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`, `Eric`, `Bob`, `Peter`, `Arnold`
 - Each person has a unique birthday month: `mar`, `april`, `sept`, `feb`, `jan`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Kailyn`, `Penny`, `Aniya`
 - Each person has an occupation: `engineer`, `doctor`, `lawyer`, `artist`, `teacher`
 - People have unique hair colors: `red`, `blonde`, `black`, `gray`, `brown`

## Clues:
1. The person whose birthday is in March is in the fifth house.
2. The person whose birthday is in February is in the first house.
3. The person who is a doctor is Eric.
4. The person whose mother's name is Janelle is in the third house.
5. The person who is an artist is the person who has brown hair.
6. The person who is an artist is in the fourth house.
7. The person whose mother's name is Penny is somewhere to the left of the person who has black hair.
8. Peter is the person who has black hair.
9. The person who has gray hair is the person who is a teacher.
10. Alice is The person whose mother's name is Kailyn.
11. Arnold is somewhere to the right of the person whose birthday is in September.
12. The person who has brown hair is the person whose birthday is in January.
13. Arnold is the person who has blonde hair.
14. The person whose mother's name is Holly is the person who has black hair.
15. Peter is the person who is a lawyer.
16. The person whose birthday is in September is somewhere to the left of The person whose mother's name is Kailyn.
17. Alice is the person who has gray hair.
 What is the value of attribute Birthday for the person whose attribute HairColor is red? Please reason step by step, and put your final answer within \boxed{}

Answer: feb


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: `Bella`, `Meredith`, `Fred`
 - People use unique phone models: `samsung galaxy s21`, `iphone 13`, `google pixel 6`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`
 - Everyone has a unique favorite cigar: `blue master`, `prince`, `pall mall`
 - People have unique heights: `short`, `very short`, `average`

## Clues:
1. Eric is directly left of the Prince smoker.
2. The person who has an average height is somewhere to the right of Peter.
3. The person who uses a Samsung Galaxy S21 is in the second house.
4. The person who smokes Blue Master is in the third house.
5. The person who uses a Google Pixel 6 is Eric.
6. Arnold is somewhere to the right of the person who loves science fiction books.
7. The person who uses an iPhone 13 is the person's child is named Bella.
8. The person who is short is the Prince smoker.
9. The person who loves mystery books is the person who uses a Google Pixel 6.
10. The person's child is named Meredith and the person who is short are next to each other.
 What is the value of attribute Cigar for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: prince


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`
 - Each person has a unique birthday month: `sept`, `april`, `feb`, `jan`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`, `stew`
 - Each person has a unique hobby: `cooking`, `gardening`, `painting`, `photography`
 - Each person has an occupation: `teacher`, `artist`, `engineer`, `doctor`
 - Each mother is accompanied by their child: `Bella`, `Samantha`, `Fred`, `Meredith`

## Clues:
1. The person who is a teacher is the person who loves cooking.
2. The person whose birthday is in January is directly left of Eric.
3. The person who is an artist is not in the fourth house.
4. The person who loves the spaghetti eater is not in the third house.
5. The person's child is named Bella is the person who paints as a hobby.
6. The person whose birthday is in January is the person who is a teacher.
7. The person who paints as a hobby is the person who is a doctor.
8. The person whose birthday is in September is directly left of Arnold.
9. The person who is a pizza lover is Peter.
10. The person who loves the stew is somewhere to the left of the photography enthusiast.
11. Arnold is the person's child is named Meredith.
12. The person whose birthday is in January is directly left of the person whose birthday is in September.
13. The person whose birthday is in April is in the fourth house.
14. The person's child is named Fred is directly left of the person whose birthday is in April.
15. Alice is in the first house.
16. The person whose birthday is in April is the person who enjoys gardening.
 What is the value of attribute House for the person whose attribute Occupation is engineer? 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: `Bob`, `Peter`, `Eric`, `Alice`, `Arnold`, `Carol`
 - People have unique hair colors: `auburn`, `blonde`, `brown`, `black`, `red`, `gray`
 - People have unique heights: `very tall`, `average`, `very short`, `tall`, `super tall`, `short`

## Clues:
1. The person who has blonde hair is directly left of Bob.
2. Alice is in the fourth house.
3. The person who is short is Arnold.
4. The person who is tall is in the sixth house.
5. The person who has black hair is not in the fourth house.
6. The person who has red hair is Eric.
7. The person who is super tall is somewhere to the right of the person who has an average height.
8. The person who has blonde hair is Carol.
9. There is one house between the person who has gray hair and the person who has red hair.
10. The person who is very short is in the fifth house.
11. Bob is the person who has brown hair.
12. The person who has gray hair is in the third house.
13. The person who has blonde hair is the person who is very tall.
 What is the value of attribute House for the person whose attribute Height is average? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `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 CarModel for the person whose attribute Children is Meredith? Please reason step by step, and put your final answer within \boxed{}

Answer: toyota camry


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`, `Bob`, `Peter`, `Alice`, `Carol`, `Eric`
 - Everyone has something unique for lunch: `stew`, `grilled cheese`, `stir fry`, `soup`, `pizza`, `spaghetti`
 - People have unique heights: `tall`, `average`, `super tall`, `very short`, `very tall`, `short`
 - Each person has a unique favorite drink: `root beer`, `boba tea`, `coffee`, `water`, `tea`, `milk`
 - Each person has a unique type of pet: `hamster`, `fish`, `cat`, `dog`, `bird`, `rabbit`
 - People use unique phone models: `samsung galaxy s21`, `xiaomi mi 11`, `google pixel 6`, `iphone 13`, `huawei p50`, `oneplus 9`

## Clues:
1. The person who uses an iPhone 13 is in the third house.
2. Bob is the person who is tall.
3. The person who loves the soup is in the second house.
4. The root beer lover is directly left of the person who uses a Xiaomi Mi 11.
5. The person who uses a Huawei P50 is directly left of the person who loves eating grilled cheese.
6. The person who loves stir fry is the person who likes milk.
7. The person who loves eating grilled cheese is the person who is tall.
8. The person who uses a Xiaomi Mi 11 is the coffee drinker.
9. The person who uses a OnePlus 9 is Arnold.
10. The person who owns a rabbit is not in the fifth house.
11. The person with a pet hamster is somewhere to the right of the person who uses a Google Pixel 6.
12. The person who is super tall is the person with an aquarium of fish.
13. The person with an aquarium of fish is Alice.
14. The tea drinker is directly left of the person who is a pizza lover.
15. The person who uses a Samsung Galaxy S21 is Carol.
16. The person who is a pizza lover is the person who is short.
17. Arnold is the person who is very tall.
18. The person who loves the spaghetti eater is the person who uses a Google Pixel 6.
19. The boba tea drinker is somewhere to the right of the person who loves the soup.
20. The person with a pet hamster is not in the fifth house.
21. The person who is very tall is not in the second house.
22. The person who is super tall is somewhere to the left of Peter.
23. The person who is very short is the person who loves the spaghetti eater.
24. The person who keeps a pet bird is somewhere to the left of the person who loves the spaghetti eater.
25. The person with an aquarium of fish is directly left of Eric.
26. The person who owns a dog is the person who likes milk.
 What is the value of attribute PhoneModel for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: iphone 13


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`, `Arnold`, `Eric`, `Bob`, `Peter`
 - People have unique heights: `average`, `short`, `tall`, `very tall`, `very short`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Kailyn`, `Janelle`, `Aniya`
 - Each person has a unique type of pet: `cat`, `dog`, `hamster`, `fish`, `bird`
 - The people keep unique animals: `bird`, `dog`, `fish`, `horse`, `cat`
 - People have unique favorite sports: `swimming`, `basketball`, `baseball`, `tennis`, `soccer`

## Clues:
1. The person who loves basketball is the person who keeps a pet bird.
2. The person who loves soccer is directly left of the person who is very short.
3. The person whose mother's name is Aniya is directly left of the person who is tall.
4. The person who loves tennis is Eric.
5. The person whose mother's name is Holly is in the first house.
6. The person who owns a dog is somewhere to the right of Bob.
7. The person who owns a dog is the cat lover.
8. The person whose mother's name is Aniya is Peter.
9. The person who is very short is directly left of Alice.
10. The person whose mother's name is Janelle is the bird keeper.
11. The person who has a cat is directly left of the person who has an average height.
12. The person whose mother's name is Penny is the person with an aquarium of fish.
13. The person with an aquarium of fish is directly left of the dog owner.
14. The person who is very tall is somewhere to the right of the fish enthusiast.
15. The person who loves tennis is in the fourth house.
16. The person who loves basketball is directly left of the person who loves baseball.
 What is the value of attribute Height for the person whose attribute Mother is Penny? Please reason step by step, and put your final answer within \boxed{}

Answer: tall


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Bob`, `Peter`, `Alice`, `Eric`, `Arnold`
 - Each person has a unique birthday month: `feb`, `may`, `mar`, `april`, `sept`, `jan`
 - Each person prefers a unique type of vacation: `city`, `beach`, `cultural`, `mountain`, `camping`, `cruise`
 - People have unique hair colors: `blonde`, `brown`, `black`, `gray`, `auburn`, `red`

## Clues:
1. Arnold is directly left of the person whose birthday is in April.
2. The person who loves beach vacations is not in the second house.
3. The person whose birthday is in January is in the second house.
4. The person who has red hair is not in the fourth house.
5. The person who likes going on cruises and the person who enjoys camping trips are next to each other.
6. Eric is the person whose birthday is in February.
7. Arnold is somewhere to the left of the person whose birthday is in February.
8. The person who prefers city breaks is not in the second house.
9. The person whose birthday is in February is the person who enjoys mountain retreats.
10. Alice and Carol are next to each other.
11. The person whose birthday is in September is the person who likes going on cruises.
12. The person who has brown hair is directly left of the person who has blonde hair.
13. The person whose birthday is in March is in the sixth house.
14. The person who has black hair is Alice.
15. The person who has gray hair is in the third house.
16. The person who has black hair is the person whose birthday is in September.
17. Peter is not in the third house.
18. There are two houses between the person who loves beach vacations and the person whose birthday is in February.
 What is the value of attribute House for the person whose attribute HairColor is brown? 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`, `Alice`, `Peter`, `Arnold`
 - Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `desert`, `watermelon`
 - People have unique favorite sports: `soccer`, `tennis`, `basketball`, `swimming`
 - People own unique car models: `tesla model 3`, `toyota camry`, `honda civic`, `ford f150`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`

## Clues:
1. The person who owns a Tesla Model 3 is the person who loves the rose bouquet.
2. Peter is the Dragonfruit smoothie lover.
3. The Desert smoothie lover is the person who owns a Toyota Camry.
4. The person who loves tennis is in the first house.
5. The person who owns a Toyota Camry and the person who loves basketball are next to each other.
6. Arnold is the person who loves basketball.
7. The person who owns a Honda Civic is the person who loves a bouquet of daffodils.
8. Eric is the person who loves the rose bouquet.
9. The Watermelon smoothie lover is not in the first house.
10. The person who owns a Honda Civic is somewhere to the right of the Desert smoothie lover.
11. The person who loves basketball is the person who loves the boquet of lilies.
12. The person who loves tennis and the person who loves soccer are next to each other.
 What is the value of attribute Flower for the person whose attribute Smoothie is watermelon? Please reason step by step, and put your final answer within \boxed{}

Answer: lilies


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Alice`, `Eric`, `Bob`, `Peter`, `Arnold`
 - Each person has a unique birthday month: `jan`, `mar`, `feb`, `may`, `april`, `sept`
 - Each person prefers a unique type of vacation: `city`, `camping`, `beach`, `cruise`, `cultural`, `mountain`
 - Each person has a unique favorite drink: `milk`, `water`, `tea`, `root beer`, `coffee`, `boba tea`

## Clues:
1. There is one house between the person who goes on cultural tours and Alice.
2. The person whose birthday is in February is Peter.
3. The person whose birthday is in January is the person who enjoys camping trips.
4. Eric and Arnold are next to each other.
5. Carol is the person who likes going on cruises.
6. The person whose birthday is in January is directly left of Arnold.
7. The root beer lover is somewhere to the right of the person whose birthday is in April.
8. The person whose birthday is in September is the tea drinker.
9. The coffee drinker is Alice.
10. The person whose birthday is in March is Alice.
11. The person who prefers city breaks is the one who only drinks water.
12. The person whose birthday is in September is in the second house.
13. Bob is directly left of the boba tea drinker.
14. There is one house between the coffee drinker and the one who only drinks water.
15. The person who loves beach vacations and the coffee drinker are next to each other.
 What is the value of attribute House for the person whose attribute Birthday is jan? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


A researcher receives a container for analysis holding approximately 150 samples of oil. Each sample has specific characteristics regarding sulfur content - either low-sulfur or high-sulfur - and density - either light or heavy. The relative frequency (statistical probability) that a randomly chosen sample is heavy oil is $\frac{2}{11}$. The relative frequency that a randomly chosen sample is light oil with low sulfur content is $\frac{7}{13}$. How many samples of high-sulfur oil are there in the container if there are no low-sulfur samples among the heavy oil samples?

Answer: 66


Let \( S = \{1, 2, 3, \ldots, 100\} \). Find the largest integer \( k \) such that \( S \) has \( k \) distinct non-empty subsets with the property that for any two distinct subsets among these \( k \) subsets, if their intersection is non-empty, then the smallest element in their intersection is different from the largest elements in each of these two subsets.

Answer: 2^{99}-1


On a board, nine quadratic trinomials are written: \(x^{2}+a_{1} x+b_{1}, x^{2}+a_{2} x+b_{2}, \ldots, x^{2}+a_{9} x+b_{9}\). It is known that the sequences \(a_{1}, a_{2}, \ldots, a_{9}\) and \(b_{1}, b_{2}, \ldots, b_{9}\) are arithmetic progressions. It turns out that the sum of all nine trinomials has at least one root. What is the maximum number of the original trinomials that may not have any roots?

Answer: 4


A square-shaped plaza with a side length of 20 meters is being renovated. If the entire plaza were paved with cheaper light-colored tiles, the material cost would be 100,000 CZK. If the entire plaza were covered with more expensive dark-colored tiles, the material cost would be 300,000 CZK. However, the architect designed a central light-colored square, bordered by a 2-meter-wide strip of dark tiles, with the rest of the plaza covered in the same light-colored tiles as the center. According to this design, the cost of materials for the dark section will be the same as the total material cost for the light sections.

Determine:

a) The cost of materials to pave the plaza according to this project,

b) The side length of the light-colored square in the central part of the plaza.

(L. Šimůnek)

Answer: 10.5


A triangle has side lengths 18, 24, and 30. Find the area of the triangle whose vertices are the incenter, circumcenter, and centroid of the original triangle.

Answer: 6


The volume of water increases by 10% upon freezing. By what percentage does the volume of ice decrease upon melting?

Answer: 9\frac{1}{11}


Using different letters to represent different digits, a five-digit number $\overline{ABCDE}$ has differences between the digits at adjacent positions represented by $F, G, H, I$. What is the maximum possible value of $\overline{ABCDE}$?

Answer: 98274


Ten men sit side by side at a long table, all facing the same direction. Each of them is either a knight (and always tells the truth) or a knave (and always lies). Each of the people announces: "There are more knaves on my left than knights on my right." How many knaves are in the line?

Answer: 5


The digits are wearing masks (identical digits wear identical masks, different digits wear different masks). Which digit is hidden behind each mask?

Under the elephant mask, the digit is:

Under the mouse mask, the digit is:

Under the pig mask, the digit is:

Under the panda mask, the digit is:

Answer: 1


Let \( f \) be a polynomial with integer coefficients such that the greatest common divisor of all its coefficients is 1. For any \( n \in \mathbb{N} \), \( f(n) \) is a multiple of 85. Find the smallest possible degree of \( f \).

Answer: 17


Suppose you live in a small town and are well-acquainted with all its residents. The townspeople, recognizing your economic literacy, propose that you open a staffing agency where you represent all employable residents to employers in the labor market. The townspeople will pay you 1% of their earned wages, with no other means to find employment. You have calculated that your costs for placing townspeople into jobs are given by \(TC(L)=\frac{1}{3200}L^{2}\), where \(L\) is the number of labor hours that the townspeople will work at enterprises. The townspeople trust you to the extent that they are willing to work for any wage you negotiate on their behalf. The town has only one major enterprise that produces a finished product using the technology \(q=16\sqrt{L}\) and sells it in a perfectly competitive market at a price of 1.

After consideration, you realize there are two possible negotiation schemes with the enterprise:
1) You propose a wage rate \(w\), and the enterprise decides the quantity of labor (in hours) it will use at that wage rate.
2) The enterprise proposes a wage rate \(w\), and you decide the volume of employment (in hours) you will offer to the townspeople.

Regardless of the negotiation scheme, all employable citizens work the same number of hours.

(a) Which scheme would you personally prefer, and what amount would your staffing agency earn under this scheme?

(b) Is it truly beneficial for the townspeople that you represent their interests? Justify your answer.

Answer: 0.24


Let \( n \) be an integer greater than 1. If all digits of \( 9997n \) are odd, find the smallest possible value of \( n \).

Answer: 3335


A $3 \times 3 \times 3$ cube composed of $27$ unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a $3 \times 3 \times 1$ block (the order is irrelevant) with the property that the line joining the centers of the two cubes makes a $45^\circ$ angle with the horizontal plane.

Answer: 60


Square $ABCD$ is divided into four rectangles by lines $EF$ and $GH$. Line $EF$ is parallel to $AB$, and line $GH$ is parallel to $BC$. It is given that $\angle BAF = 18^\circ$. The lines $EF$ and $GH$ intersect at point $P$. The area of rectangle $PFCH$ is twice that of rectangle $AGPE$. Given that the value of $\angle FAH$ in degrees is $x$, find the nearest integer to $x$.

Answer: 45


Let $ABC$ be a triangle with area $5$ and $BC = 10$. Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G$. Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2 + AC^2$.

Answer: 200


For each positive integer $n$ (written with no leading zeros), let $t(n)$ equal the number formed by reversing the digits of $n$. For example, $t(461) = 164$ and $t(560) = 65$. Determine the number of three-digit positive integers $m$ for which $m + t(t(m))$ is odd.

Answer: 50


There are 300 points in space. Four planes $A$, $B$, $C$, and $D$ each have the property that they split the 300 points into two equal sets. (No plane contains one of the 300 points.) What is the maximum number of points that can be found inside the tetrahedron whose faces are on $A$, $B$, $C$, and $D$?

Answer: 100


A function $f$ is defined on the positive integers by \[ \left\{ \begin{array}{rcl} f(1) &=& 1, \\ f(3) &=& 3, \\ f(2n) &=& f(n), \\ f(4n+1) &=& 2f(2n+1)-f(n), \\ f(4n+3) &=& 3f(2n+1)-2f(n), \end{array} \right. \] for all positive integers $n$. Determine the number of positive integers $n$, less than or equal to 1988, for which $f(n) = n$.

Answer: 92


Suppose we choose two numbers $x, y \in [0,1]$ uniformly at random. If the probability that the circle with center $(x, y)$ and radius $|x-y|$ lies entirely within the unit square $[0,1] \times [0,1]$ is written as $\frac{p}{q}$ with $p$ and $q$ relatively prime nonnegative integers, then what is $p^2+q^2$?

Answer: 10


On the number line, consider the point $x$ that corresponds to the value $10$. Consider $24$ distinct integer points $y_1, y_2, \ldots, y_{24}$ on the number line such that for all $k$ where $1 \leq k \leq 12$, $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of
\[\sum_{n=1}^{24} (|y_n-1| + |y_n+1|).\]

Answer: 494


For each positive integer $k$, let $S(k)$ be the sum of its digits. For example, $S(21) = 3$ and $S(105) = 6$. Let $n$ be the smallest integer for which $S(n) - S(5n) = 2013$. Determine the number of digits in $n$.

Answer: 504


In the quadrilateral $A B C D$ (with internal angles $\alpha, \beta, \gamma, \delta$ it holds: $\alpha=\gamma=90^{\circ},|A D|=|C D|$ and $\beta=\delta+100^{\circ}$. Let $M$ be the intersection of the angle bisectors of $\angle D A C$ and $\angle A C D$. What is the measure of the internal angle at vertex $M$ in triangle $A M D$.

(Ptáčková)

Answer: 125^\circ


## 236. How old is Willie?

- Is this problem your teacher gave you? - I asked Willie. - It looks quite boring.
- No, - Willie replied, - I came up with it myself. - See this polynomial? My age is its root. I mean, if you substitute the number of years I was when I last celebrated my birthday for \( x \), the polynomial will turn into zero.
- Hold on, - I noted, - this doesn't seem too hard. Our equation has integer coefficients, and we need to find an integer root... Let me try \( x = 7 \ldots \) No, it results in 77.
- Do I look that young? - Willie asked.
- Alright, let's try a larger integer... No, now it results in 85, not zero.
- You're always joking, - Willie said. - You know very well that I'm not a little kid.

How old is Willie?

Answer: 14


Exercise 15. Théo places tokens in the cells of a $30 \times 30$ grid while following these rules:

$\triangleright$ Each cell contains at most one token.

$\triangleright$ For each empty cell, there is, among the cells in the same row or the same column, at least one cell containing a token.

$\triangleright$ For each token, there is, among the cells in the same row or the same column, at most one other cell containing a token.

Determine the smallest integer $k$ satisfying the following property: regardless of the arrangement chosen by Théo, each $k \times k$ square of the grid contains at least one cell with a token.

Answer: 21


4. In a football tournament, only teams from Small Town and Big City participated. There were 9 more teams from Big City than from Small Town. Each team met exactly once, with the winning team getting 1 point, the losing team 0 points, and no draws were possible. Teams from Big City scored 9 times as many points as teams from Small Town. Determine the maximum possible number of wins for the best team from Small Town.

The task should be solved independently. You have 210 minutes for solving. The use of notes, literature, or a pocket calculator is not allowed.

49th Mathematical Competition for High School Students in Slovenia

Velenje, April 16, 2005

## Tasks for 4th Year

Answer: 11


13. $(x, y)=\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\left(x_{1}<x_{2}\right)$ are two integer solutions of the equation $x^{2}-y^{2}-2 x+6 y-8=0$. In the Cartesian coordinate system, point $A\left(x_{1}, y_{1}\right)$ and point $B\left(x_{2}, y_{2}\right)$ are symmetric about $P(1,3)$, and point $C(5,-1)$. If $\triangle A B C$ has exactly 21 integer points (points with integer coordinates) on its sides and inside, then the coordinates of point $A$ are $\qquad$.

Answer: (-1,1)


5. Tanya was sequentially writing down numbers of the form $n^{7}-1$ for natural numbers $n=2,3, \ldots$ and noticed that for $n=8$ the resulting number is divisible by 337. For what smallest $n>1$ will she get a number divisible by $2022?$

Answer: 79


37. Tetrahedron. There are 6 rods of different lengths, and it is known that in any order, a tetrahedron (triangular pyramid) can be formed from them. How many different tetrahedrons can result from this?

Answer: 30


3. On the surface of the cube $A B C D E F G H$, a broken line composed of four identical segments is constructed in the faces $A B F E, B C G F, C D H G$ and $G H E F$, which starts from vertex $A$ and ends at vertex $E$. Determine in what ratio this broken line divides the edge $C G$.

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


4. As shown in the figure, in the square $A B C D$ with side length 2, $E$ is the midpoint of $A B$. Now, $\triangle A E D$ and $\triangle B E C$ are folded along $E D$ and $E C$ respectively, so that $E A$ and $E B$ coincide, forming a tetrahedron. The volume of this tetrahedron is $\qquad$

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


The point $M$ lies inside the rhombus $ABCD$. It is known that $\angle DAB=110^o$, $\angle AMD=80^o$, $\angle BMC= 100^o$. What can the angle $\angle AMB$ be equal?

Answer: 80^\circ


(T. Eisenkölbl) Given is a natural number $n \geq 3$. On a board, the numbers $1,2,3, \ldots, n$ are written. In each move, one chooses two numbers and replaces them with their arithmetic mean. This is done until only one number remains on the board.

Determine the smallest integer that can be achieved at the end through a suitable sequence of moves.

Answer: 2


The 42nd question: Given infinite real sequences $a_{1}, a_{2}, \ldots$ and $b_{1}, b_{2}, \ldots$, for $n \geq 1$, we have $a_{n+1}+b_{n+1}=\frac{a_{n}+b_{n}}{2}$, $a_{n+1} b_{n+1}=\sqrt{a_{n} b_{n}}$. Suppose $b_{2016}=1$ and $a_{1}>0$, find all possible values of $a_{1}$.

---

The 42nd question: Given infinite real sequences \(a_{1}, a_{2}, \ldots\) and \(b_{1}, b_{2}, \ldots\), for \(n \geq 1\), we have \(a_{n+1} + b_{n+1} = \frac{a_{n} + b_{n}}{2}\) and \(a_{n+1} b_{n+1} = \sqrt{a_{n} b_{n}}\). Suppose \(b_{2016} = 1\) and \(a_{1} > 0\), find all possible values of \(a_{1}\).

Answer: 2^{2015}


Determine the absolute value of the sum \[ \lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor, \] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.

You may use the fact that $\sin{n^\circ}$ is irrational for positive integers $n$ not divisible by $30$.

Answer: 178


Isaac repeatedly flips a fair coin. Whenever a particular face appears for the $2n+1$th time, for any nonnegative integer $n$, he earns a point. The expected number of flips it takes for Isaac to get $10$ points is $\frac{a}{b}$ for coprime positive integers $a$ and $b$. Find $a + b$.

Answer: 39


Let $T=TNFTPP$. Points $A$ and $B$ lie on a circle centered at $O$ such that $\angle AOB$ is right. Points $C$ and $D$ lie on radii $OA$ and $OB$ respectively such that $AC = T-3$, $CD = 5$, and $BD = 6$. Determine the area of quadrilateral $ACDB$.

Answer: 44


Unit square $ABCD$ is divided into four rectangles by lines $EF$ and $GH$. Line $EF$ is parallel to $AB$, and line $GH$ is parallel to $BC$. The point $P$ is the intersection of $EF$ and $GH$. It is given that $BF = \frac{1}{4}$ and $BF + DH = FH$. Calculate the nearest integer to the degree measure of $\angle FAH$.

Answer: 45


Let $p(x)$ be a nonzero polynomial of degree less than $1992$ having no nonconstant factor in common with $x^3 - x$. Let $$ \frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 - x} \right) = \frac{f(x)}{g(x)} $$ for polynomials $f(x)$ and $g(x)$. Find the smallest possible degree of $f(x)$.

Answer: 3984


In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$, he starts at $A$ and walks to $C$. Then, he walks to the midpoint of side $AD$, say $A_1$. Then, he walks to the midpoint of side $CD$, say $C_1$, and then the midpoint of $A_1D$, which is $A_2$. He continues in this fashion indefinitely. The total length of his path, if $AB=5$ and $BC=12$, is of the form $a + b\sqrt{c}$. Find $\frac{abc}{4}$.

Answer: 793


Let $a, b, c$, and $d$ be real numbers. The six sums of two numbers $x$ and $y$, different from the previous four, are $117$, $510$, $411$, and $252$, in no particular order. Determine the maximum possible value of $x + y$.

Answer: 804


Consider a $2008 \times 2008$ chessboard. Let $M$ be the smallest number of rectangles that can be drawn on the chessboard such that the sides of every cell of the board are contained within the sides of one of the rectangles. Find the value of $M$. (For example, for a $2 \times 3$ chessboard, the value of $M$ is 3.)

Answer: 2009


A circle intersects an equilateral triangle $\triangle XYZ$ at points $A$, $B$, $C$, $D$, $E$, and $F$ such that the points $X$, $A$, $B$, $Y$, $C$, $D$, $Z$, $E$, and $F$ lie on the equilateral triangle in that order. Given:

- $AC^2 + CE^2 + EA^2 = 1900$
- $BD^2 + DF^2 + FB^2 = 2092$

Compute the positive difference between the areas of triangles $\triangle ACE$ and $\triangle BDF$.

Answer: 96


Define a sequence $\{a_{n}\}_{n=1}^{\infty}$ with $a_{1} = 1$ and $a_{n+1} = a_{n} + \lfloor \sqrt{a_{n}} \rfloor$ for all $n \geq 1$. Determine the smallest $N$ such that $a_{N} > 2017$.

Answer: 95


Given 2005 distinct numbers $a_1, a_2, \dots, a_{2005}$. By one question, we may take three different indices $1 \le i < j < k \le 2005$ and find out the set of numbers $\{a_i, a_j, a_k\}$ (unordered, of course). Find the minimal number of questions necessary to determine all numbers $a_i$.

Answer: 1003


Let $A_1B_1C_1$, $A_2B_2C_2$, and $A_3B_3C_3$ be three triangles in the plane. For $1 \le i \le 3$, let $D_i$, $E_i$, and $F_i$ be the midpoints of $B_iC_i$, $A_iC_i$, and $A_iB_i$, respectively. Furthermore, for $1 \le i \le 3$ let $G_i$ be the centroid of $A_iB_iC_i$.

Suppose that the areas of the triangles $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$, $D_1D_2D_3$, $E_1E_2E_3$, and $F_1F_2F_3$ are $2$, $3$, $4$, $20$, $21$, and $2020$, respectively. Compute the largest possible area of $G_1G_2G_3$.

Answer: 917


There is a sequence defined as follows:
- $a(2) = 0$
- $a(3) = 1$
- For $n \geq 4$, $a(n) = a\left(\left\lfloor \frac{n}{2} \right\rfloor\right) + a\left(\left\lceil \frac{n}{2} \right\rceil\right)$

Find $a(2014)$.

Note: $\left\lfloor \frac{n}{2} \right\rfloor$ and $\left\lceil \frac{n}{2} \right\rceil$ denote the floor function (largest integer $\leq \frac{n}{2}$) and the ceiling function (smallest integer $\geq \frac{n}{2}$), respectively.

Answer: 34


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`
 - The mothers' names in different houses are unique: `Janelle`, `Holly`, `Aniya`, `Kailyn`
 - Everyone has a favorite smoothie: `watermelon`, `dragonfruit`, `desert`, `cherry`
 - People have unique heights: `tall`, `average`, `short`, `very short`
 - Each person has a unique level of education: `high school`, `associate`, `master`, `bachelor`

## Clues:
1. The person whose mother's name is Janelle is in the third house.
2. The Desert smoothie lover is the person with a master's degree.
3. The Desert smoothie lover is not in the first house.
4. The person who is very short is somewhere to the left of the person with a high school diploma.
5. Eric and the person who likes Cherry smoothies are next to each other.
6. The person with a high school diploma is not in the third house.
7. The person whose mother's name is Kailyn is the person with an associate's degree.
8. The person who likes Cherry smoothies is The person whose mother's name is Aniya.
9. The person who is tall is The person whose mother's name is Janelle.
10. Arnold is somewhere to the right of the person who has an average height.
11. The Dragonfruit smoothie lover is directly left of the person who is short.
12. The person who is tall is Alice.
 What is the value of attribute Smoothie for the person whose attribute Height is very short? Please reason step by step, and put your final answer within \boxed{}

Answer: watermelon


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`, `Alice`
 - People have unique favorite book genres: `mystery`, `fantasy`, `romance`, `science fiction`
 - Each person prefers a unique type of vacation: `cruise`, `mountain`, `city`, `beach`
 - They all have a unique favorite flower: `roses`, `daffodils`, `carnations`, `lilies`
 - People have unique favorite music genres: `pop`, `rock`, `jazz`, `classical`

## Clues:
1. Alice is the person who loves jazz music.
2. Eric is the person who loves romance books.
3. The person who loves a carnations arrangement is in the first house.
4. The person who loves a bouquet of daffodils is the person who loves science fiction books.
5. The person who enjoys mountain retreats is in the third house.
6. The person who enjoys mountain retreats is the person who loves mystery books.
7. The person who loves science fiction books is Peter.
8. The person who prefers city breaks is in the second house.
9. The person who prefers city breaks and the person who loves classical music are next to each other.
10. The person who loves beach vacations is not in the first house.
11. The person who loves romance books is the person who loves pop music.
12. The person who loves romance books is somewhere to the left of Alice.
13. The person who loves the boquet of lilies is the person who loves mystery books.
14. Peter is somewhere to the left of Alice.
 What is the value of attribute MusicGenre for the person whose attribute BookGenre is science fiction? Please reason step by step, and put your final answer within \boxed{}

Answer: rock


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`
 - People have unique heights: `very tall`, `average`, `tall`, `very short`, `short`
 - Everyone has something unique for lunch: `stew`, `grilled cheese`, `spaghetti`, `pizza`, `stir fry`

## Clues:
1. Alice is the person who is short.
2. The person who is tall is in the third house.
3. The person who has an average height is not in the second house.
4. The person who has an average height is somewhere to the left of the person who loves the stew.
5. The person who loves stir fry is Arnold.
6. The person who is a pizza lover is the person who is tall.
7. Eric is the person who is tall.
8. Bob is somewhere to the right of Arnold.
9. The person who loves eating grilled cheese is somewhere to the right of Eric.
10. The person who is very short is somewhere to the left of Arnold.
 What is the value of attribute Height for the person whose attribute Food is pizza? Please reason step by step, and put your final answer within \boxed{}

Answer: tall


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`, `Arnold`, `Peter`
 - Each mother is accompanied by their child: `Meredith`, `Fred`, `Bella`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `desert`
 - Each person prefers a unique type of vacation: `mountain`, `city`, `beach`
 - Each person has a favorite color: `white`, `red`, `yellow`

## Clues:
1. There is one house between the person's child is named Bella and the person's child is named Meredith.
2. The person who likes Cherry smoothies is the person who loves beach vacations.
3. The person who prefers city breaks is Arnold.
4. The person whose favorite color is red is the person's child is named Fred.
5. The person's child is named Fred is the Watermelon smoothie lover.
6. Eric is not in the first house.
7. The person's child is named Meredith is the person who loves white.
8. The person who loves white is the Desert smoothie lover.
9. The Desert smoothie lover is Eric.
 What is the value of attribute Smoothie for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: watermelon


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`, `Arnold`, `Peter`
 - People own unique car models: `ford f150`, `tesla model 3`, `toyota camry`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `city`
 - The people are of nationalities: `dane`, `swede`, `brit`
 - People have unique hair colors: `blonde`, `black`, `brown`
 - Each person lives in a unique style of house: `colonial`, `ranch`, `victorian`

## Clues:
1. The Swedish person is the person who owns a Ford F-150.
2. The person who has blonde hair is not in the second house.
3. The person who owns a Toyota Camry and Arnold are next to each other.
4. The person who has black hair is somewhere to the right of the person in a ranch-style home.
5. The person who has black hair is the Dane.
6. The person residing in a Victorian house is not in the second house.
7. Peter is directly left of the person who loves beach vacations.
8. The person living in a colonial-style house is the person who has blonde hair.
9. The person who has black hair is Arnold.
10. The Dane is the person who loves beach vacations.
11. Peter is the person who prefers city breaks.
 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: 3


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Peter`
 - They all have a unique favorite flower: `lilies`, `roses`, `carnations`, `daffodils`
 - People have unique favorite book genres: `fantasy`, `mystery`, `science fiction`, `romance`
 - Each person prefers a unique type of vacation: `city`, `beach`, `cruise`, `mountain`
 - People have unique hair colors: `black`, `red`, `brown`, `blonde`
 - Everyone has something unique for lunch: `grilled cheese`, `pizza`, `stew`, `spaghetti`

## Clues:
1. The person who likes going on cruises is the person who has brown hair.
2. The person who loves the stew is the person who loves beach vacations.
3. Peter is the person who loves romance books.
4. There is one house between the person who has brown hair and the person who loves the spaghetti eater.
5. The person who is a pizza lover is Arnold.
6. The person who prefers city breaks is somewhere to the right of the person who loves the boquet of lilies.
7. The person who loves the stew is directly left of the person who loves mystery books.
8. The person who has black hair is directly left of the person who loves fantasy books.
9. The person who loves a carnations arrangement is directly left of the person who is a pizza lover.
10. The person who has blonde hair is not in the fourth house.
11. The person who prefers city breaks is directly left of Eric.
12. The person who loves the boquet of lilies is somewhere to the right of Alice.
13. The person who loves a bouquet of daffodils is in the fourth house.
 What is the value of attribute Flower for the person whose attribute Vacation is cruise? Please reason step by step, and put your final answer within \boxed{}

Answer: lilies


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Peter`, `Bob`
 - Each person has a unique type of pet: `fish`, `bird`, `cat`, `dog`, `hamster`
 - People use unique phone models: `samsung galaxy s21`, `iphone 13`, `google pixel 6`, `huawei p50`, `oneplus 9`
 - People have unique heights: `short`, `very tall`, `very short`, `tall`, `average`
 - Each mother is accompanied by their child: `Fred`, `Meredith`, `Samantha`, `Timothy`, `Bella`

## Clues:
1. The person with an aquarium of fish is Arnold.
2. The person who uses a Huawei P50 is not in the fifth house.
3. The person with an aquarium of fish is in the fourth house.
4. The person who uses a Samsung Galaxy S21 is in the second house.
5. The person's child is named Bella is the person who owns a dog.
6. Arnold is somewhere to the left of the person with a pet hamster.
7. The person who uses a Google Pixel 6 is directly left of the person's child is named Meredith.
8. The person's child is named Bella is directly left of the person who is short.
9. The person who keeps a pet bird is somewhere to the left of the person's child is named Bella.
10. The person who has an average height and the person who uses a OnePlus 9 are next to each other.
11. Alice is the person who has a cat.
12. Bob is somewhere to the right of the person who is very short.
13. The person who uses a Google Pixel 6 is not in the fourth house.
14. Eric is the person who uses an iPhone 13.
15. The person's child is named Fred is the person who keeps a pet bird.
16. The person who is very tall is in the first house.
17. There is one house between the person who uses an iPhone 13 and the person who is the mother of Timothy.
 What is the value of attribute PhoneModel for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: google pixel 6


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`, `Arnold`, `Eric`, `Bob`, `Peter`
 - People have unique heights: `average`, `short`, `tall`, `very tall`, `very short`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Kailyn`, `Janelle`, `Aniya`
 - Each person has a unique type of pet: `cat`, `dog`, `hamster`, `fish`, `bird`
 - The people keep unique animals: `bird`, `dog`, `fish`, `horse`, `cat`
 - People have unique favorite sports: `swimming`, `basketball`, `baseball`, `tennis`, `soccer`

## Clues:
1. The person who loves basketball is the person who keeps a pet bird.
2. The person who loves soccer is directly left of the person who is very short.
3. The person whose mother's name is Aniya is directly left of the person who is tall.
4. The person who loves tennis is Eric.
5. The person whose mother's name is Holly is in the first house.
6. The person who owns a dog is somewhere to the right of Bob.
7. The person who owns a dog is the cat lover.
8. The person whose mother's name is Aniya is Peter.
9. The person who is very short is directly left of Alice.
10. The person whose mother's name is Janelle is the bird keeper.
11. The person who has a cat is directly left of the person who has an average height.
12. The person whose mother's name is Penny is the person with an aquarium of fish.
13. The person with an aquarium of fish is directly left of the dog owner.
14. The person who is very tall is somewhere to the right of the fish enthusiast.
15. The person who loves tennis is in the fourth house.
16. The person who loves basketball is directly left of the person who loves baseball.
 What is the value of attribute Pet for the person whose attribute Animal is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: hamster


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: `Bob`, `Eric`, `Arnold`, `Carol`, `Alice`, `Peter`
 - Each person has a unique hobby: `cooking`, `painting`, `knitting`, `photography`, `woodworking`, `gardening`
 - People have unique hair colors: `brown`, `auburn`, `black`, `red`, `gray`, `blonde`
 - People have unique favorite sports: `swimming`, `volleyball`, `basketball`, `baseball`, `tennis`, `soccer`
 - The mothers' names in different houses are unique: `Sarah`, `Holly`, `Penny`, `Janelle`, `Kailyn`, `Aniya`
 - Everyone has a unique favorite cigar: `dunhill`, `blends`, `pall mall`, `prince`, `yellow monster`, `blue master`

## Clues:
1. The person who has gray hair is directly left of The person whose mother's name is Janelle.
2. The person who loves soccer is in the sixth house.
3. Arnold is the person who has blonde hair.
4. The photography enthusiast is the person who smokes Yellow Monster.
5. There is one house between the Dunhill smoker and the person who loves cooking.
6. The person who paints as a hobby is the person who has blonde hair.
7. The Dunhill smoker is directly left of Alice.
8. The person who paints as a hobby is directly left of the person who loves volleyball.
9. The person who loves soccer is The person whose mother's name is Aniya.
10. The person partial to Pall Mall is somewhere to the left of Carol.
11. The person who enjoys knitting and the person who loves basketball are next to each other.
12. The person partial to Pall Mall is directly left of The person whose mother's name is Holly.
13. The person who loves soccer is the Prince smoker.
14. The person who loves swimming is The person whose mother's name is Penny.
15. The person whose mother's name is Kailyn is the person who loves volleyball.
16. The person who loves basketball is The person whose mother's name is Janelle.
17. The person who enjoys knitting is the person who loves volleyball.
18. The person who enjoys knitting is not in the third house.
19. The person who has auburn hair is somewhere to the left of the photography enthusiast.
20. The person whose mother's name is Sarah is not in the fifth house.
21. Bob is directly left of the person who has red hair.
22. The Dunhill smoker is the person who enjoys gardening.
23. The person who loves baseball is the person who has auburn hair.
24. Eric is the person who smokes many unique blends.
25. The person who has brown hair is somewhere to the right of The person whose mother's name is Penny.
 What is the value of attribute Mother for the person whose attribute Hobby is woodworking? Please reason step by step, and put your final answer within \boxed{}

Answer: Janelle


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`, `Bob`, `Arnold`, `Peter`, `Carol`
 - People have unique favorite music genres: `classical`, `rock`, `jazz`, `pop`, `country`, `hip hop`
 - Each person has an occupation: `teacher`, `artist`, `nurse`, `engineer`, `lawyer`, `doctor`
 - Each person prefers a unique type of vacation: `cruise`, `camping`, `city`, `cultural`, `mountain`, `beach`
 - Each person has a favorite color: `blue`, `white`, `red`, `yellow`, `purple`, `green`

## Clues:
1. The person who loves beach vacations is Arnold.
2. The person who is an engineer is the person who enjoys mountain retreats.
3. Peter is not in the second house.
4. The person whose favorite color is red is in the third house.
5. The person who loves beach vacations and the person who is a teacher are next to each other.
6. The person who is a nurse is Carol.
7. The person who goes on cultural tours is the person who loves pop music.
8. The person who loves rock music and the person who loves country music are next to each other.
9. The person whose favorite color is green is in the second house.
10. The person whose favorite color is red is the person who enjoys camping trips.
11. Peter is the person who loves pop music.
12. The person who loves classical music is the person who loves purple.
13. The person who loves blue is directly left of Alice.
14. The person who loves rock music is in the fourth house.
15. The person who loves white is the person who is a doctor.
16. The person who loves jazz music is the person whose favorite color is red.
17. Carol is somewhere to the right of the person who prefers city breaks.
18. The person who loves rock music is the person who is an artist.
19. The person who enjoys mountain retreats is in the first house.
20. Bob is the person who loves classical music.
 What is the value of attribute House for the person whose attribute Occupation is teacher? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


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`, `Carol`, `Arnold`, `Bob`, `Alice`
 - Each person has a favorite color: `blue`, `red`, `green`, `yellow`, `purple`, `white`
 - Each person prefers a unique type of vacation: `city`, `beach`, `cruise`, `cultural`, `mountain`, `camping`
 - People have unique hair colors: `blonde`, `black`, `gray`, `brown`, `red`, `auburn`
 - Each person has an occupation: `artist`, `lawyer`, `engineer`, `teacher`, `doctor`, `nurse`
 - They all have a unique favorite flower: `roses`, `iris`, `daffodils`, `carnations`, `tulips`, `lilies`

## Clues:
1. The person who loves a bouquet of daffodils is the person who is an artist.
2. The person who enjoys camping trips is the person whose favorite color is green.
3. The person who has brown hair is in the first house.
4. The person who loves blue is somewhere to the right of the person who enjoys mountain retreats.
5. The person who loves the vase of tulips is the person who enjoys mountain retreats.
6. Eric is in the second house.
7. The person who likes going on cruises and Arnold are next to each other.
8. The person who loves yellow and the person who enjoys camping trips are next to each other.
9. The person who loves yellow is somewhere to the left of Peter.
10. The person who is a lawyer and Arnold are next to each other.
11. The person who has red hair is Arnold.
12. Alice is the person who is a doctor.
13. The person who loves purple is directly left of the person who loves the vase of tulips.
14. The person who has brown hair is somewhere to the left of the person who is an engineer.
15. The person who has gray hair is somewhere to the left of the person whose favorite color is red.
16. The person who is an artist is somewhere to the right of Arnold.
17. Alice is the person who loves the boquet of lilies.
18. The person whose favorite color is red is directly left of the person who prefers city breaks.
19. The person who prefers city breaks is somewhere to the right of the person who has blonde hair.
20. There is one house between Alice and Bob.
21. The person who loves the rose bouquet is directly left of the person who is a nurse.
22. The person whose favorite color is green is in the fourth house.
23. The person who loves beach vacations is somewhere to the left of the person who has red hair.
24. The person who has black hair is Alice.
25. The person who loves blue is the person who loves a carnations arrangement.
 What is the value of attribute House for the person whose attribute HairColor is brown? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


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: `Bob`, `Carol`, `Eric`, `Peter`, `Alice`, `Arnold`
 - Each person lives in a unique style of house: `victorian`, `ranch`, `modern`, `craftsman`, `mediterranean`, `colonial`
 - Everyone has a unique favorite cigar: `yellow monster`, `blue master`, `blends`, `pall mall`, `dunhill`, `prince`

## Clues:
1. The person living in a colonial-style house is not in the second house.
2. The person partial to Pall Mall is somewhere to the left of the person in a ranch-style home.
3. The person in a ranch-style home is not in the fourth house.
4. The Prince smoker is not in the fourth house.
5. The person in a Mediterranean-style villa is not in the fourth house.
6. The person who smokes many unique blends is in the third house.
7. The person in a modern-style house is directly left of Carol.
8. The person residing in a Victorian house is somewhere to the left of the person in a ranch-style home.
9. Eric is not in the sixth house.
10. Alice and the person in a Craftsman-style house are next to each other.
11. There are two houses between the person partial to Pall Mall and Bob.
12. The person in a ranch-style home is not in the third house.
13. The Prince smoker is somewhere to the right of the person in a Craftsman-style house.
14. The person who smokes Yellow Monster is not in the second house.
15. There are two houses between Arnold and Bob.
16. The person living in a colonial-style house is directly left of the person in a Mediterranean-style villa.
17. The person in a Mediterranean-style villa is not in the second house.
18. The person who smokes Blue Master is somewhere to the right of the person living in a colonial-style house.
19. The person living in a colonial-style house is directly left of Peter.
 What is the value of attribute Cigar for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: yellow monster


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 Hobby for the person whose attribute Birthday is feb? Please reason step by step, and put your final answer within \boxed{}

Answer: painting


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`, `Arnold`, `Bob`, `Alice`, `Carol`
 - The people keep unique animals: `rabbit`, `dog`, `fish`, `bird`, `cat`, `horse`
 - Each person has a unique hobby: `cooking`, `knitting`, `painting`, `woodworking`, `photography`, `gardening`
 - People have unique favorite sports: `basketball`, `soccer`, `swimming`, `tennis`, `baseball`, `volleyball`
 - Each person prefers a unique type of vacation: `cruise`, `cultural`, `city`, `camping`, `mountain`, `beach`
 - Each person has a unique favorite drink: `water`, `root beer`, `boba tea`, `tea`, `milk`, `coffee`

## Clues:
1. The person who enjoys camping trips is the person who loves cooking.
2. The person who loves baseball is the photography enthusiast.
3. The person who loves volleyball is somewhere to the left of the person who prefers city breaks.
4. Carol is the person who loves tennis.
5. The dog owner is the person who loves swimming.
6. The photography enthusiast is the tea drinker.
7. The person who loves baseball is directly left of the person who goes on cultural tours.
8. Peter is the person who enjoys camping trips.
9. The person who enjoys knitting is the person who keeps horses.
10. The coffee drinker is directly left of the person who loves soccer.
11. The person who enjoys mountain retreats is the person who loves soccer.
12. The boba tea drinker is not in the fourth house.
13. The cat lover is somewhere to the left of the dog owner.
14. The person who loves beach vacations is Bob.
15. The coffee drinker is somewhere to the right of Alice.
16. The tea drinker is the fish enthusiast.
17. The person who likes milk is Carol.
18. There is one house between the rabbit owner and the woodworking hobbyist.
19. The person who loves cooking is not in the third house.
20. The person who loves tennis is the rabbit owner.
21. The dog owner is directly left of the person who likes going on cruises.
22. The woodworking hobbyist is the dog owner.
23. There are two houses between the person who paints as a hobby and the person who keeps horses.
24. Arnold is the root beer lover.
 What is the value of attribute Drink for the person whose attribute Animal is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: root beer


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

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

Answer: ford f150


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`, `Arnold`, `Carol`, `Alice`, `Eric`, `Bob`
 - People have unique favorite book genres: `mystery`, `science fiction`, `fantasy`, `romance`, `biography`, `historical fiction`
 - Everyone has a unique favorite cigar: `dunhill`, `blends`, `yellow monster`, `prince`, `blue master`, `pall mall`
 - The people are of nationalities: `brit`, `swede`, `chinese`, `dane`, `german`, `norwegian`
 - Each person has a unique type of pet: `bird`, `rabbit`, `fish`, `cat`, `hamster`, `dog`

## Clues:
1. The person with an aquarium of fish is in the first house.
2. The person who owns a dog is not in the third house.
3. The British person is the person who loves historical fiction books.
4. Bob is the person who owns a dog.
5. The person with a pet hamster is the British person.
6. Eric is in the sixth house.
7. There is one house between the person who smokes Yellow Monster and the person who owns a dog.
8. Alice is directly left of the person who has a cat.
9. The British person is directly left of the person who smokes Blue Master.
10. Carol is in the second house.
11. The person who loves fantasy books is the Swedish person.
12. The person who loves romance books is directly left of the Dunhill smoker.
13. The person partial to Pall Mall is the Swedish person.
14. The Prince smoker is the Chinese.
15. The person who loves biography books is somewhere to the left of the British person.
16. The person who loves science fiction books is in the fourth house.
17. The British person is Alice.
18. The Norwegian is the person who loves biography books.
19. The Dane and the Dunhill smoker are next to each other.
20. Peter is not in the third house.
21. The person who owns a rabbit is the person who loves romance books.
 What is the value of attribute House for the person whose attribute BookGenre is mystery? Please reason step by step, and put your final answer within \boxed{}

Answer: 6


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Bob`, `Eric`, `Carol`
 - Everyone has something unique for lunch: `stir fry`, `grilled cheese`, `spaghetti`, `pizza`, `soup`, `stew`
 - People use unique phone models: `xiaomi mi 11`, `iphone 13`, `oneplus 9`, `google pixel 6`, `samsung galaxy s21`, `huawei p50`
 - People have unique favorite music genres: `pop`, `country`, `classical`, `jazz`, `rock`, `hip hop`
 - Each person has a unique favorite drink: `water`, `boba tea`, `milk`, `root beer`, `tea`, `coffee`
 - Each person has an occupation: `nurse`, `lawyer`, `artist`, `doctor`, `teacher`, `engineer`

## Clues:
1. The tea drinker is somewhere to the right of the person who is a teacher.
2. Eric is the person who uses a Google Pixel 6.
3. The person who loves the spaghetti eater and the person who loves hip-hop music are next to each other.
4. The boba tea drinker is somewhere to the left of the person who uses a Google Pixel 6.
5. The person who uses a Huawei P50 is somewhere to the left of the person who uses a OnePlus 9.
6. The person who is a pizza lover is the person who loves rock music.
7. The person who is a pizza lover is somewhere to the right of the person who loves the soup.
8. The person who likes milk is somewhere to the left of the person who loves hip-hop music.
9. The person who uses a OnePlus 9 is Arnold.
10. The person who is a lawyer is the person who loves stir fry.
11. The person who is an artist is Alice.
12. Bob is the person who loves classical music.
13. The person who loves rock music is directly left of the person who loves classical music.
14. The person who is a doctor is the person who uses a Huawei P50.
15. Bob is the root beer lover.
16. There are two houses between the person who is an engineer and the person who uses a OnePlus 9.
17. Eric is the person who loves jazz music.
18. The person who uses a Xiaomi Mi 11 is the person who is an artist.
19. Carol is directly left of the person who uses a Huawei P50.
20. The person who is a lawyer is the coffee drinker.
21. The person who loves eating grilled cheese is the person who loves pop music.
22. The person who loves hip-hop music is the person who uses a Samsung Galaxy S21.
23. The coffee drinker is somewhere to the left of the person who likes milk.
 What is the value of attribute Food for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: stew


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: `Bob`, `Arnold`, `Alice`, `Peter`, `Carol`, `Eric`
 - Each person prefers a unique type of vacation: `mountain`, `city`, `cultural`, `camping`, `beach`, `cruise`
 - Everyone has a favorite smoothie: `lime`, `desert`, `cherry`, `blueberry`, `dragonfruit`, `watermelon`
 - People have unique favorite sports: `tennis`, `volleyball`, `basketball`, `baseball`, `swimming`, `soccer`
 - Everyone has a unique favorite cigar: `blends`, `yellow monster`, `dunhill`, `pall mall`, `blue master`, `prince`
 - People have unique heights: `tall`, `very tall`, `average`, `super tall`, `very short`, `short`

## Clues:
1. Peter is the Prince smoker.
2. The person who smokes many unique blends is the person who enjoys mountain retreats.
3. The person who loves baseball is the person who is tall.
4. The person who goes on cultural tours is the person who drinks Blueberry smoothies.
5. The person who likes Cherry smoothies is not in the fourth house.
6. The person who loves soccer is the person who enjoys camping trips.
7. The person who is super tall is somewhere to the right of the person who likes going on cruises.
8. The person who drinks Blueberry smoothies is the person who is tall.
9. The person who is very tall is the Dunhill smoker.
10. The person who enjoys mountain retreats is Eric.
11. The person who is super tall is Alice.
12. Bob is the person who smokes Blue Master.
13. The person who loves volleyball is in the first house.
14. Peter and the person who drinks Lime smoothies are next to each other.
15. Carol is the person who likes Cherry smoothies.
16. Peter is the person who is short.
17. The person who smokes many unique blends is somewhere to the right of the Dragonfruit smoothie lover.
18. The person who loves swimming is Carol.
19. The person who is very short is the person who smokes Blue Master.
20. The person who loves tennis is somewhere to the left of the person who is very short.
21. The person who loves swimming is directly left of the person who enjoys camping trips.
22. The person who drinks Blueberry smoothies is not in the third house.
23. Alice is somewhere to the right of the person who drinks Lime smoothies.
24. The person who loves tennis is the person who is super tall.
25. The person who smokes Yellow Monster and the person who prefers city breaks are next to each other.
26. The Watermelon smoothie lover is the person who is very short.
27. The person who loves beach vacations is somewhere to the left of the person who is very tall.
 What is the value of attribute Height for the person whose attribute Cigar is dunhill? Please reason step by step, and put your final answer within \boxed{}

Answer: very tall


11. (10 points) There are 20 piles of stones, each containing 2006 stones. The rule is: taking one stone from each of any 19 piles and placing them into another pile is considered one operation. After fewer than 20 such operations, one pile has 1990 stones, and another pile has between 2080 and 2100 stones. How many stones are in this pile?

Answer: 2090


8th Putnam 1948 Problem A5 Let α 1 , α 2 , ... , α n be the nth roots of unity. Find ∏ i<j (α i - α j ) 2 .

Answer: (-1)^{\frac{(n-1)(n-2)}{2}} n^{n}


For a fixed positive integer  $n \geq 3$  we are given a  $n$   $\times$   $n$  board with all unit squares initially white. We define a *floating plus*as a  $5$ -tuple  $(M,L,R,A,B)$  of unit squares such that  $L$  is in the same row and left of  $M$ ,  $R$  is in the same row and right of  $M$ ,  $A$  is in the same column and above  $M$  and  $B$  is in the same column and below  $M$ . It is possible for  $M$  to form a floating plus with unit squares that are not next to it. Find the largest positive integer  $k$  (depending on  $n$ ) such that we can color some  $k$  squares black in such a way that there is no black colored floating plus.

*Proposed by Nikola Velov*

Answer: 4(n - 1)


Question 224, Given a positive integer $n(n \geq 2)$, choose $m$ different numbers from $1, 2, \ldots, 3n$. Among these, there must be four pairwise distinct numbers $a, b, c, d$, satisfying $a=b+c+d$. Find the minimum value of $m$.

---

The translation maintains the original format and line breaks as requested.

Answer: 2n + 2


Example $8 A, B, C, D, E$ five people participated in an exam, which has seven questions, all of which are true or false questions. The scoring criteria are: for each question, 1 point is awarded for a correct answer, 1 point is deducted for a wrong answer, and no points are awarded or deducted for unanswered questions. Table 1 records the answers given by $A, B, C, D, E$ five individuals. It is now known that $A, B, C, D$ each scored 2 points. Question: How many points should $E$ get? What is the correct answer for each question?

Answer: 4


10,11

The side of the base $ABC$ of the pyramid $TABCD$ is 4, the lateral edge $TA$ is perpendicular to the plane of the base. Find the area of the section of the pyramid by a plane passing through the midpoints of the edges $AC$ and $BT$ and parallel to the median $BD$ of the face $BCT$, given that the distance from the vertex $T$ to this plane is $\frac{1}{2}$.

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


12. Select 3 cards from five number cards $0,4,6,7,8$ to form a two-digit number, then a total of $\qquad$ different three-digit numbers can be formed (Note: Cards can be flipped).

Answer: 78


Let $R$ $S$ , and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the $x$ -axis. The left edge of $R$ and the right edge of $S$ are on the $y$ -axis, and $R$ contains $\frac{9}{4}$ as many lattice points as does $S$ . The top two vertices of $T$ are in $R \cup S$ , and $T$ contains $\frac{1}{4}$ of the lattice points contained in $R \cup S.$ See the figure (not drawn to scale). [asy] size(8cm); label(scale(.8)*"$y$", (0,60), N); label(scale(.8)*"$x$", (60,0), E); filldraw((0,0)--(55,0)--(55,55)--(0,55)--cycle, yellow+orange+white+white); label(scale(1.3)*"$R$", (55/2,55/2)); filldraw((0,0)--(0,28)--(-28,28)--(-28,0)--cycle, green+white+white); label(scale(1.3)*"$S$",(-14,14)); filldraw((-10,0)--(15,0)--(15,25)--(-10,25)--cycle, red+white+white); label(scale(1.3)*"$T$",(3.5,25/2)); draw((0,-10)--(0,60),EndArrow()); draw((-34,0)--(60,0),EndArrow()); [/asy] The fraction of lattice points in $S$ that are in $S \cap T$ is $27$ times the fraction of lattice points in $R$ that are in $R \cap T$ . What is the minimum possible value of the edge length of $R$ plus the edge length of $S$ plus the edge length of $T$

Answer: 337


10. As shown in Figure 6, in two concentric circles with center 0, the chord $AB$ of the larger circle is tangent to the smaller circle at point $C$, and $AB=8 \text{ cm}$. Then the area of the shaded part in the figure is $\qquad$

Answer: 16\pi


4B. On the sides of a square with side length 2, isosceles trapezoids are constructed outward, such that the vertices of all trapezoids are simultaneously vertices of a regular dodecagon.

What is the perimeter of the dodecagon?

Answer: 12(\sqrt{3} - 1)


In a movie theater, five friends took seats numbered 1 to 5 (with seat number 1 being the leftmost seat). During the movie, Anya left to buy popcorn. When she returned, she found that Varya had moved two seats to the right, Galya had moved one seat to the left, and Diana and Ella had switched their seats, leaving the end seat for Anya. In which seat was Anya sitting before she left?

Answer: 2


2. Let $a=\sqrt{x^{2}+x y+y^{2}}, b=p \sqrt{x y}$, $c=x+y$. If for any positive numbers $x$ and $y$, a triangle exists with sides $a$, $b$, and $c$, then the range of the real number $p$ is $\qquad$

Answer: (2 - \sqrt{3}, 2 + \sqrt{3})


Problem 2. (4 points) In a box, there are red, blue, yellow, and white balls, a total of 111. If 100 balls are drawn from the box with closed eyes, among them there will definitely be four balls, each two of which are of different colors. How many balls at least need to be drawn from the box (with closed eyes) to be able to assert with certainty that among them there are three balls, each two of which are of different colors?

Answer: 88


9. There are 50 consecutive even numbers: $2,4,6,8, \cdots \cdots, 98,100$. From these 50 numbers, the minimum number of numbers to be taken such that their sum is 2022 (each number can be taken at most once) is

Answer: 28


On an island, there are knights who always tell the truth and liars who always lie. One day, 15 natives, among whom there were both knights and liars, formed a circle. Each of them said, "Of the two people standing opposite me, one is a knight and the other is a liar." How many knights are there among them?

Answer: 10


## Task B-1.4.

In each cell of a $4 \times 4$ table, a number is written. For each cell, the sum of the numbers in its adjacent cells is equal to the same natural number $x$ (two cells are adjacent if they share a side). Determine the number $x$ such that the sum of all numbers in the table is 282.

Answer: 47


8. Consider each permutation of the nine digits $1,2, \cdots, 9$ as a nine-digit number. Among them, the number of those that are divisible by 11 is $\qquad$.

Answer: 31680


4. As shown in Figure 1, given that the two medians $A D$ and $B E$ of $\triangle A B C$ intersect at point $G$, we can obtain 8 figures:
$\triangle A B D$, $\triangle A C D$,
$\triangle B A E$, $\triangle B C E$,
$\triangle G A B$, $\triangle G A E$,
$\triangle G B D$, quadrilateral $C E G D$.
If two figures are randomly selected from these, the probability that the areas of these two figures are equal is . $\qquad$

Answer: \dfrac{2}{7}


Mr. Johnson is a retired chauffeur who now offers transportation services to elderly residents in his neighborhood. Every Monday, he drives residents to the local community center for their weekly activities. This week, he has to pick up 5 residents from their homes. Each resident lives 2 miles apart from the next one on his route. After picking up the last resident, it takes Mr. Johnson 3 miles to drive to the community center. On his return trip, he drops everyone off at their homes in the reverse order.

How many miles will Mr. Johnson drive in total for the round trip on Monday?

Answer: 22


Jamie, a university student studying journalism, is inspired by a blogger who has 120,000 followers and never self-censors their content. Jamie decides to start their own blog and aims to gain followers by writing authentic and uncensored articles. In the first month, Jamie gains 5,000 followers. The second month sees an increase of 10% in new followers compared to the first month. However, during the third month, Jamie decides to write a particularly bold article which attracts 25% more new followers than the number gained in the second month. How many followers does Jamie have at the end of the third month?

Answer: 17375


Professor Lin, a linguistics expert fluent in multiple languages, is analyzing a set of 30 literary works written by various authors. She is particularly interested in highlighting cultural nuances and idiomatic expressions. For her research, she decides to focus on 5 specific languages. In each language, she identifies 12 unique cultural expressions in each literary work. However, during her analysis, she discovers that 15 of these expressions are repeated across all languages, and 6 expressions are unique to each specific language. How many cultural expressions did Professor Lin identify in total from the 30 literary works, taking into account the repetitions and unique expressions?

Answer: 45


Lisa founded a travel platform called "Budget Wanderer" that specializes in helping budget-conscious travelers find affordable vacation packages. She is planning a promotional campaign to attract more users by offering a special deal: a 5-day trip to a popular destination that includes flights, accommodations, and a guided tour.

The original price of the trip package is $800. For the promotion, Lisa decides to offer a 15% discount to attract more travelers. Additionally, for every group of 4 travelers booking together, she offers an extra $50 off the total cost of their booking.

If a group of 4 friends decides to book this trip during the promotional campaign, what is the total cost they will pay for the entire package after applying the discount and the additional group discount?

Answer: 2670


Alex is a barista who works at a coffee shop and dreams of becoming a lawyer. Every week, Alex works 25 hours at the coffee shop and earns $12 per hour. Alex has been saving money to pay for law school, which costs $15,000 per year. Alex also seeks advice from a successful lawyer, who recommends setting aside an additional $500 each year for books and other materials.

If Alex plans to save enough money to cover one year of law school tuition and the additional cost for books and materials in 3 years, how many hours must Alex work each year to meet this goal, assuming all earnings are saved for this purpose?

Answer: 431


Sarah runs a popular Bachelor-themed podcast and records an episode every week to discuss the latest happenings on the show. Each episode is 45 minutes long. Sarah spends 20 minutes preparing notes for each contestant she discusses, and she talks about 3 contestants in each episode. After recording, she spends an additional 30 minutes editing the podcast. If Sarah discusses 3 episodes of the show in one podcast recording, how much total time in minutes does she spend each week preparing, recording, and editing the podcast?

Answer: 135


Alex is an aspiring environmental engineer interested in improving water treatment technologies. One day, Alex is tasked with analyzing the efficiency of a small water treatment plant. The plant processes 1,200 gallons of water every hour. During a test run, Alex notices that 90% of the processed water meets the required purity standards. However, Alex wants to increase the efficiency so that 95% of the water meets the purity standards. If Alex can implement a new technology that increases the purity standard by 2% every hour, how many hours will it take for the plant to achieve the desired 95% purity level?

Answer: 3


Sarah, an ICU nurse, is preparing medication for her patients during her shift. She needs to administer a total of 5 doses of medicine to her patients. Each dose consists of 2 tablets. If Sarah has 12 patients and each patient requires the same amount of medicine, how many total tablets does she need for all her patients during her shift?

Answer: 120


Alex, a fresh graduate who is excited about the latest C++ language features, decides to experiment with modern programming techniques by writing a series of small programs. Each program focuses on a new feature of C++. Over the weekend, Alex plans to write 10 programs in total. Each program takes 2 hours to write and an additional 1 hour for testing and debugging. However, Alex finds that the 4th program is particularly challenging and takes an extra 2 hours to debug. On Sunday, Alex spends 3 hours revisiting and improving the code of the first program he wrote on Saturday. If Alex spends 8 hours programming on Saturday and the rest on Sunday, how many hours in total does Alex spend programming over the weekend?

Answer: 35


Jamie is a single parent who works full-time and is also learning about child psychology to better understand and support their child, Alex. Jamie dedicates 8 hours a day to work and manages to spend 3 hours each weekday evening focusing on child psychology studies. On weekends, Jamie spends a total of 4 hours per day with Alex doing various activities and another 2 hours studying child psychology. Calculate the total number of hours Jamie dedicates to work, studying child psychology, and spending quality time with Alex over the course of one week.

Answer: 67


A group of sports bloggers are analyzing the performance of a basketball team over a recent series of games. They have collected statistics for 5 different games. In each game, the team scored the following points: 98, 105, 87, 112, and 95. To provide a more meaningful analysis, the bloggers want to find the average points scored per game. Additionally, they notice that the team's star player was responsible for scoring 30% of the total points across all games. How many points did the star player score in total during these 5 games?

Answer: 149.1


A music historian is organizing a collection of Christmas songs from the 1920s and 1930s. She has found 15 songs from the 1920s and 20 songs from the 1930s. She plans to create playlists with an equal number of songs from each decade. If each playlist contains 5 songs, how many complete playlists can she make?

Answer: 6


Let $E$ be an ellipse with foci $A$ and $B$. Suppose there exists a parabola $P$ such that:
- $P$ passes through $A$ and $B$,
- the focus $F$ of $P$ lies on $E$,
- the orthocenter $H$ of $\triangle FAB$ lies on the directrix of $P$.

If the major and minor axes of $E$ have lengths $50$ and $14$, respectively, compute $AH^2 + BH^2$.

Answer: 2402


Let $p$ be the largest prime less than $2013$ for which \[ N = 20 + p^{p^{p+1}-13} \] is also prime. Find the remainder when $N$ is divided by $10^4$.

Answer: 1781


John and Mary each select a natural number and tell them to Bill. Bill writes down their sum and product on two separate pieces of paper, hides one, and shows the other to John and Mary. John looks at the number, which is $2002$, and declares he cannot determine Mary's number. Knowing this, Mary also says she cannot determine John's number. What was Mary's number?

Answer: 1001


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

Answer: 280


The diagram below shows a $18 \times 35$ rectangle with eight points marked that divide each side into three equal parts. Four triangles are removed from each of the corners of the rectangle, leaving the shaded region. Find the area of this shaded region.

Answer: 490


Five teams participated in a commercial football tournament. Each team was supposed to play exactly one match against every other team. However, due to financial difficulties, some games were canceled.

By the end of the tournament, all teams had different scores, and no team had zero points. What is the minimum number of games that could have been played in this tournament, given that:
- A win awards 3 points,
- A draw awards 1 point,
- A defeat awards 0 points?

Answer: 6


Emma's calculator has ten buttons: one for each digit $1, 2, \ldots, 9$, and one marked "clear". When Emma presses one of the buttons marked with a digit, that digit is appended to the right of the display. When she presses the "clear" button, the display is completely erased. If Emma starts with an empty display and presses five (not necessarily distinct) buttons at random, where all ten buttons have equal probability of being chosen, the expected value of the number produced is $\frac{m}{n}$, for relatively prime positive integers $m$ and $n$. Find $100m+n$. (Take an empty display to represent the number 0.)

Answer: 6642902


Let $a_1, a_2, \ldots, a_{100}$ be non-negative integers such that:

1. There exists a positive integer $k \leq 100$ such that $a_1 \leq a_2 \leq \cdots \leq a_k$ and $a_i = 0$ for $i > k$.
2. $a_1 + a_2 + a_3 + \cdots + a_{100} = 100$.
3. $a_1 + 2a_2 + 3a_3 + \cdots + 100a_{100} = 2022$.

Find the minimum value of $a_1 + 2^2a_2 + 3^2a_3 + \cdots + 100^2a_{100}$.

Answer: 40940


Suppose 2012 reals are selected independently and at random from the unit interval $[0,1]$, and then written in nondecreasing order as $x_1 \le x_2 \le \cdots \le x_{2012}$. If the probability that $x_{i+1} - x_i \le \frac{1}{2011}$ for $i=1,2,\ldots,2011$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$, find the remainder when $m+n$ is divided by 1000.

Answer: 601


There exist two triples of real numbers \((a,b,c)\) such that \(a-\frac{1}{b}, b-\frac{1}{c}, c-\frac{1}{a}\) are the roots to the cubic equation \(x^3-5x^2-15x+3\) listed in increasing order. Denote those \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\). If \(a_1, b_1,\) and \(c_1\) are the roots to the monic cubic polynomial \(f\) and \(a_2, b_2,\) and \(c_2\) are the roots to the monic cubic polynomial \(g\), find \(f(0)^3+g(0)^3\).

Answer: -14


A circle with area $40$ is tangent to a circle with area $10$. Let $R$ be the smallest rectangle containing both circles. The area of $R$ is $\frac{n}{\pi}$. Find $n$.

Answer: 240


In $\triangle ABC$, points $E$ and $F$ lie on $\overline{AC}$ and $\overline{AB}$, respectively. Denote by $P$ the intersection of $\overline{BE}$ and $\overline{CF}$. Compute the maximum possible area of $\triangle ABC$ if $PB = 14$, $PC = 4$, $PE = 7$, and $PF = 2$.

Answer: 84


Billy and Bobby are located at points $A$ and $B$, respectively. They each walk directly toward the other point at a constant rate; once the opposite point is reached, they immediately turn around and walk back at the same rate. The first time they meet, they are located 3 units from point $A$; the second time they meet, they are located 10 units from point $B$. Find all possible values for the distance between $A$ and $B$.

Answer: 15


The diagonals of convex quadrilateral $BSCT$ meet at the midpoint $M$ of $\overline{ST}$. Lines $BT$ and $SC$ meet at $A$, and $AB = 91$, $BC = 98$, $CA = 105$. Given that $\overline{AM} \perp \overline{BC}$, find the positive difference between the areas of $\triangle SMC$ and $\triangle BMT$.

Answer: 336


A black bishop and a white king are placed randomly on a $2000 \times 2000$ chessboard (in distinct squares). Let $p$ be the probability that the bishop attacks the king (that is, the bishop and king lie on some common diagonal of the board). Then $p$ can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m$.

Answer: 1333


As shown in the figure, a large square is divided into six smaller squares. If two smaller squares share more than one common point, they are called adjacent. Fill the numbers $1, 2, 3, 4, 5, 6$ into the smaller squares in the right figure, with each smaller square containing one number such that the difference between the numbers in any two adjacent smaller squares is never 3. How many different ways can this be done?

Answer: 96


What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak?
Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.

Answer: \sqrt{3}


31. Given $\overline{A B C D E F} \times B=\overline{E F A B C D}$, the same letter represents the same digit, and different letters represent different digits. Then the possible cases for $\overline{A B C D E F}$ are $\qquad$ kinds.

Answer: 2


Given a triangle \( \triangle ABC \) where the angle bisector of \( \angle A \) intersects the A-excircle at two points \( D \) and \( E \), with \( D \) lying on the segment \( AE \). Prove that \(\frac{AD}{AE} \leq \frac{BC^2}{DE^2}\).

Answer: \frac{AD}{AE} \leq \frac{BC^2}{DE^2}


18. 8 (IMO 42 Preliminary) Let $\triangle A B C$ be an acute triangle, and construct isosceles $\triangle D A C$, $\triangle E A B$, and $\triangle F B C$ outside $\triangle A B C$ such that $D A=D C$, $E A=E B$, $F B=F C$, $\angle A D C=2 \angle B A C$, $\angle B E A=2 \angle A B C$, and $\angle C F B=2 \angle A C B$. Let $D^{\prime}$ be the intersection of line $D B$ and $E F$, $E^{\prime}$ be the intersection of $E C$ and $D F$, and $F^{\prime}$ be the intersection of $F A$ and $D E$. Find the value of $\frac{D B}{D D^{\prime}}+\frac{E C}{E E^{\prime}}+\frac{F A}{F F^{\prime}}$.

Answer: 4


$4 \cdot 25$ Introduce a relation “ $\rightarrow$ ” among the elements of set $S$, such that
(1) For any $a, b \in S$, either $a \rightarrow b$ or $b \rightarrow a$, but not both;
(2) For any $a, b, c \in S$, if $a \rightarrow b, b \rightarrow c$, then $c \rightarrow a$.

How many elements can set $S$ contain at most?

Answer: 3


Question 167, Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with left and right foci $F_{1}$ and $F_{2}$, respectively, a line $l$ is drawn through the right focus of the ellipse, intersecting the ellipse at points $\mathrm{P}$ and $\mathrm{Q}$. Try to find the maximum value of the area of the incircle of $\triangle \mathrm{F}_{1} \mathrm{PQ}$.

Answer: \dfrac{9\pi}{16}


I2.3 If $X=\sqrt{(100)(102)(103)(105)+(Q-3)}$ is an integer and $R$ is the units digit of $X$, find the value of $R$.

I2.4 If $S$ is the sum of the last 3 digits (hundreds, tens, units) of the product of the positive roots of $\sqrt{2012} \cdot x^{\log _{2012} x}=x^{R}$, find the value of $S$.

Answer: 3


B4. On a grid, a kind of spiral is drawn as follows. The spiral starts at point $(0,0)$

$1^{\text {st }}$ step: 1 to the right

$2^{\text{nd}}$ step: 1 diagonally to the top right

$3^{\text{rd}}$ step: 2 up

$4^{\text{th}}$ step: 2 diagonals to the top left

$5^{\text{th }}$ step: 3 to the left

$6^{\text{th}}$ step: 3 diagonals to the bottom left and so on.

![](https://cdn.mathpix.com/cropped/2024_04_17_0b81ff2ef64511bfe8eeg-2.jpg?height=497&width=649&top_left_y=1459&top_left_x=1246)

In the figure, the first 10 steps are drawn.

You then reach grid point $(8,-1)$.

Give the coordinates of the grid point where the spiral ends after 2003 steps.

The work of the Stichting Nederlandse Wiskunde Olympiade is made possible by financial contributions and support from:

The Ministry of Education, Culture and Science

The Dutch Association of Mathematics Teachers

The Mathematical Society

Eindhoven University of Technology
The Citogroup

Utrecht University

AKZO/NOBEL

Nature and Technology

Answer: (1502, 503)


13. Among the thirteen natural numbers from $1 \sim 13$, select twelve natural numbers to fill in the 12 empty cells in the figure, so that the sum of the four numbers in each row is equal, and the sum of the three numbers in each column is equal. What is the number that is not filled in the cells? And please fill in the other numbers in the cells.

---

The translation preserves the original format and line breaks.

Answer: 7


A26 (19-2, Vietnam) In a finite sequence of real numbers, the sum of any 7 consecutive terms is negative, while the sum of any 11 consecutive terms is positive. How many terms can such a sequence have at most?

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

Answer: 16


Let $a, b, c > 0$, and $a + b + c = A \leqslant 1$. Given $\alpha > 0$, then prove that $\left(\frac{1}{a} - a\right)^{\alpha} + \left(\frac{1}{b} - b\right)^{\alpha} + \left(\frac{1}{c} - c\right)^{\alpha} \geqslant 3\left(\frac{3}{A} - \frac{A}{3}\right)^{\alpha}$.

Answer: \left(\frac{1}{a} - a\right)^{\alpha} + \left(\frac{1}{b} - b\right)^{\alpha} + \left(\frac{1}{c} - c\right)^{\alpha} \geqslant 3\left(\frac{3}{A} - \frac{A}{3}\right)^{\alpha}


Let \( A, B, C \) be three non-collinear points corresponding to the complex numbers \( Z_{0} = ai \), \( Z_{1} = \frac{1}{2} + bi \), and \( Z_{2} = 1 + ci \) (where \( a, b, c \) are real numbers). Prove that the curve \( Z = Z_{0} \cos^{4} t + 2Z_{1} \cos^{2} t \sin^{2} t + Z_{2} \sin^{4} t \) (\( t \in \mathbf{R} \)) shares exactly one external common point with the midline in \(\triangle ABC\) that is parallel to \( AC \), and find this point.

Answer: \frac{1}{2} + \frac{a + 2b + c}{4}i