing != Movie-Genre:documentary or both
- Hobby:rock-climbing == Job:social-worker or Hobby:rock-climbing == Transport:snowmobile or both
- Movie-Genre:martial-arts and Job:musician have different parity positions
- Hobby:traveling and Transport:helicopter have different parity positions
- Hobby:rock-climbing == Transport:snowmobile or Hobby:rock-climbing == Job:musician, but not both
- Movie-Genre:mystery is not to the right of Job:accountant
- Hobby:rock-climbing is not to the right of Job:musician
- Movie-Genre:mystery is somewhere between Job:nurse and Job:social-worker
- Movie-Genre:documentary == Transport:ship or Movie-Genre:documentary == Hobby:rock-climbing, but not both
- Movie-Genre:martial-arts is not to the right of Transport:ship
- Transport:ship is somewhere between Transport:snowmobile and Hobby:traveling
- Transport:motorbike == Movie-Genre:action or Transport:motorbike == Hobby:rock-climbing or both
- Job:social-worker is somewhere between Job:accountant and Job:paramedic
- Job:nurse == Hobby:dancing or Hobby:dancing == Transport:motorbike or both

Answer the following question:

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

Answer: paramedic


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: designer, fisherman, lawyer, mechanic, videographer
- Beverage: almond-milk, iced-tea, juice, lemonade, mirinda
- Movie-Genre: adventure, musical, romance, satire, thriller
- Sport: badminton, baseball, ice-hockey, rowing, tennis
- Transport: bike, car, roller, snowmobile, tram

Given the following premises about the line of people:
- Sport:tennis is between Movie-Genre:adventure and Movie-Genre:satire
- Beverage:lemonade == Job:videographer
- Beverage:iced-tea is on the left of Transport:car
- Movie-Genre:adventure is on the right of Sport:badminton
- Beverage:juice != Sport:ice-hockey
- Sport:baseball is on the left of Transport:bike
- Job:designer is on the left or right of Movie-Genre:musical
- Beverage:iced-tea is in an even position
- Transport:bike != Sport:badminton
- Transport:roller is in the middle
- Beverage:iced-tea is not to the right of Job:mechanic
- Beverage:juice is not to the left of Beverage:lemonade
- Transport:roller is not to the right of Beverage:almond-milk
- Movie-Genre:satire is on the left or right of Movie-Genre:musical
- Movie-Genre:thriller and Job:fisherman have the same parity positions
- Beverage:lemonade is somewhere to the right of Transport:tram
- Movie-Genre:romance == Job:mechanic or Job:mechanic == Sport:ice-hockey, but not both
- Beverage:juice and Job:fisherman have different parity positions

Answer the following question:

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

Answer: fisherman


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, Food, Music-Genre, Sport, Transport.
The attributes have the following possible values:
- Job: analyst, freelancer, scientist, security-guard
- Food: cauliflower, cucumber, plum, strawberry
- Music-Genre: indie, jazz, metal, rock
- Sport: badminton, rugby, soccer, weightlifting
- Transport: airplane, car, helicopter, subway

Given the following premises about the line of people:
- Food:cauliflower is not to the left of Sport:badminton
- Music-Genre:metal is not to the right of Sport:weightlifting
- Transport:subway is somewhere to the left of Food:cucumber
- Transport:helicopter is on the left or right of Job:analyst
- Transport:subway and Music-Genre:jazz have different parity positions
- Transport:airplane and Music-Genre:jazz have different parity positions
- Sport:soccer == Transport:subway or Food:cauliflower == Sport:soccer, but not both
- Music-Genre:jazz is not to the right of Transport:subway
- Food:strawberry and Music-Genre:indie have the same parity positions
- Transport:airplane is in an odd position
- Transport:helicopter is on the left or right of Food:strawberry
- Job:freelancer and Food:strawberry have different parity positions
- Sport:rugby == Transport:subway or Job:freelancer == Sport:rugby or both
- Job:scientist is not to the right of Food:cauliflower
- Food:strawberry is somewhere to the left of Sport:badminton
- Music-Genre:metal and Job:scientist have the same parity positions

Answer the following question:

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

Answer: rock


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, Movie-Genre, Music-Genre.
The attributes have the following possible values:
- Job: architect, designer, engineer, police-officer, project-manager
- Food: broccoli, grapefruit, nectarine, peach, watermelon
- Movie-Genre: animation, musical, mystery, scientific, superhero
- Music-Genre: blues, indie, metal, r&b, reggae

Given the following premises about the line of people:
- Job:project-manager is on the left or right of Job:designer
- Job:project-manager is on the left or right of Food:watermelon
- Food:grapefruit is somewhere to the right of Movie-Genre:superhero
- Music-Genre:indie is somewhere between Job:engineer and Movie-Genre:scientific
- Movie-Genre:animation is in the middle
- Food:peach is on the far right
- Food:grapefruit != Music-Genre:indie or Food:grapefruit != Movie-Genre:animation or both
- Movie-Genre:superhero is somewhere to the right of Movie-Genre:musical
- Job:architect == Food:broccoli or Food:broccoli == Music-Genre:metal, but not both
- Music-Genre:metal == Job:architect or Music-Genre:metal == Movie-Genre:mystery, but not both
- Job:project-manager is not to the right of Job:designer
- Food:grapefruit != Job:project-manager
- Music-Genre:blues is on the left or right of Food:grapefruit
- Music-Genre:reggae is somewhere between Job:engineer and Movie-Genre:scientific

Answer the following question:

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

Answer: project-manager


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, Movie-Genre, Nationality, Pet, Sport.
The attributes have the following possible values:
- Job: accountant, entrepreneur, fisherman, musician, project-manager, scientist
- Movie-Genre: crime, disaster, documentary, family, western, zombie
- Nationality: argentine, canadian, colombian, indian, nigerian, russian
- Pet: bird, chinchilla, frog, goat, hedgehog, rat
- Sport: golf, ice-hockey, rugby, sailing, snowboarding, surfing

Given the following premises about the line of people:
- Job:musician is in an odd position
- Pet:bird and Nationality:colombian have different parity positions
- Movie-Genre:zombie is not to the left of Nationality:canadian
- Movie-Genre:documentary is in an even position
- Pet:hedgehog is somewhere between Nationality:colombian and Sport:surfing
- Nationality:colombian is not to the right of Pet:frog
- Movie-Genre:family and Pet:frog have the same parity positions
- Job:entrepreneur is somewhere to the right of Nationality:russian
- Pet:goat is not to the left of Nationality:canadian
- Job:scientist is somewhere between Job:entrepreneur and Job:project-manager
- Pet:goat is not to the right of Sport:sailing
- Nationality:nigerian == Pet:chinchilla or Nationality:nigerian == Sport:golf or both
- Nationality:canadian is somewhere to the left of Job:musician
- Job:accountant and Nationality:canadian have the same parity positions
- Job:entrepreneur is somewhere to the right of Movie-Genre:disaster
- Nationality:canadian and Sport:ice-hockey have different parity positions
- Sport:sailing and Pet:hedgehog have different parity positions
- Nationality:nigerian is not to the right of Nationality:indian
- Movie-Genre:disaster is somewhere to the right of Movie-Genre:documentary
- Job:project-manager is not to the left of Job:accountant
- Sport:ice-hockey and Job:musician have the same parity positions
- Nationality:argentine is somewhere to the left of Sport:sailing
- Job:entrepreneur and Movie-Genre:family have the same parity positions
- Sport:rugby is not to the right of Sport:snowboarding
- Job:scientist == Movie-Genre:zombie or Movie-Genre:zombie == Pet:bird or both
- Pet:bird and Movie-Genre:western have different parity positions
- Movie-Genre:crime is somewhere to the right of Movie-Genre:family
- Movie-Genre:zombie is somewhere to the left of Pet:chinchilla
- Nationality:colombian == Sport:snowboarding or Pet:goat == Nationality:colombian, but not both

Answer the following question:

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

Answer: fisherman


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Nationality, Food, Beverage, Pet, Transport.
The attributes have the following possible values:
- Nationality: brazilian, indonesian, mexican, spanish, turkish
- Food: asparagus, avocado, banana, broccoli, cucumber
- Beverage: 7up, fanta, iced-tea, mirinda, tea
- Pet: ferret, goat, hamster, horse, lizard
- Transport: bike, boat, subway, train, tram

Given the following premises about the line of people:
- Nationality:brazilian is somewhere to the right of Nationality:turkish
- Food:cucumber is on the left or right of Pet:hamster
- Food:avocado is not to the right of Transport:tram
- Transport:bike and Food:asparagus have different parity positions
- Transport:tram is not to the right of Nationality:brazilian
- Transport:bike == Pet:ferret or Food:broccoli == Transport:bike or both
- Nationality:spanish is somewhere to the left of Beverage:fanta
- Pet:goat == Food:asparagus or Pet:goat == Transport:subway, but not both
- Beverage:mirinda is on the right of Pet:goat
- Beverage:7up and Nationality:brazilian have different parity positions
- Beverage:7up is in the middle
- Nationality:turkish and Pet:ferret have different parity positions
- Beverage:iced-tea == Transport:train or Beverage:iced-tea == Nationality:indonesian or both
- Pet:lizard is on the far right
- Food:asparagus and Transport:tram have the same parity positions
- Nationality:mexican == Transport:subway
- Transport:boat == Beverage:mirinda or Pet:goat == Transport:boat, but not both
- Nationality:mexican is between Pet:hamster and Pet:lizard

Answer the following question:

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

Answer: horse


There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes:  Job, Food, Movie-Genre, Music-Genre, Nationality, Pet.
The attributes have the following possible values:
- Job: accountant, firefighter, journalist, mechanic, nurse, security-guard
- Food: avocado, cauliflower, grapes, kiwi, peas, pineapple
- Movie-Genre: adventure, disaster, documentary, epic, musical, sports
- Music-Genre: ambient, d&b, rock, salsa, soul, trance
- Nationality: british, dutch, indian, malaysian, nigerian, thai
- Pet: bird, ferret, guinea-pig, hamster, horse, lizard

Given the following premises about the line of people:
- Music-Genre:trance == Food:avocado or Movie-Genre:documentary == Music-Genre:trance, but not both
- Movie-Genre:musical is somewhere to the left of Job:nurse
- Food:grapes is somewhere to the right of Nationality:dutch
- Food:cauliflower is on the right of Job:nurse
- Job:mechanic != Pet:lizard or Job:mechanic != Music-Genre:d&b or both
- Nationality:thai == Movie-Genre:adventure
- Nationality:nigerian == Movie-Genre:documentary or Nationality:nigerian == Pet:bird or both
- Food:avocado != Job:accountant or Job:accountant != Pet:horse or both
- Music-Genre:d&b is on the left or right of Food:kiwi
- Job:firefighter != Movie-Genre:documentary
- Job:security-guard != Nationality:malaysian
- Food:avocado == Music-Genre:trance or Food:avocado == Movie-Genre:documentary or both
- Food:peas != Pet:lizard or Pet:lizard != Music-Genre:rock or both
- Movie-Genre:epic is between Nationality:british and Movie-Genre:documentary
- Movie-Genre:adventure is somewhere between Job:accountant and Food:cauliflower
- Pet:guinea-pig is on the right of Nationality:indian
- Music-Genre:ambient is somewhere to the right of Food:grapes
- Pet:guinea-pig is not to the left of Music-Genre:trance
- Music-Genre:rock is on the right of Food:grapes
- Movie-Genre:epic is not to the left of Job:mechanic
- Nationality:thai is somewhere to the left of Music-Genre:soul
- Job:firefighter is not to the left of Pet:ferret
- Job:accountant and Music-Genre:salsa have different parity positions
- Pet:bird is on the right of Pet:horse
- Music-Genre:ambient is on the left of Nationality:thai
- Job:security-guard and Food:peas have different parity positions
- Job:firefighter is not to the right of Food:avocado
- Job:journalist is on the left of Movie-Genre:musical
- Food:pineapple != Job:nurse
- Nationality:indian is on the left of Nationality:british
- Movie-Genre:sports and Music-Genre:soul have the same parity positions

Answer the following question:

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

Answer: soul


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, Movie-Genre, Music-Genre, Pet.
The attributes have the following possible values:
- Job: bartender, dressmaker, firefighter, fisherman, teacher
- Hobby: card-games, chess, fishing, singing, skydiving
- Beverage: 7up, almond-milk, hot-chocolate, mirinda, water
- Movie-Genre: crime, satire, scientific, superhero, zombie
- Music-Genre: blues, gospel, hip-hop, house, reggae
- Pet: chinchilla, ferret, goldfish, hedgehog, rabbit

Given the following premises about the line of people:
- Music-Genre:house is on the left of Movie-Genre:scientific
- Hobby:skydiving is somewhere to the right of Pet:rabbit
- Job:dressmaker is somewhere to the right of Beverage:7up
- Beverage:almond-milk is not to the right of Beverage:hot-chocolate
- Hobby:fishing is on the right of Movie-Genre:superhero
- Job:teacher is somewhere to the right of Beverage:mirinda
- Beverage:water == Pet:ferret or Beverage:water == Hobby:singing, but not both
- Hobby:singing is somewhere to the right of Music-Genre:hip-hop
- Pet:goldfish and Job:teacher have the same parity positions
- Job:teacher is not to the right of Movie-Genre:zombie
- Music-Genre:house and Hobby:skydiving have different parity positions
- Music-Genre:gospel is not to the right of Job:firefighter
- Movie-Genre:crime is somewhere to the left of Job:bartender
- Hobby:card-games and Beverage:water have different parity positions
- Movie-Genre:superhero == Music-Genre:gospel or Movie-Genre:superhero == Hobby:fishing, but not both
- Job:dressmaker is on the left of Movie-Genre:crime
- Hobby:skydiving is not to the right of Music-Genre:gospel
- Pet:hedgehog is on the right of Pet:chinchilla
- Beverage:water is on the right of Job:dressmaker
- Movie-Genre:scientific == Job:bartender or Movie-Genre:scientific == Music-Genre:blues, but not both

Answer the following question:

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

Answer: reggae


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, Nationality, Transport.
The attributes have the following possible values:
- Job: bartender, dressmaker, electrician, mechanic, social-worker
- Hobby: chess, cooking, gardening, photography, traveling
- Nationality: american, colombian, italian, thai, turkish
- Transport: helicopter, scooter, taxi, train, trike

Given the following premises about the line of people:
- Nationality:turkish and Transport:trike have the same parity positions
- Hobby:traveling == Transport:scooter
- Hobby:photography is on the right of Hobby:chess
- Hobby:chess == Nationality:turkish or Hobby:chess == Job:dressmaker, but not both
- Job:bartender is in the middle
- Transport:helicopter is on the left or right of Nationality:american
- Hobby:photography is somewhere between Hobby:chess and Job:electrician
- Job:electrician is not to the right of Nationality:colombian
- Transport:train is on the left of Job:social-worker
- Hobby:photography is somewhere to the right of Hobby:gardening
- Nationality:colombian is somewhere to the left of Job:dressmaker
- Nationality:italian is somewhere to the left of Hobby:chess
- Nationality:american is somewhere to the right of Nationality:colombian

Answer the following question:

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

Answer: turkish


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, Music-Genre, Beverage, Sport, Transport.
The attributes have the following possible values:
- Nationality: dutch, nigerian, pakistani, polish
- Food: eggplant, lemon, peach, radish
- Music-Genre: folk, funk, metal, trance
- Beverage: almond-milk, mirinda, soy-milk, tea
- Sport: biathlon, golf, parkour, rugby
- Transport: airplane, bike, ship, trike

Given the following premises about the line of people:
- Food:eggplant is somewhere between Food:peach and Transport:trike
- Music-Genre:funk is not to the right of Beverage:mirinda
- Sport:golf is not to the right of Music-Genre:folk
- Music-Genre:folk and Food:eggplant have different parity positions
- Food:lemon is not to the right of Music-Genre:funk
- Music-Genre:trance is not to the left of Sport:golf
- Transport:bike is not to the left of Sport:parkour
- Sport:rugby == Beverage:almond-milk or Sport:rugby == Nationality:dutch or both
- Music-Genre:trance == Food:eggplant or Food:eggplant == Beverage:tea, but not both
- Beverage:mirinda and Transport:airplane have different parity positions
- Music-Genre:folk == Nationality:dutch or Sport:parkour == Music-Genre:folk or both
- Music-Genre:trance and Food:lemon have the same parity positions
- Nationality:polish and Food:peach have the same parity positions
- Nationality:nigerian is somewhere between Transport:airplane and Transport:trike
- Beverage:mirinda is somewhere between Food:eggplant and Food:radish
- Beverage:soy-milk is not to the right of Food:lemon
- Food:lemon is not to the left of Nationality:pakistani

Answer the following question:

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

Answer: funk


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, Movie-Genre, Pet, Sport, Transport.
The attributes have the following possible values:
- Food: cabbage, carrot, lettuce, pepper, potato
- Movie-Genre: comedy, crime, disaster, martial-arts, zombie
- Pet: chinchilla, dog, fish, horse, snake
- Sport: cycling, sailing, surfing, volleyball, weightlifting
- Transport: boat, car, helicopter, snowmobile, subway

Given the following premises about the line of people:
- Sport:cycling is somewhere to the right of Pet:fish
- Movie-Genre:comedy == Sport:cycling or Movie-Genre:comedy == Food:potato or both
- Sport:volleyball is somewhere to the right of Movie-Genre:martial-arts
- Transport:helicopter is somewhere to the left of Food:potato
- Sport:cycling and Pet:horse have different parity positions
- Food:lettuce and Pet:snake have different parity positions
- Pet:horse is not to the left of Food:pepper
- Food:potato is somewhere between Movie-Genre:disaster and Transport:snowmobile
- Movie-Genre:comedy == Sport:surfing or Sport:surfing == Food:potato or both
- Pet:horse is somewhere to the left of Sport:surfing
- Transport:snowmobile is somewhere between Movie-Genre:zombie and Movie-Genre:martial-arts
- Movie-Genre:comedy and Transport:subway have the same parity positions
- Pet:chinchilla is somewhere to the left of Sport:weightlifting
- Food:lettuce is not to the left of Movie-Genre:zombie
- Movie-Genre:martial-arts is somewhere between Movie-Genre:comedy and Pet:snake
- Movie-Genre:zombie == Transport:car or Transport:car == Pet:snake or both
- Movie-Genre:disaster and Food:cabbage have the same parity positions
- Sport:cycling is not to the left of Sport:weightlifting

Answer the following question:

Question: Question: What is the Sport of the person who has/is comedy??
 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, Hobby, Food, Music-Genre, Pet, Transport.
The attributes have the following possible values:
- Job: architect, dancer, entrepreneur, librarian, security-guard, social-worker
- Hobby: dancing, drawing, fishing, reading, traveling, woodworking
- Food: grapefruit, lettuce, pear, pomegranate, pumpkin, spinach
- Music-Genre: ambient, d&b, folk, gospel, techno, trance
- Pet: frog, hedgehog, mouse, pony, rabbit, rat
- Transport: airplane, quad-bike, roller, subway, trike, van

Given the following premises about the line of people:
- Transport:airplane == Pet:hedgehog or Transport:airplane == Hobby:woodworking, but not both
- Job:dancer == Pet:pony or Music-Genre:ambient == Job:dancer, but not both
- Music-Genre:trance is on the right of Music-Genre:d&b
- Job:entrepreneur is on the left or right of Pet:hedgehog
- Pet:rat is not to the right of Hobby:reading
- Hobby:dancing == Transport:subway or Hobby:dancing == Food:pumpkin, but not both
- Food:spinach is not to the left of Job:librarian
- Transport:subway is on the right of Pet:pony
- Pet:pony is somewhere between Job:security-guard and Music-Genre:d&b
- Job:dancer is on the right of Food:pumpkin
- Food:spinach is on the far left or far right
- Transport:roller is not to the left of Hobby:woodworking
- Pet:frog is on the right of Pet:mouse
- Pet:rat is on the left or right of Food:lettuce
- Music-Genre:gospel is not to the right of Food:pear
- Music-Genre:techno is on the right of Music-Genre:folk
- Pet:rabbit is on the far left or far right
- Food:pomegranate and Job:architect have different parity positions
- Pet:hedgehog == Job:dancer or Pet:hedgehog == Music-Genre:trance, but not both
- Pet:mouse == Job:security-guard
- Hobby:fishing == Transport:roller or Hobby:fishing == Pet:rat, but not both
- Hobby:traveling is on the right of Pet:hedgehog
- Music-Genre:gospel is not to the left of Pet:rat
- Hobby:fishing and Music-Genre:trance have different parity positions
- Hobby:fishing is somewhere to the right of Transport:quad-bike
- Pet:mouse and Music-Genre:ambient have different parity positions
- Transport:van is not to the right of Pet:rabbit
- Food:pomegranate is on the left of Pet:frog

Answer the following question:

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

Answer: trance


$\textbf{Laura's Historical Stamps}$

Laura sorts her stamp collection by continent and era in which they were issued. She paid these amounts at an antique market: Brazil and Spain, $7$ cents each, France $9$ cents each, and Peru $6$ cents each (where Brazil and Peru are from South America and Spain and France are from Europe). The number of stamps she has by country and decade is as follows:
- $80$s: Brazil $12$, France $10$, Spain $18$, Peru $15$
- $90$s: Brazil $14$, France $12$, Peru $20$, Spain $16$

Calculate in dollars and cents, how much her European stamps from the $80\text{'s}$ and $90\text{'s}$ cost her.

Answer: \$4.36


The graph of $y = f(x)$ is given as part of a function composition. If the graph of $y = h(x)$ is obtained from the graph of $y = f(x)$ by reflecting it across the y-axis and then shifting it 6 units to the right, express $h(x)$ in terms of $f(x)$.

Answer: h(x) = f(6 - x)


Cara is sitting at a circular table with her six friends, including her friend Alex. How many different pairs of people could Cara be sitting between where one of the pairs must always be Alex?

Answer: 5


Jen rode 50 miles at 12 miles per hour, 20 miles at 40 miles per hour, and 10 miles at 20 miles per hour. What was her average speed, in miles per hour, for the entire ride?

Answer: \frac{480}{31} \text{ mph}


Rosie can make two pies from nine apples. She has thirty-six apples now. How many pies can she make, and will there be any apples left?

Answer: 0


To make lemonade, the recipe requires a ratio of $5$ parts water to $2$ parts lemon juice. If I want to make 1.5 gallons of lemonade, and there are 4 quarts in a gallon, how many cups of water does the recipe require?

Answer: \frac{120}{7} \text{ cups}


City School UK has 180 students, with 100 boys and 80 girls. If $\frac{1}{5}$ of the boys and $\frac{1}{4}$ of the girls are absent one day, what percent of the total student population is absent?

Answer: 22.22\%


The sandwich shop has five types of bread, seven types of meat, and six types of cheese. A sandwich is made up of one bread, one meat, and one cheese type. Chicken, pepper bacon, Swiss cheese, and rye bread are all available options. If Carl never orders a sandwich with a chicken/Swiss cheese combination, nor a rye bread/pepper bacon combination, and additionally avoids any sandwich containing both chicken and rye bread, how many different sandwiches could Carl order?

Answer: 194


Consider a solar system where the planets have the following numbers of moons:

| Planet   | Number of Moons |
|----------|-----------------|
| Mercury  | 0               |
| Venus    | 0               |
| Earth    | 3               |
| Mars     | 4               |
| Jupiter  | 18              |
| Saturn   | 24              |
| Uranus   | 17              |
| Neptune  | 1               |
| Pluto    | 3               |

What is the median number of moons per planet? Additionally, calculate the mean number of moons and compare it to the number of moons Earth has.

Answer: 7.78


Let $M$ be the number of ordered pairs of nonempty sets $\mathcal{A}$ and $\mathcal{B}$ that satisfy:
$\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12,13,14\}$,
$\mathcal{A} \cap \mathcal{B} = \emptyset$,
The number of elements of $\mathcal{A}$ is not an element of $\mathcal{A}$,
The number of elements of $\mathcal{B}$ is not an element of $\mathcal{B}$.
Calculate $M$.

Answer: 3172


After discovering a new group of planets, scientists identify 12 habitable planets. Out of these 12, 6 are considered "Earth-like" (requiring 3 units of colonization each due to their complexity and resource abundance) and the remaining 6 are "Mars-like" (requiring 1 unit of colonization each as they are less resourceful). Given that humanity can mobilize 15 units of colonization in total, how many different configurations of planets can be occupied, considering that all planets are distinct?

Answer: 326


A rectangular room measures 15 feet by 9 feet. How many square yards of carpet, allowing for a 1 foot overlap on each side for fitting, are needed to cover the floor of the room?

Answer: 21\text{ square yards}


A circular floor pattern includes a central point from which 12 equally spaced rays emerge. One ray points due North. Determine the measure in degrees of the smaller angle formed between the ray pointing due North and the ray pointing South-Southeast.

Answer: 150


In 2003, the average monthly rainfall in Clouddale was $37.5\text{ mm}.$ Each year, the average monthly rainfall increases by $3\text{ mm}$ more than the previous year's average. What was the total amount of rain that fell in Clouddale in 2005?

Answer: 522\text{ mm}


A comet follows an elliptical orbit around a star with the star situated at one focus. At its closest approach (perihelion), the comet is 3 astronomical units (AU) from the star, and at its farthest point (aphelion), it is 15 AU away. Calculate the distance of the comet from the star when it is exactly halfway along its orbit.

Answer: 9


A regular polygon has a perimeter of 180 cm and each side has a length of 15 cm. How many sides does this polygon have and what is its area if the apothem is 12 cm?

Answer: 1080 \text{ cm}^2


Andy and Alexa baked a total of 30 cookies. This time, Alexa ends up eating three times the number of cookies eaten by Andy. Determine the maximum number of cookies Andy could have eaten.

Answer: 7


Chandra has four bowls of different colors (red, blue, yellow, green) and five glasses, with an extra glass having no matching bowl color (purple). If she randomly chooses a bowl and a glass, how many different color pairings are possible considering that one glass color has no pairing dish color?

Answer: 20


The average age of 40 fifth-graders is 10 years. The average age of 60 of their parents is 35 years. Additionally, there are 10 teachers with an average age of 45 years. What is the average age of all these fifth-graders, parents, and teachers combined?

Answer: 26.82


A cylinder has a radius of 5 cm and a height of 12 cm. Calculate both the longest segment that would fit inside this cylinder and the volume of the cylinder.

Answer: 300\pi \text{ cm}^3


The line $y = c-x$ where $0 < c < 6$ intersects the $y$-axis at $P$ and the line $x=6$ at $S$. If the ratio of the area of triangle $QRS$ to the area of triangle $QOP$ is 4:16, what is the value of $c$?

Answer: 4


A rectangular tank holds 216 cubic feet of water. Determine the volume of this tank in cubic meters, knowing that 1 cubic meter is approximately 35.315 cubic feet.

Answer: 6.12


The Jackson High School Football Club has 24 players, including 4 goalkeepers. The club organizes a competition to discover which goalkeeper can save the most goals from penalty kicks. For each penalty kick, one goalkeeper will defend the goal while the remaining players (including the other goalkeepers) attempt to score.

How many penalty kicks are required so that every player has the opportunity to shoot against each goalkeeper?

Answer: 92


Two concentric circles have the radius of the inner circle as \( r \) feet and the outer circle as \( 3r \) feet. If the width of the gray region between these circles is \( 4 \) feet, what is the area of the gray region, expressed in terms of \( \pi \) and \( r \)?

Answer: 8\pi r^2


Circles of diameter 3 inches are lined up as shown. What is the area, in square inches, of the shaded region in a 1.5-foot length of this pattern?

Answer: 13.5\pi


A bookstore has a sale on days of the month that are multiples of 3 (such as June 3, June 6...). A shoe store has a sale every 7 days, starting from June 1. How many times in the months of June, July, and August do the two stores have sales on the same date?

Answer: 5


Alex is a software developer who specializes in creating custom publishing software for independent authors. He recently developed a new software package that helps authors publish their books more efficiently. Alex has already sold this software to 15 authors at a price of $120 each.

Now, Alex plans to introduce a new feature to the software which he believes will increase its value and allow him to sell it for $150. He estimates that with this new feature, he will be able to sell the software to 10 more authors.

How much more money will Alex make if he successfully sells the upgraded software to the additional 10 authors at the new price compared to what he earned from selling the original version to the first 15 authors?

Answer: 1500


A fearless stunt pilot named Amelia was inspired by her grandfather's stories from his time as a pilot during the war. To honor his memory, she decides to perform a series of thrilling aerial maneuvers in her plane. On one of her flights, Amelia plans to perform 3 loop-the-loops, each taking 2 minutes, followed by 4 barrel rolls, each taking 1.5 minutes. After completing these maneuvers, she flies in a straight line for 10 minutes to calm down the audience. If the whole performance lasts 30 minutes, how many minutes did Amelia spend not performing stunts?

Answer: 18


Maria is a seasoned cellist in Malta's National Orchestra and has been teaching cello lessons for 15 years. She currently teaches 4 students every week. Each student attends a one-hour lesson. This week, Maria plans to practice with the orchestra for 3 hours on Monday, 2 hours on Wednesday, and 4 hours on Friday. If she also spends 1 hour on each teaching day preparing lesson materials, how many total hours does Maria dedicate to teaching and practicing this week?

Answer: 17


Lady Amelia is a big fan of the British noble system and decides to host a grand tea party at her mansion. She invites 5 dukes, 8 marquesses, 12 earls, 15 viscounts, and 20 barons. Each duke brings 3 guests, each marquess brings 2 guests, and each of the other nobles brings 1 guest. If Lady Amelia's dining hall can seat 100 people comfortably, how many additional chairs does she need to bring in for everyone to have a seat?

Answer: 38


A veteran football player, renowned for his exceptional collection of luxury cars, decides to showcase some of his favorite vehicles. He has 8 luxury cars in his collection, each from a different manufacturer. On a sunny Saturday, he takes out 4 of these cars to display at a local car show. During the show, he learns that each of these cars has an average value of $250,000.

If he sold 2 of the cars at the show for a total of $600,000, how much is the average value of the remaining cars in his collection after these sales?

Answer: 233333.33


A flamenco teacher named Carmen is preparing a new choreography and wants to receive feedback from three dance critics. She plans to invite each critic to one of her classes on different days. Each class consists of 12 students and lasts 90 minutes. Carmen decides that for each class, she will spend 15 minutes discussing her choreography with the critic and the rest of the time teaching her students. If she teaches 5 classes a week and wants to refine her teaching methods by getting feedback from the critics in three weeks, how many total minutes will Carmen spend discussing her choreography with the critics over the three weeks?

Answer: 45


As a sports journalist with a particular interest in cue sports, you are covering a major pool tournament. You need to write an article about the total number of balls pocketed in a series of matches. In the first match, the players pocket a total of 48 balls. In the second match, the total number of balls pocketed is 15% more than in the first match. In the third match, the number of balls pocketed is 8 fewer than in the second match. How many balls were pocketed in total across all three matches?

Answer: 150


Police Chief Taylor is analyzing the crime rates in the city over the past year. In January, there were 120 reported incidents. By June, the number of incidents had decreased by 15%. Over the summer months of July and August, the incidents increased by 20% from June's numbers. However, Chief Taylor's new law enforcement strategies helped reduce the incidents by 25% in September compared to August. How many reported incidents were there in September?

Answer: 92


A member of an underground art forum loves collecting prints of a freelance graphic designer's piracy-themed creations. Each month, the designer releases 3 new piracy-themed prints. The forum member decides to buy 2 prints each month. After 6 months, the forum member decides to gift half of their collection to a friend who is also a fan of the designer's work. How many prints does the forum member have left after gifting their friend?

Answer: 6


Tommy is a curious schoolboy who loves history. He recently read about the ancient Egyptian pharaohs and learned that the Great Pyramid of Giza was completed around 2560 B.C. Tommy wonders how many years have passed since the pyramid was completed until today, which is the year 2023. To find out, he starts by calculating the total number of years from 2560 B.C. to 1 B.C., then adds the years from 1 A.D. to 2023 A.D. How many years ago was the Great Pyramid of Giza completed?

Answer: 4582


Alex, a passionate coffee shop owner, only accepts cash payments at their cozy café. One morning, Alex opens the café with $50 in the cash register. By lunchtime, Alex sells 8 cups of coffee for $2 each and 5 pastries for $3 each. In the afternoon, Alex sells 10 more cups of coffee and 7 pastries, each at the same prices as before. At the end of the day, Alex realizes that 3 customers paid with a $20 bill, and the rest paid with exact change. How much cash does Alex have in the register at closing time?

Answer: 122


An artisan named Raul, who admires the weaving skills of an elderly woman named Maria in his village, decides to learn her craft. Maria weaves beautiful tapestries using a traditional technique that has been passed down through generations. Each tapestry requires 7 days to complete, and Maria can weave 3 tapestries in a month while also teaching Raul.

Raul wants to practice his weaving skills and plans to create his own tapestries. If Raul can weave at half the speed of Maria, how many days will it take Raul to weave 5 tapestries?

Answer: 70


A policy analyst working in a government agency is supervising a student's research project. As part of the project, the student needs to analyze survey data collected from 150 local residents about their satisfaction with a new public transportation policy. The survey results show that 60% of the respondents are satisfied, 25% are neutral, and the rest are dissatisfied.

The policy analyst wants to calculate how many more residents are satisfied with the policy compared to those who are dissatisfied. Can you help the analyst determine this number?

Answer: 68


A sports commentator is analyzing West Ham's recent match and notices something interesting about their tactics. During the first half of the match, West Ham made 8 successful passes for every 3 unsuccessful passes. In the second half, they adjusted their tactics, making 12 successful passes for every 2 unsuccessful passes. If West Ham made a total of 120 successful passes during the entire match, how many unsuccessful passes did they make in total?

Answer: 30


A skilled negotiator from a non-EU country is preparing to negotiate a trade deal that will protect their country's interests. During the negotiation, they aim to increase their country's export of goods by 20%. Currently, the country exports 150,000 units of goods annually. In addition, they want to reduce imports by 15%, as the country currently imports 200,000 units of goods per year. What will be the total change in the number of units traded (exports increased plus imports decreased) after the negotiation?

Answer: 60000


Jesse is a former 80s rock musician who has overcome addiction and now supports a cause advocating for recovery and mental health awareness. He decides to organize a charity concert to raise funds for his advocacy. Jesse plans to sell tickets to the concert at $25 each. He expects to sell 200 tickets. Additionally, Jesse's band will sell 50 special edition merchandise items at $15 each.

To set up the concert, there are expenses: the venue costs $1,200, and the sound equipment rental is $800. Jesse also decides to donate $500 from his personal savings to support the cause.

Calculate the total amount of money raised after covering all expenses.

Answer: 4250


Maria is a strong-willed mother who always encourages her children to make their own choices and believes in the power of women's individual decisions. She has planned a special day to celebrate International Women's Day with her family. She wants to take her daughters, Emma and Sofia, to an inspiring women's conference and then treat them to dinner at their favorite restaurant.

The conference tickets cost $15 each, and Maria buys one for herself and one for each of her two daughters. After the conference, they head to the restaurant, where Maria lets her daughters choose their meals. Emma chooses a meal that costs $12, and Sofia chooses a meal that costs $10. Maria decides to have a meal that costs $18.

After dinner, Maria wants to treat her daughters to ice cream from the dessert menu. The ice cream costs $5 per scoop, and each of them decides to have two scoops. How much money does Maria spend in total on the conference tickets, dinner, and ice cream for herself and her daughters?

Answer: 115


Jamie, a trauma survivor, is inspired by a retired officer, Mr. Thomas, who often shares his wisdom on healing and moving forward. Jamie decides to write a journal reflecting on their journey every day for a month to track their progress. In the first week, Jamie writes 5 pages each day. Motivated by Mr. Thomas's encouragement, Jamie increases the number of pages by 2 each day for the second week. For the remaining two weeks, Jamie maintains a steady pace of writing 8 pages a day. How many pages does Jamie write in total for the entire month?

Answer: 238


Alex, a film critique with a passion for old Christmas movies, is planning a holiday movie marathon. He has a collection of 12 classic Christmas films, each with a soundtrack composed by different artists. However, 5 of these soundtracks were composed by his favorite composer, Harry Gregson-Williams.

Alex decides to watch 3 movies per day leading up to Christmas. On the first day, he randomly selects and watches 2 movies with soundtracks not composed by Harry Gregson-Williams and 1 movie with a soundtrack by him.

If Alex continues to watch the same pattern of movies (2 non-Gregson-Williams and 1 Gregson-Williams) each day, how many days will it take him to watch all 12 movies from his collection?

Answer: 5


The mayor's office staff is planning a community fair to promote local artisans and small businesses. They have decided to allocate different areas of the park for various activities: arts and crafts, food stalls, and live performances. The total area of the park is 15,000 square feet. The arts and crafts section will take up 3,000 square feet, the food stalls need twice that area, and the live performances area requires 1,500 square feet more than the food stalls. How much area will be left for other activities after these three sections are allocated?

Answer: -1500


Two acclaimed historical fiction writers, Alex and Jamie, are collaborating on a new literary project. They plan to write a book consisting of 12 chapters. They agree that Alex will write 5 chapters, Jamie will write 4 chapters, and they will co-write the remaining chapters together. If each chapter takes an average of 10 days to complete, calculate the total number of days Alex and Jamie will spend working on the book, including the chapters they co-write.

Answer: 150


$\triangle ABC$ is a triangle and $D,E,F$ are points on $BC$, $CA$, $AB$ respectively. It is given that $BF=BD$, $CD=CE$ and $\angle BAC=48^{\circ}$. If the answer is $x^\circ$, find the value of $x$ for the angle $\angle EDF$.

Answer: 66


In how many ways can $10$ distinct books be placed onto a $3$-shelf bookcase such that no shelf is empty? Express your answer in the form \( n \cdot 10! \), where \( n \) is an integer.

Answer: 36


Given the line $y = \frac{3}{4}x + 6$ and a line $L$ parallel to the given line and $4$ units from it. Find a possible equation for line $L$. The original answer is in the form $y = \frac{3}{4}x + c$, please give the value of c.

Answer: 1


A massless elastic cord (that obeys Hooke's Law) will break if the tension in the cord exceeds \(T_{max}\). One end of the cord is attached to a fixed point, the other is attached to an object of mass \(3m\). If a second, smaller object of mass \(m\) moving at an initial speed \(v_0\) strikes the larger mass and the two stick together, the cord will stretch and break, but the final kinetic energy of the two masses will be zero. If instead the two collide with a perfectly elastic one-dimensional collision, the cord will still break, and the larger mass will move off with a final speed of \(v_f\). All motion occurs on a horizontal, frictionless surface.

Find the ratio of the total kinetic energy of the system of two masses after the perfectly elastic collision and the cord has broken to the initial kinetic energy of the smaller mass prior to the collision. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 7


Real numbers \(a\) and \(b\) are chosen with \(1 < a < b\) such that no triangle with positive area has side lengths \(1, a, b\) or \(\frac{1}{b}, \frac{1}{a}, 1\). The original answer is in the form \(\frac{k+\sqrt{n}}{m}\). Find the value of \(k + m + n\).

Answer: 10


Two circles are externally tangent. Lines \(\overline{PAB}\) and \(\overline{PA'B'}\) are common tangents with \(A\) and \(A'\) on the smaller circle and \(B\) and \(B'\) on the larger circle. If \(PA = AB = 4\), then what is the value of \(x\) if the area of the smaller circle is \(x\pi\)?

Answer: 2


$A(-1,1)$, $B$, and $C$ are points on the hyperbola $x^2 - y^2 = 1$. If $\triangle ABC$ is a regular triangle, find the area of $\triangle ABC$. The original answer is in the form of k\sqrt{m}, please give the value of k + m.

Answer: 6


$ABCDE$ is a convex pentagon. Given the following measurements: \( m(\widehat{B}) = m(\widehat{D}) = 90^\circ \), \( m(\widehat{C}) = 120^\circ \), \( |AB| = 2 \), \( |BC| = |CD| = \sqrt{3} \), and \( |ED| = 1 \). Find the length of \( |AE| \). The original answer is in the form \(\sqrt{k}\), please give the value of \(k\).

Answer: 3


Let \(S\) be the set of ordered triples \((x,y,z)\) of real numbers for which
\[ \log_{10} (x + y) = z \text{ and } \log_{10} (x^2 + y^2) = z + 1. \]
There are real numbers \(a\) and \(b\) such that for all ordered triples \((x,y,z)\) in \(S\) we have
\[ x^3 + y^3 = a \cdot 10^{3z} + b \cdot 10^{2z}. \]
The original answer is in \(\frac{m}{n}\) format. Please find the value of \(m + n\).

Answer: 31


In the figure, polygons $A$, $E$, and $F$ are isosceles right triangles; $B$, $C$, and $D$ are squares with sides of length $1$; and $G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).

Answer: 11


A small block with an initial speed $v_0$ smoothly moves onto a sloped big block of mass $M$. After the small block reaches a height $h$ on the slope, it slides back down. Find the height $h$ in terms of $v_0$, $g$, $m$, and $M$. The original answer is in the form \frac{k}{m} \cdot \frac{n v_0^2}{p + M}. Please give the value of k + m + n + p.

Answer: 5


Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every \(90\) seconds, and Robert runs clockwise and completes a lap every \(80\) seconds. Both start from the same line at the same time. At some random time between \(10\) minutes and \(11\) minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m, where k and m are integers in the simplest form of the fraction.

Answer: 19


The diagram shows a heptagon with a line of three circles on each side. Each circle is to contain exactly one number. The numbers 8 to 14 are distributed as shown and the numbers 1 to 7 are to be distributed to the remaining circles. The total of the numbers in each of the lines of three circles is to be the same. What is this total?

A) 18
B) 19
C) 20
D) 21
E) 22

Answer: 19


Rachel has two indistinguishable tokens, and places them on the first and second squares of a \(1 \times 6\) grid of squares. She can move the pieces in two ways:
- If a token has a free square in front of it, then she can move this token one square to the right.
- If the square immediately to the right of a token is occupied by the other token, then she can "leapfrog" the first token; she moves the first token two squares to the right, over the other token, so that it is on the square immediately to the right of the other token.

If a token reaches the 6th square, then it cannot move forward anymore, and Rachel must move the other one until it reaches the 5th square. How many different sequences of moves for the tokens can Rachel make so that the two tokens end up on the 5th square and the 6th square?

Answer: 42


In $\triangle ABC$, it is known that $AB = AC$, $AE \perp BC$, $CD = CA$, and $AD = BD$. Find the measure of $\angle DAE$ in degrees.

Answer: 18


In an equilateral triangle \(ABC\) with side length \(a\), points \(M, N, P, Q\) are located as shown in the figure. It is known that \(MA + AN = PC + CQ = a\). Find the measure of angle \(NOQ\).

Answer: 60


Two mutually perpendicular rays, propagating in the same vertical plane, fall from the air onto a horizontal glass surface. The refractive index of the glass is $n=1.6$. The angle of refraction for the first ray is $\beta=30^{\circ}$. Determine the angle between the refracted rays.

Answer: 52


A box contains 10 identical balls. Among them, 3 balls are labeled with the number 1, 4 balls are labeled with the number 2, and 3 balls are labeled with the number 5. One ball is randomly selected from the box, recorded, and then placed back into the box. Another ball is then randomly selected. Let the sum of the numbers on the balls from the first and second selections be denoted as $\xi$.
(I) Find the distribution of the random variable $\xi$.
(II) Find the expected value $E(\xi)$ of the random variable $\xi$.

Answer: 5.20


The acute angle formed by the hands of a clock at 3:30 p.m. is \( A^\circ \). Find \( A \).

If \( \tan (3A + 15)^\circ = \sqrt{B} \), find \( B \).

If \( \log_{10} AB = C \log_{10} 15 \), find \( C \).

The points \( (1, 3) \), \( (4, 9) \), and \( (2, D) \) are collinear. Find \( D \).

Answer: 5


An acute-angled isosceles triangle and a trapezoid are inscribed in a circle. One base of the trapezoid is the diameter of the circle, and the lateral sides of the trapezoid are parallel to the lateral sides of the triangle. Find the ratio of the areas of the trapezoid and the triangle.

Answer: 1


A talented writer focusing on psychological horror is drafting a new story that combines eerie elements with futuristic technology. In the story, they introduce a haunted AI that sends messages to 5 people every 6 hours. Each message contains 9 words, and each word increases in complexity with each consecutive message. The writer plans to write a total of 72 messages in one day to fully develop the storyline. If the writer wants to calculate how many total words they will have written by the end of the day, how many words will the haunted AI have sent in 24 hours?

Answer: 648


Maria is a regional sales representative for a company that sells electronic gadgets. Each week, she needs to manage the inventory levels and ensure that the delivery schedules are met for her region. This week, she starts with 150 gadgets in her inventory. She receives a shipment of 200 more gadgets on Monday. By Wednesday, she sells 120 gadgets. On Thursday, she sends out a delivery of 50 gadgets to a nearby store. On Friday, another delivery of 80 gadgets is scheduled to be sent out, but she discovers that 30 gadgets are defective and cannot be sold or delivered. How many gadgets does Maria have left in her inventory at the end of the week?

Answer: 70


Mr. Johnson, a retired bank security officer, often reminisces about his days of protecting the bank from criminals. One day, he decides to organize his old case files. He has 28 files, each representing a crime he helped prevent. He wants to divide these files into 4 equal stacks so he can easily review them one at a time each week. Additionally, he discovers 3 extra files that were misplaced, and decides to add them evenly to the stacks. How many files will each stack contain after he adds the extra files?

Answer: 8


Mr. Smith is a workers' compensation lawyer with 20 years of experience. Over his career, he has successfully handled an average of 45 cases per year. This year, he decided to take on 10% more cases than his yearly average to offer more help to injured workers. How many cases did Mr. Smith decide to handle this year?

Answer: 50


John is a middle-class factory worker who supports the local businessman's economic initiatives. The businessman has proposed building a new factory that will create 150 new jobs but will increase local pollution levels. John is concerned about the environment but sees the benefit of additional jobs for the community. If the new factory increases local pollution by 20% but also increases the local employment rate by 10%, how many more jobs will be created if the original number of jobs in the community was 1,200?

Answer: 120


Sara, a homesteader living off the grid, has a sustainable farm where she grows vegetables and raises chickens. She collects eggs from her chickens every morning. This week, Sara collected the following number of eggs each day: Monday - 12, Tuesday - 15, Wednesday - 10, Thursday - 13, Friday - 11. On Saturday and Sunday, she collects twice the average number of eggs she collects on weekdays. How many eggs does Sara collect in total during the week?

Answer: 110


A visual artist is creating a painting that incorporates elements of light and shadow. The artist uses 12 different shades of gray to depict the shadows in the painting and 8 different shades of yellow to depict the light. If the artist decides to use 3 shades of gray for each shadowed area and 2 shades of yellow for each lit area, how many total areas of light and shadow can the artist create in the painting?

Answer: 248


A dedicated student is studying postcolonial literature and decides to analyze Margaret Atwood's works over the course of a month. The student plans to read three of Atwood's novels. The first novel has 320 pages, the second novel has 275 pages, and the third novel has 400 pages. The student wants to complete the reading in 30 days, reading the same number of pages each day. How many pages does the student need to read per day to finish all three novels in 30 days?

Answer: 34


Dr. Star, a professional astronomer, is giving a special tour to a group of 12 interns at the renowned Celestial Heights Observatory. During the tour, Dr. Star explains that they will observe 4 different planets through a powerful telescope. Each intern will have 10 minutes to observe each planet. After the observation, the interns will gather for a 30-minute discussion session led by Dr. Star. How much total time, in minutes, will the entire session (observation plus discussion) take for all the interns?

Answer: 510


Nathaniel Kong is a famous musician, and you are his biggest fan! You decide to create a playlist of all his songs to listen to every day. Nathaniel has released 4 albums, each with 10 songs. He recently released a new album with 12 songs, and you bought a special edition of one of his previous albums that includes 3 bonus tracks. How many songs in total do you have in your Nathaniel Kong playlist now?

Answer: 55


An art magazine columnist is reviewing a new typeface designed by a famous typographer. The typeface includes 5 different styles: regular, italic, bold, bold italic, and light. Each style has a unique number of glyphs: regular has 128 glyphs, italic has 140 glyphs, bold has 150 glyphs, bold italic has 135 glyphs, and light has 120 glyphs. The columnist decides to analyze a sample set consisting of 3% of the total glyphs from each style to write a detailed critique.

How many total glyphs does the columnist analyze from all the styles combined?

Answer: 21


Mr. Thompson is a high school teacher in San Francisco who is planning a field trip for his class to the Exploratorium, a famous science museum in the city. He has 25 students in his class and plans to rent a bus for transportation. The bus rental company charges $150 for the first hour and $100 for each additional hour. The museum is 20 miles from the school, and the bus travels at an average speed of 30 miles per hour. If the class plans to spend 3 hours at the museum and the round trip takes the bus a total of 1 hour and 20 minutes of driving time, how much will Mr. Thompson need to pay for the bus rental?

Answer: 550


Alex, a non-religious skeptic living in Kansas, loves to explore unusual facts about the state. Recently, Alex discovered that Kansas has a curious number of counties: 105. Wanting to dig deeper into the state's history, Alex decides to visit 3 counties each weekend. If Alex also plans to take a 2-week break from visiting counties, how many weekends will it take for Alex to visit all the counties in Kansas?

Answer: 37


Mr. Thompson, the owner of Green Acres Farm, oversees several workers who help him manage the farm's daily operations. On Monday, Mr. Thompson asked his workers to plant vegetable patches. Each worker can plant 3 vegetable patches in a day, and Mr. Thompson has 8 workers currently available. How many vegetable patches can all the workers plant together in one day?

On Tuesday, Mr. Thompson decided to split the workers into two equal groups to handle different tasks. The first group continued planting vegetable patches, while the second group harvested the ripe vegetables. If each group worked for 2 days, how many vegetable patches did the first group plant by the end of Wednesday?

Finally, Mr. Thompson realized he had 24 vegetable patches already planted before the week started. By the end of Wednesday, how many total vegetable patches does Mr. Thompson have planted on the farm?

Answer: 72


* Four, the sequence $1,1,3,3,3^{2}, 3^{2}, \cdots, 3^{1992}, 3^{1992}$ consists of two 1s, two 3s, two $3^{2}$s, ..., two $3^{1992}$s arranged in ascending order. The sum of the terms in the sequence is denoted as $S$. For a given natural number $n$, if it is possible to select some terms from different positions in the sequence such that their sum is exactly $n$, it is called a selection scheme, and the number of all selection schemes with the sum $n$ is denoted as $f(n)$. Try to find:
$$
f(1)+f(2)+\cdots+f(s) \text { . }
$$

Answer: 4^{1993}-1


8.2. Points $A, B, C, D$ are the centers of the circles inscribed in the faces $P Q S, P R S$, $Q R S$ and $P Q R$ of the tetrahedron $P Q R S$. It is known that $P Q=R S=7, P R=Q S=8, P S=Q R=9$. Find the volume of the tetrahedron $A B C D$. Round your answer to the nearest hundredth if necessary.

Answer: 1.84


5. In a regular $1000-$gon, all diagonals are drawn. What is the maximum number of diagonals that can be selected such that among any three of the selected diagonals, at least two have the same length?

Answer: 2000


## Zadatak A-4.3.

Odredi sve prirodne brojeve $m$ i $n$ za koje je $2^{n}+5 \cdot 3^{m}$ kvadrat nekog prirodnog broja.

Answer: (2,2)


3. Village Chief Slow Sheep writes down the natural numbers $1 \sim 100$ on the blackboard in sequence. Happy Sheep first erases the numbers that can be divided by 5, then Boiling Sheep erases the numbers that contain the digit 5, and Lazy Sheep erases the numbers whose sum of digits can be divided by 5. At this point, there are $\qquad$ numbers left on the blackboard.

Answer: 56


21. The formula for the volume of a sphere is $V=\frac{4}{3} \pi r^{3}$, where $r$ is the radius of the sphere. In a cylindrical container, several solid iron spheres with the same radius as the cylinder's base can be placed. When water is poured into the container, the volume of water is six times the volume of one iron sphere when the water level just reaches the top of the container. The number of iron spheres in the container is $\qquad$.

Answer: 12


17. The monster's battle mode is: 7 seconds preparation -2 seconds fire breathing - 7 seconds preparation -2 seconds fire breathing….. Ultraman Tiga's battle mode is: 11 seconds preparation - an instant laser - 11 seconds preparation - an instant laser....
The laser can only be effective when the monster is fire breathing, and hitting it will defeat the monster. Once, the monster and Ultraman Tiga met, and they started preparing at the same time, so the monster will be defeated by Ultraman Tiga in $\qquad$ seconds.

Answer: 44


1614. 600 corn seeds are planted with a probability of 0.9 for each seed to germinate. Find the boundary of the absolute value of the deviation of the frequency of germinated seeds from the probability $p=0.9$, if this boundary is to be guaranteed with a probability $P=0.995$.

Answer: 0.034


Example 2 As shown in Figure 4, in a $7 \times 8$ rectangular chessboard, a chess piece is placed at the center of each small square. If two chess pieces are in small squares that share an edge or a vertex, then these two chess pieces are said to be "connected". Now, some of the 56 chess pieces are removed so that no 5 remaining chess pieces are in a straight line (horizontally, vertically, or diagonally). What is the minimum number of chess pieces that need to be removed to meet this requirement? Explain your reasoning $^{\mid 2\rfloor}$.

---

The translation preserves the original text's formatting and structure.

Answer: 11


In a small pond, there are eleven lily pads in a row labeled $0$ through $10$. A frog is sitting on pad $1$. When the frog is on pad $N$, where $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad $0$, it will be eaten by a patiently waiting snake. If the frog reaches pad $10$, it will exit the pond, never to return. Find the probability that the frog will escape being eaten by the snake. The original answer is in \frac{m}{n} format, please give the value of m + n.

Answer: 209


The circumradius of an acute triangle \(ABC\) is twice the distance from its circumcenter to \(AB\). Given that \(|AC|=2\) and \(|BC|=3\), what is the altitude passing through \(C\)? The original answer is in the form \(\frac{k}{m}\sqrt{n}\), where \(\frac{k}{m}\) is in its simplest form. Please provide the value of \(k + m + n\).

Answer: 31


Jerry and Aaron both pick two integers from $1$ to $6$, inclusive, and independently and secretly tell their numbers to Dennis.

Dennis then announces, "Aaron's number is at least three times Jerry's number."

Aaron says, "I still don't know Jerry's number."

Jerry then replies, "Oh, now I know Aaron's number."

What is the sum of their numbers?

Answer: 7


Five unit squares are arranged in the coordinate plane with the lower left corner at the origin. The slanted line, extending from \((a,0)\) to \((3,3)\), divides the entire region into two regions of equal area. The original answer is in the form \(\frac{k}{m}\). Please give the value of k + m.

Answer: 5


How many values of $\theta$ in the interval $0 < \theta \le 2\pi$ satisfy $1 - 3\sin\theta + 5\cos3\theta = 0$? Provide your answer as an integer.

Answer: 6


$ABCD$ is a cyclic quadrilateral. If $\angle B = \angle D$, $AC \cap BD = \{E\}$, $\angle BCD = 150^\circ$, $|BE| = x$, $|AC| = z$, then find $|ED|$ in terms of $x$ and $z$. The original answer is in the form \(\frac{kx + mz}{n}\), where n cannot be divisible by 2. Please provide the value of \(k + m + n\).

Answer: 1


The circle with radius $4$ passing through points $A$ and $B$ is tangent to another circle with radius $3$ passing through points $A$ and $C$, where the distance $|AB| = 2$. If the line $BC$ is tangent to the second circle, what is the value of $|BC|^2$?

Answer: 7


Professor Smith, a history professor, invited Dr. Lopez, an expert in colonial history, to guest speak in their class about the impacts of colonization. Dr. Lopez plans to distribute 240 pamphlets to the students in the class, which summarize key points from their talk. If there are 30 students in the class, and Dr. Lopez wants each student to receive an equal number of pamphlets, how many pamphlets will each student receive? Additionally, if 5 students are absent on the day of the talk, how many extra pamphlets will Dr. Lopez have after distributing them to the students who are present?

Answer: 40


A former teammate who played football alongside Bo Bonnheim once played a memorable game where they scored a total of 35 points. During the game, Bo scored three times as many points as his teammate. Their team scored an additional 21 points from other players. How many points did Bo's former teammate score in that game?

Answer: 3.5


DJ Analog is preparing for a music festival performance and has set up his collection of vintage synthesizers and drum machines. He plans to use 5 different synthesizers and 3 drum machines for his set. Each synthesizer can run for 2 hours before needing to rest for 15 minutes, and each drum machine can run for 1.5 hours before needing to rest for 10 minutes.

DJ Analog starts his performance at 6:00 PM and plans to play continuously without any breaks for himself. If he wants to keep all his equipment running as long as possible without exceeding their operating limits, how many minutes of music can DJ Analog play before he needs to rest any of his equipment?

Answer: 90


Samuel is a PhD candidate working on a thesis about the history of economic thought within the church. During his research, he decides to analyze the growth of economic writings by church scholars over the centuries. He finds that in the 14th century, there were 15 significant economic manuscripts written. In the 15th century, the number of manuscripts increased by 40%. By the 16th century, there was a further increase of 25% in the number of manuscripts compared to the 15th century.

How many significant economic manuscripts were there in total by the end of the 16th century?

Answer: 62


A movie producer specializing in CGI technology is working on a new film that features realistic animal characters. For one scene, the producer needs to create 8 different animals. Each animal requires 5 hours of CGI work to look realistic. Additionally, each animal character needs a 2-hour post-CGI review to ensure quality. If the producer can allocate 2 CGI artists working at the same time, how many total hours will it take for the CGI work and review of all the animals?

Answer: 28


Alex is a Xamarin developer who spends a significant amount of time debugging cross-platform issues. On a typical day, Alex fixes 3 issues on Android, 4 issues on iOS, and 2 issues that are common to both platforms. Last week, Alex worked for 5 days. How many total issues did Alex fix last week?

Answer: 45


Grandma Eleanor is hosting a family dinner to celebrate her grandson's graduation. She plans to invite her children, grandchildren, and a few close neighbors to maintain a community spirit. In total, she expects 5 children, 8 grandchildren, and 3 neighbors to attend. Grandma Eleanor wants to ensure everyone feels welcome and has enough to eat, so she decides to prepare 3 dishes per person. Each dish can serve 2 people. How many dishes does Grandma Eleanor need to prepare for the dinner?

Answer: 24


A local historian is giving a tour of a World War II museum in Normandy. During the tour, the historian tells the group that this week, they have conducted 5 tours each day from Monday to Friday. Each tour lasts 90 minutes and starts at a different hour to accommodate different groups. On Saturday, the historian decided to add 2 more tours to make the weekend more exciting and also conducted 5 tours on that day, each lasting 90 minutes. If the historian wants to calculate the total number of hours spent giving tours throughout the week, how many hours did they spend?

Answer: 45


The neighborhood association leader, Alex, surveyed their community to collect data on recent crime incidents. In the first week, there were 12 reported incidents. In the second week, the number of incidents decreased by 25%. In the third week, the number of incidents increased by 10% compared to the second week. Alex wants to present this data to the association meeting to discuss strategies for further reducing crime.

How many crime incidents were reported in the third week?

Answer: 10


Alex, a responsible older cousin, is a role model for his teenage cousin, Jamie. Alex spends quality time with Jamie by helping him with his studies and planning fun but educational activities. This weekend, Alex decided to mentor Jamie on budgeting and managing time. They started with a simple project where they planned a day out with a budget of $50.

They decided to spend money on three activities:
1. Going to the movies, which costs $12 per person.
2. Having lunch at a local café, which costs $8 per person.
3. Visiting an arcade, where they plan to spend $10 in total.

Alex also wants to set aside $5 from the budget for emergency expenses. How much money will they have left after participating in these activities and setting aside the emergency fund?

Answer: -5


Dr. Smith, a cognitive psychologist, wants to conduct an experiment to explore how perception affects immersion in virtual reality. She plans to test different durations of VR sessions with a group of participants. Dr. Smith has chosen three different durations for the sessions: 15 minutes, 30 minutes, and 45 minutes. She decides to have 4 sessions of each duration, but 3 participants can only do the 15-minute sessions due to scheduling conflicts.

In total, how many VR session minutes will Dr. Smith conduct if she has 10 participants in the experiment and each participates in all the sessions they can attend?

Answer: 2700


Evariste has drawn twelve triangles such that two consecutive triangles share exactly one edge.

Sophie colors every triangle side in red, green, or blue. Among the $3^{24}$ possible colorings, how many have the property that every triangle has one edge of each color?

Answer: 4098


The figure is created by arranging four squares, each with a side length of 5 units, on each side of a central square with a side length of 20 units. Calculate the perimeter of the entire figure.

Answer: 120


Let $S_1$, $S_2$, $\dots$, $S_{125}$ be 125 sets of 5 numbers each, comprising 625 distinct numbers. Let $m_i$ be the median of $S_i$. Let $M$ be the median of $m_1$, $m_2$, $\dots$, $m_{125}$. What is the greatest possible number of the 625 numbers that are less than $M$?

Answer: 436


Alison is eating $2401$ grains of rice for lunch. She eats the rice in a peculiar manner:

1. If she has only one grain of rice remaining, she eats it.
2. Otherwise, she finds the smallest positive integer $d > 1$ for which she can group the rice into equal groups of size $d$ with none left over.
3. She then groups the rice into groups of size $d$, eats one grain from each group, and puts the rice back into a single pile.

How many steps does it take her to finish all her rice?

Answer: 17


5 people stand in a line facing one direction. In every round, the person at the front moves randomly to any position in the line, including the front or the end. Suppose that \( \frac{m}{n} \) is the expected number of rounds needed for the last person of the initial line to appear at the front of the line, where \( m \) and \( n \) are relatively prime positive integers. What is \( m + n \)?

Answer: 137


Let $k$ be a positive integer. A sequence $a_0, a_1, \ldots, a_n$, where $n > 0$, of positive integers satisfies the following conditions:

1. $a_0 = a_n = 1$;
2. $2 \leq a_i \leq k$ for each $i = 1, 2, \ldots, n-1$;
3. For each $j = 2, 3, \ldots, k$, the number $j$ appears $\phi(j)$ times in the sequence $a_0, a_1, \ldots, a_n$, where $\phi(j)$ is the number of positive integers that do not exceed $j$ and are coprime to $j$;
4. For any $i = 1, 2, \ldots, n-1$, $\gcd(a_i, a_{i-1}) = 1 = \gcd(a_i, a_{i+1})$, and $a_i$ divides $a_{i-1} + a_{i+1}$.

Suppose there is another sequence $b_0, b_1, \ldots, b_n$ of integers such that $\frac{b_{i+1}}{a_{i+1}} > \frac{b_i}{a_i}$ for all $i = 0, 1, \ldots, n-1$. Find the minimum value of $b_n - b_0$.

Answer: 1


There are 300 children in a camp. Everyone has no more than $k-1$ friends. What is the smallest $k$ for which it might be impossible to create some new friendships so that everyone has exactly $k$ friends?

Answer: 151


Yang has a standard $6$-sided die, a standard $8$-sided die, and a standard $10$-sided die. He tosses these three dice simultaneously. The probability that the three numbers that show up form the side lengths of a right triangle can be expressed as $\frac{m}{n}$, for relatively prime positive integers $m$ and $n$. Find $100m+n$.

Answer: 1180


Suppose that $ABCDE$ is a convex pentagon with $\angle A = 90^\circ$, $\angle B = 105^\circ$, $\angle C = 90^\circ$ and $AB = 2$, $BC = CD = DE = \sqrt{2}$. If the length of $AE$ is $\sqrt{a} - b$ where $a$ and $b$ are integers, what is the value of $a + b$?

Answer: 4


$P$ is a point inside $\triangle ABC$, line $AC$ and $BP$ intersect at $Q$, line $AB$ and $CP$ intersect at $R$. It is known that $AR = RB = CP$ and $OQ = PQ$. Find the measure of $\angle BRC$.

Answer: 120


In a rectangular coordinate system, \( O \) is the origin, and points \( A \) and \( B \) are in the first quadrant. \( A \) lies on the line \( y = \tan \theta x \), where \( \theta \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) \) and \( |OA| = \frac{1}{\sqrt{2} - \cos \theta} \). Point \( B \) is on the hyperbola \( x^2 - y^2 = 1 \) such that the area of triangle \( \triangle OAB \) is minimized. Determine the value of \( \theta \) within \( \left( \frac{\pi}{4}, \frac{\pi}{2} \right) \) that maximizes the area of \( \triangle OAB \), and find the maximum area.

Answer: \frac{\sqrt{6}}{6}


Let \( q(x) = q^1(x) = 2x^2 + 2x - 1 \), and let \( q^n(x) = q(q^{n-1}(x)) \) for \( n > 1 \). How many negative real roots does \( q^{2016}(x) \) have?

Answer: \frac{2^{2017}+1}{3}


Let \( P_{1} P_{2} \ldots P_{8} \) be a convex octagon. An integer \( i \) is chosen uniformly at random from 1 to 7, inclusive. For each vertex of the octagon, the line between that vertex and the vertex \( i \) vertices to the right is painted red. What is the expected number of times two red lines intersect at a point that is not one of the vertices, given that no three diagonals are concurrent?

Answer: \frac{54}{7}


Points $A, B, C, D, E, F$ are located clockwise on a circle as shown in the figure. Chords $AD$ and $CE$ intersect at point $X$ forming a right angle, and chords $AD$ and $BF$ intersect at point $Y$. Given that $CX=12$, $XE=27$, $XY=15$, $BY=9$, and $YF=11$.

(a) (2 points) Find the length of segment $AD$.

(b) (2 points) Find the radius of the circle.

Answer: 19.5


A truncated cone is circumscribed around a sphere, with the area of one base being 4 times larger than the area of the other base. Find the angle between the generatrix of the cone and the plane of its base.

Answer: \arccos(\frac{1}{3})


The coordinates of the vertices of a triangle are $A(2, -1)$, $B(3, 1)$, and $C\left(2^{1999}, 2^{2000}\right)$. Calculate the area of the triangle $ABC$.

Answer: 2.5


Let $O$ be an arbitrary point inside an equilateral $\triangle ABC$. From point $O$, perpendiculars are dropped to $BC$, $CA$, and $AB$, with footpoints $D$, $E$, and $F$, respectively. Prove that $AF + BD + CE$ is a constant.

Answer: \frac{3}{2}


A secant line splits a circle into two segments. Inside those segments, one draws two squares such that both squares has two corners on a secant line and two on the circumference. The ratio of the square's side lengths is $5:9$. Compute the ratio of the secant line versus circle radius.

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


A function $f$ satisfies the equation
\[f\left(x\right)+f\left(1-\frac{1}{x}\right)=1+x\]
for every real number $x$ except for $x = 0$ and $x = 1$. Find a closed formula for $f$.

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


3. Let $S$ be an arbitrary set of $n$ points $P_{1}, P_{2}, \ldots, P_{n}$ in the plane such that no three are collinear, and let $\alpha$ be the smallest of the angles $P_{i} P_{j} P_{k}$ where
$i \neq j \neq k \neq i$ and $i, j, k \in\{1,2, \ldots, n\}$. Determine $\max _{S} \alpha$ and find the sets $S$ for which this maximum value of the angle $\alpha$ is achieved.

Answer: \dfrac{180^\circ}{n}


13.438 By mixing $2 \mathrm{~cm}^{3}$ of three substances, 16 g of the mixture was obtained. It is known that $4 \mathrm{r}$ of the second substance occupies a volume that is $0.5 \mathrm{~cm}^{3}$ larger than $4 \mathrm{r}$ of the third substance. Find the density of the third substance, given that the mass of the second substance in the mixture is twice the mass of the first.

Answer: 4


Answer: (1,2), (1,4), (2,1), (2,3), (3,2), (3,4), (4,1), (4,3)


Let  $p$  be a prime number,  $p \ge 5$ , and  $k$  be a digit in the  $p$ -adic representation of positive integers. Find the maximal length of a non constant arithmetic progression whose terms do not contain the digit  $k$  in their  $p$ -adic representation.

Answer: p - 1


7. In a $5 \times 5$ grid, place chess pieces such that each cell contains 1 or 2 pieces, black cells contain black pieces, and white cells contain white pieces. The number of white pieces in each row must be the same, and the number of black pieces in each column must also be the same. There are $\qquad$ different ways to place the pieces. (The board cannot be rotated or flipped)

Answer: 527


9.1. There are two ingots of different copper and tin alloys weighing 6 and 12 kg respectively. From each of them, a piece of the same weight was cut off and the first piece was alloyed with the remainder of the second ingot, and the second piece - with the remainder of the first ingot, after which the ratio of copper to tin in the two resulting new ingots turned out to be the same. Find the weight of each of the cut-off pieces.

Answer: 4


3. The numbers $1,2,3,4,5,6,7,8,9$ are written into the cells of a $3 \times 3$ table. After that, all possible sums of numbers standing in adjacent (by side) cells are written down in a notebook. What is the smallest number of different numbers that could have been written in the notebook?

Answer: 4


6. Let $n$ be a given positive integer not less than 5. If there are $n$ lines in the plane, among which exactly 3 lines are pairwise parallel, and in addition, no two lines are parallel, and no three lines are concurrent. Then the number of intersection points of these lines is $\qquad$ .

Answer: \dfrac{n(n - 1)}{2} - 3


257. If it is known that the equation

$$
12 x^{5}-8 x^{4}-45 x^{3}+45 x^{2}+8 x-12=0
$$

admits the roots $+1 ; 1.5 ; -2$, write, without any calculations, the two missing roots.

Answer: \frac{2}{3}


8. Let's call the efficiency of a natural number $n$ the fraction of all natural numbers from 1 to $n$ inclusive that have a common divisor with $n$ greater than 1. For example, the efficiency of the number 6 is $\frac{2}{3}$.

a) Does there exist a number with an efficiency greater than $80\%$? If so, find the smallest such number.

b) Does there exist a number with the maximum efficiency (i.e., not less than the efficiency of any other number)? If so, find the smallest such number.

Answer: 30030


Rice grains were placed on the cells of a chessboard. The number of grains on any two adjacent cells differed by exactly

1. At the same time, there were three grains on one of the cells of the board, and 17 grains on another. A rooster pecked all the grains from one of the main diagonals of the board, and a hen - from the other. How many grains did the rooster get and how many did the hen get?

#

Answer: 80


6. Xiao Ming reports numbers in a 1 3 cycle, while Xiao Hong reports numbers in a 1 5 cycle. Both start reporting numbers at the same speed simultaneously. When both have reported 100 numbers, they have reported the same number $\qquad$ times.

Answer: 21


37. In the hundred-digit number $12345678901234567890 \ldots 1234567890$, all digits in odd positions were erased. In the resulting fifty-digit number, digits in odd positions were erased again. The erasing continued until nothing was left. What was the last digit to be erased?

Answer: 4


4. (8 points) There is a sequence, the first term is 12, the second term is 19, starting from the third term, if the sum of its previous two terms is odd, then this term equals the sum of the previous two terms, if the sum of the previous two terms is even, this term equals the difference of the previous two terms (the larger number minus the smaller number). Then, in this sequence, the $\qquad$ term first exceeds 2016.

Answer: 252


5. Three Thieves, Bingo, Bunko, and Balko, robbed a bank and carried away 22 bags of banknotes. They placed them in a row so that the first bag contained the least amount of money, and each subsequent bag contained one more stack of banknotes than the previous one. Chief Bingo divided the stolen bags of money according to the principle: the first one for me (Bingo), the second one for you (Bunko), the third one for me (Bingo), the fourth one for you (Balko), the fifth one for me (Bingo), the sixth one for you (Bunko)... Then they counted the money. Bunko and Balko found that together they had a whopping total of 6710000 SIT. They were also delighted to find that together they had 110000 SIT more than their chief. Your task is to determine how much money was in the first bag. Write down the answer.[^3]

## Solutions to the Problems and Scoring

 A contestant who arrives at the solution by any correct method (even if the scoring does not provide for it) receives all possible points.

For a correct method, any procedure that

- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete.

A contestant who has only partially solved the problem, from otherwise correct solving procedures but does not show the way to the final solution of the problem, cannot receive more than half of the possible points.

## First Year

Answer: 500000


3. Two spheres with a radius of 1 are externally tangent to each other, and both are internally tangent to a cylindrical surface with a radius of 1. Another smaller sphere is externally tangent to both of these spheres and internally tangent to the cylindrical surface. A plane passing through the center of the smaller sphere and the center of one of the larger spheres intersects the cylindrical surface to form an ellipse. The maximum possible value of the eccentricity \( e \) of this ellipse is \(\qquad\)

Answer: \dfrac{4}{5}


Alice has six magical pies in her pocket - two that increase in size (eat - grow), and the remaining are decreasing (eat - shrink). When Alice met Mary Ann, she blindly took three pies from her pocket and gave them to Mary. Find the probability that one of the girls does not have a single size-increasing pie.

Answer: 0.4


In a rectangular table, the letters of the word "olympiada" are arranged in a specific order.

| O | Л | И | M | П | И | A | Д | A |
| - | - | - | - | - | - | - | - | - |
| Л | И | M | П | И | A | Д | A | O |
| И | M | П | И | A | Д | A | O | Л |
| M | П | И | A | Д | A | O | Л | И |

Starting from the letter "O" located in the top left corner of the table, one must read the word "olympiada", ending with the letter "A". The transition between letters is allowed between adjacent cells in the table. Find the number of different ways to read the word "olympiada" according to the given table.

Answer: 93


Suppose that the function

$$
y = ax^3 + bx^2 + cx + d
$$

has both extrema. What is the relationship among the coefficients if the straight line connecting the points of the curve corresponding to the extrema passes through the origin?

Answer: 9ad = bc


Several even natural numbers are written in a row. Each number, except for the leftmost one, is obtained by subtracting the largest digit of its left neighbor from its left neighbor. Can the row contain more than 12 numbers?

Answer: Yes


In the image, there is a square \(A B C D\) with a side length of \(10 \, \text{cm}\) and a smaller square \(E F G H\), whose vertices \(E, F, G, H\) lie on the diagonals \(A C, B D\) of the square \(A B C D\) (see the image). The area that lies inside the square \(A B C D\) and outside the square \(E F G H\) is shaded. A line \(p\), which is parallel to \(A B\) at a distance of \(6.5 \, \text{cm}\) and passes through the square \(E F G H\), divides the shaded area into two parts. The area of one of these parts is \(13.8 \, \text{cm}^2\) larger than the area of the other. Calculate the side length of \(E F\).

Answer: 5.4 \ \text{cm}


As shown in the figure, a small square of the same size is cut from the corner and the edge of two identical large rectangular sheets of paper, respectively. If the perimeter of the shaded part in figure (2) is 17 centimeters more than the perimeter of the shaded part in figure (1), what is the perimeter of the cut-out small square in centimeters?

Answer: 34


Two circles touch externally. The line segment \(AB\) is the diameter of one circle and has a length of \(6 \ \text{cm}\), and the line segment \(CD\) is the diameter of the other circle and has a length of \(14 \ \text{cm}\). The quadrilateral \(ABCD\) is a trapezoid with bases \(AB\) and \(CD\).

Determine the maximum possible area of the trapezoid \(ABCD\), and explain why it cannot be larger.

Answer: 100 \text{ cm}^2


On a game show, Merble will be presented with a series of 2013 marbles, each of which is either red or blue. Each time he sees a marble, he can either keep it or pass, but cannot return to a previous marble; he receives 3 points for keeping a red marble, loses 2 points for keeping a blue marble, and gains 0 points for passing. All distributions of colors are equally likely, and Merble can only see the color of his current marble. If his goal is to end with exactly one point and he plays optimally, what is the probability that he fails?

Answer: \frac{1}{2^{2012}}


As shown in the figure, in triangle \(ABC\), \(AF : FB = BD : DC = CE : AE = 3 : 2\). Given that the area of triangle \(ABC\) is 1, find the area of triangle \(GHI\).

Answer: \frac{1}{19}


Lenočka decided to bake a pie for her birthday. She rolled out the dough into an isosceles right triangle $\triangle A B C$. Then she thought that there was enough dough for two pies and made two straight cuts parallel to the legs of the triangle.

This resulted in two triangles and one rectangle. Lenočka baked a strawberry pie from the rectangle, and combined the triangles, rolled the dough, and made a cabbage pie. Could it be that there is more dough in the strawberry pie than in the cabbage pie?

Answer: \text{No}


The inequality \( x^2 < n \) has exactly \( n \) integer solutions. Find \( n \). If there are several possible answers, list them in ascending order separated by commas or semicolons.

Answer: 0, 1, 3, 5


Karina took several matches from a box and assembled from them a $3 \times 7$ grid of squares with sides equal to one match, as shown in the picture below.

What is the minimum number of matches she needs to take from the box so that with all the matches she can assemble a square grid? (The squares in the grid must again have a side equal to one match. There should not be any extra matches left.)

Answer: 8


A city is laid out as a rectangle, divided into cells: $n$ streets are parallel to each other, and $m$ other streets intersect them at right angles. Policemen stand on the streets of the city, but not at intersections. Each policeman reports the number of the passing car, its direction of travel, and the time it passed. What is the minimum number of policemen needed to uniquely determine the path of any car traveling on a closed route (the route does not pass the same street segment twice)?

Answer: (m-1)(n-1)


The natural numbers from $1$ up to $300$ are evenly located around a circle. We say that such an ordering is "alternate" if each number is either less than its two neighbors or greater than its two neighbors. We will call a pair of neighboring numbers a "good" pair if, by removing that pair from the circumference, the remaining numbers form an alternate ordering. Determine the least possible number of good pairs in which there can be an alternate ordering of the numbers from $1$ to $300$ inclusive.

Answer: 150


Let $ABC$ be an acute triangle with incenter $I$ and circumcenter $O$. Assume that $\angle OIA = 90^{\circ}$. Given that $AI = 97$ and $BC = 144$, compute the area of $\triangle ABC$.

Answer: 14040


Michael is celebrating his fifteenth birthday today. Calculate the total number of Sundays that have occurred in his lifetime.

Answer: 783


A positive integer is called \textit{superprime} if the difference between any two of its consecutive positive divisors is a prime number. Determine all superprime integers.

Answer: 3


On Monday, a school library was attended by 5 students; on Tuesday, by 6; on Wednesday, by 4; on Thursday, by 8; and on Friday, by 7. None of them attended the library two days in a row. What is the least possible number of students who visited the library during the week?

Answer: 15


Let $n$ be the 200th smallest positive real solution to the equation $x - \frac{\pi}{2} = \tan x$. Find the greatest integer that does not exceed \(\frac{n}{2}\).

Answer: 314


A hundred silver coins are laid down in a line. A wizard can convert a silver coin into a golden one in $3$ seconds. Each adjacent golden coin reduces this time by $1$ second. What is the minimal time required for the wizard to convert all coins to gold?

Answer: 201


The cells of an $8 \times 8$ table are colored either black or white such that each row has a distinct number of black squares, and each column also has a distinct number of black squares. Determine the maximum number of pairs of adjacent cells that have different colors.

Answer: 62


13 fractions are formed using each of the numbers $1, 2, \ldots, 26$ exactly once. For example: $\frac{12}{5}, \frac{18}{26}, \ldots$. What is the maximum number of these fractions that can be integers?

Answer: 12


Rthea, a distant planet, is home to creatures whose DNA consists of two distinguishable strands of bases with a fixed orientation. Each base is one of the letters \(H, M, N, T\), and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Additionally, the bases \(H\) and \(M\) cannot appear next to each other on the same strand; the same restriction applies to \(N\) and \(T\). Determine the number of possible DNA sequences on Rthea.

Answer: 28812


Ten test papers are to be prepared for the National Olympiad. Each paper contains 4 problems, and no two papers have more than one problem in common. What is the minimum number of problems needed?

Answer: 13


Two knights placed on distinct squares of an $8 \times 8$ chessboard, where each square is a unit square, are said to attack each other if the distance between the centers of the squares on which the knights lie is $\sqrt{5}$. In how many ways can two identical knights be placed on distinct squares of an $8 \times 8$ chessboard such that they do NOT attack each other?

Answer: 1848


Suppose $a_1$, $a_2$, $\ldots$, $a_{10}$ are nonnegative integers such that \[\sum_{k=1}^{10}a_k=15 \text{ and } \sum_{k=1}^{10}ka_k = 80.\] Let $M$ and $m$ denote the maximum and minimum respectively of $\sum_{k=1}^{10}k^2a_k$. Compute $M-m$.

Answer: 286


Let $x, y, z$ be positive real numbers such that $x + y + z = 1399$. Find the maximum value of $\lfloor x \rfloor y + \lfloor y \rfloor z + \lfloor z \rfloor x$, where $\lfloor a \rfloor$ is the greatest integer not exceeding $a$.

Answer: 652400


We take $100$ consecutive natural numbers $a_{1}, a_{2}, \ldots, a_{100}$. Determine the last two digits of the number $a_{1}^{8} + a_{2}^{8} + \cdots + a_{100}^{8}$.

Answer: 30