$7 \cdot 89$ On 400 cards, the natural numbers $1,2,3, \cdots, 400$ are written. A and B play the following game: In the first step, A arbitrarily takes 200 cards for himself. B then takes 100 cards each from the remaining 200 cards and the 200 cards in A's hand for himself, leaving the remaining 200 cards for A. In the next step, A takes 100 cards each from the cards in both of their hands for himself, leaving the remaining 200 cards for B. This continues until B completes the 200th step, after which both calculate the sum of the numbers on the cards in their hands, $C_{A}, C_{B}$, and then A pays B the difference $C_{B}-C_{A}$. What is the maximum difference B can obtain if both play with the correct strategy?

Answer: 20000


G8.1 If the area of the rectangle $A B C D$ is $a+4 \sqrt{3}$, find the value of $a$.
The lines joining the centres form an equilateral triangle, side $=2$.

Answer: 8


A boarding school installed a shower room for 100 students. How many shower heads are economical, if the boiler preheating takes 3 minutes for each head; in addition, it must also run during the shower. We allocate 12 minutes for each group.

Answer: 20


\left.\begin{array}{l}{[\text { Theorem of the sum of squares of diagonals }]} \\ {[\text { Pythagorean Theorem (direct and inverse) }}\end{array}\right]

In parallelogram $A B C D$, the height drawn from vertex $B$ of the obtuse angle to side $D A$ divides it in the ratio 5:3, counting from vertex $D$. Find the ratio $A C: B D$, if $A D: A B=2$.

Answer: 2:1


6. To guard the object, the client agreed with the guards on the following: all of them will indicate the time intervals of their proposed shifts with the only condition that their union should form a predetermined time interval set by the client, and he will choose any set of these intervals that also satisfies the same condition, and pay for the work at a rate of 300 rubles per hour for each guard. What is the longest time interval the client can set to ensure that he definitely stays within 90000 rubles?

Answer: 150


【Question 4】
Three candles can burn for $30$, $40$, and $50$ minutes respectively (but they are not lit at the same time). It is known that the three candles are burning simultaneously for 10 minutes, and only one candle is burning for 20 minutes. Then, the time when exactly two candles are burning simultaneously is $\qquad$ minutes.

Answer: 35


# Problem 6. (3 points)

Three runners are moving along a circular track at constant equal speeds. When two runners meet, they instantly turn around and start running in opposite directions.

At some point, the first runner meets the second. After 15 minutes, the second runner meets the third for the first time. Another 25 minutes later, the third runner meets the first for the first time.

How many minutes does it take for one runner to complete the entire track?

Answer: 80


2. From the numbers 1, 2, 3, ..., 2016, $k$ numbers are chosen. What is the smallest $k$ such that among the chosen numbers, there will definitely be two numbers whose difference is greater than 672 and less than 1344?

Answer: 674


Last time, I was in a truck going from town $A$ to town $C$ via $B$. After five minutes of travel, I asked the driver how far we were from $A$. "Half as far as from B," was the answer. I complained about the slow speed of the truck, to which the driver noted that the truck was not capable of greater speed, but it would maintain its current speed for the rest of the journey.

$13 \mathrm{~km}$ from $B$, I asked how far we were from $C$. I got the same answer, letter for letter, as to my first question. A quarter of an hour later, we arrived at our destination. How many kilometers is the route from $A$ to $C$?

Answer: 26


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Bob`, `Arnold`
 - People have unique favorite music genres: `rock`, `classical`, `hip hop`, `pop`, `jazz`
 - Each person has a unique hobby: `photography`, `knitting`, `cooking`, `painting`, `gardening`
 - Each person prefers a unique type of vacation: `camping`, `city`, `beach`, `mountain`, `cruise`
 - Each mother is accompanied by their child: `Timothy`, `Samantha`, `Fred`, `Bella`, `Meredith`
 - They all have a unique favorite flower: `lilies`, `roses`, `tulips`, `carnations`, `daffodils`

## Clues:
1. The person who is the mother of Timothy is somewhere to the right of the person who likes going on cruises.
2. The photography enthusiast is somewhere to the right of the person who loves hip-hop music.
3. The person who paints as a hobby is the person who enjoys camping trips.
4. The person who loves the boquet of lilies is somewhere to the right of the person who prefers city breaks.
5. The person who loves a carnations arrangement is the person who loves cooking.
6. The person who loves a bouquet of daffodils is the person who loves classical music.
7. The person's child is named Samantha is somewhere to the right of Arnold.
8. The person who loves the rose bouquet is the person who likes going on cruises.
9. The person who loves hip-hop music is the person who loves the vase of tulips.
10. The person who is the mother of Timothy is somewhere to the left of the person who prefers city breaks.
11. The person who loves jazz music is the person's child is named Bella.
12. Eric is the person who likes going on cruises.
13. The person who loves a carnations arrangement is directly left of the person who loves jazz music.
14. The person who enjoys mountain retreats is the person who is the mother of Timothy.
15. The person who loves the boquet of lilies is Bob.
16. The person who enjoys gardening is directly left of the person who is the mother of Timothy.
17. The person's child is named Fred is the person who loves the boquet of lilies.
18. Alice is the person who loves rock music.
 What is the value of attribute Hobby for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: gardening


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`, `Bob`
 - Each person has a unique type of pet: `cat`, `dog`, `bird`, `fish`, `hamster`
 - Everyone has a unique favorite cigar: `pall mall`, `blends`, `prince`, `dunhill`, `blue master`
 - Everyone has a favorite smoothie: `watermelon`, `desert`, `dragonfruit`, `cherry`, `lime`
 - People use unique phone models: `google pixel 6`, `oneplus 9`, `samsung galaxy s21`, `iphone 13`, `huawei p50`
 - People have unique favorite sports: `baseball`, `soccer`, `basketball`, `swimming`, `tennis`

## Clues:
1. The person who loves swimming is directly left of the Prince smoker.
2. The person who smokes Blue Master is somewhere to the right of Eric.
3. The person who uses a OnePlus 9 is Arnold.
4. Arnold is somewhere to the right of the person with a pet hamster.
5. The person who owns a dog is Peter.
6. Eric is the Desert smoothie lover.
7. The person who loves basketball is not in the fifth house.
8. The person who has a cat is the person who uses a Samsung Galaxy S21.
9. The person who loves baseball is directly left of Peter.
10. There is one house between the Desert smoothie lover and the Dunhill smoker.
11. The person who loves baseball and the person who uses a Google Pixel 6 are next to each other.
12. The person who keeps a pet bird is in the second house.
13. The person who uses a Huawei P50 is in the fourth house.
14. The person who drinks Lime smoothies is in the fourth house.
15. The person partial to Pall Mall is in the fifth house.
16. The Dragonfruit smoothie lover is in the second house.
17. The person who loves tennis is the person with a pet hamster.
18. Bob is somewhere to the right of the person who owns a dog.
19. The Watermelon smoothie lover is Arnold.
20. The Prince smoker is somewhere to the left of the Dunhill smoker.
 What is the value of attribute PhoneModel for the person whose attribute FavoriteSport is tennis? Please reason step by step, and put your final answer within \boxed{}

Answer: huawei p50


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`
 - People use unique phone models: `huawei p50`, `iphone 13`, `samsung galaxy s21`, `google pixel 6`, `oneplus 9`
 - People have unique hair colors: `red`, `brown`, `gray`, `blonde`, `black`
 - Everyone has a favorite smoothie: `dragonfruit`, `watermelon`, `cherry`, `desert`, `lime`
 - Each person has a unique hobby: `knitting`, `photography`, `gardening`, `painting`, `cooking`
 - They all have a unique favorite flower: `daffodils`, `tulips`, `roses`, `carnations`, `lilies`

## Clues:
1. Bob is the person who loves the vase of tulips.
2. The person who has red hair is Bob.
3. The person who loves a bouquet of daffodils is the person who uses a Google Pixel 6.
4. The person who likes Cherry smoothies is the person who enjoys gardening.
5. The person who uses a Google Pixel 6 is somewhere to the right of the person who loves a carnations arrangement.
6. The person who has brown hair and the person who loves a carnations arrangement are next to each other.
7. The person who paints as a hobby is somewhere to the left of the person who loves cooking.
8. The person who uses an iPhone 13 is somewhere to the left of the person who loves a carnations arrangement.
9. The person who has brown hair is somewhere to the right of the person who uses a OnePlus 9.
10. The person who paints as a hobby is Peter.
11. Eric is directly left of the person who uses an iPhone 13.
12. The person who uses a OnePlus 9 is directly left of the person who uses a Samsung Galaxy S21.
13. Arnold is the person who drinks Lime smoothies.
14. The person who has red hair is the person who enjoys gardening.
15. The Dragonfruit smoothie lover and the Desert smoothie lover are next to each other.
16. The person who has brown hair is directly left of the Desert smoothie lover.
17. The person who loves the boquet of lilies is directly left of the person who loves a carnations arrangement.
18. The person who drinks Lime smoothies is the person who has black hair.
19. The person who uses a Huawei P50 is the person who has gray hair.
20. The person who drinks Lime smoothies is the photography enthusiast.
 What is the value of attribute Flower for the person whose attribute PhoneModel is iphone 13? Please reason step by step, and put your final answer within \boxed{}

Answer: lilies


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Alice`, `Peter`, `Arnold`, `Carol`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `roses`, `tulips`, `iris`, `lilies`
 - People have unique heights: `very tall`, `super tall`, `average`, `short`, `very short`, `tall`
 - Each person has a favorite color: `blue`, `yellow`, `purple`, `red`, `green`, `white`
 - Everyone has a unique favorite cigar: `yellow monster`, `pall mall`, `prince`, `blue master`, `blends`, `dunhill`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `xiaomi mi 11`, `samsung galaxy s21`, `huawei p50`, `oneplus 9`

## Clues:
1. The person who uses an iPhone 13 is the person who loves white.
2. Peter is the person whose favorite color is red.
3. The Prince smoker is directly left of the person partial to Pall Mall.
4. The person who smokes Blue Master and the Dunhill smoker are next to each other.
5. The person who loves blue is the person who uses a Xiaomi Mi 11.
6. The person who loves white is Alice.
7. The person who uses a OnePlus 9 is somewhere to the left of Arnold.
8. The person who is tall is the person who smokes Blue Master.
9. The person who uses a Samsung Galaxy S21 is the Dunhill smoker.
10. Bob is directly left of the person who is very short.
11. The person who loves a bouquet of daffodils is Eric.
12. The person who loves the boquet of iris is the person who uses a OnePlus 9.
13. The person who uses a Huawei P50 is the person who is very short.
14. The person who smokes many unique blends is directly left of Peter.
15. The Prince smoker is not in the first house.
16. The person who is very tall is the person who loves a bouquet of daffodils.
17. The person who uses a Google Pixel 6 is not in the fifth house.
18. The person who is tall is not in the third house.
19. The person who uses a Huawei P50 is somewhere to the left of the person who uses a Samsung Galaxy S21.
20. Arnold is somewhere to the left of the person who is short.
21. The person who loves the vase of tulips is not in the first house.
22. The person who loves the boquet of lilies is directly left of the person who loves yellow.
23. Arnold is the person who loves purple.
24. The person who uses a OnePlus 9 and the person who uses a Samsung Galaxy S21 are next to each other.
25. The person who has an average height is not in the first house.
26. The person who loves the rose bouquet is Alice.
 What is the value of attribute House for the person whose attribute Flower is iris? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`
 - Each mother is accompanied by their child: `Meredith`, `Fred`, `Bella`
 - Each person has a unique birthday month: `jan`, `april`, `sept`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`
 - Everyone has something unique for lunch: `spaghetti`, `grilled cheese`, `pizza`
 - The mothers' names in different houses are unique: `Janelle`, `Aniya`, `Holly`

## Clues:
1. Arnold is the person who loves eating grilled cheese.
2. The person who loves the spaghetti eater and The person whose mother's name is Holly are next to each other.
3. Peter is directly left of the person whose birthday is in September.
4. The person who loves the spaghetti eater is the person whose birthday is in January.
5. The person who likes Cherry smoothies is not in the first house.
6. The person's child is named Bella is the Desert smoothie lover.
7. The person whose birthday is in September is Eric.
8. The person whose mother's name is Aniya is in the first house.
9. The person's child is named Fred is somewhere to the right of the person who likes Cherry smoothies.
10. The person who likes Cherry smoothies is the person whose birthday is in January.
 What is the value of attribute Food for the person whose attribute Smoothie is desert? Please reason step by step, and put your final answer within \boxed{}

Answer: grilled cheese


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Peter`, `Alice`, `Eric`, `Arnold`, `Bob`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `victorian`, `mediterranean`, `craftsman`, `modern`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`, `chevrolet silverado`, `bmw 3 series`, `honda civic`
 - The mothers' names in different houses are unique: `Janelle`, `Aniya`, `Kailyn`, `Penny`, `Sarah`, `Holly`

## Clues:
1. The person whose mother's name is Penny is directly left of The person whose mother's name is Aniya.
2. Peter is in the sixth house.
3. The person who owns a Chevrolet Silverado is The person whose mother's name is Penny.
4. The person whose mother's name is Sarah is not in the fifth house.
5. There is one house between the person who owns a Honda Civic and the person who owns a Toyota Camry.
6. The person in a Craftsman-style house is somewhere to the right of the person in a ranch-style home.
7. Eric is somewhere to the left of The person whose mother's name is Sarah.
8. The person residing in a Victorian house is Carol.
9. The person who owns a Ford F-150 is the person in a Mediterranean-style villa.
10. The person whose mother's name is Kailyn is somewhere to the left of the person who owns a Ford F-150.
11. Bob is The person whose mother's name is Janelle.
12. The person who owns a BMW 3 Series is somewhere to the left of Eric.
13. The person whose mother's name is Holly is the person in a modern-style house.
14. The person who owns a Tesla Model 3 is not in the second house.
15. The person who owns a Toyota Camry is somewhere to the right of Eric.
16. The person who owns a Honda Civic is in the fourth house.
17. The person in a Craftsman-style house is the person who owns a Chevrolet Silverado.
18. Arnold is the person who owns a Ford F-150.
 What is the value of attribute House for the person whose attribute Mother is Penny? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Peter`, `Eric`, `Alice`
 - They all have a unique favorite flower: `lilies`, `daffodils`, `carnations`, `roses`, `tulips`
 - Each person has an occupation: `artist`, `doctor`, `lawyer`, `teacher`, `engineer`
 - Everyone has a favorite smoothie: `lime`, `watermelon`, `cherry`, `dragonfruit`, `desert`
 - People own unique car models: `honda civic`, `ford f150`, `bmw 3 series`, `toyota camry`, `tesla model 3`

## Clues:
1. Arnold is the person who is an artist.
2. The person who is a doctor is the person who loves the rose bouquet.
3. Bob is the person who loves a carnations arrangement.
4. The person who owns a BMW 3 Series is Arnold.
5. The person who loves the boquet of lilies is somewhere to the right of Alice.
6. The Watermelon smoothie lover is somewhere to the left of the person who drinks Lime smoothies.
7. The person who is a lawyer is the person who owns a Toyota Camry.
8. The person who loves the boquet of lilies is Arnold.
9. There are two houses between Bob and the person who owns a Honda Civic.
10. The person who owns a Tesla Model 3 is in the third house.
11. Peter is directly left of the person who loves the boquet of lilies.
12. The Desert smoothie lover is the person who is a teacher.
13. The person who is an artist and the Dragonfruit smoothie lover are next to each other.
14. The person who is a doctor is the Watermelon smoothie lover.
15. The person who loves the vase of tulips is directly left of the person who owns a Toyota Camry.
 What is the value of attribute CarModel for the person whose attribute Occupation is lawyer? Please reason step by step, and put your final answer within \boxed{}

Answer: toyota camry


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

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

Answer: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Arnold`, `Eric`, `Carol`, `Bob`
 - People own unique car models: `toyota camry`, `ford f150`, `chevrolet silverado`, `tesla model 3`, `bmw 3 series`, `honda civic`
 - The people keep unique animals: `horse`, `fish`, `cat`, `rabbit`, `bird`, `dog`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `desert`, `dragonfruit`, `lime`, `blueberry`

## Clues:
1. The fish enthusiast is in the first house.
2. Peter is in the second house.
3. The person who owns a Chevrolet Silverado is Eric.
4. Alice is directly left of the person who owns a Toyota Camry.
5. The Watermelon smoothie lover is directly left of Bob.
6. The cat lover is Peter.
7. There are two houses between the dog owner and the person who drinks Blueberry smoothies.
8. The Watermelon smoothie lover is directly left of the person who keeps horses.
9. The person who owns a Chevrolet Silverado and the person who owns a Ford F-150 are next to each other.
10. The person who drinks Lime smoothies is not in the second house.
11. The cat lover is the person who owns a Tesla Model 3.
12. The person who likes Cherry smoothies is the rabbit owner.
13. The person who drinks Blueberry smoothies is somewhere to the right of Alice.
14. The person who owns a Chevrolet Silverado is not in the fifth house.
15. The person who keeps horses is the Desert smoothie lover.
16. Carol is the bird keeper.
17. The person who owns a Honda Civic is Arnold.
 What is the value of attribute House for the person whose attribute CarModel is chevrolet silverado? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


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

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

Answer: stew


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`, `Alice`, `Bob`
 - Each person has a unique level of education: `associate`, `doctorate`, `high school`, `bachelor`, `master`
 - Each mother is accompanied by their child: `Fred`, `Meredith`, `Bella`, `Timothy`, `Samantha`
 - Everyone has a unique favorite cigar: `prince`, `pall mall`, `blue master`, `dunhill`, `blends`
 - The people are of nationalities: `swede`, `brit`, `dane`, `german`, `norwegian`
 - People have unique favorite book genres: `romance`, `mystery`, `biography`, `science fiction`, `fantasy`

## Clues:
1. The person with a master's degree is the person partial to Pall Mall.
2. The person with a high school diploma is in the fourth house.
3. The person's child is named Samantha is in the first house.
4. The Norwegian and the person with a master's degree are next to each other.
5. The German is the Dunhill smoker.
6. The person who is the mother of Timothy is directly left of the person with a master's degree.
7. Arnold and the Dane are next to each other.
8. The Swedish person is the person who loves mystery books.
9. The Swedish person is Alice.
10. Arnold is the person with an associate's degree.
11. The person with a doctorate is in the second house.
12. Peter is somewhere to the right of the person who loves biography books.
13. The person who smokes many unique blends is Alice.
14. The person who loves fantasy books is in the first house.
15. Peter is the Norwegian.
16. The person with a doctorate is the person's child is named Meredith.
17. The Prince smoker is somewhere to the left of the person who smokes many unique blends.
18. The person partial to Pall Mall is the person's child is named Bella.
19. The person who loves romance books is Bob.
 What is the value of attribute BookGenre for the person whose attribute Cigar is blends? Please reason step by step, and put your final answer within \boxed{}

Answer: mystery


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Alice`, `Arnold`
 - Each person has a unique hobby: `photography`, `cooking`, `painting`, `gardening`
 - Everyone has a unique favorite cigar: `dunhill`, `blue master`, `prince`, `pall mall`

## Clues:
1. The person who loves cooking is not in the fourth house.
2. The person who paints as a hobby is somewhere to the left of the photography enthusiast.
3. The person partial to Pall Mall is Arnold.
4. There is one house between Peter and the photography enthusiast.
5. The Dunhill smoker is directly left of the person who loves cooking.
6. The person partial to Pall Mall is directly left of Alice.
7. The person partial to Pall Mall is somewhere to the left of the person who smokes Blue Master.
8. The Prince smoker is the photography enthusiast.
 What is the value of attribute House for the person whose attribute Cigar is pall mall? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`
 - Each person has a favorite color: `white`, `yellow`, `red`
 - Each person prefers a unique type of vacation: `city`, `mountain`, `beach`
 - Each person has a unique birthday month: `sept`, `april`, `jan`
 - Each mother is accompanied by their child: `Meredith`, `Fred`, `Bella`

## Clues:
1. The person whose birthday is in January is the person's child is named Bella.
2. The person who loves white is Arnold.
3. The person who loves beach vacations is Arnold.
4. The person whose birthday is in September is somewhere to the left of Arnold.
5. The person whose favorite color is red is directly left of the person who enjoys mountain retreats.
6. Peter is somewhere to the right of the person's child is named Fred.
7. The person who loves yellow is the person's child is named Bella.
 What is the value of attribute House for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


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

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

Answer: water


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`
 - Each person has a unique birthday month: `jan`, `sept`, `april`
 - The people are of nationalities: `brit`, `swede`, `dane`
 - Each person has a favorite color: `yellow`, `white`, `red`

## Clues:
1. The Dane is the person whose favorite color is red.
2. The person who loves yellow is the person whose birthday is in September.
3. The person whose birthday is in September is directly left of the person whose birthday is in April.
4. The person whose birthday is in September is somewhere to the left of Arnold.
5. The Dane is in the second house.
6. The person whose birthday is in January is Peter.
7. The British person is the person who loves white.
 What is the value of attribute Birthday for the person whose attribute Nationality is brit? Please reason step by step, and put your final answer within \boxed{}

Answer: jan


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

## Clues:
1. The Chinese is the person who loves pop music.
2. The person with a bachelor's degree is directly left of Arnold.
3. The person with a master's degree is in the first house.
4. Carol is somewhere to the left of Peter.
5. The Dane is the person with a doctorate.
6. The person who loves classical music is the person with a master's degree.
7. The person with a bachelor's degree is in the fifth house.
8. Eric is somewhere to the right of Bob.
9. Carol is the British person.
10. The person with a doctorate is somewhere to the right of the Norwegian.
11. The person with a doctorate is the person who loves rock music.
12. The Norwegian is somewhere to the right of the German.
13. The person who attended trade school is somewhere to the left of the person with a high school diploma.
14. The person who attended trade school and the Chinese are next to each other.
15. The person who loves jazz music is Arnold.
16. The person who loves hip-hop music is directly left of Alice.
 What is the value of attribute House for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Peter`
 - The mothers' names in different houses are unique: `Holly`, `Kailyn`, `Aniya`, `Janelle`
 - Each person has a unique birthday month: `april`, `feb`, `sept`, `jan`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `victorian`, `craftsman`

## Clues:
1. Peter is the person whose birthday is in February.
2. The person whose birthday is in February is not in the fourth house.
3. The person whose birthday is in January is somewhere to the right of the person in a ranch-style home.
4. The person whose birthday is in January is directly left of Eric.
5. The person in a ranch-style home is the person whose birthday is in April.
6. The person whose mother's name is Janelle is the person whose birthday is in September.
7. The person in a ranch-style home and The person whose mother's name is Kailyn are next to each other.
8. Arnold is The person whose mother's name is Holly.
9. The person whose mother's name is Kailyn is directly left of the person residing in a Victorian house.
10. The person whose birthday is in January is the person in a Craftsman-style house.
 What is the value of attribute Name for the person whose attribute Birthday is feb? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Bob`, `Peter`, `Alice`
 - Each person has a unique birthday month: `feb`, `sept`, `april`, `mar`, `jan`

## Clues:
1. The person whose birthday is in March is somewhere to the left of the person whose birthday is in January.
2. Arnold is the person whose birthday is in January.
3. The person whose birthday is in March is directly left of Alice.
4. The person whose birthday is in September and the person whose birthday is in January are next to each other.
5. Eric is not in the fourth house.
6. The person whose birthday is in January is not in the fourth house.
7. Peter is in the first house.
8. Alice is directly left of the person whose birthday is in April.
 What is the value of attribute Birthday for the person whose attribute House is 5? Please reason step by step, and put your final answer within \boxed{}

Answer: jan


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`
 - Everyone has a unique favorite cigar: `blue master`, `prince`, `pall mall`
 - Each person has a unique hobby: `photography`, `gardening`, `cooking`
 - Each person has a unique level of education: `high school`, `associate`, `bachelor`
 - Each person has a unique favorite drink: `tea`, `milk`, `water`

## Clues:
1. The person partial to Pall Mall is Peter.
2. The person who likes milk is directly left of the person with a high school diploma.
3. Eric is the tea drinker.
4. Arnold and the Prince smoker are next to each other.
5. The person who enjoys gardening is somewhere to the left of the Prince smoker.
6. The person who likes milk is the person with an associate's degree.
7. The person with a bachelor's degree is directly left of the photography enthusiast.
 What is the value of attribute House for the person whose attribute Education is high school? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


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

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

Answer: 5


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`
 - Each mother is accompanied by their child: `Fred`, `Meredith`, `Bella`
 - The people keep unique animals: `cat`, `bird`, `horse`
 - Each person has a unique favorite drink: `tea`, `water`, `milk`

## Clues:
1. Peter is somewhere to the left of the person's child is named Meredith.
2. The person's child is named Fred is directly left of the person who keeps horses.
3. The person's child is named Fred is in the second house.
4. The person who likes milk is the person who keeps horses.
5. The person whose mother's name is Holly is Eric.
6. The tea drinker is the cat lover.
7. The person's child is named Bella is Eric.
8. The tea drinker and Arnold are next to each other.
9. The person whose mother's name is Janelle is in the second house.
 What is the value of attribute Name for the person whose attribute Mother is Janelle? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Peter`, `Alice`, `Arnold`, `Carol`
 - People own unique car models: `ford f150`, `honda civic`, `toyota camry`, `tesla model 3`, `chevrolet silverado`, `bmw 3 series`
 - The mothers' names in different houses are unique: `Sarah`, `Penny`, `Holly`, `Aniya`, `Kailyn`, `Janelle`
 - Each person has a unique hobby: `photography`, `cooking`, `knitting`, `gardening`, `woodworking`, `painting`

## Clues:
1. The person who owns a Toyota Camry is in the sixth house.
2. Carol is the photography enthusiast.
3. The person who owns a Chevrolet Silverado is The person whose mother's name is Aniya.
4. The person who owns a Chevrolet Silverado is not in the second house.
5. The person who owns a Ford F-150 is The person whose mother's name is Sarah.
6. The person who owns a BMW 3 Series is Bob.
7. The person whose mother's name is Kailyn is in the sixth house.
8. Eric is directly left of the person who enjoys knitting.
9. There is one house between The person whose mother's name is Sarah and the person who owns a Toyota Camry.
10. The person whose mother's name is Penny is somewhere to the right of the person who enjoys knitting.
11. The person whose mother's name is Aniya is somewhere to the right of the person who owns a Honda Civic.
12. Alice is somewhere to the right of the person who owns a Ford F-150.
13. Eric is the person who enjoys gardening.
14. The woodworking hobbyist is somewhere to the left of the person who enjoys knitting.
15. There is one house between The person whose mother's name is Sarah and the person who loves cooking.
16. The person who owns a Honda Civic is Arnold.
17. The person whose mother's name is Holly is directly left of the person who enjoys knitting.
 What is the value of attribute CarModel for the person whose attribute Mother is Janelle? Please reason step by step, and put your final answer within \boxed{}

Answer: honda civic


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Peter`, `Bob`, `Eric`, `Carol`
 - Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `watermelon`, `lime`, `desert`, `blueberry`
 - Each person has a unique birthday month: `mar`, `april`, `jan`, `may`, `sept`, `feb`
 - The people keep unique animals: `cat`, `rabbit`, `horse`, `bird`, `fish`, `dog`

## Clues:
1. The bird keeper is in the third house.
2. The person whose birthday is in May is somewhere to the left of Eric.
3. Eric is the Desert smoothie lover.
4. The rabbit owner is the person whose birthday is in January.
5. The fish enthusiast is directly left of the person who likes Cherry smoothies.
6. The person whose birthday is in September is Peter.
7. There is one house between Peter and the rabbit owner.
8. The person who drinks Lime smoothies is not in the sixth house.
9. The Watermelon smoothie lover is the dog owner.
10. The person whose birthday is in April is somewhere to the right of Carol.
11. The bird keeper is somewhere to the left of the person who keeps horses.
12. The person who drinks Lime smoothies is Alice.
13. The Dragonfruit smoothie lover is directly left of Carol.
14. The person who drinks Blueberry smoothies is the person whose birthday is in February.
15. The person whose birthday is in September is in the third house.
16. Bob is the Dragonfruit smoothie lover.
17. The person who likes Cherry smoothies is somewhere to the left of the person whose birthday is in May.
18. The rabbit owner is the Desert smoothie lover.
 What is the value of attribute Smoothie for the person whose attribute Birthday is april? Please reason step by step, and put your final answer within \boxed{}

Answer: watermelon


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Alice`, `Arnold`, `Bob`, `Carol`
 - Each person has a unique hobby: `cooking`, `knitting`, `painting`, `gardening`, `woodworking`, `photography`
 - Everyone has a favorite smoothie: `lime`, `cherry`, `desert`, `blueberry`, `watermelon`, `dragonfruit`
 - The people are of nationalities: `chinese`, `dane`, `german`, `norwegian`, `swede`, `brit`
 - People have unique favorite music genres: `hip hop`, `country`, `rock`, `jazz`, `classical`, `pop`

## Clues:
1. The person who loves rock music is Bob.
2. The person who enjoys knitting is the German.
3. The woodworking hobbyist is in the sixth house.
4. The person who loves rock music is directly left of the woodworking hobbyist.
5. There is one house between the Watermelon smoothie lover and the person who loves hip-hop music.
6. The Chinese is the Watermelon smoothie lover.
7. The Desert smoothie lover is the Norwegian.
8. The person who drinks Blueberry smoothies is not in the sixth house.
9. The person who loves pop music is somewhere to the left of the person who enjoys gardening.
10. Carol is somewhere to the left of the Swedish person.
11. Peter is somewhere to the right of the person who loves pop music.
12. The Norwegian is somewhere to the right of the Dragonfruit smoothie lover.
13. The German is the person who loves country music.
14. The person who paints as a hobby is directly left of the person who loves hip-hop music.
15. The Dane is Arnold.
16. The person who loves jazz music is the Chinese.
17. The British person is not in the third house.
18. The Swedish person is the person who loves pop music.
19. The person who drinks Lime smoothies is the photography enthusiast.
20. The person who loves hip-hop music is directly left of Alice.
21. The Dane is the woodworking hobbyist.
 What is the value of attribute Hobby for the person whose attribute Smoothie is lime? Please reason step by step, and put your final answer within \boxed{}

Answer: photography


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Alice`
 - People have unique favorite music genres: `pop`, `classical`, `jazz`, `rock`
 - Each person has a unique hobby: `cooking`, `gardening`, `painting`, `photography`
 - Each person has a unique level of education: `master`, `associate`, `high school`, `bachelor`
 - Each person has a unique type of pet: `fish`, `bird`, `dog`, `cat`

## Clues:
1. The person who loves classical music is not in the fourth house.
2. Arnold is the person with a master's degree.
3. Alice is the person who owns a dog.
4. The person who enjoys gardening is not in the third house.
5. The person with a bachelor's degree is Alice.
6. Alice is the photography enthusiast.
7. Arnold is directly left of the person who loves pop music.
8. The person with an associate's degree is directly left of the photography enthusiast.
9. The person who loves rock music is the person who loves cooking.
10. The person with an associate's degree is Eric.
11. The person who has a cat is the person who loves jazz music.
12. Arnold is somewhere to the right of the person with an aquarium of fish.
 What is the value of attribute House for the person whose attribute MusicGenre is jazz? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`
 - People use unique phone models: `huawei p50`, `iphone 13`, `samsung galaxy s21`, `google pixel 6`, `oneplus 9`
 - People have unique hair colors: `red`, `brown`, `gray`, `blonde`, `black`
 - Everyone has a favorite smoothie: `dragonfruit`, `watermelon`, `cherry`, `desert`, `lime`
 - Each person has a unique hobby: `knitting`, `photography`, `gardening`, `painting`, `cooking`
 - They all have a unique favorite flower: `daffodils`, `tulips`, `roses`, `carnations`, `lilies`

## Clues:
1. Bob is the person who loves the vase of tulips.
2. The person who has red hair is Bob.
3. The person who loves a bouquet of daffodils is the person who uses a Google Pixel 6.
4. The person who likes Cherry smoothies is the person who enjoys gardening.
5. The person who uses a Google Pixel 6 is somewhere to the right of the person who loves a carnations arrangement.
6. The person who has brown hair and the person who loves a carnations arrangement are next to each other.
7. The person who paints as a hobby is somewhere to the left of the person who loves cooking.
8. The person who uses an iPhone 13 is somewhere to the left of the person who loves a carnations arrangement.
9. The person who has brown hair is somewhere to the right of the person who uses a OnePlus 9.
10. The person who paints as a hobby is Peter.
11. Eric is directly left of the person who uses an iPhone 13.
12. The person who uses a OnePlus 9 is directly left of the person who uses a Samsung Galaxy S21.
13. Arnold is the person who drinks Lime smoothies.
14. The person who has red hair is the person who enjoys gardening.
15. The Dragonfruit smoothie lover and the Desert smoothie lover are next to each other.
16. The person who has brown hair is directly left of the Desert smoothie lover.
17. The person who loves the boquet of lilies is directly left of the person who loves a carnations arrangement.
18. The person who drinks Lime smoothies is the person who has black hair.
19. The person who uses a Huawei P50 is the person who has gray hair.
20. The person who drinks Lime smoothies is the photography enthusiast.
 What is the value of attribute House for the person whose attribute Smoothie is lime? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`
 - Each mother is accompanied by their child: `Meredith`, `Fred`, `Bella`
 - Each person has a unique birthday month: `jan`, `april`, `sept`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`
 - Everyone has something unique for lunch: `spaghetti`, `grilled cheese`, `pizza`
 - The mothers' names in different houses are unique: `Janelle`, `Aniya`, `Holly`

## Clues:
1. Arnold is the person who loves eating grilled cheese.
2. The person who loves the spaghetti eater and The person whose mother's name is Holly are next to each other.
3. Peter is directly left of the person whose birthday is in September.
4. The person who loves the spaghetti eater is the person whose birthday is in January.
5. The person who likes Cherry smoothies is not in the first house.
6. The person's child is named Bella is the Desert smoothie lover.
7. The person whose birthday is in September is Eric.
8. The person whose mother's name is Aniya is in the first house.
9. The person's child is named Fred is somewhere to the right of the person who likes Cherry smoothies.
10. The person who likes Cherry smoothies is the person whose birthday is in January.
 What is the value of attribute Mother for the person whose attribute Food is spaghetti? Please reason step by step, and put your final answer within \boxed{}

Answer: Janelle


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - Each person has a unique birthday month: `sept`, `april`, `jan`

## Clues:
1. Arnold is in the third house.
2. The person whose birthday is in January is directly left of Arnold.
3. Peter is the person whose birthday is in September.
 What is the value of attribute Birthday for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: jan


4. Vasya, a bibliophile, has 1300 books arranged on $k$ bookshelves in his cabinets. One day, Vasya moved the cabinets to a new place, taking all the books out beforehand. In the process, he completely forgot how the books were arranged before and placed them in the cabinets in a different order. For what maximum $k$ will there definitely be 5 books that were on the same shelf both before and after the rearrangement?

Answer: 18


Let the decimal representation of the number \( N \) be \( \overline{a_{n} a_{n-1} \ldots a_{1} a_{0}} \), and let \( r_{i} \) be the remainder when \( 10^{i} \) is divided by \( m \) for \( i = 0, \ldots, n \).

Prove that the number \( N \) is divisible by \( m \) if and only if the number \( M = a_{n} r_{n} + a_{n-1} r_{n-1} + \ldots + a_{1} r_{1} + a_{0} \) is divisible by \( m \).

Answer: N \text{ is divisible by } m \iff M \text{ is divisible by } m


6. There exists a pair of positive integers $(a, b)$ satisfying
$$
\frac{k}{2015}(a+b)=[a, b]
$$

The number of positive integers $k$ is $\qquad$ ( $[a, b]$ denotes the least common multiple of the positive integers $a$ and $b$).

Answer: 1007


8. There are two people, A and B. A, on a car, notices that B is walking in the opposite direction. $1 \mathrm{~min}$ later, A gets off the car to chase B. If A's speed is twice that of B, but slower than the car's speed by $\frac{4}{5}$, then the time it takes for A to catch up with B after getting off the car is $\mathrm{min}$.

Answer: 11


5.3. Solve the inequality

$$
12 \cdot \frac{|x+10|-|x-20|}{|4 x-25|-|4 x-15|}-\frac{|x+10|+|x-20|}{|4 x-25|+|4 x-15|} \geqslant-6
$$

In the answer, write the sum of its integer solutions that satisfy the condition $|x|<100$.

Answer: 10


18. (10 points) As shown in the figure, the side length of square $A B C D$ is 20 cm, and $E$ and $F$ are the midpoints of $A B$ and $B C$ respectively. What is the area of quadrilateral $B E G F$ in square centimeters?

Answer: 80


## Task 3.

Let $A B C$ be a right triangle with a right angle at vertex $C$. Circle $k$ with diameter $\overline{A C}$ intersects side $\overline{A B}$ at point $D$, and the tangent to circle $k$ at point $D$ intersects side $\overline{B C}$ at point $E$. The circumcircle of triangle $C D E$ intersects side $\overline{A B}$ at points $D$ and $F$. Determine the ratio of the areas of triangles $A B C$ and $B E F$.

Answer: 4


Given a triangle \( \triangle ABC \) with an incircle that touches sides \( BC \), \( AC \), and \( AB \) at points \( D \), \( E \), and \( F \) respectively. Lines are drawn from points \( F \) and \( E \) perpendicular to \( BC \), and the feet of these perpendiculars are \( K \) and \( L \) respectively. These perpendicular lines intersect the incircle again at points \( M \) and \( N \) respectively. Prove that \( \frac{S_{\triangle BMD}}{S_{\triangle CND}} = \frac{DK}{DL} \).

Answer: \frac{S_{\triangle BMD}}{S_{\triangle CND}} = \frac{DK}{DL}


Let \( n \) numbers all equal to 1 be written on a blackboard. A move consists of replacing two numbers on the board with two copies of their sum. It happens that after \( h \) moves all \( n \) numbers on the blackboard are equal to \( m \). Prove that \( h \leq \frac{1}{2} n \log _{2} m \).

Answer: h \leq \frac{1}{2} n \log_{2} m


5. A teacher fills in the cells of a class journal of size $7 \times 8$ (7 rows, 8 columns). In each cell, she puts one of three grades: 3, 4, or 5. After filling in the entire journal, it turns out that in each row, the number of threes is not less than the number of fours and not less than the number of fives, and in each column, the number of fours is not less than the number of fives and not less than the number of threes. How many fives could the teacher have put?

Answer: 8


10. (12 points) Huihui and Shanshan are teachers in the senior and junior classes of a kindergarten, respectively. Huihui prepared a large bag of apples to distribute to the children in her class, with each child getting exactly 3 apples; Shanshan prepared a large bag of oranges to distribute to the children in her class, with each child getting exactly 5 oranges. However, they took the wrong bags. In the end, Huihui gave each child 3 oranges, and was short of 12; Shanshan gave each child 6 apples, and they were exactly enough. How many apples did Huihui prepare? $\qquad$

Answer: 72


8. Let the set $T=\left\{x_{1}, x_{2}, \cdots, x_{10}\right\}$ have five-element subsets such that any two elements of $T$ appear in at most two subsets. The maximum number of such subsets is $\qquad$ .

Answer: 18


3. In an $8 \times 8$ grid, fill in different natural numbers so that each cell contains only one number. If a number in a cell is greater than at least 6 numbers in its row and at least 6 numbers in its column, then this cell is called a "good cell". Therefore, the maximum number of "good cells" is $\qquad$.

Answer: 16


19th Balkan 2002 Problem 4 N is the set of positive integers. Find all functions f: N → N such that f( f(n) ) + f(n) = 2n + 2001 or 2n + 2002.

Answer: f(n) = n + 2001


## Task B-3.7.

Let $A B C D$ be a convex quadrilateral such that $|A B|=2 \sqrt{3}, |B C|=3, |C D|=1$, $\angle D A B=90^{\circ}$ and $\angle A B C=30^{\circ}$. Determine $\angle B C D, \angle C D A$ and $|A D|$.

Answer: 120^{\circ}


254. $f(x)=\ln ^{3}\left(x^{2}-1\right)$.

254. $f(x)=\ln ^{3}\left(x^{2}-1\right)$.

The translation is the same as the original text because it is a mathematical expression which is universal and does not change when translated into English. However, if you meant to have the problem number and the function description translated, it would be:

254. $f(x)=\ln ^{3}\left(x^{2}-1\right)$.

254. $f(x)=\ln ^{3}\left(x^{2}-1\right)$.

In this case, the text is already in a form that is commonly used in English mathematical literature.

Answer: \dfrac{6x \left[\ln(x^2 - 1)\right]^2}{x^2 - 1}


20. Given the ellipse $\frac{x^{2}}{5^{2}}+\frac{y^{2}}{4^{2}}=1$, a line is drawn through its left focus $F_{1}$ intersecting the ellipse at points $\mathrm{A}, \mathrm{B}$. $\mathrm{D}(a, \theta)$ is a point to the right of $F_{1}$, and lines $\mathrm{AD}$ and $\mathrm{BD}$ intersect the left directrix of the ellipse at points $M, N$. If the circle with diameter $M N$ passes exactly through $F_{1}$, find the value of $a$.

Answer: 5


5. Let $x, y$ be positive numbers, and $x+y=k$. Try to find the range of $k$ such that the inequality
$$
\left(x+\frac{1}{x}\right)\left(y+\frac{1}{y}\right) \geqslant\left(\frac{k}{2}+\frac{2}{k}\right)^{2}
$$

always holds.

Answer: (0, 2\sqrt{2 + \sqrt{5}}]


67. Person A, B, and C are sawing wooden sticks of different lengths but the same thickness. A saws sticks that are all 8 meters long, B saws sticks that are all 10 meters long, and C saws sticks that are all 6 meters long. They are required to saw them into 2-meter segments. If in the same amount of time, A, B, and C sawed 24, 25, and 27 segments respectively, then the person with the fastest sawing speed saw $\qquad$ more times than the person with the slowest sawing speed.

Answer: 2


12. For the parabola $y=a x^{2}+b x+1$, the parameters $a$ and $b$ satisfy $8 a^{2}+4 a b=b^{3}$. Then, as $a$ and $b$ vary, the trajectory equation of the vertex $(s, t)$ of the parabola is $\qquad$

Answer: xy = 1


8. There are 10 cards, each card has two different numbers from $1,2,3,4,5$, and no two cards have the same pair of numbers. These 10 cards are to be placed into five boxes labeled $1,2,3,4,5$, with the rule that a card with numbers $i, j$ can only be placed in box $i$ or box $j$. A placement is called "good" if the number of cards in box 1 is more than the number of cards in any other box. The number of "good" placements is $\qquad$.

Answer: 120


Let \( n \) be a positive integer and consider an \( n \times n \) grid of real numbers. Determine the greatest possible number of cells \( c \) in the grid such that the entry in \( c \) is both strictly greater than the average of \( c \)'s column and strictly less than the average of \( c \)'s row.

Answer: (n-1)^2


Task 1. Determine all pairs of prime numbers whose difference of squares is equal to 120.

Answer: (13, 7)


5. Arrange the 5 numbers $2,0,1,9,2019$ in any order in a row, to form an 8-digit number (the first digit cannot be 0), then the number of different 8-digit numbers that can be produced is $\qquad$ .

Answer: 95


4. (4 points) In a parallelogram with sides 4 and 7, the bisectors of the four interior angles are drawn. Find the ratio of the area of the quadrilateral formed by the intersection of the bisectors to the area of the parallelogram.

Answer: \dfrac{9}{56}


39. Two ants, a red one and a black one, crawl back and forth between points $A$ and $B$ on a ruler. The red ant starts from point $A$, and the black ant starts from point $B$ at the same time. The black ant's speed is 1.25 times that of the red ant. They meet for the second time at the $124 \mathrm{~cm}$ mark on the ruler, and for the third time at the $96 \mathrm{~cm}$ mark. Therefore, point $A$ is at $\qquad$ $\mathrm{cm}$ mark.

Answer: 82


## 10. Light Bulbs

In the room, there are two light bulbs. When the switch of the first light bulb is turned on, it lights up after 6 seconds and stays on for 5 seconds, then it turns off for 6 seconds and lights up for 5 seconds again, and this cycle repeats continuously. When the switch of the second light bulb is turned on, it lights up after 4 seconds and stays on for 3 seconds, then it turns off for 4 seconds and lights up for 3 seconds again, and this cycle repeats continuously. Linda turned on both switches at the same time and turned them off after 2021 seconds. How many seconds did both light bulbs light up simultaneously during this time?

Result: 392

Answer: 392


Problem 5. From the digits $a, b, c, d, e$, a five-digit number $\overline{a b c d e}$ is formed. For the two-digit numbers $\overline{a b}, \overline{b c}, \overline{c d}, \overline{d e}$, formed from the same digits, it is known that

$$
(\overline{a b}+\overline{b c})(\overline{b c}+\overline{c d})(\overline{c d}+\overline{d e})=157605
$$

Find the number $\overline{a b c d e}$. Multi-digit numbers cannot start with zero.

Answer: 12345


A point $P$ lies on the arc $AB$ of the circumcircle of triangle $ABC$ that does not contain $C$. Prove that:

$$
\frac{a^{2}}{t_{a}}+\frac{b^{2}}{t_{b}}=\frac{c^{2}}{t_{c}}
$$

where $t_{a}, t_{b}, t_{c}$ are the areas of the triangles $BCP, ACP, ABP$, respectively.

Answer: \frac{a^{2}}{t_{a}}+\frac{b^{2}}{t_{b}}=\frac{c^{2}}{t_{c}}


87. The natural numbers $1,2,3,4,5 \cdots \cdots$ are arranged in sequence, and the multiples of 2 and 3 are crossed out, with all multiples of 7 being retained. The 2007th remaining number is $\qquad$.

Answer: 4683


【Question 12】
Use $1,2,3,4,5$ to form a five-digit number, such that the difference between any two adjacent digits is at least 2. How many such five-digit numbers are there? $\qquad$个.

Answer: 14


II. (25 points) As shown in Figure 4, in the Cartesian coordinate system $x O y$, the vertices of the polygon $O A B C D E$ are
$$
\begin{array}{l}
O(0,0), A(0,6), B(4,6), \\
C(4,4), D(6,4), E(6,0) .
\end{array}
$$

It is known that the line $l$ passes through point $M$, intersects sides $O A$ and $D E$, and divides the polygon $O A B C D E$ into two parts of equal area.
(1) If point $M\left(\frac{7}{2}, \frac{5}{2}\right)$, find the function expression of the line $l$;
(2) Prove that if point $M\left(3, \frac{8}{3}\right)$, then there are infinitely many lines $l$ that divide the polygon $O A B C D E$ into two parts of equal area.

Answer: y = -\dfrac{1}{3}x + \dfrac{11}{3}


8,9 |  |

Segments $A M$ and $B H$ are the median and altitude of an acute triangle $A B C$, respectively. It is known that $A H=1$ and $2 \angle M A C=\angle M C A$. Find the side $B C$.

Answer: 2


Some unit-radius circles are drawn on the plane, and we color the center of each circle blue. We mark some points on the circumference of the circles in red such that exactly 2 red points lie on each circumference. What is the maximum number of blue points if there are a total of 25 colored points?

Answer: 20


Carlos, a Mexican history enthusiast, is creating a timeline of events from the Second Mexican Empire and the French intervention in Mexico. He wants to include the following key events:

1. The French army's victory at the Battle of Puebla on May 5, 1862.
2. The establishment of the Second Mexican Empire on April 10, 1864.
3. The execution of Emperor Maximilian I on June 19, 1867.

Carlos decides to calculate the total number of days between these historical events. First, he calculates the number of days from the Battle of Puebla to the establishment of the Second Mexican Empire, and then from the establishment of the empire to the execution of Maximilian I. Finally, he wants to know the total number of days from the Battle of Puebla to Maximilian I's execution.

How many days are there in total from the Battle of Puebla (May 5, 1862) to the execution of Maximilian I (June 19, 1867)?

Answer: 1871


Professor Smith, a political science expert, is studying the impact of right-wing populism on marginalized communities. She visits 3 different communities to collect data. In the first community, she finds that 120 people are affected, while in the second community, 85 people face challenges due to these political changes. In the third community, the number of affected individuals is 50% more than in the second community. How many people, in total, are affected in all three communities?

Answer: 333


Alex, a busy corporate lawyer, works 60 hours each week and wants to find time for a meaningful relationship. Alex decides to allocate some hours from their workweek to spend with a potential partner. If Alex manages to reduce their work hours by 10% each week, how many hours will Alex have available to spend with a potential partner every week?

Answer: 6


Alex, a tech entrepreneur, loves combining his passion for technology with the golden era of music he grew up hearing about from his parents. He has decided to create a unique music app that streams classic hits from the 1960s and 1970s. To start his app, Alex wants to offer a collection of 1000 songs. He plans to source 40% of these songs from the 1960s and the remaining from the 1970s. Additionally, for every 3 songs he sources from the 1960s, he wants to add 2 songs from a list of remastered classics, which are not counted in the initial 1000 songs. How many remastered classics should Alex plan to include in his app?

Answer: 266


Alex is a high school student who loves design and is passionate about creating environmentally friendly products. Alex is working on a project to design a new type of reusable water bottle. To ensure the design is ethical and sustainable, Alex wants to calculate the total reduction in plastic waste if their bottle is used instead of single-use plastic bottles.

Alex discovers that each reusable bottle can replace 150 single-use bottles over its lifespan. Currently, Alex's school uses 2,000 single-use bottles every month. If Alex's design is adopted and each student in the school switches to the reusable bottle, how many single-use bottles will be saved in one year?

Answer: 24000


Dr. Smith, a renowned economic historian, is studying the development of trade routes in ancient civilizations. She discovers that a particular trade route was used to transport goods between two cities. The distance between these cities is 300 miles. Merchants traveling this route stopped at rest points every 60 miles to exchange goods and gather resources. If each merchant journeyed back and forth between the cities 5 times in a year, how many rest points in total did they stop at during the entire year?

Answer: 50


Aisha is a passionate Qatari woman advocating for women's rights. She plans to organize a series of workshops across 5 different cities in Qatar to promote gender equality. In each city, she aims to have 3 workshops. Each workshop can accommodate 25 participants. To ensure the workshops are successful, Aisha plans to distribute 2 informational booklets and 1 promotional badge to each participant. How many informational booklets and promotional badges does Aisha need in total for all the workshops she is organizing?

Answer: 1125


Alex is an agricultural business owner who grows a single type of crop, corn, on a large farm. This season, Alex has planted corn on 150 acres of land. Each acre of Alex's farm typically produces about 120 bushels of corn.

This year, due to favorable weather conditions, the yield increased by 10% per acre. However, a small section of the farm, covering 10 acres, was affected by pests, reducing the yield on those acres by 20%.

Calculate the total number of bushels of corn Alex harvested this season.

Answer: 19536


A coastal management officer is tasked with building a series of breakwaters to protect a 2-kilometer stretch of coastline. Each breakwater is 100 meters long and needs to be placed every 200 meters along the coast. If the officer has already placed 3 breakwaters, how many more breakwaters does the officer need to install to complete the project?

Answer: 7


A doctoral student is analyzing political communication strategies used by different leaders. She decides to study two leaders, Leader A and Leader B, over a month. Leader A gives 3 speeches per week, each containing an average of 2000 words. Leader B gives 2 speeches per week, each containing an average of 2500 words. At the end of the month, the student wants to know how many more words Leader B has spoken compared to Leader A. Assume the month has exactly 4 weeks. How many more words has Leader B spoken than Leader A in that month?

Answer: -4000


11. In the triangle $A B C$ the points $M$ and $N$ lie on the side $A B$ such that $A N=A C$ and $B M=B C$.
We know that $\angle M C N=43^{\circ}$.
Find the size in degrees of $\angle A C B$.

Answer: 94


A set of $8$ problems was prepared for an examination. Each student was given $3$ of them. No two students received more than one common problem. What is the largest possible number of students?

Answer: 8


[ Parallelism of lines and planes $]$  [ Systems of segments, lines, and circles ]

In space, there are $n$ segments, no three of which are parallel to the same plane. For any two segments, the line connecting their midpoints is perpendicular to both segments. What is the largest $n$ for which this is possible?

Answer: 2


86. At some point on a straight line, there is a particle. In the first second, it splits in half, and the halves move in opposite directions to a distance I from the previous position. In the next second, the resulting particles again split in half, and the halves move in opposite directions to a distance 1 from their previous positions. Upon colliding, any two particles are destroyed, so, for example, after two seconds, only two particles remain. How many particles will remain after 129 seconds?

Answer: 4


The cosine of the angle between two adjacent lateral faces of a regular quadrilateral pyramid is given as $k$. Find the cosine of the angle between a lateral face and the base plane, and determine the permissible values of $k$.

Answer: \sqrt{-k}


26. In the following diagram, $\angle A C B=90^{\circ}, D E \perp B C, B E=A C, B D=\frac{1}{2} \mathrm{~cm}$, and $D E+B C=1 \mathrm{~cm}$. Suppose $\angle A B C=x^{\circ}$. Find the value of $x$.

Answer: 30


5. In $\triangle A B C$, $\angle C=90^{\circ}, \angle B=30^{\circ}$, $A C=1$, $M$ is the midpoint of $A B$. Fold $\triangle A C M$ along $C M$ so that the distance between points $A$ and $B$ is $\sqrt{2}$. Then the distance from point $A$ to the plane $B C M$ is $\qquad$

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


There are eight identical Black Queens in the first row of a chessboard and eight identical White Queens in the last row. The Queens move one at a time, horizontally, vertically or diagonally by any number of squares as long as no other Queens are in the way. Black and White Queens move alternately. What is the minimal number of moves required for interchanging the Black and White Queens?

[i](5 points)[/i]

Answer: 24


24. 300 monkeys form a circle, preparing to select one monkey as the king. The rules are as follows: First, number these monkeys clockwise, from 1 to 300, and start operating counterclockwise from the 300th monkey: leave the 300th monkey, eliminate the 299th monkey, leave the 298th monkey, eliminate the 297th monkey, $\qquad$ continue this process until only one monkey remains, and this remaining monkey will become the king. The king's number is $\qquad$.

Answer: 212


10. (20 points) Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
\begin{array}{l}
a_{1}=p, a_{2}=p+1, \\
a_{n+2}-2 a_{n+1}+a_{n}=n-20,
\end{array}
$$

where $p$ is a given real number, and $n$ is a positive integer. Try to find the value of $n$ that minimizes $a_{n}$.

Answer: 40


Terry the Tiger lives on a cube-shaped world with edge length $2$. Thus he walks on the outer surface. He is tied, with a leash of length $2$, to a post located at the center of one of the faces of the cube. The surface area of the region that Terry can roam on the cube can be represented as $\frac{p \pi}{q} + a\sqrt{b}+c$ for integers $a, b, c, p, q$ where no integer square greater than $1$ divides $b, p$ and $q$ are coprime, and $q > 0$. What is $p + q + a + b + c$? (Terry can be at a location if the shortest distance along the surface of the cube between that point and the post is less than or equal to $2$.)

Answer: 14


8. 【Question 8】As shown in the figure, given that the radius of the large circle is 2, then the area of the shaded part is $\qquad$ (use $\pi$ to represent pi).

The translation maintains the original text's line breaks and format.

Answer: 4\pi - 8


[formulas for abbreviated multiplication (other)] [Problems on maximum and minimum]

On the board, 101 numbers are written: $1^{2}, 2^{2}, \ldots, 101^{2}$. In one operation, it is allowed to erase any two numbers and write down the absolute value of their difference instead.

What is the smallest number that can result after 100 operations?

Answer: 1


18. Four different prime numbers $a, b, c, d$ satisfy the following properties:
(1) $a+b+c+d$ is also a prime number;
(2) The sum of two of $a, b, c, d$ is also a prime number:
(3) The sum of three of $a, b, c, d$ is also a prime number. The smallest value of $a+b+c+d$ that satisfies the conditions is $\qquad$

Answer: 31


We say that a rook is "attacking" another rook on a chessboard if the two rooks are in the same row or column of the chessboard and there is no piece directly between them.  Let $n$ be the maximum number of rooks that can be placed on a $6\times 6$ chessboard such that each rook is attacking at most one other.  How many ways can $n$ rooks be placed on a $6\times 6$ chessboard such that each rook is attacking at most one other?

Answer: 8100


Let
\[f_1(x) = \frac{1}{x}\quad\text{and}\quad f_2(x) = 1 - x\]
Let $H$ be the set of all compositions of the form $h_1 \circ h_2 \circ \ldots \circ h_k$, where each $h_i$ is either $f_1$ or $f_2$.  For all $h$ in $H$, let $h^{(n)}$ denote $h$ composed with itself $n$ times.  Find the greatest integer $N$ such that $\pi, h(\pi), \ldots, h^{(N)}(\pi)$ are all distinct for some $h$ in $H$.

Answer: 2


Suppose the polynomial $x^3 - x^2 + bx + c$ has real roots $a, b, c$. What is the square of the minimum value of $abc$?

Answer: 1


Let $\ell_1$ and $\ell_2$ be two parallel lines, a distance of 15 apart. Points $A$ and $B$ lie on $\ell_1$ while points $C$  and $D$ lie on $\ell_2$ such that $\angle BAC = 30^\circ$ and $\angle ABD = 60^\circ$. The minimum value of $AD + BC$ is $a\sqrt b$, where $a$ and $b$ are integers and $b$ is squarefree. Find $a + b$.

Answer: 23


Shirley has a magical machine.  If she inputs a positive even integer $n$, the machine will output $n/2$, but if she inputs a positive odd integer $m$, the machine will output $m+3$. The machine keeps going by automatically using its output as a new input, stopping immediately before it obtains a number already processed. Shirley wants to create the longest possible output sequence possible with initial input at most $100$. What number should she input?

Answer: 67


Let $ABC$ be a triangle with $AB = 15, BC = 17$, $CA = 21$, and incenter $I$. If the circumcircle of triangle $IBC$ intersects side $AC$ again at $P$, find $CP$.

Answer: 6


Consider the sum $\overline{a b} + \overline{ c d e}$, where each of the letters is a distinct digit between $1$ and $5$. How many values are possible for this sum?

Answer: 27


A function $f:\{1,2, \ldots, n\} \to \{1, \ldots, m\}$ is [i]multiplication-preserving[/i] if $f(i)f(j) = f(ij)$ for all $1 \le i \le j \le ij \le n$, and [i]injective[/i] if $f(i) = f(j)$ only when $i = j$. For $n = 9, m = 88$, the number of injective, multiplication-preserving functions is $N$. Find the sum of the prime factors of $N$, including multiplicity. (For example, if $N = 12$, the answer would be $2 + 2 + 3 = 7$.)

Answer: 33


A rectangular piece of paper has corners labeled $A, B, C$, and $D$, with $BC = 80$ and $CD = 120$. Let $M$ be the midpoint of side $AB$. The corner labeled $A$ is folded along line $MD$ and the corner labeled $B$ is folded along line $MC$ until the segments $AM$ and $MB$ coincide. Let $S$ denote the point in space where $A$ and $B$ meet. If $H$ is the foot of the perpendicular from $S$ to the original plane of the paper, find $HM$.

Answer: 45


373. Cutting a Cube. In a mathematical journal, the following question and answer appeared.

Question: "A carpenter wants to cut a wooden cube with a side of 3 inches into 27 cubes with a side of 1 inch. He can easily do this by making 6 cuts while keeping the pieces together so they don't fall apart. What is the minimum number of cuts if the carpenter is allowed to rearrange the pieces in any way during the cutting process $?$"

Answer: "The minimum number of cuts is 6, since it takes 6 cuts, one for each face, to separate the inner cube."

Now, suppose the inner cube is missing. What is the minimum number of cuts in this case if the pieces can be rearranged in any way during the cutting process $?$

Answer: 6


Inside a rectangular grid with a perimeter of 50 cells, a rectangular hole with a perimeter of 32 cells is cut along the boundaries of the cells (the hole does not contain boundary cells). If you cut this figure along all the horizontal lines of the grid, you get 20 strips each 1 cell wide. How many strips will you get if you cut it along all the vertical lines of the grid instead? (A $1 \times 1$ square is also considered a strip!)

Answer: 21


1. (15 points) A small rubber ball moves between two massive vertical walls, colliding with them. One of the walls is stationary, while the other is moving away from it at a constant speed \( u = 100 \, \text{cm} / \text{s} \). Assuming the ball's motion is always horizontal and the collisions are perfectly elastic, find its final speed if the initial speed of the ball was \( v_{0} = 2017 \, \text{cm} / \text{s} \).

Answer: 17


4. The center of the hyperbola is at the origin $O$, with foci on the $x$-axis. A line passing through the right focus of the hyperbola and with a slope of $\sqrt{\frac{3}{5}}$ intersects the hyperbola at points $P$ and $Q$. If $P O \perp O Q,|P Q|$ $=4$, find the equation of the hyperbola.

Answer: \dfrac{x^2}{1} - \dfrac{y^2}{3} = 1


Six children stood in a circle and whispered their favorite numbers to their neighbors on the left and right. Each child added the two numbers they heard and said the result out loud. The drawing shows what three of the children said. What number did the girl next to the question mark think of?

Answer: 6


Hooligans Vasya and Petya tore up a wall newspaper, where Petya tore each piece into 5 parts, and Vasya into 9 parts. When attempting to gather the newspaper, 1988 scraps were found. Prove that not all pieces were found.

Answer: \text{Not all pieces were found.}


In the infinite sequence \(\left\{x_n\right\}(n \geqslant 1)\), for every odd \(n\), \(x_n\), \(x_{n+1}\), and \(x_{n+2}\) form a geometric progression, while for every even \(n\), \(x_n\), \(x_{n+1}\), and \(x_{n+2}\) form an arithmetic progression. Given that \(x_1 = a\) and \(x_2 = b\):

1. Find the general term formula of the sequence. What necessary and sufficient conditions on real numbers \(a\) and \(b\) ensure the existence of such an infinite sequence?
2. Find the harmonic mean of \(x_2, x_4, \cdots, x_{2n}\), i.e., \(\frac{n}{\sum_{k=1}^{n} \frac{1}{x_{2k}}}\).

Answer: nb - (n-1)a


[ Touching circles [ $\underline{\text { C }}$ median proportionals in a right triangle ]

In a parallelogram, there are two circles touching each other and three sides of the parallelogram each. The radius of one of the circles is 1. It is known that one of the segments of a side of the parallelogram from a vertex to the point of tangency is $\sqrt{3}$. Find the area of the parallelogram.

Answer: \dfrac{8\sqrt{3} + 12}{3}


Consider $a_{1}, a_{2}, \cdots, a_{1979}$ as an arbitrary permutation of the natural numbers $1, 2, \cdots, 1979$. To facilitate pairing, let $a_{1980}=0$. Calculate $\left|a_{1}-a_{2}\right|, \left|a_{3}-a_{1}\right|, \cdots, \left|a_{1979}-a_{1980}\right|$ and then arbitrarily rearrange these 990 numbers into $b_{1}, b_{2}, \cdots, b_{990}$. Next, calculate $\left|b_{1}-b_{2}\right|, \left|b_{3}-b_{4}\right|, \cdots, \left|b_{989} - b_{990}\right|$. Continue similarly (adding a zero when there is an odd number to make a pair). Eventually, this process results in a single number $x$. Prove that $x$ is an even number.

Answer: x \text{ is even}


10. In the race between the tortoise and the hare, the hare's and tortoise's speed ratio at the start was $3: 2$. When the hare reached the pavilion along the way, it found that running another 3 kilometers would complete half the race, so it had a good sleep. When the tortoise passed the pavilion, it increased its speed to twice the original and continued on. When the hare woke up, it found that the distance the tortoise had exceeded it was equal to the distance it had exceeded the tortoise when it went to sleep. The hare regretted it and immediately increased its speed to 1.5 times the original to catch up with the tortoise, and they both arrived at the finish line at the same time. The total length of the race was $\qquad$ kilometers.

Answer: 12


In a hockey tournament with \( n \) teams, each team plays against each other exactly once. A winning team earns 2 points, a tie earns 1 point, and a loss earns 0 points. What is the maximum points difference that can occur between teams placed next to each other in the rankings?

Answer: n


A magic square is a square with side 3 consisting of 9 unit squares, such that the numbers written in the unit squares (one number in each square) satisfy the following property: the sum of the numbers in each row is equal to the sum of the numbers in each column and is equal to the sum of all the numbers written in any of the two diagonals.

A rectangle with sides $m\ge3$ and $n\ge3$ consists of $mn$ unit squares. If in each of those unit squares exactly one number is written, such that any square with side $3$ is a magic square, then find the number of most different numbers that can be written in that rectangle.

Answer: 9


4. In the region
$$
\left\{\begin{array}{l}
\frac{x^{2}}{9}+y^{2} \leqslant 1 \\
x \geqslant 0
\end{array}\right.
$$

the maximum radius of the circle that can be contained is $\qquad$ .

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


Answer: 2000


3. There are 64 small cubic blocks with an edge length of 1, each having two faces painted red and the other faces white. Among them, 20 small cubes have two adjacent faces painted red, and 44 small cubes have two opposite faces painted red. Now, they are assembled into a large cube. The maximum area of the red part on the surface of the large cube is $\qquad$

Answer: 76


Let $ABC$ be an acute-angled, nonisosceles triangle. Altitudes $AA'$ and $BB' $meet at point $H$, and the medians of triangle $AHB$ meet at point $M$. Line $CM$ bisects segment $A'B'$. Find angle $C$.

(D. Krekov)

Answer: 45^\circ


Pythagorean Theorem (direct and inverse). Area of a quadrilateral

A quadrilateral is inscribed in a circle of radius 13, with diagonals that are perpendicular to each other. One of the diagonals is 18, and the distance from the center of the circle to the point of intersection of the diagonals is \(4 \sqrt{6}\).

Find the area of the quadrilateral.

Answer: 18\sqrt{161}


n+2 students played a tournament. Each pair played each other once. A player scored 1 for a win, 1/2 for a draw and nil for a loss. Two students scored a total of 8 and the other players all had equal total scores. Find n.

Answer: 14


In parallelogram $A B C D$, diagonal $B D$ is equal to 2, angle $C$ is $45^{\circ}$, and line $C D$ is tangent to the circumcircle of triangle $A B D$. Find the area of parallelogram $A B C D$.

Answer: 4


ерешин D.A.

Once, Rabbit was in a hurry to meet with Donkey Eeyore, but Winnie-the-Pooh and Piglet unexpectedly came to him. Being well-mannered, Rabbit offered his guests some refreshments. Pooh tied a napkin around Piglet's mouth and single-handedly ate 10 pots of honey and 22 cans of condensed milk, with each pot of honey taking him 2 minutes and each can of milk taking 1 minute. Upon learning that there was no more sweet food in the house, Pooh said goodbye and took Piglet away. Rabbit sadly thought that he wouldn't have been late for the meeting with the donkey if Pooh had shared with Piglet. Knowing that Piglet eats a pot of honey in 5 minutes and a can of milk in 3 minutes, Rabbit calculated the minimum time it would take for the guests to finish his supplies. What is this time? (A can of milk and a pot of honey can be divided into any parts.)

#

Answer: 30


(7) A bag contains $n(n>1)$ red balls and 5 white balls. Each time a prize is drawn, two balls are taken out of the bag at once. If the two balls are of different colors, it is considered a win. After each draw, the balls are put back into the bag. Let the probability of winning exactly once in three draws be $p$. Then $n=$ $\qquad$ when $p$ reaches its maximum value $\qquad$ .

Answer: 20


The quadratic function \( y = f(x) \) has a quadratic term coefficient of 1, and both the linear term coefficient and the constant term are integers. If \( f(f(x)) = 0 \) has four distinct real roots, which can be arranged in an arithmetic sequence, find \( f(x) \) such that the sum of its coefficients is minimized.

Answer: x^2 + 22x + 105


20. As shown in the figure, $P$ is a point on the side $AB$ of a regular hexagon $A B C D E F$, $P M / / C D$ intersects $E F$ at $M$, and $P N / / B C$ intersects $C D$ at $N$. A red and a blue sprite start from point $N$ at the same time and walk along the perimeter of pentagon $N P M E D$ and hexagon $C B A F E D$ at a uniform speed, respectively, and return to point $N$ at the same time after each completes one round. What is the ratio of the blue sprite's speed to the red sprite's speed?

Answer: \dfrac{6}{5}


8. For the $k$-element subset $T$ of the set $\{1,2, \cdots, 242\}$, such that the sum of any two elements of $T$ (which can be the same) is not a cube of 3. Then the maximum value of $k$ is $\qquad$

Answer: 121


In the sides of a 30-degree angle, we drew triangles as shown in the figure. The points $A_{i}$ are on one side, and the points $B_{i}$ are on the other side: the angles $O A_{i} B_{i}$ are 120 degrees, and the angles $O B_{i} A_{i+1}$ are right angles. Determine the ratio of the areas of the triangles $A_{1} B_{1} A_{2}$ and $A_{48} B_{48} A_{49}$.

![](https://cdn.mathpix.com/cropped/2024_05_02_5df2143229b99329f40cg-1.jpg?height=415&width=702&top_left_y=302&top_left_x=690)

Answer: \dfrac{1}{2^{94}}


81. Let positive real numbers $x, y, z$ satisfy the condition $2 x y z=3 x^{2}+4 y^{2}+5 z^{2}$, find the minimum value of the expression $P=3 x+2 y+z$.

Answer: 36


The vertices \( A, B \) and \( D_1 \) of a cube \( ABCD A_1 B_1 C_1 D_1 \) lie on the lateral surface of a cylinder, the axis of which is parallel to the line \( DC_1 \). Find the radius of the base of the cylinder, if the edge of the cube is equal to \( a \).

Answer: \dfrac{3\sqrt{2}}{4}a


3. Let points $A$ and $B$ lie on the parabola $y^{2}=6 x$ and the circle $\odot C:(x-4)^{2}+y^{2}=1$, respectively. Then the range of $|A B|$ is

Answer: [\sqrt{15} - 1, +\infty)


There are $800$ marbles in a bag. Each marble is colored with one of $100$ colors, and there are eight marbles of each color. Anna draws one marble at a time from the bag, without replacement, until she gets eight marbles of the same color, and then she immediately stops.

Suppose Anna has not stopped after drawing $699$ marbles. Compute the probability that she stops immediately after drawing the $700$th marble.

Answer: \dfrac{99}{101}


3. Given $\triangle A B C$ in which the lengths of the sides are consecutive natural numbers. The median from vertex $A$ is perpendicular to the angle bisector of the angle at vertex $B$. Calculate the perimeter of triangle $\triangle A B C$.

Answer: 9


13.326. Three cars are dispatched from $A$ to $B$ at equal time intervals. They arrive in $B$ simultaneously, then proceed to point $C$, which is 120 km away from $B$. The first car arrives there one hour after the second. The third car, upon arriving at $C$, immediately turns back and meets the first car 40 km from $C$. Determine the speed of the first car, assuming that the speed of each car was constant throughout the route.

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-1207.jpg?height=518&width=618&top_left_y=76&top_left_x=93)

Fig. 13.16

Answer: 30


56. In triangle $ABC$, $R$ and $r$ are the radii of the circumcircle and the incircle, respectively. Let $A_{1}, B_{1}$, $C_{1}$ be the points of intersection of the angle bisectors of triangle $ABC$ with the circumcircle. Find the ratio of the areas of triangles $ABC$ and $A_{1} B_{1} C_{1}$.

Answer: \dfrac{2r}{R}


Example 9 Let $S$ be the unit circle in the complex plane (i.e., the set of complex numbers with modulus 1), and let $f$ be a mapping from $S$ to $S$. For any $z \in S$, define
$$
\begin{array}{l}
f^{(1)}(z)=f(z), f^{(2)}(z)=f(f(z)), \cdots, \\
f^{(k)}(z)=\underbrace{f(\cdots(f(z)) \cdots) .}_{k \uparrow}
\end{array}
$$

If $C \in S$ and a natural number $n$ exist such that
$$
\begin{array}{l}
f^{(1)}(C) \neq C, f^{(2)}(C) \neq C, \cdots, \\
f^{(n-1)}(C) \neq C, f^{(n)}(C)=C,
\end{array}
$$

then $C$ is called an $n$-periodic point of $f$.
Let $m$ be a natural number greater than 1, and $f$ is defined as follows:
$$
f(z)=z^{m}(z \in S) \text {. }
$$

Try to calculate the number of 1989-periodic points of $f$.
(4th Russian Mathematical Olympiad)

Answer: m^{1989} - m^{663} - m^{153} - m^{117} + m^{51} + m^{39} + m^9 - m^3


There are 2 houses, numbered 1 to 2 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`
 - People have unique favorite music genres: `rock`, `pop`
 - Each person lives in a unique style of house: `colonial`, `victorian`
 - Everyone has something unique for lunch: `grilled cheese`, `pizza`
 - People have unique favorite sports: `basketball`, `soccer`
 - Each mother is accompanied by their child: `Bella`, `Fred`

## Clues:
1. The person living in a colonial-style house is the person who is a pizza lover.
2. Arnold is the person living in a colonial-style house.
3. The person who loves pop music is the person living in a colonial-style house.
4. The person who is a pizza lover is not in the second house.
5. Eric is the person who loves soccer.
6. The person who loves eating grilled cheese is the person's child is named Bella.
 What is the value of attribute MusicGenre for the person whose attribute Children is Fred? Please reason step by step, and put your final answer within \boxed{}

Answer: pop


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `iphone 13`, `huawei p50`, `samsung galaxy s21`
 - Each person has a favorite color: `blue`, `red`, `yellow`, `green`, `white`
 - The mothers' names in different houses are unique: `Janelle`, `Penny`, `Holly`, `Aniya`, `Kailyn`
 - The people are of nationalities: `german`, `norwegian`, `swede`, `dane`, `brit`
 - Everyone has something unique for lunch: `spaghetti`, `grilled cheese`, `stew`, `pizza`, `stir fry`

## Clues:
1. The person who uses a OnePlus 9 and Arnold are next to each other.
2. There are two houses between the person who uses a OnePlus 9 and The person whose mother's name is Kailyn.
3. The person whose mother's name is Penny is somewhere to the left of the Swedish person.
4. The person who loves yellow is the person who uses a Google Pixel 6.
5. The person who loves eating grilled cheese is The person whose mother's name is Holly.
6. Arnold is the person who loves the spaghetti eater.
7. Bob is the person who uses a Huawei P50.
8. The German is the person who is a pizza lover.
9. Eric is directly left of the person who loves blue.
10. Alice is The person whose mother's name is Holly.
11. The person who loves stir fry is the British person.
12. The person whose mother's name is Penny is not in the third house.
13. Arnold is directly left of the person who uses a Huawei P50.
14. The British person is somewhere to the left of the Dane.
15. The person who uses a Google Pixel 6 is Arnold.
16. The person whose mother's name is Aniya is the person who uses a Samsung Galaxy S21.
17. The person who loves blue is the Norwegian.
18. The person who loves white is the person who uses a Huawei P50.
19. The person whose favorite color is red is directly left of the person who loves blue.
 What is the value of attribute Food for the person whose attribute Mother is Holly? Please reason step by step, and put your final answer within \boxed{}

Answer: grilled cheese


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

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

Answer: german