There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Beverage, Food, Movie-Genre, Music-Genre.
The attributes have the following possible values:
- Beverage: milk, soy-milk, tea, water
- Food: asparagus, avocado, kale, pear
- Movie-Genre: adventure, romance, scientific, superhero
- Music-Genre: blues, pop, punk, trance

Given the following premises about the line of people:
- Movie-Genre:superhero is somewhere to the left of Beverage:soy-milk
- Food:avocado is between Beverage:soy-milk and Movie-Genre:adventure
- Food:asparagus is between Food:kale and Music-Genre:pop
- Beverage:tea is somewhere to the right of Music-Genre:blues
- Movie-Genre:scientific is somewhere to the left of Beverage:milk
- Beverage:soy-milk is on the far right
- Food:kale is not to the left of Music-Genre:trance

Answer the following question:

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

Answer: kale


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, Movie-Genre, Music-Genre, Food, Sport.
The attributes have the following possible values:
- Nationality: argentine, colombian, dutch, polish
- Hobby: camping, card-games, chess, filmmaking
- Movie-Genre: comedy, documentary, martial-arts, mystery
- Music-Genre: country, folk, indie, metal
- Food: avocado, eggplant, grapefruit, lemon
- Sport: golf, rugby, skiing, weightlifting

Given the following premises about the line of people:
- Sport:skiing is somewhere to the right of Food:eggplant
- Music-Genre:country is on the left of Hobby:chess
- Movie-Genre:comedy == Hobby:chess
- Nationality:dutch is on the left of Nationality:colombian
- Sport:rugby == Movie-Genre:martial-arts
- Food:lemon == Nationality:argentine
- Nationality:dutch is somewhere to the right of Sport:rugby
- Hobby:chess == Music-Genre:indie
- Movie-Genre:documentary is somewhere to the left of Music-Genre:folk
- Food:grapefruit is on the right of Music-Genre:country
- Nationality:polish is somewhere between Food:eggplant and Sport:golf
- Food:avocado == Hobby:camping
- Nationality:dutch is somewhere to the right of Hobby:filmmaking

Answer the following question:

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

Answer: indie


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, Music-Genre, Pet, Transport.
The attributes have the following possible values:
- Nationality: argentine, french, indonesian, malaysian, pakistani
- Music-Genre: classical, dubstep, r&b, soul, techno
- Pet: cat, chinchilla, fish, frog, mouse
- Transport: airplane, bike, motorbike, roller, ship

Given the following premises about the line of people:
- Pet:cat is not to the left of Music-Genre:r&b
- Nationality:pakistani is on the right of Pet:cat
- Music-Genre:r&b is between Music-Genre:classical and Music-Genre:techno
- Transport:bike and Nationality:pakistani have different parity positions
- Transport:ship == Pet:mouse or Pet:mouse == Music-Genre:soul or both
- Nationality:pakistani is on the right of Music-Genre:techno
- Transport:roller is on the left or right of Nationality:french
- Music-Genre:dubstep is on the left of Pet:chinchilla
- Nationality:indonesian and Transport:motorbike have different parity positions
- Nationality:malaysian is on the left of Music-Genre:r&b
- Nationality:indonesian is on the left of Transport:bike
- Transport:motorbike is not to the right of Music-Genre:r&b
- Pet:fish and Music-Genre:soul have different parity positions
- Pet:fish is not to the right of Transport:bike

Answer the following question:

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

Answer: malaysian


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, Music-Genre, Pet, Sport.
The attributes have the following possible values:
- Food: corn, cucumber, orange, radish
- Hobby: dancing, filmmaking, photography, reading
- Movie-Genre: disaster, sports, time-travel, zombie
- Music-Genre: hip-hop, jazz, rock, trance
- Pet: fish, frog, goldfish, snake
- Sport: badminton, basketball, golf, handball

Given the following premises about the line of people:
- Hobby:photography is somewhere to the left of Pet:goldfish
- Movie-Genre:sports == Sport:handball
- Hobby:photography is on the far left or far right
- Movie-Genre:disaster is on the left of Movie-Genre:time-travel
- Food:cucumber is on the left of Sport:golf
- Food:orange is somewhere to the right of Music-Genre:hip-hop
- Food:cucumber == Music-Genre:rock
- Pet:snake == Music-Genre:rock
- Music-Genre:trance is on the far left
- Sport:badminton is on the right of Food:corn
- Music-Genre:hip-hop is on the left of Hobby:dancing
- Music-Genre:trance is on the left or right of Music-Genre:hip-hop
- Pet:frog == Food:corn
- Hobby:reading is somewhere to the left of Hobby:filmmaking
- Movie-Genre:zombie is on the left of Sport:handball

Answer the following question:

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

Answer: time-travel


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Beverage, Movie-Genre, Nationality, Sport.
The attributes have the following possible values:
- Job: freelancer, photographer, police-officer, teacher
- Beverage: hot-chocolate, iced-tea, mirinda, tea
- Movie-Genre: adventure, crime, documentary, thriller
- Nationality: australian, mexican, nigerian, thai
- Sport: ice-hockey, rowing, sailing, tennis

Given the following premises about the line of people:
- Movie-Genre:thriller and Sport:tennis have the same parity positions
- Sport:ice-hockey is not to the left of Sport:tennis
- Movie-Genre:adventure and Job:photographer have different parity positions
- Job:freelancer is on the far right
- Beverage:tea is on the far right
- Sport:rowing is on the left or right of Beverage:mirinda
- Nationality:nigerian is somewhere to the left of Nationality:thai
- Job:freelancer and Nationality:thai have different parity positions
- Movie-Genre:adventure != Sport:sailing
- Movie-Genre:adventure is not to the right of Movie-Genre:thriller
- Job:photographer is not to the right of Sport:tennis
- Job:photographer is somewhere to the right of Sport:sailing
- Job:police-officer and Nationality:nigerian have the same parity positions
- Movie-Genre:crime and Sport:sailing have the same parity positions
- Nationality:mexican == Movie-Genre:adventure or Job:teacher == Nationality:mexican or both
- Beverage:hot-chocolate is somewhere to the left of Nationality:mexican

Answer the following question:

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

Answer: tea


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Beverage, Hobby, Movie-Genre, Pet.
The attributes have the following possible values:
- Beverage: 7up, cola, iced-tea, tea, water
- Hobby: card-games, dancing, reading, sudoku, traveling
- Movie-Genre: action, adventure, documentary, drama, scientific
- Pet: bird, ferret, hamster, pony, rabbit

Given the following premises about the line of people:
- Hobby:reading is on the right of Pet:pony
- Beverage:7up is on the left or right of Hobby:card-games
- Hobby:dancing == Beverage:cola
- Beverage:7up is on the left of Pet:bird
- Pet:bird is somewhere to the left of Beverage:iced-tea
- Movie-Genre:action is in the middle
- Movie-Genre:scientific is on the left or right of Movie-Genre:action
- Movie-Genre:drama is on the left or right of Hobby:traveling
- Pet:ferret is on the left of Hobby:dancing
- Pet:rabbit is somewhere to the right of Pet:hamster
- Hobby:reading == Movie-Genre:drama
- Movie-Genre:documentary is on the right of Movie-Genre:drama
- Pet:pony is somewhere to the left of Beverage:water

Answer the following question:

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

Answer: water


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Beverage, Hobby, Pet, Sport.
The attributes have the following possible values:
- Beverage: lemonade, mirinda, sprite, tea
- Hobby: camping, fishing, magic-tricks, sudoku
- Pet: dog, ferret, frog, rat
- Sport: biathlon, cricket, sailing, soccer

Given the following premises about the line of people:
- Hobby:sudoku is on the left or right of Sport:cricket
- Pet:rat is not to the left of Beverage:tea
- Beverage:mirinda is on the far left
- Sport:sailing is on the left of Beverage:tea
- Pet:dog is somewhere to the right of Hobby:sudoku
- Pet:ferret is on the left of Beverage:sprite
- Hobby:fishing is somewhere to the left of Pet:frog
- Hobby:sudoku is somewhere to the left of Hobby:fishing
- Sport:soccer is somewhere to the right of Hobby:magic-tricks

Answer the following question:

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

Answer: mirinda


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Nationality, Movie-Genre, Hobby, Sport.
The attributes have the following possible values:
- Nationality: american, malaysian, spanish, thai
- Movie-Genre: animation, musical, scientific, thriller
- Hobby: collecting, filmmaking, fishing, hiking
- Sport: badminton, biathlon, cycling, handball

Given the following premises about the line of people:
- Sport:cycling and Hobby:hiking have different parity positions
- Hobby:filmmaking is somewhere to the right of Movie-Genre:scientific
- Hobby:collecting is on the far right
- Sport:badminton is on the left of Nationality:american
- Sport:biathlon is somewhere between Hobby:hiking and Sport:badminton
- Nationality:malaysian is on the right of Movie-Genre:thriller
- Movie-Genre:musical and Nationality:malaysian have the same parity positions
- Nationality:thai is not to the left of Nationality:malaysian
- Sport:cycling != Nationality:american

Answer the following question:

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

Answer: fishing


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Beverage, Food, Pet, Sport, Transport.
The attributes have the following possible values:
- Beverage: almond-milk, coffee, iced-tea, juice
- Food: cherry, grapes, plum, strawberry
- Pet: bird, cat, fish, hedgehog
- Sport: cycling, soccer, swimming, weightlifting
- Transport: boat, quad-bike, skateboard, taxi

Given the following premises about the line of people:
- Transport:taxi is on the left of Sport:swimming
- Food:plum is on the left or right of Beverage:coffee
- Sport:weightlifting != Food:cherry
- Transport:skateboard is somewhere between Beverage:almond-milk and Food:plum
- Transport:quad-bike is between Beverage:iced-tea and Pet:bird
- Pet:bird is somewhere to the right of Food:grapes
- Sport:cycling is on the left of Transport:boat
- Sport:cycling is not to the right of Food:grapes
- Sport:soccer is somewhere to the right of Pet:hedgehog
- Pet:fish is not to the right of Sport:swimming
- Pet:cat == Beverage:iced-tea

Answer the following question:

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

Answer: hedgehog


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, Beverage, Transport.
The attributes have the following possible values:
- Job: bartender, journalist, pilot, police-officer, social-worker
- Food: asparagus, cucumber, nectarine, pumpkin, tomato
- Beverage: almond-milk, hot-chocolate, soy-milk, sprite, tea
- Transport: bike, bus, ship, tram, van

Given the following premises about the line of people:
- Food:tomato is somewhere to the right of Job:social-worker
- Beverage:sprite == Transport:tram or Food:pumpkin == Beverage:sprite or both
- Transport:bus and Beverage:sprite have the same parity positions
- Job:pilot is not to the left of Food:tomato
- Food:nectarine is somewhere to the right of Transport:bike
- Food:cucumber is not to the right of Beverage:almond-milk
- Transport:van and Food:pumpkin have the same parity positions
- Food:cucumber is on the far left or far right
- Job:journalist is on the left or right of Job:social-worker
- Beverage:hot-chocolate is somewhere to the right of Food:tomato
- Transport:tram is not to the left of Beverage:soy-milk
- Transport:bus is somewhere to the right of Transport:van
- Job:police-officer is somewhere to the left of Job:journalist
- Job:police-officer and Food:cucumber have different parity positions
- Food:nectarine is somewhere to the left of Transport:van
- Transport:tram is on the left or right of Beverage:almond-milk
- Food:cucumber and Beverage:almond-milk have the same parity positions
- Transport:bus != Job:journalist

Answer the following question:

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

Answer: hot-chocolate


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Nationality, Food, Hobby, Music-Genre, Beverage.
The attributes have the following possible values:
- Nationality: egyptian, german, malaysian, nigerian
- Food: grapefruit, lime, mango, raspberry
- Hobby: camping, collecting, gardening, rock-climbing
- Music-Genre: country, indie, jazz, reggae
- Beverage: iced-tea, juice, soy-milk, water

Given the following premises about the line of people:
- Nationality:nigerian is somewhere to the left of Hobby:rock-climbing
- Food:lime is on the left or right of Music-Genre:jazz
- Hobby:collecting is somewhere between Food:mango and Hobby:gardening
- Food:mango is not to the left of Hobby:collecting
- Hobby:collecting is somewhere to the right of Food:grapefruit
- Music-Genre:indie and Beverage:water have the same parity positions
- Hobby:camping is not to the right of Beverage:iced-tea
- Music-Genre:country is somewhere to the left of Music-Genre:reggae
- Music-Genre:indie is somewhere to the right of Food:raspberry
- Hobby:gardening != Music-Genre:country
- Music-Genre:reggae is somewhere to the right of Beverage:juice
- Hobby:collecting is not to the left of Hobby:rock-climbing
- Nationality:german is on the far right
- Nationality:malaysian is in an odd position

Answer the following question:

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

Answer: nigerian


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, Beverage, Music-Genre, Nationality.
The attributes have the following possible values:
- Job: architect, electrician, fisherman, videographer
- Food: lime, pear, pineapple, radish
- Beverage: fanta, juice, milk, water
- Music-Genre: country, house, punk, techno
- Nationality: chinese, german, indonesian, polish

Given the following premises about the line of people:
- Beverage:juice is somewhere to the right of Beverage:fanta
- Nationality:polish and Music-Genre:punk have the same parity positions
- Nationality:polish == Music-Genre:country or Nationality:polish == Food:lime, but not both
- Nationality:chinese is on the left of Food:pineapple
- Job:architect is in an even position
- Food:radish != Job:architect
- Job:architect is on the left or right of Beverage:water
- Beverage:milk is somewhere to the right of Nationality:polish
- Beverage:milk is in an odd position
- Nationality:german is not to the right of Job:fisherman
- Nationality:german is in an even position
- Food:radish and Job:electrician have different parity positions
- Job:electrician == Beverage:water or Beverage:water == Food:lime, but not both
- Job:electrician is on the far left
- Job:electrician == Beverage:fanta or Music-Genre:techno == Job:electrician or both

Answer the following question:

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

Answer: videographer


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, Nationality.
The attributes have the following possible values:
- Job: chef, designer, entrepreneur, manager, project-manager
- Hobby: camping, card-games, filmmaking, magic-tricks, video-games
- Beverage: coffee, hot-chocolate, soy-milk, tea, water
- Movie-Genre: documentary, drama, family, thriller, western
- Nationality: american, australian, egyptian, indian, mexican

Given the following premises about the line of people:
- Nationality:indian is on the right of Movie-Genre:family
- Hobby:card-games is between Movie-Genre:family and Movie-Genre:documentary
- Movie-Genre:thriller is on the left of Beverage:water
- Hobby:camping is on the left of Movie-Genre:thriller
- Movie-Genre:family is on the right of Nationality:australian
- Beverage:tea is not to the left of Nationality:australian
- Movie-Genre:western is on the right of Movie-Genre:drama
- Hobby:filmmaking is in the middle
- Job:chef is on the left of Job:project-manager
- Hobby:magic-tricks is on the right of Nationality:egyptian
- Beverage:tea is on the left or right of Job:entrepreneur
- Nationality:mexican is not to the left of Job:designer
- Hobby:camping is somewhere to the left of Beverage:soy-milk
- Beverage:tea is on the left of Beverage:hot-chocolate
- Job:manager is on the right of Job:entrepreneur

Answer the following question:

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

Answer: soy-milk


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, Hobby, Movie-Genre, Music-Genre, Beverage.
The attributes have the following possible values:
- Nationality: american, dutch, mexican, turkish
- Food: avocado, mango, nectarine, onion
- Hobby: baking, card-games, dancing, gardening
- Movie-Genre: crime, scientific, sports, western
- Music-Genre: classical, gospel, house, punk
- Beverage: coffee, juice, milk, soy-milk

Given the following premises about the line of people:
- Hobby:card-games is on the right of Nationality:mexican
- Nationality:mexican == Hobby:baking
- Food:mango is on the far left
- Movie-Genre:crime == Food:mango
- Hobby:dancing is in an odd position
- Movie-Genre:sports == Music-Genre:punk
- Nationality:mexican is somewhere to the right of Music-Genre:house
- Beverage:coffee is on the left of Beverage:milk
- Movie-Genre:scientific is on the left or right of Food:onion
- Music-Genre:classical is on the right of Movie-Genre:sports
- Hobby:baking is on the left or right of Beverage:soy-milk
- Movie-Genre:western is somewhere to the left of Music-Genre:punk
- Nationality:american is on the right of Beverage:soy-milk
- Food:nectarine is somewhere to the left of Nationality:turkish

Answer the following question:

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

Answer: scientific


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Job, Hobby, Music-Genre, Pet.
The attributes have the following possible values:
- Job: coach, engineer, mechanic, musician, paramedic
- Hobby: camping, rock-climbing, singing, traveling, video-games
- Music-Genre: folk, gospel, pop, salsa, techno
- Pet: chinchilla, dog, horse, pony, rat

Given the following premises about the line of people:
- Music-Genre:salsa is not to the left of Hobby:singing
- Music-Genre:pop == Pet:chinchilla or Job:paramedic == Music-Genre:pop or both
- Job:paramedic and Music-Genre:salsa have different parity positions
- Hobby:rock-climbing is somewhere to the left of Music-Genre:gospel
- Job:engineer is not to the left of Music-Genre:gospel
- Pet:pony is on the left or right of Job:coach
- Job:mechanic is on the left or right of Music-Genre:pop
- Hobby:camping is somewhere to the left of Hobby:traveling
- Pet:horse is on the left or right of Music-Genre:salsa
- Job:mechanic is on the far left or far right
- Pet:dog is somewhere between Pet:rat and Pet:chinchilla
- Pet:chinchilla is somewhere to the left of Hobby:singing
- Hobby:singing != Pet:dog
- Hobby:traveling is somewhere to the left of Hobby:rock-climbing
- Hobby:rock-climbing is somewhere to the right of Music-Genre:techno
- Pet:horse is on the left or right of Pet:rat
- Job:paramedic and Pet:chinchilla have different parity positions
- Music-Genre:pop is somewhere to the left of Job:paramedic

Answer the following question:

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

Answer: salsa


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Beverage, Food, Movie-Genre, Music-Genre, Sport, Transport.
The attributes have the following possible values:
- Beverage: cola, lemonade, mirinda, water
- Food: garlic, lettuce, mango, peach
- Movie-Genre: drama, family, musical, spy
- Music-Genre: d&b, dubstep, gospel, punk
- Sport: cricket, parkour, skateboarding, volleyball
- Transport: helicopter, roller, ship, tram

Given the following premises about the line of people:
- Beverage:lemonade == Movie-Genre:spy or Food:peach == Beverage:lemonade, but not both
- Food:peach is on the far left or far right
- Music-Genre:d&b == Beverage:water or Music-Genre:d&b == Sport:parkour, but not both
- Transport:roller is somewhere to the left of Movie-Genre:family
- Movie-Genre:musical is not to the left of Movie-Genre:family
- Beverage:cola is not to the right of Music-Genre:punk
- Food:mango is not to the left of Music-Genre:punk
- Sport:parkour is somewhere to the left of Beverage:mirinda
- Transport:roller != Music-Genre:punk
- Food:peach is on the left or right of Transport:helicopter
- Movie-Genre:spy is somewhere to the right of Movie-Genre:drama
- Transport:ship is somewhere to the left of Sport:skateboarding
- Sport:skateboarding == Movie-Genre:spy or Sport:skateboarding == Food:peach or both
- Sport:parkour == Movie-Genre:family or Music-Genre:gospel == Sport:parkour or both
- Food:garlic is somewhere to the left of Transport:roller
- Music-Genre:punk is on the left or right of Beverage:cola
- Sport:cricket is on the left or right of Sport:skateboarding

Answer the following question:

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

Answer: parkour


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, Music-Genre, Pet, Transport.
The attributes have the following possible values:
- Nationality: brazilian, egyptian, indonesian, spanish
- Music-Genre: gospel, jazz, punk, reggae
- Pet: hamster, lizard, mouse, snake
- Transport: airplane, boat, bus, van

Given the following premises about the line of people:
- Nationality:indonesian is on the right of Pet:snake
- Nationality:brazilian is somewhere between Transport:boat and Nationality:egyptian
- Transport:bus is somewhere to the left of Music-Genre:gospel
- Music-Genre:punk is on the left or right of Nationality:indonesian
- Music-Genre:jazz is on the right of Transport:bus
- Pet:hamster == Nationality:spanish
- Nationality:spanish is on the left of Pet:mouse
- Music-Genre:punk != Transport:van

Answer the following question:

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

Answer: indonesian


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Job, Food, Movie-Genre, Nationality, Transport.
The attributes have the following possible values:
- Job: accountant, journalist, photographer, security-guard, social-worker
- Food: apple, blueberry, corn, lime, pear
- Movie-Genre: comedy, fantasy, scientific, superhero, time-travel
- Nationality: argentine, brazilian, indian, pakistani, russian
- Transport: airplane, bike, bus, subway, taxi

Given the following premises about the line of people:
- Job:security-guard is on the far left or far right
- Transport:taxi is not to the left of Movie-Genre:time-travel
- Transport:bike is somewhere to the right of Transport:taxi
- Movie-Genre:superhero is on the left of Nationality:pakistani
- Job:accountant == Movie-Genre:time-travel
- Nationality:argentine is somewhere to the right of Job:accountant
- Movie-Genre:superhero is not to the left of Food:corn
- Movie-Genre:time-travel is in an even position
- Food:blueberry is not to the left of Food:apple
- Nationality:brazilian is somewhere between Job:security-guard and Food:blueberry
- Transport:bus is not to the left of Movie-Genre:scientific
- Food:corn is somewhere to the left of Job:security-guard
- Movie-Genre:comedy is on the far left
- Job:photographer is on the far left or far right
- Transport:subway is on the left of Nationality:russian
- Job:journalist is not to the right of Food:apple
- Food:pear is not to the left of Job:security-guard

Answer the following question:

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

Answer: journalist


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, Sport.
The attributes have the following possible values:
- Job: designer, dressmaker, engineer, firefighter, fisherman
- Hobby: card-games, chess, gardening, singing, sudoku
- Beverage: 7up, coffee, fanta, hot-chocolate, soy-milk
- Movie-Genre: action, disaster, epic, martial-arts, western
- Sport: badminton, climbing, cricket, rugby, volleyball

Given the following premises about the line of people:
- Hobby:singing is in an odd position
- Hobby:singing and Job:engineer have the same parity positions
- Sport:badminton == Movie-Genre:martial-arts or Sport:badminton == Job:designer, but not both
- Job:dressmaker is not to the left of Beverage:fanta
- Job:dressmaker is on the left or right of Movie-Genre:action
- Hobby:sudoku is not to the left of Beverage:7up
- Movie-Genre:disaster is between Beverage:soy-milk and Job:designer
- Job:fisherman is not to the right of Movie-Genre:epic
- Movie-Genre:western is somewhere to the right of Movie-Genre:disaster
- Hobby:singing != Job:fisherman
- Beverage:hot-chocolate is on the left or right of Sport:climbing
- Hobby:chess == Beverage:fanta or Movie-Genre:western == Hobby:chess or both
- Sport:rugby is on the left or right of Beverage:7up
- Job:designer is somewhere to the left of Beverage:hot-chocolate
- Hobby:gardening is between Beverage:hot-chocolate and Hobby:chess
- Beverage:fanta is on the left or right of Sport:volleyball
- Beverage:hot-chocolate is somewhere to the left of Beverage:fanta

Answer the following question:

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

Answer: fanta


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.
The attributes have the following possible values:
- Nationality: colombian, mexican, nigerian, polish, thai
- Food: lemon, onion, peas, pepper, potato
- Beverage: 7up, almond-milk, coffee, cola, iced-tea
- Pet: frog, goat, hamster, horse, snake

Given the following premises about the line of people:
- Beverage:iced-tea is on the left or right of Beverage:cola
- Nationality:colombian is not to the left of Food:lemon
- Beverage:coffee and Nationality:colombian have the same parity positions
- Food:onion is on the right of Beverage:almond-milk
- Pet:frog != Food:peas
- Pet:snake and Nationality:thai have different parity positions
- Food:potato is on the far left
- Nationality:thai is on the right of Nationality:nigerian
- Food:lemon is on the far right
- Beverage:almond-milk is not to the left of Pet:hamster
- Nationality:thai is between Nationality:nigerian and Nationality:mexican
- Beverage:cola and Food:onion have the same parity positions
- Nationality:thai is not to the right of Pet:goat
- Nationality:thai is on the left or right of Beverage:cola
- Nationality:thai is on the right of Pet:frog
- Beverage:7up is not to the right of Pet:hamster
- Beverage:cola is on the left of Food:lemon

Answer the following question:

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

Answer: 7up


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Hobby, Food, Music-Genre, Pet, Transport.
The attributes have the following possible values:
- Job: dancer, dressmaker, engineer, nurse
- Hobby: baking, filmmaking, hiking, puzzles
- Food: carrot, cauliflower, lettuce, mango
- Music-Genre: ambient, classical, jazz, rock
- Pet: fish, frog, lizard, turtle
- Transport: boat, car, helicopter, trike

Given the following premises about the line of people:
- Music-Genre:ambient == Hobby:puzzles
- Job:nurse is somewhere between Job:dancer and Job:engineer
- Hobby:filmmaking is on the left of Music-Genre:rock
- Pet:lizard is on the right of Music-Genre:jazz
- Job:dancer == Music-Genre:classical
- Pet:fish is on the right of Food:cauliflower
- Pet:frog is somewhere to the left of Transport:trike
- Transport:helicopter is somewhere to the right of Pet:turtle
- Job:dressmaker != Food:lettuce
- Transport:boat is on the left or right of Pet:turtle
- Transport:helicopter is on the far left or far right
- Music-Genre:rock is somewhere to the left of Hobby:hiking
- Food:carrot is on the left of Food:cauliflower
- Hobby:hiking == Food:cauliflower

Answer the following question:

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

Answer: jazz


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, Movie-Genre, Music-Genre, Sport, Transport.
The attributes have the following possible values:
- Food: apricot, raspberry, spinach, watermelon
- Movie-Genre: action, adventure, thriller, time-travel
- Music-Genre: dubstep, folk, house, reggae
- Sport: cricket, handball, skiing, weightlifting
- Transport: quad-bike, roller, subway, van

Given the following premises about the line of people:
- Sport:cricket is somewhere to the right of Sport:skiing
- Transport:quad-bike is somewhere to the right of Sport:weightlifting
- Food:apricot is in an even position
- Sport:handball is somewhere between Music-Genre:dubstep and Transport:roller
- Music-Genre:folk is not to the left of Transport:quad-bike
- Music-Genre:house is not to the left of Movie-Genre:adventure
- Food:raspberry is not to the left of Movie-Genre:action
- Transport:roller and Music-Genre:house have the same parity positions
- Movie-Genre:time-travel is not to the left of Music-Genre:folk
- Movie-Genre:adventure is on the left or right of Food:raspberry
- Food:raspberry is somewhere to the left of Food:watermelon
- Transport:van is somewhere to the right of Sport:handball
- Sport:weightlifting is somewhere to the right of Sport:handball

Answer the following question:

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

Answer: skiing


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, Movie-Genre, Food.
The attributes have the following possible values:
- Nationality: dutch, french, indonesian, russian
- Hobby: drawing, fishing, hiking, reading
- Movie-Genre: adventure, disaster, drama, musical
- Food: cabbage, cauliflower, papaya, tomato

Given the following premises about the line of people:
- Hobby:hiking is not to the left of Movie-Genre:adventure
- Nationality:dutch is not to the right of Food:cauliflower
- Hobby:reading is on the left of Hobby:hiking
- Nationality:russian is between Food:tomato and Nationality:indonesian
- Hobby:drawing is on the right of Food:cabbage
- Food:tomato is on the left of Hobby:hiking
- Movie-Genre:disaster == Nationality:dutch
- Movie-Genre:adventure is on the left or right of Nationality:dutch
- Nationality:french is somewhere to the left of Food:papaya
- Movie-Genre:musical is not to the right of Hobby:hiking

Answer the following question:

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

Answer: musical


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, Food, Sport, Transport.
The attributes have the following possible values:
- Nationality: british, dutch, indian, indonesian
- Hobby: board-games, filmmaking, photography, woodworking
- Music-Genre: d&b, jazz, metal, r&b
- Food: pepper, pineapple, pomegranate, pumpkin
- Sport: climbing, lacrosse, skateboarding, swimming
- Transport: bus, helicopter, scooter, train

Given the following premises about the line of people:
- Sport:swimming is not to the left of Music-Genre:jazz
- Hobby:filmmaking is not to the right of Hobby:woodworking
- Nationality:dutch is on the left or right of Transport:helicopter
- Music-Genre:jazz != Food:pomegranate
- Hobby:woodworking is somewhere to the right of Food:pumpkin
- Music-Genre:jazz is somewhere to the right of Nationality:british
- Food:pepper is somewhere to the right of Music-Genre:d&b
- Nationality:indonesian is somewhere to the right of Hobby:woodworking
- Nationality:british is on the far left or far right
- Sport:climbing != Food:pineapple
- Sport:swimming is on the left of Sport:skateboarding
- Music-Genre:r&b == Food:pineapple
- Hobby:photography is on the left or right of Transport:scooter
- Transport:train is not to the left of Sport:lacrosse
- Transport:train is somewhere to the left of Nationality:indonesian
- Music-Genre:r&b is somewhere to the left of Food:pepper
- Nationality:dutch is on the far right

Answer the following question:

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

Answer: pomegranate


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Beverage, Hobby, Music-Genre, Transport.
The attributes have the following possible values:
- Beverage: cola, fanta, iced-tea, juice, tea
- Hobby: baking, chess, drawing, filmmaking, woodworking
- Music-Genre: country, d&b, electronic, jazz, r&b
- Transport: bus, motorbike, roller, taxi, van

Given the following premises about the line of people:
- Music-Genre:r&b is not to the left of Transport:taxi
- Transport:taxi is somewhere to the left of Beverage:fanta
- Beverage:fanta is not to the right of Beverage:juice
- Beverage:tea is in an even position
- Hobby:filmmaking is somewhere to the right of Hobby:drawing
- Hobby:woodworking is in an even position
- Hobby:chess is in the middle
- Transport:roller is on the left or right of Transport:motorbike
- Beverage:fanta is somewhere to the left of Music-Genre:d&b
- Music-Genre:jazz and Beverage:cola have different parity positions
- Transport:motorbike is on the left of Hobby:drawing
- Transport:taxi is on the right of Music-Genre:jazz
- Music-Genre:electronic is in the middle
- Beverage:cola is somewhere to the left of Beverage:iced-tea
- Transport:van and Beverage:iced-tea have the same parity positions

Answer the following question:

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

Answer: fanta


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, Movie-Genre, Music-Genre, Beverage, Sport.
The attributes have the following possible values:
- Nationality: american, french, mexican, polish
- Hobby: board-games, card-games, gardening, writing
- Movie-Genre: epic, martial-arts, musical, spy
- Music-Genre: electronic, metal, soul, trance
- Beverage: iced-tea, milk, mirinda, sprite
- Sport: baseball, swimming, tennis, volleyball

Given the following premises about the line of people:
- Movie-Genre:epic is on the left or right of Beverage:milk
- Music-Genre:soul is on the left of Sport:volleyball
- Nationality:polish is not to the left of Sport:tennis
- Beverage:milk == Nationality:american
- Beverage:milk != Sport:baseball
- Hobby:gardening == Movie-Genre:spy or Movie-Genre:spy == Beverage:milk, but not both
- Nationality:french and Sport:swimming have different parity positions
- Sport:tennis is on the left or right of Beverage:sprite
- Sport:baseball != Beverage:iced-tea or Sport:baseball != Nationality:french or both
- Movie-Genre:epic and Hobby:gardening have the same parity positions
- Hobby:writing != Nationality:polish
- Nationality:french and Movie-Genre:martial-arts have different parity positions
- Beverage:mirinda is on the left or right of Hobby:writing
- Hobby:card-games is somewhere between Sport:volleyball and Music-Genre:metal
- Hobby:gardening is on the left of Nationality:french
- Sport:volleyball is somewhere to the left of Music-Genre:trance

Answer the following question:

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

Answer: trance


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Beverage, Food, Movie-Genre, Music-Genre, Transport.
The attributes have the following possible values:
- Beverage: coffee, lemonade, soy-milk, sprite, tea
- Food: avocado, cauliflower, orange, peach, peas
- Movie-Genre: disaster, fantasy, horror, satire, sports
- Music-Genre: d&b, disco, dubstep, jazz, punk
- Transport: airplane, bus, scooter, skateboard, van

Given the following premises about the line of people:
- Movie-Genre:satire is somewhere to the right of Movie-Genre:sports
- Food:peas is on the left or right of Music-Genre:d&b
- Movie-Genre:fantasy is on the left of Beverage:coffee
- Music-Genre:dubstep is on the right of Food:peach
- Movie-Genre:horror is on the far left
- Transport:airplane is on the right of Music-Genre:jazz
- Transport:van is somewhere to the left of Beverage:lemonade
- Food:orange is on the far left
- Food:cauliflower is on the left or right of Beverage:soy-milk
- Transport:scooter is on the left or right of Music-Genre:punk
- Music-Genre:dubstep == Transport:airplane
- Transport:bus is on the far left or far right
- Music-Genre:d&b is in an odd position
- Transport:skateboard is in an odd position
- Beverage:sprite is on the far right
- Beverage:coffee is on the left of Movie-Genre:disaster
- Music-Genre:punk is in an odd position
- Transport:scooter is on the left or right of Transport:airplane
- Movie-Genre:fantasy is in an odd position

Answer the following question:

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

Answer: orange


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Beverage, Movie-Genre, Music-Genre, Nationality, Pet.
The attributes have the following possible values:
- Job: dancer, entrepreneur, librarian, teacher
- Beverage: coffee, juice, sprite, water
- Movie-Genre: family, martial-arts, sports, zombie
- Music-Genre: blues, classical, house, r&b
- Nationality: chinese, indonesian, nigerian, thai
- Pet: ferret, hamster, hedgehog, snake

Given the following premises about the line of people:
- Music-Genre:blues and Nationality:chinese have the same parity positions
- Pet:snake and Nationality:nigerian have different parity positions
- Job:entrepreneur and Nationality:chinese have different parity positions
- Job:librarian == Nationality:indonesian or Pet:ferret == Job:librarian, but not both
- Music-Genre:classical is not to the right of Music-Genre:r&b
- Music-Genre:r&b is not to the left of Movie-Genre:zombie
- Movie-Genre:zombie and Music-Genre:blues have the same parity positions
- Pet:hamster == Nationality:nigerian or Pet:hamster == Movie-Genre:martial-arts, but not both
- Beverage:juice and Movie-Genre:sports have different parity positions
- Movie-Genre:sports is somewhere to the left of Nationality:nigerian
- Movie-Genre:zombie is somewhere between Nationality:chinese and Beverage:sprite
- Movie-Genre:sports is somewhere to the right of Movie-Genre:family
- Nationality:nigerian == Job:entrepreneur or Nationality:nigerian == Pet:ferret or both
- Nationality:thai is somewhere to the right of Movie-Genre:zombie
- Job:entrepreneur is somewhere to the left of Beverage:water
- Music-Genre:r&b is somewhere to the left of Job:teacher

Answer the following question:

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

Answer: family


The members of a distinguished committee were choosing a president, and each member gave one vote to one of the $27$ candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least $1$ than the number of votes for that candidate. What is the smallest possible number of members of the committee?
Please give the answer directly without any intermediate steps.

Answer: 134


A convex pentagon  $ ABCDE$  is inscribed in a circle. The distances of  $ A$  from the lines  $ BC,CD,DE$  are  $ a,b,c,$  respectively. Compute the distance of  $ A$  from the line  $ BE$ .

Answer: \dfrac{ac}{b}


[u]Set 4 [/u]

[b]4.1[/b] Triangle $T$ has side lengths $1$, $2$, and $\sqrt7$. It turns out that one can arrange three copies of triangle $T$ to form two equilateral triangles, one inside the other, as shown below. Compute the ratio of the area of the outer equilaterial triangle to the area of the inner equilateral triangle.
[img]https://cdn.artofproblemsolving.com/attachments/0/a/4a3bcf4762b97501a9575fc6972e234ffa648b.png[/img]

[b]4.2[/b]  Let $T$ be the answer from the previous problem. The diagram below features two concentric circles of radius $1$ and $T$ (not necessarily to scale). Four equally spaced points are chosen on the smaller circle, and rays are drawn from these points to the larger circle such that all of the rays are tangent to the smaller circle and no two rays intersect. If the area of the shaded region can be expressed as $k\pi$ for some integer $k$, find $k$.
[img]https://cdn.artofproblemsolving.com/attachments/a/5/168d1aa812210fd9d60a3bb4a768e8272742d7.png[/img]

[b]4.3[/b] Let $T$ be the answer from the previous problem. $T^2$ congruent squares are arranged in the configuration below (shown for $T = 3$), where the squares are tilted in alternating fashion such that they form congruent rhombuses between them. If all of the rhombuses have long diagonal twice the length of their short diagonal, compute the ratio of the total area of all of the rhombuses to the total area of all of the squares. (Hint: Rather than waiting for $T$, consider small cases and try to find a general formula in terms of $T$, such a formula does exist.)
[img]https://cdn.artofproblemsolving.com/attachments/1/d/56ef60c47592fa979bfedd782e5385e7d139eb.png[/img]

PS. You should use hide for answers.

Answer: 7


10. From the 100 natural numbers 1 to 100, choose any two different natural numbers so that their product is a perfect square. The number of different ways to choose them is $\qquad$ kinds.

Answer: 105


Let \( h \) be a semicircle with diameter \( AB \). A point \( P \) is chosen arbitrarily on the segment \( AB \). The line perpendicular to \( AB \) through \( P \) intersects the semicircle \( h \) at point \( C \). The segment \( PC \) divides the area of the semicircle into two parts. In each of these parts, a circle is inscribed that is tangent to \( AB \), \( PC \), and \( h \). The points where these two circles touch \( AB \) are labeled \( D \) and \( E \), with \( D \) lying between \( A \) and \( P \).

Prove that the measure of the angle \( \angle DCE \) does not depend on the choice of \( P \).

(Walther Janous)

Answer: 45^\circ


[ $\left.\frac{\text { Pairings and Groups; Bijections }}{\text { Regular Polygons }}\right]$
[ Examples and Counterexamples. Constructions ]

A circus arena is illuminated by $n$ different spotlights. Each spotlight illuminates a certain convex shape. It is known that if one arbitrary spotlight is turned off, the arena will still be fully illuminated, but if any two arbitrary spotlights are turned off, the arena will no longer be fully illuminated. For which values of $n$ is this possible?

Answer: n \geq 2


## Zadatak B-4.2.

Odredite rješenja jednadžbe

$$
\log _{4}\left(\cos ^{2} x\right)-\log _{2}\left(\frac{\sin x}{4}\right)=2
$$

u intervalu $[0,2 \pi]$.

Answer: \dfrac{\pi}{4}


6. [20] Andrew has a fair six sided die labeled with 1 through 6 as usual. He tosses it repeatedly, and on every third roll writes down the number facing up as long as it is not the 6 . He stops as soon as the last two numbers he has written down are squares or one is a prime and the other is a square. What is the probability that he stops after writing squares consecutively?

Answer: \dfrac{4}{25}


Suppose we have $10$ balls and $10$ colors. For each ball, we (independently) color it one of the $10$ colors, then group the balls together by color at the end. If $S$ is the expected value of the square of the number of distinct colors used on the balls, find the sum of the digits of $S$ written as a decimal.

[i]Proposed by Michael Kural[/i]

Answer: 55


Problem 10.3. Let $n$ be a natural number. Find the number of sequences $a_{1} a_{2} \ldots a_{2 n}$, where $a_{i}=+1$ or $a_{i}=-1$ for $i=1,2 \ldots, 2 n$, such that

$$
\left|\sum_{i=2 k-1}^{2 l} a_{i}\right| \leq 2
$$

for all $k$ and $l$ for which $1 \leq k \leq l \leq n$.

Answer: 2 \cdot 3^n - 2^n


5.5. After the hockey game, Anton said that he scored 3 goals, while Ilya only scored one. Ilya said that he scored 4 goals, and Seryozha scored as many as 5. Seryozha said that he scored 6 goals, and Anton only two. Could it be that together they scored 10 goals, given that each of them told the truth once and lied once? Explain your answer.

Answer: No


One, (20 points) Class A and Class B set out from School $A$ simultaneously to a military camp $B$ for military training, which is $75 \mathrm{~km}$ away from the school. The walking speed of Class A students is $4 \mathrm{~km} / \mathrm{h}$, and the walking speed of Class B students is $5 \mathrm{~km} / \mathrm{h}$. The school has a car, which travels at a speed of $40 \mathrm{~km} / \mathrm{h}$ when empty and $20 \mathrm{~km} / \mathrm{h}$ when loaded, and this car can carry only one class of students at a time. Now, it is required that both classes of students arrive at the military camp simultaneously. How much time do they need at least to arrive?

Answer: \dfrac{27}{4}


A rectangular prism has dimensions with height twice the width and depth thrice the width. Similar to a cube's diagonals, a segment connecting non-edge vertices is also a diagonal in a rectangular prism. Compute the total number of such diagonals in the prism.

Answer: 16


When the digits in the number $3086$ are reversed, we obtain the number $6803$. Factorize $6803$ into $p \cdot q \cdot r$, such that $p$, $q$, and $r$ are three distinct primes. How many other positive integers are the products of exactly three distinct primes $p_1$, $p_2$, and $p_3$ such that $p_1 + p_2 + p_3 = p+q+r$?

Answer: 0 \text{ distinct integers}


The median of the set of numbers $\{15, 35, 50, x, 18\}$ is eight less than the mean. If $x$ is a positive integer, what is the value of $x$?

Answer: 97


Consider the function $y = \frac{3-x}{4x+5}$. For which value of $x$, excluding any values making the denominator zero, is $y$ unattainable?

Answer: -\frac{1}{4}


A circle is inscribed around a quadrilateral $WXYZ$ such that $\angle WYZ = 20^\circ$ and $\angle WXY = 70^\circ$. Determine the sum of degrees in $\angle YWX + \angle YZW$.

Answer: 90^\circ


A circle with radius 2 rolls around the inside of a right triangle with sides 9, 12, and 15. The circle remains tangent to at least one side of the triangle at all times. Determine the total distance traveled by the center of the circle when it returns to its starting position.

Answer: 12


Let $f(x)$ be a differentiable function that satisfies the equation
\[ f(x+y) = f(x) + f(y) \]
for all real numbers $x$ and $y$. Find the value of $f'(0)$.

Answer: 0


There are two stores offering discounts on Camera $Y$. The list price of Camera $Y$ is $\$59.99$. Store A offers a $15\%$ discount, while Store B offers a flat $\$12$ discount. How much cheaper, in cents, is the cheaper store's price for Camera $Y$ compared to the more expensive store?

Answer: 300


At Unity Community Club, a survey was conducted to find out the average age of its members. The average age of the female members was found to be 35 years old. The average age of the male members was 30 years old. Additionally, the average age of the child members was 10 years old. The overall average age of all the members combined was 25 years old. What is the ratio of female to male to child members? Express your answer as a common fraction in the form $\frac{f:m:c}$.

Answer: 1:1:1


In right $\Delta XYZ$, $\angle XYZ$ is a right angle. Point $N$ is the midpoint of $\overline{XZ}$. Length $\overline{XY}$ is 5 cm and $\overline{YZ}$ is 12 cm. Calculate the lengths of the median $\overline{YN}$ and the hypotenuse $\overline{XZ}$. Express your answers as decimals to the nearest tenth.

Answer: 6.5 \text{ cm}


In Ms. Johnson's class, each student averages one day absent out of fifteen. What is the probability that out of any three students chosen at random, exactly one student will be absent while the other two are present? Express your answer as a percent rounded to the nearest tenth.

Answer: 17.4\%


Each point in an octagonal lattice is one unit from its nearest neighbor. How many equilateral triangles have all vertices in this lattice? Assume the shape is a regular octagon with a center point.

Answer: 0


16. A rectangular thin wooden board, 7 meters long and 5 meters wide, has a small rectangular area at one corner damaged by insects, the damaged part is 2 meters long and 1 meter wide. Please design a method to saw off the damaged area in one straight cut, making the area removed as small as possible. What is the minimum area to be removed? Explain your reasoning.

Answer: 4


3. Define the function $f$ on the positive integers such that

$$
f(1)=1, \quad f(3)=3
$$

and

$$
\begin{aligned}
f(2 n) & =f(n) \\
f(4 n+1) & =2 f(2 n+1)-f(n) \\
f(4 n+3) & =3 f(2 n+1)-2 f(n)
\end{aligned}
$$

for all positive integers $n$.

Determine the number of integers $n$ in the range $1 \leq n \leq 1988$ for which $f(n)=n$ holds.

Answer: 92


8. Let $F$ be the set of all sequences of the form $\left(A_{1}, A_{2}, \cdots, A_{n}\right)$, where $A_{i}$ is a subset of $B=\{1,2, \cdots, 10\}$, and let $|A|$ denote the number of elements in the set $A$. Then $\sum_{F}\left|\bigcup_{i=1}^{n} A_{i}\right|=$ $\qquad$ .

Answer: 10 \cdot (2^{10n} - 2^{9n})


13.24 A pile of stones has a total weight of 100 kilograms, where the weight of each stone does not exceed 2 kilograms. By taking out some of these stones in various ways, and finding the difference between the sum of the weights of these stones and 10 kilograms, let the minimum value of the absolute values of these differences be \(d\), find the maximum value of \(d\).

Answer: \dfrac{10}{11}


Consider a square \(A B C D\) with center \(O\). Let \(E, F, G,\) and \(H\) be points on the interior of sides \(AB, BC, CD,\) and \(DA\), respectively, such that \(AE = BF = CG = DH\). It is known that \(OA\) intersects \(HE\) at point \(X\), \(OB\) intersects \(EF\) at point \(Y\), \(OC\) intersects \(FG\) at point \(Z\), and \(OD\) intersects \(GH\) at point \(W\). Let \(x\) and \(y\) be the lengths of \(AE\) and \(AH\), respectively.

a) Given that the Area of \(EFGH\) is \(1 \text{ cm}^2\), calculate the value of \(x^2 + y^2\).

b) Verify that \(HX = \frac{y}{x+y}\). Then, conclude that \(X, Y, Z,\) and \(W\) are the vertices of a square.

c) Calculate the product:

    \[\text{Area}(ABCD) \cdot \text{Area}(XYZW).\]

Answer: 1


2. The little squirrel dried mushrooms: $85 \%$ were white mushrooms, and the rest were russulas. Then he ate some of the white mushrooms, and now the russulas made up $30 \%$ of the remaining mushrooms. What part of the mushrooms did the little squirrel eat?

Answer: \dfrac{1}{2}


Tokarev S.I.

A set of five-digit numbers $\left\{N_{1}, N_{k}\right\}$ is such that any five-digit number, all digits of which are in non-decreasing order, coincides in at least one digit with at least one of the numbers $N_{1}, N_{k}$. Find the smallest possible value of $k$.

Answer: 2


2. In $\triangle A B C$, $I$ is the incenter, and $C A+A I=$ $B C$. If $\angle A=80^{\circ}$, then $\angle B=$ $\qquad$

Answer: 40^\circ


10.224. Given an isosceles triangle with a base of 12 cm and a lateral side of $18 \mathrm{~cm}$. What segments need to be laid off from the vertex of the triangle on its lateral sides, so that by connecting their ends, a trapezoid with a perimeter of 40 cm is obtained?

Answer: 6


9.90 To color some of the squares in a $2 \times n$ strip of squares such that no $2 \times 2$ square is completely colored, let $P_{n}$ denote the number of different coloring methods that satisfy this condition. Prove that $P_{1989}$ is divisible by 3, and find the highest power of 3 that divides $P_{1989}$.

Answer: 3^{994}


My friend and I are playing a game with the following rules: If one of us says an integer \( n \), the opponent then says an integer of their choice between \( 2n \) and \( 3n \), inclusive. Whoever first says 2007 or greater loses the game, and their opponent wins. I must begin the game by saying a positive integer less than 10. With how many of them can I guarantee a win?

Answer: 6


Misha drew a triangle with a perimeter of 11 and cut it into parts with three straight cuts parallel to the sides, as shown in the figure. The perimeters of the three shaded figures (trapezoids) were 5, 7, and 9. Find the perimeter of the small triangle that resulted from the cutting.

Answer: 10


From 2 kg 600 grams of copper, three items were made: a spoon, a candlestick, and a teapot. Each item is three times heavier than the previous one. What is the weight of the teapot?

Answer: 1800\,


There are 195 different cards with the numbers \(1, 5, 7, 5^{2}, 7^{2}, \ldots, 5^{97}, 7^{97}\) (each card has exactly one number, and each number appears exactly once). In how many ways can you choose 3 cards so that the product of the numbers on the chosen cards is the square of an integer divisible by 35?

Answer: 223488


The angles of a triangle are A, B, and C. Find the smallest possible value of \(\tan \frac{A}{2} + \tan \frac{B}{2} + \tan \frac{C}{2}\) and the largest possible value of \(\tan \frac{A}{2} \tan \frac{B}{2} \tan \frac{C}{2}\).

Answer: \frac{1}{3\sqrt{3}}


Find all functions $f\colon(0,\infty)\to(0,\infty)$ such that $\frac{f(p)^2+f(q)^2}{f(r^2)+f(s^2)}=\frac{p^2+q^2}{r^2+s^2}$ for all $p,q, r, s > 0$ with $pq = rs$. There are multiple solutions for $f(x)$, represented by their respective function numbers. Please provide the sum of these numbers.

Answer: 0


Let $n$ and $k$ be fixed positive integers of the same parity, $k \ge n$. We are given $2n$ lamps numbered $1$ through $2n$; each of them can be on or off. At the beginning all lamps are off. We consider sequences of $k$ steps. At each step one of the lamps is switched (from off to on or from on to off). Let $N$ be the number of $k$-step sequences ending in the state: lamps $1,....,n$ on, lamps $n+1,...,2n$ off. Let $M$ be the number of $k$-step sequences leading to the same state and not touching lamps $n+1,....2n$ at all. Find the ratio $N/M$ when $k = n + 1$. Please provide the integer value of the ratio.

Answer: 1


For any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers with sum $s_A = a_1 + a_2 + a_3 + a_4$, let $p_A$ denote the number of pairs $(i, j)$ with $1 \leq i < j \leq 4$ for which $a_i + a_j$ divides $s_A$. Determine the maximum value of $p_A$ among all sets of four distinct positive integers.

Answer: 4


Let $n \geq 1$ be an odd integer. Determine all functions $f$ from the set of integers to itself such that for all integers $x$ and $y$, the difference $f(x) − f(y)$ divides $x^n − y^n$. The original answer involves parameters \varepsilon, d, and c. Provide the sum of all distinct possible values of \varepsilon + d + c.

Answer: 0


For each positive integer $k$, let $t(k)$ be the largest odd divisor of $k.$ Determine all positive integers $a$ for which there exists a positive integer $n$ such that all the differences $\begin{aligned}t(n+a)-t(n), t(n+a+1)-t(n+1),\cdots,t(n+2a-1)-t(n+a-1)\end{aligned}$ are divisible by $4$. Please provide the sum of all possible values of $a$.

Answer: 9


Call admissible a set $A$ of integers that has the following property: If $x,y\in A$ (possibly $x=y)$ then $x^2+kxy+y^2\in A$ for every integer $k.$ Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. Please provide the sum of the pair $(m, n)$.

Answer: 1


Find all triples $(x, y, z)$ of positive integers such that $x ≤ y ≤ z$ and $x^3(y^3 + z^3) = 2012(xyz + 2)$. Please provide the sum of the values of x, y, and z for all such triples.

Answer: 505


$A \pm1-sequence$ is a sequence of 2022 numbers $a_{1},\ldots,a_{2022}$, each equal to either +1 or -1. Determine the largest $C$ so that, for any $\pm1$ sequence, there exists an integer $k$ and indices $1\leqslant t_{1}<\ldots<t_{k}\leqslant2022$ so that $t_{i+1}-t_{i}\leqslant2$ for all $i$, and $$\left|\sum_{i=1}^ka_{t_i}\right|\geqslant C.$$ Please provide the value of C.

Answer: 506


Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has $|y^2-P(x)|\leqslant2|x|$ if and only if $|x^2-P(y)|\leqslant2|y|$. Determines the positive integer value in the set of possible values for P(0).

Answer: 1


Let $n\geqslant2$ be an integer. Consider an $n \times n$ chessboard divided into $n^ 2$ unit squares. We call a configuration of n rooks on this board happy if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that for every happy configuration of rooks, we can find a $k \times k$ square without a rook on any of its $k ^2$ unit squares. Please provide the value of $k$ when $n = 10$. The answer should be a single integer.

Answer: 3


$A$ card deck consists of $1024$ cards. On each card, a set of distinct decimal digits is written in such a way that no two of these sets coincide (thus, one of the cards is empty). Two players alternately take cards from the deck, one card per turn. After the deck is empty, each player checks if he can throw out one of his cards so that each of the ten digits occurs on an even number of his remaining cards. If one player can do this but the other one cannot, the one who can is the winner; otherwise a draw is declared. Determine the number of possible first moves of the first player after which he has a winning strategy.What is the total number of possible moves

Answer: 1023


Let $n$ be a fixed positive integer. Find the maximum possible value of $$\sum\limits_{1\leqslant r<s\leqslant2n}(s-r-n)x_rx_s,$$ where $-1\leqslant x_{i}\leqslant1$ for all $i=1,2,\cdots,2n.$ Please provide the value of the expression when $n$ is a specific integer, such as $n=5$. The answer should be given in integer form, which is the product of n and (n-1).

Answer: 20


Let $a_{1}, a_{2} \ldots, a_{51}$ be non-zero elements of a field. We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence $b_{1} \ldots, b_{51}$. If this new sequence is a permutation of the original one, what can be the characteristic of the field? Please provide the sum of all possible characteristics.

Answer: 9


Find all $r>0$ such that whenever $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a differentiable function such that $|\operatorname{grad} f(0,0)|=1$ and $|\operatorname{grad} f(u)-\operatorname{grad} f(v)| \leq|u-v|$ for all $u, v \in \mathbb{R}^{2}$, then the maximum of $f$ on the disk $\left\{u \in \mathbb{R}^{2}:|u| \leq r\right\}$ is attained at exactly one point. $\left(\operatorname{grad} f(u)=\left(\partial_{1} f(u), \partial_{2} f(u)\right)\right.$ is the gradient vector of $f$ at the point $u$. For a vector $\left.u=(a, b),|u|=\sqrt{a^{2}+b^{2}}.\right)$ The original answer is in \frac{m}{n} format, please give the value of m + n.

Answer: 3


Call a polynomial $P\left(x_{1}, \ldots, x_{k}\right)$ good if there exist $2 \times 2$ real matrices $A_{1}, \ldots, A_{k}$ such that $$ P\left(x_{1}, \ldots, x_{k}\right)=\operatorname{det}\left(\sum_{i=1}^{k} x_{i} A_{i}\right) $$ Find all values of $k$ for which all homogeneous polynomials with $k$ variables of degree 2 are good. Please provide the sum of all possible values of $k$.

Answer: 3


How many nonzero coefficients can a polynomial $P(z)$ have if its coefficients are integers and $|P(z)| \leq 2$ for any complex number $z$ of unit length? Please provide the sum of all possible values.

Answer: 3


Determine the value of
$$
\sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n}\right) \cdot \ln \left(1+\frac{1}{2 n}\right) \cdot \ln \left(1+\frac{1}{2 n+1}\right)
$$
The original answer is in the form of \frac{m}{n} \ln^3(2). Please provide the value of m + n.

Answer: 3


Let $S$ be the set of all subsets of $\{2,3,\ldots,2016\}$ with size $1007$, and for a nonempty set $T$ of numbers, let $f(T)$ be the product of the elements in $T$. Determine the remainder when \[ \sum_{T\in S}\left(f(T)-f(T)^{-1}\right)^2\] is divided by $2017$. Note: For $b$ relatively prime to $2017$, $b^{-1}$ is the unique positive integer less than $2017$ for which $2017$ divides $bb^{-1} -1$.

Answer: 2014


Farmer James invents a new currency such that for every positive integer $n \le 6$, there exists an $n$-coin worth $n!$ cents. Furthermore, he has exactly $n$ copies of each $n$-coin. An integer $k$ is said to be \textit{nice} if Farmer James can make $k$ cents using at least one copy of each type of coin. How many positive integers less than 2018 are nice?

Answer: 210


Suppose there are $160$ pigeons and $n$ holes. The $1$st pigeon flies to the $1$st hole, the $2$nd pigeon flies to the $4$th hole, and so on, such that the $i$th pigeon flies to the $(i^2 \mod n)$th hole, where $k \mod n$ is the remainder when $k$ is divided by $n$. What is the minimum $n$ such that there is at most one pigeon per hole?

Answer: 326


Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$. Let $O$ be an arbitrary point in the interior of $\triangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell_2$, and $\ell_3$ with the sides of $\triangle ABC$ form a hexagon whose area is $A$. Compute the minimum value of $A$.

Answer: 280


A cardboard box in the shape of a rectangular parallelepiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimeters and its base has a radius of $30$ centimeters, find the volume of the largest box that can be completely enclosed inside the container with the lid on.

Answer: 64000


Answer: 4608


There are $110$ guinea pigs for each of the $110$ species, arranged in a $110 \times 110$ array. Find the maximum integer $n$ such that, no matter how the guinea pigs are arranged, we can always find a column or a row of $110$ guinea pigs containing at least $n$ different species.

Answer: 11


S is the set of all $(a, b, c, d, e, f)$ where $a$, $b$, $c$, $d$, $e$, $f$ are integers such that $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$. Find the largest $k$ which divides $abcdef$ for all members of $S$.

Answer: 24


The eight points $A, B, \ldots, G, H$ lie on five circles as shown. Each of these letters is represented by one of the eight numbers $1, 2, \ldots, 7, 8$, replaced such that the following conditions are met:

1. Each of the eight numbers is used exactly once.
2. The sum of the numbers on each of the five circles is the same.

How many ways are there to substitute the letters with the numbers in this way?

Answer: 8


Shirley has a magical machine. If she inputs a positive even integer $n$, the machine will output $\frac{n}{2}$, but if she inputs a positive odd integer $m$, the machine will output $m+3$. The machine continues operating by automatically using its output as a new input, stopping immediately before it processes a number that has already been processed. Shirley wants to create the longest possible output sequence with an initial input at most $100$. What number should she input?

Answer: 67


An infinite sequence of real numbers $a_1, a_2, \dots$ satisfies the recurrence \[ a_{n+3} = a_{n+2} - 2a_{n+1} + a_n \] for every positive integer $n$. Given that $a_1 = a_3 = 1$ and $a_{98} = a_{99}$, compute $a_1 + a_2 + \dots + a_{100}$.

Answer: 3


Let $A$ and $B$ be two non-empty subsets of $X = \{1, 2, \ldots, 8 \}$ with $A \cup B = X$ and $A \cap B = \emptyset$. Let $P_A$ be the product of all elements of $A$ and let $P_B$ be the product of all elements of $B$. Find the minimum possible value of the sum $P_A + P_B$.

Answer: 402


## Task B-4.7.

In an equilateral triangle $A B C$, point $D$ is on side $\overline{A B}$, point $E$ is on side $\overline{B C}$, and point $F$ is on side $\overline{A C}$. Given that $|A D|=3$, $|C E|=2$, and $\varangle D F E=60^{\circ}$. If the area of triangle $D E F$ is $\frac{3}{2} \sqrt{3}$, determine the area of triangle $A B C$.

Answer: \dfrac{25\sqrt{3}}{4}


6. Let $a, b, c, d \in\{-1,0,1\}$. If the ordered array $(a, b, c, d)$ satisfies that $a+b$, $c+d$, $a+c$, and $b+d$ are all distinct, then $(a, b, c, d)$ is called a "good array". The probability that an ordered array $(a, b, c, d)$ is a good array among all possible ordered arrays $(a, b, c, d)$ is $\qquad$

Answer: \dfrac{16}{81}


A [i]chaisri[/i] figure is a triangle which the three vertices are vertices of a regular $2019$-gon. Two different chaisri figure may be formed by different regular $2019$-gon.
A [i]thubkaew[/i] figure is a convex polygon which can be dissected into multiple chaisri figure where each vertex of a dissected chaisri figure does not necessarily lie on the border of the convex polygon.

Determine the maximum number of vertices that a thubkaew figure may have.

Answer: 4038


6. If in the expression $9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ some of the “ $\times$ ” are changed to “ $\div$ ” so that the final result is still a natural number, denoted as $N$, then the smallest $N$ is $\qquad$ .

Answer: 70


B3. With a broad brush, the diagonals of a square tile are painted, as shown in the figure. Exactly half of the tile's surface is painted. The width of the paintbrush is 1, as indicated. Calculate the length of the side of the tile.

![](https://cdn.mathpix.com/cropped/2024_04_17_8dd363468b37c79bb3bag-2.jpg?height=284&width=330&top_left_y=823&top_left_x=1662)

Answer: 2 + 2\sqrt{2}


10-4-1. Anya and Borya are playing rock-paper-scissors. In this game, each player chooses one of the figures: rock, scissors, or paper. Rock beats scissors, scissors beat paper, and paper beats rock. If the players choose the same figure, the game ends in a tie.

Anya and Borya played 25 rounds. Anya chose rock 12 times, scissors 6 times, and paper 7 times. Borya chose rock 13 times, scissors 9 times, and paper 3 times. There were no ties in any of the rounds. How many times could Anya have won? List all possible answers.

Answer: 16


2. How many six-digit numbers of the form $\overline{a b c a c b}$ are divisible by 23?

Answer: 42


Let $\omega_1$ and $\omega_2$ be circles with radii $3$ and $12$ and externally tangent at point $P$. Let a common external tangent intersect $\omega_1$, $\omega_2$ at $S$, $T$ respectively and the common internal tangent at point $Q$. Define $X$ to be the point on $\overrightarrow{QP}$ such that $QX=10$. If $XS$, $XT$ intersect $\omega_1$, $\omega_2$ a second time at $A$ and $B$, determine $\tan\angle APB$.
.
[i]2015 CCA Math Bonanza Individual Round #15[/i]

Answer: \dfrac{2}{3}


16.15. (England, 68). Find the maximum number of points that can be placed on a sphere of radius 1 so that the distance between any two of them is: a) not less than $\sqrt{2} ;$ b) greater than $\sqrt{2}$.

Answer: 6


Consider a paper punch that can be centered at any point
of the plane and that, when operated, removes from the
plane precisely those points whose distance from the
center is irrational. How many punches are needed to
remove every point?

Answer: 3


A rectangular storage bin measures $10$ feet by $12$ feet, is $3$ feet tall, and sits on a flat plane.  A pile of dirt is pushed up against the outside of the storage bin so that it slants down from the top of the storage bin to points on the ground $4$ feet away from the base of the storage bin as shown.  The number of cubic feet of dirt needed to form the pile can be written as  $m + n \pi$ where  $m$ and $n$ are positive integers.  Find $m + n.$

Answer: 280


Let $x$ be a positive integer less than $200$, and let $y$ be obtained by writing the base 10 digits of $x$ in reverse order. Given that $x$ and $y$ satisfy $11x^2+363y=7xy+6571$, find $x$.

Answer: 133


The alphabet in its natural order $\text{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ is $T_0$.  We apply a permutation to $T_0$ to get $T_1$ which is $\text{JQOWIPANTZRCVMYEGSHUFDKBLX}$.  If we apply the same permutation to $T_1$, we get $T_2$ which is $\text{ZGYKTEJMUXSODVLIAHNFPWRQCB}$.  We continually apply this permutation to each $T_m$ to get $T_{m+1}$.  Find the smallest positive integer $n$ so that $T_n=T_0$.

Answer: 252


We have some identical paper squares which are black on one side of the sheet and white on the other side.  We can join nine squares together to make a $3$ by $3$ sheet of squares by placing each of the nine squares either white side up or black side up.  Two of these $3$ by $3$ sheets are distinguishable if neither can be made to look like the other by rotating the sheet or by turning it over.  How many distinguishable $3$ by $3$ squares can we form?

Answer: 70


Starting with a sequence of $n 1's$, you can insert plus signs to get various sums. For example, when $n = 10$,  you can get the sum $1 + 1 + 1 + 11 + 11 + 111 = 136$, and the sum $1 + 1 + 11 + 111 + 111 = 235$.  Find the number of values of $n$ so that the sum of $1111$ is possible.

Answer: 121


Three congruent ellipses are mutually tangent. Their major axes are parallel.  Two of the ellipses are tangent at the end points of their minor axes as shown. The distance between the centers of these two ellipses is $4$. The distances from those two centers to the center of the third ellipse are both $14$.  There are positive integers m and n so that the area between these three ellipses is $\sqrt{n}-m \pi$. Find $m+n$.

Answer: 776


A city is laid out with a rectangular grid of roads with 10 streets numbered from 1 to 10 running east-west and 16 avenues numbered from 1 to 16 running northsouth.  All streets end at First and Sixteenth Avenues, and all avenues end at First and Tenth Streets.  A rectangular city park is bounded on the north and south by Sixth Street and Eighth Street, and bounded on the east and west by Fourth Avenue and Twelfth Avenue. Although there are no breaks in the roads that bound the park, no road goes through the park.  The city paints a crosswalk from every street corner across any adjacent road.  Thus, where two roads cross such as at Second Street and Second Avenue, there are four crosswalks painted, while at corners such as First Street and First Avenue, there are only two crosswalks painted.
How many crosswalks are there painted on the roads of this city?

Answer: 544


Let $S$ S be a point chosen at random from the interior of the square $ABCD$, which has side $AB$ and diagonal $AC$.  Let $P$ be the probability that the segments $AS$, $SB$, and $AC$ are congruent to the sides of a triangle.  Then $P$ can be written as $\dfrac{a-\pi\sqrt{b}-\sqrt{c}}{d}$ where $a,b,c,$ and $d$ are all positive integers and $d$ is as small as possible.  Find $ab+cd$.

Answer: 160


For natural number $n$, define the function $f(n)$ to be the number you get by $f(n)$ adding the digits of the number $n$. For example, $f(16)=7$, $f(f(78))=6$, and $f(f(f(5978)))=2$.
Find the least natural number $n$ such that $f(f(f(n)))$ is not a one-digit number.


Example 78. A triangular pyramid is cut by a plane into two polyhedra. We will find the ratio of the volumes of these polyhedra, given that the cutting plane divides the edges converging at one vertex of the pyramid in the ratio $1: 2, 1: 2, 2: 1$, counting from this vertex.

Construction of the image. Let the quadrilateral $S A B C$ with its diagonals be the image of the given pyramid (Fig. 87), $\triangle P Q R$ - the image of the given section. This image is complete, and what is quite curious, no parameter has been spent on it.

Given: $\quad S A B C$ - pyramid, $\quad S$ - vertex, $\quad P \in[S A]$, $Q \in[A B]$, $\quad R \in[A C]$, $\quad|A Q|:|Q B|=1: 2,|A R|:|R C|=1: 2$, $|A P|:|P S|=2: 1,(P Q R)$ - cutting plane.

To find: $V_{1}: V_{2}$ - the ratio of the volumes of the resulting polyhedra.

Answer: \dfrac{25}{2}


12. Let \( \left.T=\left|9^{k}\right| k \in \mathbf{Z}, 0 \leqslant k \leqslant 4000\right\} \), given that \( 9^{4000} \) is a 3817-digit number. And its leading digit is 9, then in \( T \), there are \(\qquad\) numbers whose leading digit is 9.

Answer: 184


4. Let $D$ and $E$ be the midpoints of sides $A C$ and $B C$ of an equilateral triangle $A B C$. The ray $D E$ intersects the circumcircle of triangle $A B C$ at point $F$. In what ratio does point $E$ divide the segment $D F$?

Answer: \dfrac{1+\sqrt{5}}{2}


## Task 6 - 321236

Let $k_{1}, k_{2}$, and $k_{3}$ be three concentric circles with radii $r_{1}=5, r_{2}=3 \sqrt{2}$, and $r_{3}=1$, respectively.

Determine the maximum possible value for the area of a triangle $A B C$ with the property that $A$ lies on $k_{1}$, $B$ lies on $k_{2}$, and $C$ lies on $k_{3}$.

Hint: It can be assumed as a known fact that among all triangles with the mentioned property, there exists a triangle with the maximum possible area.

Answer: 14


60. (See [34].) Let $\mathrm{P}(|X| \leqslant b)=1, \mathrm{E} X=0$ and $\mathrm{D} X=\sigma^{2}$ for some $\sigma, b>0$. What is the maximum possible value of $\mathrm{E} e^{X}$?

Answer: \dfrac{\sigma^2 e^{b} + b^2 e^{-\sigma^2 / b}}{\sigma^2 + b^2}


10th Putnam 1950 Problem A3 The sequence a n is defined by a 0 = α, a 1 = β, a n+1 = a n + (a n-1 - a n )/(2n). Find lim a n .

Answer: \alpha + (\beta - \alpha) e^{-1/2}


Let \( a, b, m, n \) be natural numbers, with \( a \) and \( b \) being coprime and \( a > 1 \).

Prove that if \( a^{m} + b^{m} \) is divisible by \( a^{n} + b^{n} \), then \( m \) is divisible by \( n \).

Answer: m \text{ is divisible by } n


31. How many planes of symmetry do the following have: a) a cube; b) a regular tetrahedron?

Answer: 9


For a prime number $p(p \geqslant 3)$, define $F(p)=$ $\sum_{k=1}^{p-1} k^{r}$, where $r$ is a positive odd number not exceeding $p$; $f(p)=$ $\frac{1}{2}-\left\{\frac{F(p)}{p^{2}}\right\}$, where $\{x\}=x-[x]$ represents the fractional part of $x$. Find the value of $f(p)$.

Answer: \dfrac{1}{2}


In a regular hexagon with an area of \(144 \, \text{cm}^2\), all its diagonals were marked. The hexagon thus "broke apart" into several triangles and quadrilaterals.

a) Determine into how many parts the diagonals divided it.

b) Determine the area of a regular hexagon formed by combining all the quadrilateral parts of the given hexagon.

Answer: 24


$\left.\begin{array}{l}\text { Tangent circles }\end{array}\right]$

Two circles touch each other internally at point $A$. From the center $O$ of the larger circle, a radius $O B$ is drawn, touching the smaller circle at point $C$. Find $\angle B A C$.

Answer: 45^\circ


Five. (20 points) A scientific expedition team went to the upper reaches of a certain river to investigate an ecological area. After setting out, they advanced at a speed of $17 \mathrm{~km}$ per day, traveling upstream along the riverbank for several days before reaching their destination. They then spent several days investigating the ecological area. After completing their mission, they returned at a speed of 25 $\mathrm{km}$ per day. On the 60th day after departure, the expedition team traveled $24 \mathrm{~km}$ and returned to the starting point. How many days did the scientific expedition team spend investigating the ecological area?

Answer: 23


14. Given $A\left(x_{1}, y_{1}\right) 、 B\left(x_{2}, y_{2}\right)$ are two moving points on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=$ $1(a>b>0)$, $O$ is the origin, and $O A \perp$ $O B$. Find the minimum length of segment $A B$.

Answer: \dfrac{2ab}{\sqrt{a^2 + b^2}}


3. In trapezoid $ABCD$, $BC \parallel AD$, $AB=AD$, $\angle ABC=\frac{2\pi}{3}$, $\angle BCD=\frac{\pi}{2}$. $\triangle ABD$ is folded along $BD$, and the projection of point $A$ on plane $BCD$ is point $P$. It is known that the cosine of the angle between $AB$ and $CD$ is $\frac{\sqrt{3}}{6}$. Then the cosine of the angle between $BP$ and $CD$ is

Answer: \dfrac{1}{2}


K2) Let $n$ be a natural number. Roger has a square garden of size $(2 n+1) \times(2 n+1)$. He erects fences to divide it into rectangular beds. He wants exactly two horizontal $k \times 1$ beds and exactly two vertical $1 \times k$ beds for every even number $k$ between 1 and $2 n+1$, as well as a single square bed of size $1 \times 1$ when he is finished. How many different ways can Roger divide his garden?

Answer: 2^n


Let $a$ and $b$ be positive real numbers with $a\ge b$. Let $\rho$ be the maximum possible value of $\frac {a}{b}$ for which the system of equations $$
a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2
$$has a solution in $(x,y)$ satisfying $0\le x < a$ and $0\le y < b$.  Find $\rho^2.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 7


In parallelogram $ABCD$, let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$. Angles $CAB$ and $DBC$ are each twice as large as angle $DBA$, and angle $ACB$ is $r$ times as large as angle $AOB$. Find $r.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 16


In triangle $ABC$, $AB=\sqrt{30}$, $AC=\sqrt{6}$, and $BC=\sqrt{15}$. There is a point $D$ for which $\overline{AD}$ bisects $\overline{BC}$, and $\angle ADB$ is a right angle. Find the value of the ratio
\[\dfrac{[ADB]}{[ABC]}.\]In other words, compute the ratio of the area of triangle $ADB$ to the area of triangle $ABC.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 65


If \(b > 1\), \(\sin x > 0\), \(\cos x > 0\), and \(\log_b \sin x = a\), find the value of \(\log_b \cos x\). The original answer is in the form \(\frac{m}{n} \log_b(1-b^{p})\). Please provide the value of \(m + n + p\).

Answer: 5


On Halloween, 31 children walked into the principal's office asking for candy. They can be classified into three types: some always lie, some always tell the truth, and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order:

1. "Are you a truth-teller?" The principal gave a piece of candy to each of the 22 children who answered yes.
2. "Are you an alternater?" The principal gave a piece of candy to each of the 15 children who answered yes.
3. "Are you a liar?" The principal gave a piece of candy to each of the 9 children who answered yes.

How many pieces of candy in all did the principal give to the children who always tell the truth?

Answer: 7


$\underline{\text { Folklore }}$Among the actors of Karabas Barabas theater, a chess tournament was held. Each participant played exactly one game with each of the others. One solido was given for a win, half a solido for a draw, and nothing for a loss. It turned out that among any three participants, there would be a chess player who earned exactly 1.5 solido in games with the other two. What is the maximum number of actors that could have participated in such a tournament?

Answer: 5


The polynomial \(12x^3 + ax^2 + bx + c\) has three distinct real roots, but \((x^2 + x + 2001)^3 + a(x^2 + x + 2001)^2 + b(x^2 + x + 2001) + c\) has no real roots. Show that \(2001^3 + a \cdot 2001^2 + b \cdot 2001 + c > \frac{1}{64}\).

Answer: 2001^3 + a \cdot 2001^2 + b \cdot 2001 + c > \frac{1}{64}


Let $\Gamma_1, \Gamma_2, \Gamma_3$ be three pairwise externally tangent circles with radii $1,2,3,$ respectively. A circle passes through the centers of $\Gamma_2$ and $\Gamma_3$ and is externally tangent to $\Gamma_1$ at a point $P.$ Suppose $A$ and $B$ are the centers of $\Gamma_2$ and $\Gamma_3,$ respectively. What is the value of $\frac{{PA}^2}{{PB}^2}?$

[i]Proposed by Kyle Lee[/i]

Answer: \dfrac{8}{15}


33rd Eötvös 1929 Problem 1 Coins denomination 1, 2, 10, 20 and 50 are available. How many ways are there of paying 100?

Answer: 784


87. Bobo has 640 points, Guagua has 1000 points. Now both of them exchange their points for the same watch. As a result, the points Guagua has left are 4 times the points Bobo has left. Therefore, exchanging for such a watch requires $\qquad$ points.

Answer: 520


1. (3 points) The left figure below is the recently discovered Archimedes' "Stomachion" puzzle, dividing a square into 14 polygonal pieces: After research, experts found that these 14 pieces can be correctly drawn on a square with a side length of $12 \mathrm{~cm}$, as shown in the figure. Question: The area of the gray piece is $\qquad$ square centimeters.

Answer: 12


An equilateral triangle is tiled with $n^2$ smaller congruent equilateral triangles such that there are $n$ smaller triangles along each of the sides of the original triangle.  For each of the small equilateral triangles, we randomly choose a vertex $V$ of the triangle and draw an arc with that vertex as center connecting the midpoints of the two sides of the small triangle with $V$ as an endpoint. Find, with proof, the expected value of the number of full circles formed, in terms of $n.$

[img]http://s3.amazonaws.com/classroom.artofproblemsolving.com/Images/Transcripts/497b4e1ef5043a84b433a5c52c4be3ae.png[/img]

Answer: \dfrac{(n-1)(n-2)}{1458}


Problem 6.4. Misha made himself a homemade dartboard during the summer at the cottage. The round board is divided by circles into several sectors - darts can be thrown into it. Points are awarded for hitting a sector as indicated on the board.

Misha threw 8 darts 3 times. The second time he scored twice as many points as the first time, and the third time he scored 1.5 times as many points as the second time. How many points did he score the second time?

![](https://cdn.mathpix.com/cropped/2024_05_06_df4c0436ab68970fab15g-15.jpg?height=363&width=361&top_left_y=1494&top_left_x=546)

Answer: 48


Problem 11.4. A positive integer $a$ whose decimal representation has 2006 digits is called "bad" if 3 does not divide any integer formed by three consecutive digits of $a$.

a) Find the number of all bad integers whose digits are equal to 1,2 or 3 .

b) Let $a$ and $b$ be different bad integers such that $a+b$ is also a bad integer. Denote by $k$ the number of positions, where the digits of $a$ and $b$ coincide. Find all possible values of $k$.

Emil Kolev

Answer: 9 \times 2^{2004}


From a point \(M\) inside a regular tetrahedron, perpendiculars \(M A_{i} (i=1,2,3,4)\) are dropped to its faces. Prove that \(\overrightarrow{M A}_{1}+\overrightarrow{M A}_{2}+\overrightarrow{M A_{3}}+\overrightarrow{M A_{4}}= \frac{4 \overrightarrow{M O}}{3}\), where \(O\) is the center of the tetrahedron.

Answer: \overrightarrow{M A}_{1} + \overrightarrow{M A}_{2} + \overrightarrow{M A}_{3} + \overrightarrow{M A}_{4} = \frac{4 \overrightarrow{M O}}{3}


2. Given points
$$
A\left(x-\frac{3-\sqrt{5}}{2} \cdot y, 1\right) 、 B\left(x-\frac{3+\sqrt{5}}{2} \cdot y,-1\right)
$$

satisfy $\overrightarrow{O A} \cdot \overrightarrow{O B}=0$. Then the range of $S=x^{2}-x y+y^{2}-1$ is $\qquad$ .

Answer: [-\dfrac{2}{5}, +\infty)


5. Given $\odot O: x^{2}+y^{2}=1$ with a moving chord $A B=\sqrt{3}$, and a moving point $P$ on $\odot C:(x-2)^{2}+(y-3)^{2}=2$. Then the minimum value of $\sqrt{\overrightarrow{P A} \cdot \overrightarrow{P B}+\frac{3}{4}}$ is $\qquad$.

Answer: \sqrt{13} - \sqrt{2} - \dfrac{1}{2}


Mereduth has many red boxes and many blue boxes. Coloon has placed five green boxes in a row on the ground, and Mereduth wants to arrange some number of her boxes on top of this row. Assume that each box must be placed so that it straddles two lower boxes. Including the arrangement with no boxes, how many arrangements can Mereduth make?

Answer: 21


We are given weights ranging from 1 to 5771, i.e., 1, 2, 3, ..., 5770, 5771. These weights are to be partitioned into $n$ sets such that each set has an equal total weight. Determine the maximal value of $n$ for which this partitioning is possible.

Answer: 2886


Numbers $a$, $b$, and $c$ form an arithmetic sequence if $b - a = c - b$. Let $a$, $b$, and $c$ be positive integers forming an arithmetic sequence with $a < b < c$. Let $f(x) = ax^2 + bx + c$. Two distinct real numbers $r$ and $s$ satisfy $f(r) = s$ and $f(s) = r$. If $rs = 2017$, determine the smallest possible value of $a$.

Answer: 9


There are $64$ booths arranged around a circular table, each containing a chip. The chips and booths are numbered from $1$ to $64$. At the center of the table, there are $1996$ light bulbs, all initially turned off.

Every minute, the chips move simultaneously in a circular pattern as follows: chip $1$ moves one booth forward, chip $2$ moves two booths forward, and so on, such that multiple chips can occupy the same booth at the same time.

A bulb is lit whenever chip $1$ shares a booth with any other chip.

Determine the position of chip $1$ on the first minute when all $1996$ bulbs are lit.

Answer: 32


Find the number of positive integer solutions $(a,b,c,d)$ to the equation $(a^2+b^2)(c^2-d^2)=2020$. Note: The solutions $(10,1,6,4)$ and $(1,10,6,4)$ are considered different.

Answer: 8


Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:

1. In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin.

2. In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are two (not necessarily distinct) numbers from the first line.

3. In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line.

Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.

Answer: 0


Inside a circle with radius $6$ lie four smaller circles with centers $A$, $B$, $C$, and $D$. These circles touch each other as shown. The point where the circles with centers $A$ and $C$ touch each other is the center of the big circle. Calculate the area of quadrilateral $ABCD$.

Answer: 24


Let $D$, $E$, and $F$ respectively be the feet of the altitudes from $A$, $B$, and $C$ of acute triangle $\triangle ABC$ such that $AF = 28$, $FB = 35$, and $BD = 45$. Let $P$ be the point on segment $BE$ such that $AP = 42$. Find the length of $CP$.

Answer: 14


Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where the order of the summation is important. For example, $f(4) = 6$ because $4$ can be written as:
- $4$
- $2 + 2$
- $2 + 1 + 1$
- $1 + 2 + 1$
- $1 + 1 + 2$
- $1 + 1 + 1 + 1$

Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd.

Answer: 2047


Five airway companies operate in a country consisting of 36 cities. Between any pair of cities, exactly one company operates two-way flights. If some air company operates between cities $A, B$ and $B, C$, we say that the triple $A, B, C$ is properly-connected. Determine the largest possible value of $k$ such that no matter how these flights are arranged, there are at least $k$ properly-connected triples.

Answer: 3780


In each cell, a strip of length $100$ is worth a chip. You can change any $2$ neighboring chips and pay $1$ rouble, and you can also swap any $2$ chips for free, between which there are exactly $4$ chips. What is the smallest amount of rubles you can spend to rearrange the chips in reverse order?

Answer: 50


In the game of *Winners Make Zeros*, a pair of positive integers \((m,n)\) is written on a sheet of paper. Then the game begins, as the players make the following legal moves:

- If \(m \geq n\), the player chooses a positive integer \(c\) such that \(m-cn \geq 0\), and replaces \((m,n)\) with \((m-cn,n)\).
- If \(m < n\), the player chooses a positive integer \(c\) such that \(n-cm \geq 0\), and replaces \((m,n)\) with \((m,n-cm)\).

When \(m\) or \(n\) becomes \(0\), the game ends, and the last player to have moved is declared the winner. If \(m\) and \(n\) are originally \(2007777\) and \(2007\), find the largest choice the first player can make for \(c\) (on his first move) such that the first player has a winning strategy after that first move.

Answer: 999


One hundred and one of the squares of an $n \times n$ table are colored blue. It is known that there exists a unique way to cut the table into rectangles along the boundaries of its squares with the following property: every rectangle contains exactly one blue square. Find the smallest possible $n$.

Answer: 101


A convex 2011-gon is drawn on the board. Peter keeps drawing its diagonals in such a way that each newly drawn diagonal intersects no more than one of the already drawn diagonals. What is the greatest number of diagonals that Peter can draw?

Answer: 4016


Bob starts with an empty whiteboard. He then repeatedly chooses one of the digits $1, 2, \ldots, 9$ (uniformly at random) and appends it to the end of the currently written number. Bob stops when the number on the board is a multiple of $25$. Let $E$ be the expected number of digits that Bob writes. If $E = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $100m+n$.

Answer: 8102


Triangle $ABC$ with $\angle A = 90^\circ$ has incenter $I$. A circle passing through $A$ with center $I$ is drawn, intersecting $\overline{BC}$ at $E$ and $F$ such that $BE < BF$. If $\frac{BE}{EF} = \frac{2}{3}$, then $\frac{CF}{FE} = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 7


As shown in the figure, three circles intersect to create seven regions. Fill these regions with the integers from $1$ to $7$, ensuring that the sum of the four numbers in each circle is equal. What is the maximum possible sum for each circle?

Answer: 19


You are given an \( m \times n \) chocolate bar divided into \( 1 \times 1 \) squares. You can break a piece of chocolate by splitting it into two pieces along a straight line that does not cut through any of the \( 1 \times 1 \) squares. What is the minimum number of times you have to break the bar in order to separate all the \( 1 \times 1 \) squares?

Answer: m \times n - 1


It is known that \( f(x) \) is a continuous monotonically increasing function. It is also known that \( f(0) = 0 \) and \( f(1) = 1 \). Find the area of the figure bounded by the graphs of the functions \( f(x / 4) \), \( 4f(x) \), and the line \( x + y = 5 \).

Answer: 7.5


Let \( a \) and \( b \) be positive integers where \( a \) is less than \( b \). Calculate the sum of the irreducible fractions between \( a \) and \( b \) with denominator 7.

Answer: 3(b^2 - a^2)


Two squares, $ABCD$ and $BEFG$, with side lengths of $8 \text{ cm}$ and $6 \text{ cm}$ respectively, are placed next to each other as shown in the diagram. Line $DE$ intersects $BG$ at point $P$. What is the area of the shaded region $APEG$ in the diagram?

Answer: 18 \text{ cm}^2


In a math competition, the possible scores for each problem for each participating team are 0 points, 3 points, or 5 points. By the end of the competition, the sum of the total scores of three teams is 32 points. If the total score of any single team can reach 32 points, how many different possible combinations are there for the total scores of these three teams?

Answer: 255


Petya glued a paper cube and wrote numbers from 1 to 6 on its faces such that the sums of the numbers on any two opposite faces are equal. Vasya wants to cut this cube to obtain the net shown in the figure. He tries to make the sums of the numbers in the horizontal and vertical directions on this net as close as possible. What is the smallest positive difference he can get, regardless of how Petya arranged the numbers?

Answer: 1


From 24 identical wooden cubes, a "pipe" was glued - a cube $3 \times 3 \times 3$ with the "core" of three cubes removed. Can a diagonal be drawn in each square on the surface of the "pipe" to form a closed path that does not pass through any vertex more than once?

Answer: \text{No}


A point $M$ is taken on the side $AB$ of the trapezoid $ABCD$ such that $AM: BM = 2: 3$. A point $N$ is taken on the opposite side $CD$ such that the segment $MN$ divides the trapezoid into parts, one of which has an area three times greater than the other. Find the ratio $CN: DN$ if $BC: AD = 1: 2$.

Answer: 3:29


In triangle $ABC$, $AB = c$, $AC = b > c$, and $AD$ is the angle bisector. A line through point $D$ perpendicular to $AD$ intersects $AC$ at point $E$. Find $AE$.

Answer: \frac{2bc}{b+c}


The histogram shows the distribution of passenger cars by the number of passengers (in $4\%$ of cars there are from 10 to 19 passengers, in $6\%$ of cars - from 20 to 29, etc.). If a car has 60 or more passengers, we will call such a car overcrowded.

a) Find the proportion of overcrowded cars.

b) Find the minimum possible proportion of passengers traveling in overcrowded cars.

c) Can the proportion of passengers traveling in overcrowded cars be less than the proportion of overcrowded cars?

Answer: \text{No}


As shown in the figure, \(ABCD\) is a trapezoid with its diagonals intersecting at point \(O\). Extend \(AC\) to point \(E\) such that \(CE = AO\). Extend \(DB\) to point \(F\) such that \(BF = DO\). If the area of \(\triangle BFG\) is 2015 square centimeters, find the area of \(\triangle CGE\).

Answer: 2015


The tower clock strikes three times in 12 seconds. For how long will it strike six times?

Answer: 30\,


A bag contains eight yellow marbles, seven red marbles, and five black marbles. Without looking in the bag, Igor removes $N$ marbles all at once. If he is to be sure that, no matter which choice of $N$ marbles he removes, there are at least four marbles of one color and at least three marbles of another color left in the bag, what is the maximum possible value of $N$ ?

(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

Answer: 7


For \( n \geq 2 \), an equilateral triangle is divided into \( n^2 \) congruent smaller equilateral triangles. Determine all ways in which real numbers can be assigned to the \(\frac{(n+1)(n+2)}{2}\) vertices so that three such numbers sum to zero whenever the three vertices form a triangle with edges parallel to the sides of the big triangle.

Answer: 0


The side of the rhombus $ABCD$ is 6. The distance between the centers of the circumcircles of triangles $ABC$ and $BCD$ is 8. Find the radii of these circumcircles.

Answer: 3\sqrt{10}


Estimate the number of 2016-digit squares. There are at least

$$
10^{2016 / 2} - 10^{2015 / 2} - 1 > 10^{1000}.
$$

The number of different sets of 2016 digits is not more than the sets where each digit appears no more than 2016 times, i.e., $2017^{10}$. Therefore, there exists such a set of 2016 digits, permutations of which can produce at least $10^{1000} / 2017^{10} > 10^{1000} / 10^{100} = 10^{900}$ squares, and certainly 2016.

Answer: 2016


9. Let \( n \) be an integer greater than 2, and \( a_{n} \) be the largest \( n \)-digit number that is neither the sum nor the difference of two perfect squares.
(1) Find \( a_{n} \) as a function of \( n \).
(2) Find the smallest \( n \) such that the sum of the squares of its digits is a perfect square.

Answer: 66


A triangular piece of paper with an area of 1 is folded along a line parallel to one of its sides and then pressed flat. What is the minimum possible area of the resulting figure?

Answer: \frac{2}{3}