11.5. A rectangular parallelepiped with edge lengths $\sqrt{70}, \sqrt{99}, \sqrt{126}$ is orthogonally projected onto all possible planes. Find the maximum value of the projection area. Justify your answer.

Answer: 168


It is allowed to multiply the number written on the board by 5 or to rearrange its digits (putting zero in the first position is not allowed). Is it possible to obtain the 100-digit number $222 \ldots 2225$ from the number 1 using these operations?

Answer: \text{No}


4. Let's calculate the number of ways to write 3 digits:

$$
10 \cdot 10 \cdot 10=1000
$$

We will derive the formula for the number $A$, which is divisible by 5 and has a remainder of 3 when divided by 7. This number has the form

$$
A=5 k=7 n+3
$$

Solving the equation $5 k-7 n=3$ in integers, we find

$$
\left\{\begin{array}{l}
k=7 t+2 \\
n=5 t+1
\end{array}\right.
$$

Therefore, $A=35 t+10, t \in Z, t \geq 0$. Since by the problem's condition $A$ is a three-digit number, we have $100 \leq 35 t+10 \leq 999$. From this, taking into account the integer nature of $t$, we get $t=2,3, \ldots, 28$. This means that the number of three-digit numbers satisfying the problem's condition is 26. Therefore, the probability
that a three-digit number divisible by 5 and having a remainder of 3 when divided by 7 can be seen on the table is $\frac{26}{1000}$.

Answer: \dfrac{13}{500}


5. Given that $T$ is a moving point on the surface of the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, and satisfies $|T A|=2|T B|$. Let the angle between $T D_{1}$ and the plane $A B C D$ be $\theta$. Then the maximum value of $\theta$ is $\qquad$

Answer: \dfrac{\pi}{4}


17. There are 8 weights of different masses, each with an integer gram value. If any 4 of these weights are taken, two are placed on the left pan of a balance, and the other two on the right pan, the balance will never be in equilibrium, and the heaviest of these 4 weights will always be on the heavier side. What is the minimum weight of the heaviest weight among the 8 weights? $\qquad$

Answer: 34


5.85 It is known that a safe is managed by an 11-member committee, and several locks are added to the safe. The keys to these locks are distributed among the committee members. To ensure that any 6 members present can open the safe, but any 5 cannot, what is the minimum number of locks that should be added to the safe?

---

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

Answer: 462


8. Given $I$ is the incenter of $\triangle A B C$, and $5 \overrightarrow{I A}=4(\overrightarrow{B I}+\overrightarrow{C I})$. Let $R, r$ be the circumradius and inradius of $\triangle A B C$, respectively. If $r=15$, then $R=$ $\qquad$ .

Answer: 32


In a certain kingdom, there were 32 knights. Some of them were vassals of others (a vassal could have only one suzerain, and a suzerain was always richer than his vassal). A knight who had at least four vassals bore the title of baron. What is the maximum number of barons that could be under these conditions?

(The kingdom had a law: "a vassal of my vassal is not my vassal".)

#

Answer: 7


83. At a school evening, there are 12 girls and 10 boys present. In how many ways can 4 pairs for dancing be formed from them?

Answer: 2494800


p1. How many $10$-digit strings are there, such that all its digits are only zeros or ones and the sum of its even-place digits equals the sum of the odd-place digits.

p2. Find all pairs $(x, y)$ of nonnegative integers, such that $x! + 24 = y^2$.

p3. Consider a function $f: Z \to Q$ such that $$f(1) = 2015 \,\,\, \,\, f (1) + f (2) + ...+ f (n) = n^2 f (n).$$
Determine the value of $f(2015)$.

p4. Let $ABC$ be an isosceles triangle with $AB = AC$, and let $D$ be the midpoint of $BC$, $E$ the foot of the perpendicular on $AB$ from $D$, and F the midpoint of $DE$. Show that $AF$ is perpendicular to $CE$.

p5. On an island there are only two tribes: the Mienteretes who always lie and the Veritas who always tell the truth. On a certain day there is an assembly attended by $2015$ inhabitants of the island. They sit at random in a circle and each one declares: "the two people next to me are Mienteretes." The next day the assembly continues but one of them became ill, for which $2014$ inhabitants attend, again they sit at random in a circle and each one declares: “The two people next to me are from the same tribe, which is not mine ”.
Deduct the number of inhabitants of each tribe and the type of inhabitant to which the patient belongs.

Answer: 252


As shown in the figure, in $\triangle ABC$, $BC > AC$ and $\angle A = 60^\circ$. Points $D$ and $E$ are the midpoints of $AB$ and $AC$ respectively. If $PC$ bisects $\angle ACB$ and $PD$ bisects $\angle ADE$, then $\angle DPC$ equals $\qquad$.

Answer: 120^\circ


## Task 1 - 330631

Find all possible ways to combine three natural numbers $a, b, c$ such that $a+b+c=12$ and $c-b=3$!

Hint:

1. Zero should also be considered a natural number.
2. It is also allowed that among the numbers $a, b, c$, some are equal to each other.

Answer: (9, 0, 3)


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: `sept`, `april`, `jan`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`
 - People have unique hair colors: `brown`, `black`, `blonde`
 - The mothers' names in different houses are unique: `Aniya`, `Janelle`, `Holly`

## Clues:
1. Arnold is the person whose birthday is in January.
2. The person whose mother's name is Janelle is the person who has brown hair.
3. The person who has black hair is not in the second house.
4. The person who has black hair is the person whose birthday is in April.
5. The person whose mother's name is Aniya is Eric.
6. The person whose mother's name is Janelle is the person who owns a Toyota Camry.
7. The person who has black hair is The person whose mother's name is Aniya.
8. The person whose birthday is in September and the person who owns a Tesla Model 3 are next to each other.
9. Arnold is the person who has brown hair.
10. The person who owns a Toyota Camry is somewhere to the left of the person who owns a Ford F-150.
 What is the value of attribute CarModel for the person whose attribute House is 1? 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`, `Eric`, `Arnold`
 - Each person has a unique hobby: `cooking`, `painting`, `gardening`, `photography`
 - The people keep unique animals: `horse`, `fish`, `cat`, `bird`
 - People have unique favorite book genres: `fantasy`, `mystery`, `romance`, `science fiction`
 - Each person has a unique birthday month: `april`, `jan`, `sept`, `feb`
 - People have unique favorite music genres: `pop`, `rock`, `classical`, `jazz`

## Clues:
1. The person who loves cooking is the person who loves romance books.
2. The person whose birthday is in February is the person who loves pop music.
3. Eric is not in the second house.
4. The person who loves romance books is not in the fourth house.
5. The person whose birthday is in February is the fish enthusiast.
6. Alice is somewhere to the right of the person who loves fantasy books.
7. The person who keeps horses is the person who loves rock music.
8. The person who enjoys gardening is the person whose birthday is in April.
9. The person who loves jazz music is the person who loves cooking.
10. The person who loves rock music is the person who loves mystery books.
11. The person who paints as a hobby is directly left of the person who loves romance books.
12. Peter is the person who loves pop music.
13. The person who enjoys gardening is Arnold.
14. The person who loves rock music is directly left of the person whose birthday is in January.
15. The person who loves cooking is not in the third house.
16. The cat lover is somewhere to the right of the person who keeps horses.
 What is the value of attribute Birthday for the person whose attribute Hobby is photography? Please reason step by step, and put your final answer within \boxed{}

Answer: jan


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`, `Alice`
 - People have unique heights: `very short`, `short`, `tall`, `average`
 - Each person prefers a unique type of vacation: `cruise`, `beach`, `city`, `mountain`
 - Each person has a favorite color: `green`, `red`, `yellow`, `white`
 - Each mother is accompanied by their child: `Fred`, `Bella`, `Meredith`, `Samantha`

## Clues:
1. The person who loves beach vacations is the person who is very short.
2. The person who is very short is the person who loves white.
3. The person who has an average height is the person's child is named Bella.
4. The person who has an average height is the person whose favorite color is red.
5. Peter is the person's child is named Samantha.
6. The person whose favorite color is green is directly left of the person who loves white.
7. Alice is in the third house.
8. The person who likes going on cruises is the person's child is named Bella.
9. The person who is short is the person who loves yellow.
10. The person who is very short is the person's child is named Fred.
11. Eric is the person who loves yellow.
12. The person who prefers city breaks is Eric.
13. The person who is short is somewhere to the left of the person who is tall.
 What is the value of attribute Vacation for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: mountain


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`, `Alice`
 - Each person has an occupation: `doctor`, `engineer`, `artist`, `teacher`

## Clues:
1. There are two houses between Eric and Peter.
2. The person who is a teacher is Peter.
3. Peter is not in the first house.
4. There is one house between the person who is a doctor and Alice.
5. The person who is an artist is Alice.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`
 - They all have a unique favorite flower: `lilies`, `carnations`, `roses`, `daffodils`
 - Each person has a favorite color: `white`, `green`, `yellow`, `red`
 - Each person has a unique level of education: `bachelor`, `associate`, `master`, `high school`
 - Each person has a unique hobby: `painting`, `photography`, `gardening`, `cooking`
 - Everyone has a unique favorite cigar: `dunhill`, `blue master`, `prince`, `pall mall`

## Clues:
1. The person partial to Pall Mall is the person with an associate's degree.
2. The person who loves white is the Prince smoker.
3. The person with a high school diploma is not in the fourth house.
4. The person with a master's degree is not in the second house.
5. Arnold is not in the fourth house.
6. The person whose favorite color is red is not in the third house.
7. The person whose favorite color is red is somewhere to the left of Eric.
8. The person who loves cooking is somewhere to the right of Peter.
9. The person who loves a bouquet of daffodils is somewhere to the left of the person who paints as a hobby.
10. The person who loves white is the person who loves the boquet of lilies.
11. Alice is the photography enthusiast.
12. Arnold is somewhere to the right of the person partial to Pall Mall.
13. The person who loves white is in the second house.
14. The person whose favorite color is green is directly left of the person who enjoys gardening.
15. The Dunhill smoker is the person with a high school diploma.
16. The person who loves the rose bouquet is somewhere to the right of the person whose favorite color is green.
 What is the value of attribute Hobby for the person whose attribute Flower is daffodils? 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`, `Alice`, `Arnold`, `Eric`
 - The people keep unique animals: `cat`, `fish`, `bird`, `horse`
 - People have unique favorite sports: `soccer`, `swimming`, `tennis`, `basketball`
 - People have unique hair colors: `blonde`, `brown`, `black`, `red`
 - Each person has a unique type of pet: `bird`, `fish`, `dog`, `cat`

## Clues:
1. The person who loves tennis is in the second house.
2. The person who has brown hair and the person who loves basketball are next to each other.
3. Alice is the person who keeps a pet bird.
4. Arnold is the person who loves soccer.
5. The person who has blonde hair is the person with an aquarium of fish.
6. The person who keeps a pet bird is in the second house.
7. Arnold is in the third house.
8. The person who loves swimming is in the first house.
9. The person who has red hair is not in the first house.
10. The person who keeps horses is in the fourth house.
11. Eric is somewhere to the left of the fish enthusiast.
12. The person who has a cat is directly left of the bird keeper.
 What is the value of attribute House for the person whose attribute FavoriteSport is swimming? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


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`, `Peter`, `Arnold`, `Alice`
 - Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `desert`, `watermelon`
 - Everyone has a unique favorite cigar: `blue master`, `pall mall`, `dunhill`, `prince`
 - People have unique heights: `tall`, `average`, `short`, `very short`
 - People use unique phone models: `google pixel 6`, `samsung galaxy s21`, `iphone 13`, `oneplus 9`

## Clues:
1. The Dragonfruit smoothie lover is Eric.
2. The Dunhill smoker is the person who likes Cherry smoothies.
3. The person who uses a Samsung Galaxy S21 is directly left of the person who uses an iPhone 13.
4. The Dunhill smoker is somewhere to the right of the person who is very short.
5. The Watermelon smoothie lover is somewhere to the right of the Desert smoothie lover.
6. The Prince smoker is the person who uses a OnePlus 9.
7. The person who is tall is in the third house.
8. The person who is very short is the person who uses an iPhone 13.
9. The person who smokes Blue Master is not in the first house.
10. The Dunhill smoker is the person who is short.
11. Peter is not in the third house.
12. Arnold is the person who uses a Google Pixel 6.
13. The Dragonfruit smoothie lover is the person partial to Pall Mall.
 What is the value of attribute House for the person whose attribute Smoothie is desert? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
 - The people are of nationalities: `swede`, `german`, `dane`, `brit`, `chinese`, `norwegian`
 - Each person has a unique type of pet: `bird`, `cat`, `dog`, `rabbit`, `fish`, `hamster`
 - Each person has a favorite color: `red`, `green`, `purple`, `yellow`, `white`, `blue`

## Clues:
1. The person who has a cat is the British person.
2. The person with a pet hamster is in the fourth house.
3. The person who owns a rabbit is the person who loves purple.
4. Arnold is the person who loves white.
5. The Dane is somewhere to the left of the person who owns a dog.
6. Carol and Peter are next to each other.
7. The Swedish person is somewhere to the right of the person whose favorite color is green.
8. The person who loves blue is the person with a pet hamster.
9. There are two houses between the person whose favorite color is red and Bob.
10. The person who keeps a pet bird is in the second house.
11. Carol is in the first house.
12. The person who loves yellow and the German are next to each other.
13. Alice is the person who loves yellow.
14. The Dane is the person who loves yellow.
15. Bob is the Norwegian.
16. Alice is directly left of the person who loves white.
 What is the value of attribute Pet for the person whose attribute Color is purple? Please reason step by step, and put your final answer within \boxed{}

Answer: rabbit


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`, `Arnold`, `Eric`, `Alice`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `roses`, `lilies`
 - People have unique heights: `very short`, `short`, `tall`, `average`
 - The mothers' names in different houses are unique: `Janelle`, `Kailyn`, `Holly`, `Aniya`
 - Each person has an occupation: `engineer`, `doctor`, `teacher`, `artist`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `soccer`

## Clues:
1. The person who loves swimming is the person who loves the rose bouquet.
2. The person who loves the rose bouquet is Eric.
3. Arnold is the person who is tall.
4. The person who loves a bouquet of daffodils is somewhere to the right of the person who is an engineer.
5. The person who loves soccer is the person who is short.
6. The person who is a teacher is in the first house.
7. The person whose mother's name is Janelle is the person who loves a carnations arrangement.
8. The person who loves basketball is the person who has an average height.
9. Arnold is not in the third house.
10. The person whose mother's name is Holly is somewhere to the right of the person who has an average height.
11. Peter is the person who is a doctor.
12. The person whose mother's name is Aniya is Alice.
13. Arnold is the person who loves the boquet of lilies.
 What is the value of attribute House for the person whose attribute Name is Alice? 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: `Alice`, `Eric`, `Peter`, `Carol`, `Bob`, `Arnold`
 - Everyone has a favorite smoothie: `watermelon`, `blueberry`, `desert`, `cherry`, `dragonfruit`, `lime`
 - Everyone has something unique for lunch: `stew`, `pizza`, `grilled cheese`, `stir fry`, `soup`, `spaghetti`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `xiaomi mi 11`, `huawei p50`, `samsung galaxy s21`, `oneplus 9`
 - People own unique car models: `tesla model 3`, `honda civic`, `toyota camry`, `ford f150`, `chevrolet silverado`, `bmw 3 series`
 - Each person lives in a unique style of house: `craftsman`, `ranch`, `modern`, `victorian`, `mediterranean`, `colonial`

## Clues:
1. Bob is the person who uses a Xiaomi Mi 11.
2. The person who loves the soup is in the fourth house.
3. The Dragonfruit smoothie lover is somewhere to the left of the person in a ranch-style home.
4. There is one house between the person who owns a Chevrolet Silverado and the person residing in a Victorian house.
5. The person in a Mediterranean-style villa is the person who drinks Lime smoothies.
6. Eric is in the sixth house.
7. The Desert smoothie lover is the person who is a pizza lover.
8. The person living in a colonial-style house is the person who drinks Blueberry smoothies.
9. The Dragonfruit smoothie lover and the person who uses a Google Pixel 6 are next to each other.
10. The person who loves the soup is Peter.
11. Alice is somewhere to the right of the person who owns a BMW 3 Series.
12. The person who loves stir fry is the person in a ranch-style home.
13. The person who owns a Ford F-150 is the person living in a colonial-style house.
14. The person in a Craftsman-style house is somewhere to the right of the person in a modern-style house.
15. The person who loves the stew is directly left of the person in a ranch-style home.
16. The person who owns a Tesla Model 3 is directly left of the person who loves stir fry.
17. The person who loves eating grilled cheese is the person who owns a Honda Civic.
18. The person in a Mediterranean-style villa is the person who uses a Google Pixel 6.
19. The person in a Craftsman-style house is the Watermelon smoothie lover.
20. The person in a modern-style house is Carol.
21. The person who uses an iPhone 13 is in the first house.
22. The person who owns a Honda Civic is somewhere to the left of the person who owns a BMW 3 Series.
23. The person who uses a Xiaomi Mi 11 is the person who is a pizza lover.
24. The person who uses a Samsung Galaxy S21 is the person who loves stir fry.
25. The person residing in a Victorian house is directly left of the person who uses a OnePlus 9.
 What is the value of attribute Food for the person whose attribute PhoneModel is xiaomi mi 11? Please reason step by step, and put your final answer within \boxed{}

Answer: pizza


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: `Alice`, `Arnold`, `Eric`, `Peter`
 - Each person has a unique favorite drink: `water`, `tea`, `coffee`, `milk`
 - Each person has a unique level of education: `master`, `associate`, `high school`, `bachelor`
 - The mothers' names in different houses are unique: `Kailyn`, `Holly`, `Aniya`, `Janelle`
 - Each person has a favorite color: `yellow`, `white`, `green`, `red`

## Clues:
1. Eric is the person with an associate's degree.
2. The person whose mother's name is Kailyn is the person who loves yellow.
3. The person who likes milk is in the third house.
4. The person who likes milk is somewhere to the right of the person who loves white.
5. The person with a bachelor's degree is in the first house.
6. The person whose mother's name is Aniya is not in the second house.
7. The one who only drinks water is Arnold.
8. The person whose favorite color is green is Peter.
9. The person whose favorite color is red is somewhere to the left of the person who loves white.
10. The person with a high school diploma is directly left of The person whose mother's name is Janelle.
11. The person with a high school diploma is the tea drinker.
 What is the value of attribute House for the person whose attribute Drink is milk? 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: `Carol`, `Eric`, `Peter`, `Alice`, `Bob`, `Arnold`
 - Each person has a unique level of education: `doctorate`, `master`, `associate`, `high school`, `bachelor`, `trade school`
 - The people keep unique animals: `rabbit`, `fish`, `horse`, `dog`, `bird`, `cat`
 - People have unique favorite sports: `soccer`, `baseball`, `volleyball`, `basketball`, `swimming`, `tennis`

## Clues:
1. The fish enthusiast is the person with an associate's degree.
2. The cat lover is the person who attended trade school.
3. The person who loves soccer is the bird keeper.
4. Arnold is directly left of the rabbit owner.
5. The person who keeps horses is the person who loves volleyball.
6. Peter is not in the fifth house.
7. Arnold is the person who attended trade school.
8. The person who loves swimming and the person with a doctorate are next to each other.
9. Alice is in the second house.
10. The person with a doctorate and the person who loves basketball are next to each other.
11. Bob is the person with a high school diploma.
12. The person with an associate's degree is in the second house.
13. The person who loves baseball is in the fifth house.
14. The person with a master's degree is the person who loves basketball.
15. The person who keeps horses is somewhere to the left of Bob.
16. The cat lover is in the fourth house.
17. Carol is somewhere to the left of the fish enthusiast.
 What is the value of attribute FavoriteSport for the person whose attribute Education is trade school? Please reason step by step, and put your final answer within \boxed{}

Answer: swimming


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`, `Peter`, `Carol`, `Bob`, `Alice`
 - Everyone has a favorite smoothie: `blueberry`, `watermelon`, `cherry`, `lime`, `dragonfruit`, `desert`
 - The people keep unique animals: `bird`, `cat`, `fish`, `horse`, `dog`, `rabbit`

## Clues:
1. The Desert smoothie lover is directly left of Arnold.
2. Carol is the rabbit owner.
3. The Watermelon smoothie lover is the cat lover.
4. The dog owner is somewhere to the right of Peter.
5. The person who keeps horses is directly left of Carol.
6. Eric is not in the fifth house.
7. The person who drinks Lime smoothies is in the sixth house.
8. Carol is in the sixth house.
9. The bird keeper and the person who keeps horses are next to each other.
10. Alice is in the second house.
11. Arnold is in the third house.
12. The Dragonfruit smoothie lover is the dog owner.
13. The person who drinks Blueberry smoothies is not in the fifth house.
 What is the value of attribute House for the person whose attribute Animal is bird? 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: `Eric`, `Carol`, `Bob`, `Peter`, `Arnold`, `Alice`
 - Everyone has something unique for lunch: `stew`, `stir fry`, `soup`, `grilled cheese`, `pizza`, `spaghetti`
 - Each person has a unique favorite drink: `root beer`, `milk`, `boba tea`, `water`, `coffee`, `tea`
 - People have unique heights: `super tall`, `short`, `tall`, `very short`, `very tall`, `average`
 - Each person has a unique birthday month: `sept`, `feb`, `may`, `mar`, `jan`, `april`

## Clues:
1. The person whose birthday is in January is Peter.
2. The person who loves eating grilled cheese is the person whose birthday is in May.
3. The person whose birthday is in April is the person who loves stir fry.
4. The person who loves the stew is the one who only drinks water.
5. The person whose birthday is in September is not in the fourth house.
6. The person who likes milk is somewhere to the left of the person who loves stir fry.
7. The coffee drinker is in the fifth house.
8. The person who loves eating grilled cheese is directly left of the one who only drinks water.
9. The person whose birthday is in May is directly left of Arnold.
10. The root beer lover is somewhere to the left of Alice.
11. There are two houses between Eric and the person who is a pizza lover.
12. The person who loves the spaghetti eater is the person who is very short.
13. The person who is super tall is the tea drinker.
14. The person who is tall is not in the third house.
15. Peter is directly left of the person who is a pizza lover.
16. The person who likes milk is somewhere to the right of the one who only drinks water.
17. The person who is very short is the coffee drinker.
18. The person who is super tall is Bob.
19. The person who is super tall and the person whose birthday is in March are next to each other.
20. Carol is somewhere to the left of the person who has an average height.
21. The person who is very tall is the person who loves the stew.
22. There is one house between the person who loves eating grilled cheese and Peter.
 What is the value of attribute Height for the person whose attribute House is 5? Please reason step by step, and put your final answer within \boxed{}

Answer: very short


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Eric`, `Peter`, `Arnold`
 - Each person has a unique type of pet: `hamster`, `bird`, `fish`, `dog`, `cat`
 - Each person has a unique favorite drink: `coffee`, `milk`, `root beer`, `water`, `tea`
 - People have unique favorite sports: `soccer`, `swimming`, `tennis`, `baseball`, `basketball`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `iphone 13`, `huawei p50`, `samsung galaxy s21`

## Clues:
1. The root beer lover is the person who owns a dog.
2. The person who uses an iPhone 13 is the root beer lover.
3. There is one house between the person who owns a dog and the person who uses a Google Pixel 6.
4. Bob is somewhere to the right of the person who uses a OnePlus 9.
5. Alice is not in the fourth house.
6. Eric is in the fifth house.
7. The person who likes milk is the person who uses a Samsung Galaxy S21.
8. The person who loves baseball is the person who uses a Google Pixel 6.
9. The coffee drinker is in the third house.
10. Peter is the tea drinker.
11. The person who loves basketball is the person who owns a dog.
12. The person with a pet hamster is the person who loves tennis.
13. There are two houses between Alice and the tea drinker.
14. The person who loves soccer is the tea drinker.
15. The person who keeps a pet bird is directly left of the person with an aquarium of fish.
16. The person with an aquarium of fish is the person who likes milk.
 What is the value of attribute Name for the person whose attribute PhoneModel is huawei p50? 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: `Peter`, `Eric`, `Alice`, `Arnold`, `Carol`, `Bob`
 - People have unique heights: `very tall`, `very short`, `tall`, `super tall`, `short`, `average`
 - Each person prefers a unique type of vacation: `cruise`, `mountain`, `camping`, `beach`, `city`, `cultural`
 - People have unique hair colors: `blonde`, `red`, `gray`, `brown`, `auburn`, `black`
 - Each person has a unique favorite drink: `tea`, `boba tea`, `milk`, `water`, `coffee`, `root beer`
 - They all have a unique favorite flower: `lilies`, `carnations`, `iris`, `tulips`, `roses`, `daffodils`

## Clues:
1. The person who loves the vase of tulips is the person who has auburn hair.
2. The person who is short is somewhere to the right of Arnold.
3. The person who goes on cultural tours is Arnold.
4. The person who has auburn hair is Bob.
5. The person who is very short is the person who loves a bouquet of daffodils.
6. The person who loves the boquet of iris is the person who is very tall.
7. Peter is somewhere to the right of the root beer lover.
8. The person who loves beach vacations is directly left of the person who prefers city breaks.
9. Bob is directly left of the coffee drinker.
10. The person who is very short is the person who has gray hair.
11. Eric is somewhere to the right of the tea drinker.
12. The person who has red hair is the person who is tall.
13. The person who is tall is somewhere to the right of the person who is short.
14. The person who has black hair is the person who enjoys mountain retreats.
15. The one who only drinks water is the person who prefers city breaks.
16. The person who has brown hair is Carol.
17. The person who has an average height is Arnold.
18. The person who has brown hair is directly left of the person who likes milk.
19. The root beer lover is the person who is very short.
20. The person who is short is somewhere to the left of the coffee drinker.
21. The person who enjoys camping trips is the person who loves a carnations arrangement.
22. The person who is super tall and Arnold are next to each other.
23. There is one house between the person who loves the boquet of lilies and Peter.
24. The person who loves the vase of tulips is directly left of the person who has black hair.
 What is the value of attribute Flower for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: tulips


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`
 - The people keep unique animals: `bird`, `horse`, `cat`
 - Each person has a unique birthday month: `jan`, `sept`, `april`
 - Each person has a unique hobby: `photography`, `cooking`, `gardening`
 - Each person has a unique favorite drink: `milk`, `water`, `tea`
 - People have unique hair colors: `black`, `brown`, `blonde`

## Clues:
1. The person who has brown hair is the person who loves cooking.
2. The person whose birthday is in April is in the third house.
3. Eric is not in the first house.
4. The cat lover is in the second house.
5. The person who has blonde hair is somewhere to the left of the person who likes milk.
6. The person who enjoys gardening is the person who likes milk.
7. The cat lover is the person who has brown hair.
8. Arnold is the bird keeper.
9. The one who only drinks water is the photography enthusiast.
10. The person whose birthday is in September is directly left of Arnold.
 What is the value of attribute Birthday for the person whose attribute Name is Peter? 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: `Eric`, `Arnold`, `Peter`, `Bob`, `Alice`, `Carol`
 - People have unique favorite music genres: `jazz`, `country`, `pop`, `classical`, `rock`, `hip hop`
 - Each person lives in a unique style of house: `mediterranean`, `modern`, `colonial`, `ranch`, `victorian`, `craftsman`
 - People have unique favorite sports: `tennis`, `baseball`, `soccer`, `volleyball`, `swimming`, `basketball`
 - People have unique favorite book genres: `science fiction`, `historical fiction`, `mystery`, `biography`, `fantasy`, `romance`

## Clues:
1. The person who loves baseball is Arnold.
2. Alice is somewhere to the left of the person living in a colonial-style house.
3. The person who loves country music is directly left of the person who loves hip-hop music.
4. Peter is the person residing in a Victorian house.
5. The person who loves fantasy books is Peter.
6. The person who loves baseball is the person who loves romance books.
7. The person who loves hip-hop music is somewhere to the left of the person who loves biography books.
8. The person in a ranch-style home is somewhere to the left of the person who loves romance books.
9. Bob is the person who loves swimming.
10. The person who loves romance books is somewhere to the left of the person who loves mystery books.
11. The person who loves basketball is in the fourth house.
12. Bob is the person who loves jazz music.
13. The person in a Craftsman-style house is the person who loves mystery books.
14. Peter is somewhere to the left of the person who loves basketball.
15. The person who loves volleyball is somewhere to the right of the person in a Mediterranean-style villa.
16. The person in a Mediterranean-style villa is Alice.
17. The person who loves historical fiction books is directly left of the person living in a colonial-style house.
18. Carol is directly left of the person who loves pop music.
19. There is one house between the person who loves pop music and the person who loves classical music.
20. There are two houses between the person who loves tennis and Alice.
 What is the value of attribute Name for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - People have unique favorite music genres: `classical`, `rock`, `pop`
 - Each person has a unique type of pet: `cat`, `fish`, `dog`
 - The mothers' names in different houses are unique: `Janelle`, `Aniya`, `Holly`
 - The people keep unique animals: `cat`, `bird`, `horse`

## Clues:
1. The person whose mother's name is Aniya is not in the first house.
2. The person whose mother's name is Aniya is directly left of The person whose mother's name is Holly.
3. The person whose mother's name is Janelle is Peter.
4. The person who loves pop music is directly left of the person who owns a dog.
5. The person who keeps horses is in the first house.
6. Peter and the cat lover are next to each other.
7. Eric is the person who loves rock music.
8. The person whose mother's name is Janelle and the person who has a cat are next to each other.
 What is the value of attribute Pet for the person whose attribute Mother is Janelle? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Bob`, `Alice`, `Eric`, `Arnold`, `Peter`
 - Each person has a unique type of pet: `hamster`, `cat`, `fish`, `rabbit`, `dog`, `bird`
 - Each person has a favorite color: `blue`, `purple`, `yellow`, `white`, `red`, `green`

## Clues:
1. The person who has a cat is not in the third house.
2. Arnold is not in the first house.
3. The person who owns a dog is the person whose favorite color is green.
4. The person who loves yellow is somewhere to the left of Bob.
5. The person who loves blue is the person who has a cat.
6. There are two houses between the person with an aquarium of fish and Eric.
7. The person who owns a dog is Carol.
8. The person who loves yellow is the person with a pet hamster.
9. The person who loves white and the person who keeps a pet bird are next to each other.
10. The person who owns a rabbit is somewhere to the left of the person who loves white.
11. The person who owns a dog is directly left of the person who loves blue.
12. The person who loves purple is directly left of Carol.
13. Bob is not in the third house.
14. Alice is the person who loves purple.
 What is the value of attribute House for the person whose attribute Pet is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


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`
 - The people are of nationalities: `dane`, `swede`, `german`, `brit`, `norwegian`
 - Each person has a unique hobby: `gardening`, `painting`, `knitting`, `photography`, `cooking`
 - Everyone has a favorite smoothie: `dragonfruit`, `lime`, `watermelon`, `desert`, `cherry`
 - People have unique hair colors: `brown`, `gray`, `black`, `blonde`, `red`
 - People have unique favorite sports: `baseball`, `soccer`, `swimming`, `tennis`, `basketball`

## Clues:
1. The person who loves cooking is not in the second house.
2. The Watermelon smoothie lover is not in the fourth house.
3. The person who loves baseball is somewhere to the right of the Desert smoothie lover.
4. The person who drinks Lime smoothies is the person who has brown hair.
5. The person who has gray hair is somewhere to the right of the Swedish person.
6. The photography enthusiast is not in the fifth house.
7. The Swedish person is not in the first house.
8. The German is the person who has blonde hair.
9. There is one house between the Watermelon smoothie lover and the photography enthusiast.
10. Alice and the person who drinks Lime smoothies are next to each other.
11. The British person and the person who enjoys knitting are next to each other.
12. The person who has black hair is the Swedish person.
13. The British person is the person who enjoys gardening.
14. The person who loves swimming is the person who loves cooking.
15. The Dragonfruit smoothie lover is not in the third house.
16. The Watermelon smoothie lover is not in the second house.
17. Eric is somewhere to the left of the person who drinks Lime smoothies.
18. The person who loves soccer is Alice.
19. Peter is the photography enthusiast.
20. The person who enjoys gardening is directly left of the person who loves tennis.
21. The Norwegian and the photography enthusiast are next to each other.
22. The British person is Alice.
23. Bob is the person who has blonde hair.
 What is the value of attribute Nationality for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: norwegian


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`
 - Each person has a unique level of education: `bachelor`, `high school`, `associate`, `master`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`, `Kailyn`
 - The people keep unique animals: `fish`, `horse`, `cat`, `bird`
 - Each person has a unique birthday month: `jan`, `feb`, `sept`, `april`
 - People have unique favorite music genres: `rock`, `pop`, `jazz`, `classical`

## Clues:
1. The person with an associate's degree is The person whose mother's name is Aniya.
2. Peter is the person whose birthday is in September.
3. The person with a bachelor's degree is directly left of the person who loves pop music.
4. Arnold is The person whose mother's name is Janelle.
5. The person with a master's degree is Arnold.
6. The person whose mother's name is Janelle is the person who loves classical music.
7. The person whose birthday is in April is somewhere to the right of the person who loves rock music.
8. The cat lover is in the second house.
9. The fish enthusiast is in the third house.
10. Eric is the person whose birthday is in April.
11. The person who loves pop music is the person whose birthday is in January.
12. The person whose mother's name is Holly is the person with a bachelor's degree.
13. The person who keeps horses is the person with a master's degree.
14. The person whose mother's name is Aniya is the fish enthusiast.
15. Arnold is somewhere to the left of the fish enthusiast.
 What is the value of attribute Education for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: high school


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`
 - The people keep unique animals: `bird`, `horse`, `cat`
 - Each person has a unique birthday month: `jan`, `sept`, `april`
 - Each person has a unique hobby: `photography`, `cooking`, `gardening`
 - Each person has a unique favorite drink: `milk`, `water`, `tea`
 - People have unique hair colors: `black`, `brown`, `blonde`

## Clues:
1. The person who has brown hair is the person who loves cooking.
2. The person whose birthday is in April is in the third house.
3. Eric is not in the first house.
4. The cat lover is in the second house.
5. The person who has blonde hair is somewhere to the left of the person who likes milk.
6. The person who enjoys gardening is the person who likes milk.
7. The cat lover is the person who has brown hair.
8. Arnold is the bird keeper.
9. The one who only drinks water is the photography enthusiast.
10. The person whose birthday is in September is directly left of Arnold.
 What is the value of attribute Drink for the person whose attribute HairColor is brown? Please reason step by step, and put your final answer within \boxed{}

Answer: tea


In triangle \(ABC\), the angle bisector \(AL\) is drawn. Points \(E\) and \(D\) are marked on segments \(AB\) and \(BL\) respectively such that \(DL = LC\) and \(ED \parallel AC\). Find the length of segment \(ED\) given that \(AE = 15\) and \(AC = 12\).

Answer: 3


As shown in the figure, an "L"-shaped paper with a perimeter of 52 cm can be split along the dashed line into two identical rectangles. If the longest side length is 16 cm, what is the area of the "L"-shaped paper in square centimeters?

Answer: 120


In a right triangle, the sine of the smaller angle is $\frac{1}{3}$. A line is drawn perpendicular to the hypotenuse, dividing the triangle into two equal areas. In what ratio does this line divide the hypotenuse?

Answer: 2:1


Three circles with radii \( 0 < c < b < a \) are tangent to each other and to a line.

(a) Let radii \( a = 16 \) and \( b = 4 \). Find the distance \( d \) between the points of tangency of these two circles to the line.

(b) Let radii \( a = 16 \) and \( b = 4 \). Find the radius \( c \).

(c) The configuration is called nice if \( a \), \( b \), and \( c \) are all integers. Among all nice configurations, find the smallest possible value of \( c \).

Answer: 2


Three distinct vertices are randomly selected among the five vertices of a regular pentagon. Let \( p \) be the probability that the triangle formed by the chosen vertices is acute. Compute \( 10p \).

Answer: 5


In the figure below, all the small squares on the grid are equal. What is the measure of the angle $\angle A E F$? Justify your answer.

![grid figure](https://cdn.mathpix.com/cropped/2024_05_01_a50b5476db779c3986d5g-31.jpg?height=391&width=343&top_left_y=1901&top_left_x=931)

Answer: 90


We are considering numbers that can be expressed as \(10 \times n + 1\), where \( \mathbf{n} \) is a positive integer. For example: \(11 = 10 \times 1 + 1\) and \(331 = 10 \times 33 + 1\). If such a number cannot be expressed as the product of two smaller numbers of the form \(10 \times n + 1\) (these two smaller numbers can be equal), we call this number a "Zhonghuan number". For example, \(341 = 11 \times 31\), since it can be expressed as the product of two numbers of the form \(10 \times n + 1\), it is not a "Zhonghuan number". However, 11 cannot be expressed as the product of two smaller numbers of the form \(10 \times n + 1\), so it is a "Zhonghuan number". How many "Zhonghuan numbers" are there among \(11, 21, 31, \ldots, 991\)?

Answer: 87


1. How many different ways are there to choose 4 letters from $a, b, b, c, c, c, d, d, d, d$?
2. How many different four-digit numbers can be formed by choosing 4 digits from $1, 2, 2, 3, 3, 3, 4, 4, 4, 4$?

Answer: 175


Given the sequence \(\left\{a_{n}\right\}\) satisfying \(a_{1}=a > 2\), \(a_{2017} = 2017\), and for any positive integer \(n\), \(a_{n+1} = a_{n}^{2} - 2\). Compute \(\left\lfloor \frac{\sqrt{a-2}}{10^{6}} a_{1} a_{2} \cdots a_{2017} \right\rfloor\), where \(\lfloor x \rfloor\) denotes the greatest integer not exceeding the real number \(x\).

Answer: 2


There were 15 buns on a plate. Karlsson took three times as many buns as Little Boy, and Little Boy's dog Bimbo took three times fewer buns than Little Boy. How many buns are left on the plate? Explain your answer.

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: `Peter`, `Arnold`, `Eric`
 - Each mother is accompanied by their child: `Meredith`, `Bella`, `Fred`
 - The people keep unique animals: `bird`, `cat`, `horse`
 - People own unique car models: `toyota camry`, `tesla model 3`, `ford f150`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `cherry`

## Clues:
1. The person who owns a Toyota Camry is in the first house.
2. The person's child is named Bella is in the second house.
3. The Desert smoothie lover is directly left of Peter.
4. The person who likes Cherry smoothies is the person who owns a Ford F-150.
5. The person who loves mystery books is the person's child is named Meredith.
6. The person who loves science fiction books is directly left of Eric.
7. Eric is somewhere to the right of the person who loves mystery books.
8. The cat lover is the person who loves science fiction books.
9. There is one house between the person who keeps horses and the Watermelon smoothie lover.
 What is the value of attribute House for the person whose attribute Animal is bird? 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: `Carol`, `Bob`, `Alice`, `Arnold`, `Eric`, `Peter`
 - People use unique phone models: `samsung galaxy s21`, `google pixel 6`, `iphone 13`, `huawei p50`, `oneplus 9`, `xiaomi mi 11`
 - The people are of nationalities: `swede`, `chinese`, `norwegian`, `dane`, `german`, `brit`
 - Each person has a favorite color: `blue`, `red`, `yellow`, `green`, `white`, `purple`

## Clues:
1. Carol is not in the third house.
2. There is one house between the Dane and the British person.
3. Carol is the person whose favorite color is green.
4. Arnold is directly left of Alice.
5. Alice is the German.
6. The person who uses a OnePlus 9 is the person who loves purple.
7. The person who uses a Huawei P50 is not in the third house.
8. The person who uses a Samsung Galaxy S21 is in the fifth house.
9. The person who loves white is somewhere to the right of the person whose favorite color is red.
10. The person who uses a Samsung Galaxy S21 is Bob.
11. The Dane is the person who loves yellow.
12. The person who uses a Samsung Galaxy S21 is somewhere to the left of Peter.
13. The person who loves blue is Peter.
14. Peter is the British person.
15. The person who uses a Samsung Galaxy S21 is directly left of the person who uses an iPhone 13.
16. The Norwegian is the person who loves purple.
17. The person who uses a Xiaomi Mi 11 is the Chinese.
 What is the value of attribute Color for the person whose attribute Nationality is swede? Please reason step by step, and put your final answer within \boxed{}

Answer: white


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`, `Alice`
 - The people keep unique animals: `fish`, `cat`, `horse`, `bird`
 - Everyone has a unique favorite cigar: `pall mall`, `blue master`, `dunhill`, `prince`
 - They all have a unique favorite flower: `lilies`, `roses`, `daffodils`, `carnations`
 - People have unique favorite music genres: `classical`, `jazz`, `rock`, `pop`

## Clues:
1. The Dunhill smoker is Eric.
2. Eric is the cat lover.
3. The person who loves the boquet of lilies is the person who loves pop music.
4. The person partial to Pall Mall is the fish enthusiast.
5. The person who smokes Blue Master is the person who keeps horses.
6. Arnold is not in the third house.
7. The person who loves rock music is somewhere to the left of the person who loves a carnations arrangement.
8. The person who loves classical music is somewhere to the left of the fish enthusiast.
9. The person who loves rock music is Peter.
10. The person who loves jazz music is the person who smokes Blue Master.
11. The person who loves a bouquet of daffodils is in the fourth house.
 What is the value of attribute House for the person whose attribute Animal is cat? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Arnold`, `Peter`
 - People have unique favorite sports: `basketball`, `soccer`, `tennis`, `swimming`
 - Each person lives in a unique style of house: `craftsman`, `victorian`, `ranch`, `colonial`
 - People have unique favorite book genres: `mystery`, `science fiction`, `romance`, `fantasy`

## Clues:
1. The person in a Craftsman-style house is somewhere to the left of the person who loves science fiction books.
2. The person residing in a Victorian house is somewhere to the right of the person living in a colonial-style house.
3. Eric is somewhere to the left of Arnold.
4. The person who loves basketball is the person who loves romance books.
5. The person who loves soccer is somewhere to the right of Arnold.
6. Alice is the person who loves tennis.
7. The person who loves science fiction books is the person in a ranch-style home.
8. The person who loves tennis is somewhere to the left of the person in a Craftsman-style house.
9. Arnold is directly left of the person who loves mystery books.
 What is the value of attribute FavoriteSport for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: soccer


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`, `Bob`, `Alice`, `Peter`, `Arnold`
 - They all have a unique favorite flower: `carnations`, `daffodils`, `tulips`, `roses`, `lilies`
 - The people are of nationalities: `dane`, `swede`, `brit`, `norwegian`, `german`
 - Each person has an occupation: `artist`, `lawyer`, `engineer`, `teacher`, `doctor`

## Clues:
1. The person who is an engineer is Bob.
2. The person who is a lawyer is not in the second house.
3. The Dane is in the third house.
4. The British person is Eric.
5. The person who is a doctor is the Norwegian.
6. The person who loves the rose bouquet is not in the second house.
7. There is one house between Peter and the person who loves a bouquet of daffodils.
8. The German is the person who is a teacher.
9. The Swedish person is not in the first house.
10. Peter is directly left of the person who loves the boquet of lilies.
11. The person who loves the boquet of lilies is directly left of Bob.
12. The person who loves the rose bouquet is somewhere to the right of the person who is a lawyer.
13. Peter is somewhere to the left of Arnold.
14. There are two houses between the person who loves a carnations arrangement and the person who loves the boquet of lilies.
15. The person who is a teacher is not in the fourth house.
 What is the value of attribute Name for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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`, `Arnold`, `Eric`, `Alice`
 - Each person prefers a unique type of vacation: `mountain`, `cruise`, `beach`, `city`
 - Each person lives in a unique style of house: `craftsman`, `ranch`, `victorian`, `colonial`
 - The mothers' names in different houses are unique: `Kailyn`, `Holly`, `Aniya`, `Janelle`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`
 - Each person has a favorite color: `green`, `white`, `yellow`, `red`

## Clues:
1. Alice is directly left of the person whose favorite color is red.
2. The person who loves a bouquet of daffodils is The person whose mother's name is Janelle.
3. Eric is not in the first house.
4. The person who loves the boquet of lilies is directly left of the person living in a colonial-style house.
5. The person residing in a Victorian house is in the second house.
6. The person who loves the rose bouquet is the person who loves yellow.
7. The person whose mother's name is Aniya is the person who loves yellow.
8. The person who enjoys mountain retreats is in the first house.
9. The person who likes going on cruises is the person whose favorite color is red.
10. The person who loves yellow is in the second house.
11. The person who loves white is the person who prefers city breaks.
12. The person whose mother's name is Holly is not in the fourth house.
13. The person whose favorite color is red is directly left of Peter.
14. The person who loves a bouquet of daffodils is the person in a ranch-style home.
 What is the value of attribute Color for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: yellow


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`, `Peter`, `Arnold`, `Bob`, `Alice`
 - Each person lives in a unique style of house: `modern`, `craftsman`, `ranch`, `victorian`, `colonial`
 - The mothers' names in different houses are unique: `Penny`, `Kailyn`, `Holly`, `Janelle`, `Aniya`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `huawei p50`, `iphone 13`, `samsung galaxy s21`
 - Each person has a unique favorite drink: `coffee`, `water`, `root beer`, `tea`, `milk`
 - The people keep unique animals: `fish`, `dog`, `horse`, `bird`, `cat`

## Clues:
1. The person who uses a Google Pixel 6 is not in the first house.
2. The one who only drinks water is Alice.
3. The person living in a colonial-style house is somewhere to the right of the person who uses a Huawei P50.
4. The person who keeps horses is the person who uses a OnePlus 9.
5. The person in a ranch-style home is The person whose mother's name is Kailyn.
6. The root beer lover is the cat lover.
7. The person living in a colonial-style house is not in the fourth house.
8. The bird keeper is in the fourth house.
9. The tea drinker is Bob.
10. The tea drinker is somewhere to the right of The person whose mother's name is Kailyn.
11. The root beer lover is somewhere to the left of The person whose mother's name is Kailyn.
12. The person who keeps horses is the person in a modern-style house.
13. The person who uses an iPhone 13 is the person who likes milk.
14. The dog owner is the person who likes milk.
15. The person who uses a Google Pixel 6 is the person in a Craftsman-style house.
16. Eric is not in the second house.
17. The tea drinker is in the fourth house.
18. The person who keeps horses is in the third house.
19. The person in a modern-style house is The person whose mother's name is Penny.
20. The root beer lover is Peter.
21. The person whose mother's name is Aniya is not in the fourth house.
22. The person whose mother's name is Janelle is the one who only drinks water.
 What is the value of attribute House for the person whose attribute Drink is tea? 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: `Alice`, `Arnold`, `Carol`, `Bob`, `Eric`, `Peter`
 - People use unique phone models: `iphone 13`, `samsung galaxy s21`, `xiaomi mi 11`, `oneplus 9`, `google pixel 6`, `huawei p50`
 - Each person has a unique birthday month: `april`, `mar`, `feb`, `may`, `jan`, `sept`
 - Each person has an occupation: `nurse`, `doctor`, `engineer`, `teacher`, `artist`, `lawyer`

## Clues:
1. The person who uses a Google Pixel 6 is directly left of the person whose birthday is in March.
2. The person who is a nurse is Eric.
3. Arnold is directly left of the person whose birthday is in April.
4. Alice and the person whose birthday is in January are next to each other.
5. The person who is an artist is the person whose birthday is in May.
6. The person whose birthday is in September is not in the sixth house.
7. There is one house between the person whose birthday is in March and the person who is a nurse.
8. The person whose birthday is in March is the person who is a lawyer.
9. The person who uses a Xiaomi Mi 11 is directly left of the person who is a nurse.
10. The person who is a teacher is not in the sixth house.
11. Eric is the person who uses an iPhone 13.
12. The person who is an engineer and Peter are next to each other.
13. The person whose birthday is in January is in the third house.
14. The person who uses a Samsung Galaxy S21 is Bob.
15. The person who is a lawyer and the person whose birthday is in April are next to each other.
16. The person who uses a OnePlus 9 is somewhere to the right of the person who is a doctor.
 What is the value of attribute House for the person whose attribute Occupation is nurse? Please reason step by step, and put your final answer within \boxed{}

Answer: 6


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Eric`
 - Each person has a favorite color: `yellow`, `red`, `white`, `green`
 - Everyone has a unique favorite cigar: `dunhill`, `blue master`, `pall mall`, `prince`
 - Each person has a unique level of education: `high school`, `associate`, `master`, `bachelor`

## Clues:
1. The person whose favorite color is green is somewhere to the left of the person with a high school diploma.
2. Eric is the person whose favorite color is red.
3. There are two houses between Alice and the person who loves white.
4. The person who smokes Blue Master and the person partial to Pall Mall are next to each other.
5. The person who loves white is the person with a master's degree.
6. Arnold is in the fourth house.
7. The person who loves yellow is the person who smokes Blue Master.
8. The Dunhill smoker is Alice.
9. The person partial to Pall Mall is the person with a master's degree.
10. The person who loves yellow is the person with a bachelor's degree.
 What is the value of attribute Color for the person whose attribute Cigar is dunhill? Please reason step by step, and put your final answer within \boxed{}

Answer: green


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 person has an occupation: `engineer`, `teacher`, `doctor`
 - Each person has a favorite color: `white`, `yellow`, `red`
 - They all have a unique favorite flower: `carnations`, `lilies`, `daffodils`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`
 - People have unique heights: `average`, `short`, `very short`

## Clues:
1. The person who is a pizza lover is the person who is short.
2. The person who loves a carnations arrangement is in the first house.
3. There is one house between Eric and the person who loves eating grilled cheese.
4. The person who loves eating grilled cheese is somewhere to the right of Arnold.
5. The person who loves yellow is the person who is an engineer.
6. The person who loves a bouquet of daffodils is the person who is a teacher.
7. The person who is a pizza lover is the person who is an engineer.
8. The person who loves the spaghetti eater is the person who has an average height.
9. Arnold is somewhere to the right of the person who loves white.
 What is the value of attribute House for the person whose attribute Occupation is doctor? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


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`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`
 - Each person has a unique type of pet: `cat`, `fish`, `dog`
 - People have unique favorite book genres: `science fiction`, `mystery`, `romance`
 - Everyone has something unique for lunch: `spaghetti`, `pizza`, `grilled cheese`

## Clues:
1. The person who owns a Ford F-150 is not in the first house.
2. The person who owns a Ford F-150 is not in the third house.
3. The person who loves science fiction books is somewhere to the right of the person who is a pizza lover.
4. The person with an aquarium of fish is the person who is a pizza lover.
5. The person who is a pizza lover is somewhere to the left of Peter.
6. Peter is the person who owns a Tesla Model 3.
7. The person who loves science fiction books is directly left of the person who loves mystery books.
8. Arnold is the person who loves the spaghetti eater.
9. The person who has a cat is the person who loves the spaghetti eater.
 What is the value of attribute House for the person whose attribute CarModel is tesla model 3? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 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`, `Arnold`, `Alice`, `Eric`
 - Everyone has a unique favorite cigar: `blue master`, `dunhill`, `prince`, `blends`, `pall mall`
 - People have unique favorite music genres: `rock`, `classical`, `hip hop`, `pop`, `jazz`
 - Each person has a unique birthday month: `feb`, `april`, `sept`, `jan`, `mar`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`, `biography`, `fantasy`

## Clues:
1. The person who loves pop music is the person who loves romance books.
2. The person who loves fantasy books is in the second house.
3. The person who smokes Blue Master is the person who loves science fiction books.
4. The person who loves hip-hop music is the person whose birthday is in September.
5. The person who loves biography books is Arnold.
6. Alice is the person who loves science fiction books.
7. The person who loves jazz music is somewhere to the left of Eric.
8. The person who smokes Blue Master is the person whose birthday is in April.
9. The person who smokes many unique blends is not in the third house.
10. The Dunhill smoker is the person whose birthday is in February.
11. The person who loves science fiction books is in the fifth house.
12. The person who loves classical music is Bob.
13. The person who loves pop music is Eric.
14. The person who loves mystery books is the person partial to Pall Mall.
15. Peter is the person whose birthday is in February.
16. The person whose birthday is in March is the person who smokes many unique blends.
17. The person who loves classical music is not in the first house.
 What is the value of attribute House for the person whose attribute BookGenre is science fiction? 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: `Arnold`, `Eric`, `Alice`, `Peter`
 - Everyone has a unique favorite cigar: `prince`, `pall mall`, `blue master`, `dunhill`
 - Each person has a unique favorite drink: `milk`, `coffee`, `water`, `tea`
 - People have unique hair colors: `blonde`, `black`, `red`, `brown`

## Clues:
1. The person who has brown hair is not in the fourth house.
2. The person who likes milk is the person who has brown hair.
3. The person partial to Pall Mall is somewhere to the right of the person who has black hair.
4. Peter is somewhere to the left of the Dunhill smoker.
5. The person who smokes Blue Master is directly left of Alice.
6. The tea drinker is Peter.
7. Eric is the person who has brown hair.
8. Eric is not in the third house.
9. Alice is the Dunhill smoker.
10. The person who has blonde hair is the Prince smoker.
11. The one who only drinks water is not in the fourth house.
 What is the value of attribute House for the person whose attribute HairColor is black? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


In a tetrahedron  $ABCD$ , the four altitudes are concurrent at  $H$ . The line  $DH$  intersects the plane  $ABC$  at  $P$  and the circumsphere of  $ABCD$  at  $Q\neq D$ . Prove that  $PQ=2HP$ .

Answer: PQ = 2 HP


If the sum of the digits of a natural number $\alpha$ equals 7, then $a$ is called an "auspicious number". Arrange all "auspicious numbers" in ascending order $a_1,a_2,a_3,\cdots$, if $a_n=2005$, then $a_{5n}$=?

Answer: 52000


Let $n,k$, $1\le k\le n$ be fixed integers. Alice has $n$ cards in a row, where the card has position $i$ has the label $i+k$ (or $i+k-n$ if $i+k>n$). Alice starts by colouring each card either red or blue. Afterwards, she is allowed to make several moves, where each move consists of choosing two cards of different colours and swapping them. Find the minimum number of moves she has to make (given that she chooses the colouring optimally) to put the cards in order (i.e. card $i$ is at position $i$).

NOTE: edited from original phrasing, which was ambiguous.

Answer: n - \gcd(n, k)


Problem 8.7. In an acute-angled triangle $A B C$, the altitude $B H$ is drawn. It turns out that $C H=A B+A H$. How many degrees does the angle $B A C$ measure if $\angle A B C=84^{\circ}$ ?

![](https://cdn.mathpix.com/cropped/2024_05_06_b135fe79f2e6ed0cbffag-4.jpg?height=367&width=466&top_left_y=1395&top_left_x=502)

Answer: 64


2. Having walked $4 / 9$ of the length of the bridge, the traveler noticed that a car was catching up to him, but it had not yet entered the bridge. Then he turned back and met the car at the beginning of the bridge. If he had continued his movement, the car would have caught up with him at the end of the bridge. Find the ratio of the car's speed to the traveler's speed.

#

Answer: 9


Problem 12. (6 points)

Victor received a large sum of money as a birthday gift in the amount of 45 thousand rubles. The young man decided to save this part of his savings in dollars on a currency deposit. The term of the deposit agreement was 2 years, with an interest rate of 4.7% per annum, compounded quarterly. On the day the deposit was opened, the commercial bank bought dollars at a rate of 59.60 rubles per 1 US dollar and sold them at a rate of 56.60 rubles per 1 US dollar. What amount in US dollars will be on Victor's account at the end of the term of the deposit agreement (rounding to the nearest whole number)?

Answer: 873


## Task $5 / 83$

Let $n=\sum_{i=0}^{5} a_{i} \cdot 10^{i}$ be a 6-digit natural number, where $0 \leq a_{i} \leq 9, a_{i} \in N$, and $Q(n)=\sum_{i=0}^{5} a_{i}$ be its digit sum, and it holds that: $1. a_{i}>a_{k}$ for $i>k ; 2.10^{98}<n^{Q(n)}<10^{99}$.

Which numbers $n$ satisfy these conditions?

Answer: 653210


Example 2 In $\triangle A B C$, $\angle A=70^{\circ}$, point $I$ is the incenter. Given $A C+A I=B C$. Find the degree measure of $\angle B$.

---

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

Answer: 35


7. Given the set $M=\left\{(a, b) \mid\left(y^{2}+4\right) a^{2}-\right.$ $2(x y+b y+8) a+x^{2}+2 b x+2 b^{2}+12$ is a square of a linear expression in $x$ and $y$ $\}$. When $(a, b)$ takes all elements in the set $M$, the maximum distance from the point $(a, b)$ to the origin $O$ is $\qquad$

Answer: \dfrac{2\sqrt{21}}{3}


6. From the set of positive integers not exceeding 2013, $\{1,2, \cdots, 2013\}$, any two positive integers are chosen. Then the probability that they are exactly the solutions to the equation $x^{3}+y^{3}=x^{2} y^{2}$ is

Answer: \dfrac{1}{2013^2}


For the vertices of an $n$-sided polygon, we wrote different real numbers such that any written number is the product of the numbers written at the two adjacent vertices.

Determine the value of $n$!

Answer: 6


## Task 6 - 190936

For suitable natural numbers $n$, there are polyhedra with $n$ vertices and fewer than $n$ faces. For example, for $n=8$, a cuboid is such a polyhedron, as it has exactly 8 vertices and is bounded by exactly 6 planar faces (rectangles).

Investigate whether there exists a natural number $N$ such that for every natural number $n \geq N$, there is a polyhedron with $n$ vertices that is bounded by fewer than $n$ planar faces!

If this is the case, determine the smallest natural number $N$ with this property!

Answer: 6


An $m \times n$ chessboard where $m \le n$ has several black squares such that no two rows have the same pattern. Determine the largest integer $k$ such that we can always color $k$ columns red while still no two rows have the same pattern.

Answer: n - m + 1


$P T_{1}$ and $P T_{2}$ are secants of circle $\odot O$, intersecting $\odot O$ at $S_{1}$, $S_{2}$, respectively, and $P T_{1}=P T_{2}$. A line through $P$ intersects $\odot O$ at $Q$ and $R$ (with $Q$ between $R$ and $P$) and intersects $T_{1} T_{2}$ and $S_{1} S_{2}$ at $T$ and $S$. Prove that $\frac{1}{P Q}+\frac{1}{P R}=\frac{1}{P S}+\frac{1}{P T}$.

Answer: \frac{1}{PQ} + \frac{1}{PR} = \frac{1}{PS} + \frac{1}{PT}


5. The number $S$ is the result of the following sum:

$$
1+10+19+28+37+\cdots+10^{2013} .
$$

If the number $S$ is written out, how many times does the digit ' 5 ' appear in the result?

Answer: 4022


6. (12 points) Person A and Person B walk from location A to location B. B starts at 6:00 AM and walks at a constant speed; A starts at 8:00 AM and also walks at a constant speed. A's speed is 2.5 times that of B, but A needs to rest for half an hour after every half an hour of walking. After A starts, it will take $\qquad$ minutes for A to catch up with B.

Answer: 330


The acute angles $\alpha, \beta, \gamma$ satisfy $\sin^{2} \alpha + \sin^{2} \beta + \sin^{2} \gamma = 1$. Prove that $\frac{\pi}{2} < \alpha + \beta + \gamma < \frac{3\pi}{4}$.

Answer: \frac{\pi}{2} < \alpha + \beta + \gamma < \frac{3\pi}{4}


11.4. The lateral surface of a right circular cone is a circle of radius $R$, from which a sector corresponding to the central angle $(2-\sqrt{3}) \pi$ has been removed. Find the maximum value of the area of a plane section of the cone passing through its vertex.

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


13.3.6 ** Given $P(3,4)$ is a point inside the circle $x^{2}+y^{2}=64$, and two moving points $A$ and $B$ on the circumference of the circle satisfy $\angle A P B-\frac{\pi}{2}$. A rectangle $A P B Q$ is constructed with $A P$ and $B P$ as adjacent sides. Find the equation of the trajectory of point $Q$.

Answer: x^2 + y^2 = 103


Let $Q$ be a unit cube. We say a tetrahedron is "good" if all its edges are equal and all its vertices lie on the boundary of $Q$. Find all possible volumes of "good" tetrahedra.

Answer: \dfrac{1}{3}


## Task A-2.7.

The rows of a $50 \times 50$ table are labeled with numbers $a_{1}, \ldots, a_{50}$, and the columns with numbers $b_{1}, \ldots, b_{50}$. These 100 numbers are all distinct, and exactly 50 of them are rational numbers. The table is filled such that for $i, j=1,2, \ldots, 50$, the number $a_{i}+b_{j}$ is written in the cell $(i, j)$. Determine the maximum possible number of rational numbers written in the cells of the table.

Answer: 1250


2. Find all integer solutions to the equation $a^{b}=a b+2$.

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

However, it seems there's a redundancy in my response. The text has already been translated. Here is the translation:

2. Find all integer solutions to the equation $a^{b}=a b+2$.

Answer: (-1, 3)


14. $A, B, C$ three people went to Teacher $D$'s house to play. Teacher $D$ gave each of them a hat and wrote a four-digit number on each hat. It is known that these three four-digit numbers are all perfect squares (for example, $4=2^{2}, 100=10^{2}, 4.100$ are all squares of some number, such numbers are called perfect squares), and the tens digit of these three four-digit numbers is 0, and the units digit is not 0. Each child can only see the numbers on others' hats. These three children are very smart and honest, and the following conversation took place:
$A$ said: “The units digit of the numbers on $B$ and $C$'s hats are the same.”
$B$ and $C$ said at the same time: “After hearing what $A$ said, I know my number.”
$A$ said: “After hearing what $B$ and $C$ said, I also know my number, and the units digit of my number is an even number.”
Find: The sum of the numbers on $A, B, C$'s hats.

Answer: 14612


1. A palindrome is a natural number that reads the same from left to right as it does from right to left. Does there exist a five-digit palindrome that is equal to the sum of two four-digit palindromes?

Answer: 11011


3. The slant height of a cone is $l$, and the angle it forms with the base is $\theta$. The edge length of the inscribed cube in this cone is $\qquad$ (the cube has 4 vertices on the base of the cone and 4 vertices on the lateral surface of the cone).

Answer: \dfrac{\sqrt{2} l \sin\theta \cos\theta}{\sin\theta + \sqrt{2} \cos\theta}


## Task 3 - 170623

In the turning shop of a company, individual parts are turned from lead rods. Each lead rod produces one individual part.

The swarf obtained from the production of every 6 individual parts can be melted down to make one lead rod. (Any smaller amount of swarf is insufficient for this purpose.)

What is the maximum number of individual parts that can be produced in total from 36 rods according to this process?

Answer: 43


A magician and his assistant have a stack of cards, all with the same back and one of 2017 possible colors on the front (with one million cards for each color). The trick is as follows: the magician leaves the room, and the audience picks $n$ cards and places them face up in a row on the table. The assistant then turns over $n-1$ of these cards, leaving only one card face up. The magician re-enters the room, observes the cards on the table, and must guess the color of the face-down card. Find the minimum value of $n$ such that the magician and the assistant can perform this trick using a predetermined strategy.

Answer: 2018


15. For a positive integer $n$, let $S_{n}$ be
$$
\sum_{k=1}^{n} \sqrt{(2 k-1)^{2}+a_{k}^{2}}
$$

the minimum value. Where $a_{1}, a_{2}, \cdots, a_{\mathrm{n}}$ are positive integers, and their sum is 17. There is a unique value of $n$ for which $S_{n}$ is an integer, find $n$.

Answer: 12


7. In the Cartesian coordinate system $x O y$, the ellipse $\Gamma: \frac{x^{2}}{3}+y^{2}=1$ has its left vertex at $P$, and $A, B$ are two moving points on $\Gamma$. Then the range of $\overrightarrow{P A} \cdot \overrightarrow{P B}$ is $\qquad$.

Answer: [-\frac{1}{4}, 12]


For given integer $n \geq 3$, set $S =\{p_1, p_2, \cdots, p_m\}$ consists of permutations $p_i$ of $(1, 2, \cdots, n)$. Suppose that among every three distinct numbers in $\{1, 2, \cdots, n\}$, one of these number does not lie in between the other two numbers in every permutations $p_i$ ($1 \leq i \leq m$). (For example, in the permutation $(1, 3, 2, 4)$, $3$ lies in between $1$ and $4$, and $4$ does not lie in between $1$ and $2$.) Determine the maximum value of $m$.

Answer: 2^{n-1}


29. Let $p$ be a prime, $a, b \in \mathbf{N}^{*}$, satisfying: $p>a>b>1$. Find the largest integer $c$, such that for all $(p, a, b)$ satisfying the conditions, we have
$$p^{c} \mid\left(\mathrm{C}_{a p}^{b p}-\mathrm{C}_{a}^{b}\right)$$

Answer: 3


Example 8 $n$ teams are to hold a home-and-away double round-robin match (each pair of teams plays two matches, each having one home match), and each team can play multiple away matches within a week (from Sunday to Saturday). However, if a team has a home match in a certain week, no away matches for that team can be scheduled in the same week. If all matches can be completed within 4 weeks, find the maximum value of $n$.
(Note: A match held at Team A's venue is called Team A's home match and Team B's away match.)

Answer: 6


14. (6 points) As shown in the figure, even numbers starting from 2 are arranged in a clockwise right-angle spiral, with 4, 6, 10, 14, 20, 26, 34, ... appearing at the corners of the spiral. Then 2010 $\qquad$ (fill in “will” or “will not”) appear at the corner of the spiral.

Answer: will \ not


Problem 5. Small Island and Big Island have a rectangular shape and are divided into rectangular counties. In each county, a road is laid along one of the diagonals. On each island, these roads form a closed path that does not pass through any point more than once. Here is how Small Island is organized, with a total of 6 counties (see Fig. 1). Draw how Big Island might be organized if it has an odd number of counties. How many counties did you get?

[7 points] (A. V. Shapovalov)

![](https://cdn.mathpix.com/cropped/2024_05_06_611ab840a0f62c05d9afg-05.jpg?height=307&width=313&top_left_y=1453&top_left_x=300)

Fig. 1

![](https://cdn.mathpix.com/cropped/2024_05_06_611ab840a0f62c05d9afg-05.jpg?height=307&width=360&top_left_y=1453&top_left_x=791)

Fig. 2

Answer: 9


## Task B-1.7.

In a right-angled triangle $ABC$, the right angle is at vertex $C$, $|AB|=8$, and $\angle ABC=60^{\circ}$. The triangle $ABC$ is rotated around vertex $C$ by $30^{\circ}$ counterclockwise, mapping it to triangle $A'B'C$. What is the area of the common part of triangles $ABC$ and $A'B'C$?

Answer: 24 - 10\sqrt{3}


Once in the autumn, the Absent-Minded Scholar glanced at his old wall clock and saw that three flies had fallen asleep on the dial. The first was sleeping precisely at the 12 o'clock mark, while the other two were neatly positioned at the 2 o'clock and 5 o'clock marks. The Scholar made some measurements and determined that the hour hand posed no threat to the flies, but the minute hand would sweep them all away one by one. Find the probability that exactly 40 minutes after the Scholar noticed the flies, exactly two out of the three flies were swept away by the minute hand.

Answer: \dfrac{1}{2}


1. How many distinct permutations of the letters of the word REDDER are there that do not contain a palindromic substring of length at least two? (A substring is a contiguous block of letters that is part of the string. A string is palindromic if it is the same when read backwards.)

Answer: 6


38. It's the New Year, and the panda named Bao is performing on stilts. Bao stands on the stilts, and his height only accounts for $\frac{1}{4}$ of the total height of him and the stilts. During the performance, Bao accidentally breaks off a 20 dm section from each stilt. At this point, Bao stands on the broken stilts, and his height accounts for $\frac{1}{3}$ of the total height. What was the initial total height of Bao and the stilts in dm?

Answer: 80


1.3. Rex, Dzhulbars, Tarzan, Barbos and Sharik are frolicking on the lawn. Rex has bitten the one who has bitten Dzhulbars, Dzhulbars - the one who has bitten Tarzan, Tarzan - the one who has bitten Barbos, Barbos - the one who has bitten Sharik, Sharik - the one who has bitten Rex. Who, then, has bitten Rex?

## Second Round

Answer: Tarzan


2. Let the real-coefficient quadratic equation $x^{2}+a x+2 b-$ $2=0$ have two distinct real roots, one of which lies in the interval $(0,1)$, and the other in the interval $(1,2)$. Then the range of $\frac{b-4}{a-1}$ is . $\qquad$

Answer: \left( \dfrac{1}{2}, \dfrac{3}{2} \right)


The efficiency of an ideal heat engine is $40\%$. What will it become if the temperature of the heater is increased by $40\%$ and the temperature of the cooler is decreased by $40\%$?

Answer: 74\%


Example 1. As shown in the figure, in $\triangle A B C$, $A C=B C, \angle C$ $=20^{\circ}$. Also, $M$ is on $A C$, $N$ is on $B C$, and it satisfies $\angle B A N=50^{\circ}, \angle A B M=60^{\circ}$. Then $\angle N M B=$ $\qquad$ . (1991, Qinfen Cup Mathematics Invitational Competition)

Answer: 30


In the word PAMPELIŠKA, the same letters are to be replaced by the same non-zero digits, and different letters by different non-zero digits. Moreover, it must hold that the product of the digits of the resulting number is the square of some natural number. Find the largest number that can be obtained by replacing the letters under the given conditions.

(E. Patáková)

Answer: 8798643217


22. The chain has broken into five parts,

But I hope you will

Join them as quickly as you can,

Doing the minimum work.

To cut and solder a link-

It takes two minutes...

Remember, the task is given

For a very thoughtful look.

Note. Each link has three rings.

Answer: 6


25. Find the polynomial expression in $Z=x-\frac{1}{x}$ of $x^{5}-\frac{1}{x^{5}}$.

Answer: Z^5 + 5Z^3 + 5Z


3. Points $K, L$ and $M$ are the midpoints of sides $AB, BC$ and $CD$ of parallelogram $ABCD$. It turned out that quadrilaterals $KBLM$ and $BCDK$ are cyclic. Find the ratio $AC: AD$.

---

The translation is provided as requested, maintaining the original formatting and structure.

Answer: 2


A square is divided into \( n^{2} \) equal smaller squares. For a certain polyline, it is known that it passes through the centers of all the smaller squares (the polyline may intersect itself). What is the minimum number of segments of this polyline?

Answer: 2n - 2


Michelle is at the bottom-left corner of a $6 \times 6$ lattice grid, at $(0,0)$. The grid also contains a pair of one-time-use teleportation devices at $(2,2)$ and $(3,3)$. The first time Michelle moves to one of these points, she is instantly teleported to the other point and the devices disappear. If she can only move up or to the right in unit increments, in how many ways can she reach the point $(5,5)$?

Answer: 280


For every pair of real numbers $0 < a < b < 1$, we define sequences $\{x_n\}_{n \ge 0}$ and $\{y_n\}_{n \ge 0}$ by $x_0 = 0$, $y_0 = 1$, and for each integer $n \ge 1$:
\[
   x_n = (1 - a) x_{n - 1} + a y_{n - 1},
   \\ y_n = (1 - b) x_{n - 1} + b y_{n - 1}.
\]
The supermean of $a$ and $b$ is the limit of $\{x_n\}$ as $n$ approaches infinity. Over all pairs of real numbers $(p, q)$ satisfying:
\[
\left (p - \frac{1}{2} \right)^2 + \left (q - \frac{1}{2} \right)^2 \le \left(\frac{1}{10}\right)^2,
\]
the minimum possible value of the supermean of $p$ and $q$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$.

Answer: 307


A **Beaver-number** is a positive 5-digit integer whose digit sum is divisible by 17. Call a pair of **Beaver-numbers** differing by exactly $1$ a **Beaver-pair**. The smaller number in a **Beaver-pair** is called an **MIT Beaver**, while the larger number is called a **CIT Beaver**. Find the positive difference between the largest and smallest **CIT Beavers** (over all **Beaver-pairs**).

Answer: 79200


A group of $25$ friends were discussing a large positive integer. The first friend said, "It can be divided by $1$." The second friend said, "It can be divided by $2$." The third friend said, "And by $3$." The fourth friend added, "And by $4$." This pattern continued until everyone had made such a comment. If exactly $2$ friends were incorrect, and those two friends said consecutive numbers, what was the least possible integer they were discussing?

Answer: 787386600


An triangle with coordinates $(x_1,y_1)$, $(x_2, y_2)$, $(x_3,y_3)$ has centroid at $(1,1)$. The ratio between the lengths of the sides of the triangle is $3:4:5$. Given that \[x_1^3+x_2^3+x_3^3=3x_1x_2x_3+20\] and \[y_1^3+y_2^3+y_3^3=3y_1y_2y_3+21,\] the area of the triangle can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

Answer: 107


II. (25 points) Given that $D$ is a point inside $\triangle A B C$, $E$ is the midpoint of side $A C$, $A B=6, B C=10, \angle B A D=$ $\angle B C D, \angle E D C=\angle A B D$. Find the length of $D E$.

---

The above text has been translated into English, preserving the original text's line breaks and format.

Answer: 4


11.2. Several married couples came to the New Year's Eve party, each of whom had from 1 to 10 children. Santa Claus chose one child, one mother, and one father from three different families and took them for a ride in his sleigh. It turned out that he had exactly 3630 ways to choose the necessary trio of people. How many children could there be in total at this party?

(S. Volchonkov)

Answer: 33


In a particular system of units, a physical quantity can be expressed in terms of the electric charge $e$, electron mass $m_{e}$, Planck's constant $h$, and Coulomb's constant $k=\frac{1}{4 \pi \epsilon_{0}}$, where $\epsilon_{0}$ is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is $[B]=[e]^{\alpha}\left[m_{e}\right]^{\beta}[h]^{\gamma}[k]^{\delta}$. The value of $\alpha+\beta+\gamma+\delta$ is
Please give the answer directly without any intermediate steps.

Answer: 4


Let $ABCD$ be a cyclic quadrilateral with $AB = 5$, $BC = 10$, $CD = 11$, and $DA = 14$. The value of $AC + BD$ can be written as $\tfrac{n}{\sqrt{pq}}$, where $n$ is a positive integer and $p$ and $q$ are distinct primes. Find $n + p + q$.

Answer: 446


8.5. In the list $1,2, \ldots, 2016$, two numbers $a<b$ were marked, dividing the sequence into 3 parts (some of these parts might not contain any numbers at all). After that, the list was shuffled in such a way that $a$ and $b$ remained in their places, and no other of the 2014 numbers remained in the same part where they were initially. In how many ways could $a$ and $b$ have been chosen?

Answer: 508536


4- 195 Two drops of water fall successively from a steep cliff that is 300 meters high. When the first drop has fallen 0.001 millimeters, the second drop begins to fall. What is the distance between the two drops when the first drop reaches the bottom of the cliff (the answer should be accurate to 0.1 millimeters; air resistance is negligible)?

Answer: 34.6


3. Given $f(x)=a x^{3}+b x^{2}+c x+d$, if $1 \leqslant f(-1) \leqslant 2,1 \leqslant f(1) \leqslant 3,2 \leqslant f(2) \leqslant 4$, $-1 \leqslant f(3) \leqslant 1$, find the range of values for $f(4)$.

Answer: \left[ -\dfrac{87}{4}, 1 \right]


Let \( S=\left\{x_{1}, x_{2}, \ldots, x_{k+\ell}\right\} \) be a \((k+\ell)\)-element set of real numbers contained in the interval \([0,1]\); \( k \) and \(\ell\) are positive integers. A \( k \)-element subset \( A \subset S \) is called nice if
$$
\left|\frac{1}{k} \sum_{x_{i} \in A} x_{i} - \frac{1}{\ell} \sum_{x_{j} \in S \backslash A} x_{j} \right| \leq \frac{k+\ell}{2 k \ell} .
$$

Prove that the number of nice subsets is at least \(\frac{2}{k+\ell}\binom{k+\ell}{k}\).

Answer: \frac{2}{k+\ell}\binom{k+\ell}{k}


Cells of a $n*n$ square are filled with positive integers in the way that in the intersection of the $i-$th column and $j-$th row, the number $i+j$ is written. In every step, we can choose two non-intersecting equal rectangles with one dimension equal to $n$ and swap all the numbers inside these two rectangles with one another. ( without reflection or rotation ) Find the minimum number of moves one should do to reach the position where the intersection of the $i-$th column and $j-$row is written $2n+2-i-j$.

Answer: n


Problem 8. Seryozha placed numbers from 1 to 8 in the circles so that each number, except one, was used exactly once. It turned out that the sums of the numbers on each of the five lines are equal. Which number did Seryozha not use?

![](https://cdn.mathpix.com/cropped/2024_05_06_03b105aa15b6031ee02fg-3.jpg?height=326&width=443&top_left_y=174&top_left_x=812)

Answer: 6


7.5 In triangle ABC, the median BM is drawn. It is known that $\angle \mathrm{BAC}=30^{\circ}, \angle \mathrm{BMC}=45^{\circ}$. Find angle BAC.

Answer: 45^\circ


4. In the Cartesian coordinate plane $x O y$,
$\odot O: x^{2}+y^{2}=1$,
$\odot O_{1}:(x-3)^{2}+y^{2}=4$.
A tangent line is drawn from a point $M$ on the left half-axis of the $x$-axis to $\odot O$, touching $\odot O$ at point $A$, and intersecting $\odot O_{1}$ at points $B$ and $C$. If $A B=B C$, then the coordinates of point $M$ are $\qquad$

Answer: (-4, 0)


4. Let the sequence of real numbers $\left(x_{n}\right)_{n \geq 0}$ be such that $x_{0}=a>0$ and $x_{n+1}=x_{n}+\sqrt{1+x_{n}^{2}}, \forall n \in \mathbb{N}$.

Study the existence of the limit of the sequence $\left(y^{n} x_{n}\right)_{n \geq 1}$, where $y$ is a fixed real number.

Is it possible for the limit of the sequence $\left(y^{n} x_{n}\right)_{n \geq 1}$ to be 2015?

Dan Popescu

Answer: 2015


4. Given a circle centered at the origin with radius $R(R>0)$ intersects the line segment connecting $M(1,1)$ and $N\left(\frac{7}{4}, 0\right)$. Then the range of values for $R$ is $\qquad$.

Answer: \left[ \dfrac{7}{5}, \dfrac{7}{4} \right]


4.2. Find the sum of all roots of the equation $x^{2}-41 x+330=3^{x}\left(41-2 x-3^{x}\right)$.

Answer: 5


【Question 6】
A bottle contains 1001 blue balls, 1000 red balls, and 1000 green balls. At the same time, Xiaoming has enough balls of these three colors in his hand. Next, Xiaoming takes out two balls from the bottle each time, and then puts one or two balls back into the bottle according to the following rules:
(1) If a blue ball and a green ball are taken out, then put back one red ball;
(2) If a red ball and a green ball are taken out, then put back one red ball;
(3) If two red balls are taken out, then put back two blue balls;
(4) If the two balls taken out do not match the above three cases, then put back one green ball.

The above operations are repeated until only one ball is left in the bottle. The last remaining ball is $\qquad$ (fill in "red", "blue", "green", or "uncertain")

Answer: red


11. (5 points) The Yirong ferry has a speed of 40 kilometers per hour. On odd days, it travels downstream from $A$ to $B$, and on even days, it travels upstream from $B$ to $A$. (The water speed is 24 kilometers per hour.) On an odd day, the ferry lost power while traveling to a point $C$ and could only drift to $B$. The captain noted that the time taken on that day was $\frac{43}{18}$ times the original time on odd days. On an even day, the ferry lost power while traveling to a point $C$ and, during the drift, the repair crew worked for 1 hour to fix the ferry, after which it traveled to $A$ at twice its speed. The captain found that the time taken on that day was exactly the same as the original time on even days. What is the distance between $A$ and $B$ in kilometers?

Answer: 192


14. Line $A B$

and $A C$ are
tangent to circle $O$ at
points $B$ and $C$,
respectively. $P$ is a
point on the circle,
and the distances from $P$ to $A B$ and $A C$
are 4 cm and 6 cm, respectively. The distance from $P$ to $B C$ is $\qquad$ cm.

Answer: 2\sqrt{6}


6. Given that the graph of the function $y=f(x)$ is centrally symmetric about the point $(1,1)$ and axially symmetric about the line $x+y=0$. If $x \in(0,1)$, $f(x)=\log _{2}(x+1)$, then the value of $f\left(\log _{2} 10\right)$ is $\qquad$

Answer: \dfrac{17}{5}


On a $5 \times 7$ grid, 9 cells are marked. A pair of cells sharing a side is called interesting if at least one cell in the pair is marked. What is the maximum number of interesting pairs?

Answer: 35


5. Given that $P$ is a point on the right branch of the hyperbola $C: \frac{x^{2}}{2}-y^{2}=1$, and $l$ is an asymptote of the hyperbola $C$. The projection of $P$ onto $l$ is $Q$, and $F_{1}$ is the left focus of the hyperbola $C$. Then the minimum value of $\left|P F_{1}\right|+|P Q|$ is $\qquad$ .

Answer: 1 + 2\sqrt{2}


4. Solve the numerical riddle: TETA+BETA=GAMMA. (Different letters - different digits.)

#

Answer: 4940 + 5940 = 10880


Example 2 Solve the equation $3 x^{3}-[x]=3$

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

However, it seems the translation request was for the provided content itself. Here is the translation:

Example 2 Solve the equation $3 x^{3}-[x]=3$

Answer: \sqrt[3]{\dfrac{4}{3}}


$\left.\begin{array}{l}{\left[\begin{array}{l}\text { Cone }\end{array}\right]} \\ {[\underline{\text { Scrap (other) }}]}\end{array}\right]$

The axial section of the cone is an equilateral triangle with a side length of 1. Find the radius of the sphere that touches the axis of the cone, its base, and the lateral surface.

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


Two 10-digit integers are called neighbours if they diﬀer in exactly one digit (for example, integers $1234567890$ and $1234507890$ are neighbours). Find the maximal number of elements in the set of 10-digit integers with no two integers being neighbours.

Answer: 900000000