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`, `Alice`, `Carol`, `Arnold`, `Eric`
 - Each person has a unique favorite drink: `milk`, `coffee`, `tea`, `root beer`, `boba tea`, `water`
 - They all have a unique favorite flower: `iris`, `roses`, `carnations`, `daffodils`, `lilies`, `tulips`
 - Each person has a unique birthday month: `sept`, `jan`, `feb`, `april`, `may`, `mar`
 - Each person has a unique hobby: `photography`, `cooking`, `woodworking`, `gardening`, `knitting`, `painting`

## Clues:
1. The root beer lover is in the fifth house.
2. The person whose birthday is in February is Peter.
3. The boba tea drinker is not in the fourth house.
4. The person who loves the boquet of iris is somewhere to the right of the person whose birthday is in April.
5. The coffee drinker is Arnold.
6. Eric is the photography enthusiast.
7. The person who paints as a hobby is the person who loves the vase of tulips.
8. The person whose birthday is in April is in the third house.
9. The tea drinker is the person who loves a bouquet of daffodils.
10. There are two houses between the person who loves the boquet of lilies and the person who likes milk.
11. The person whose birthday is in March is somewhere to the left of the person who enjoys gardening.
12. There are two houses between the person who loves a bouquet of daffodils and the person who paints as a hobby.
13. The person whose birthday is in February is in the fourth house.
14. The person who loves cooking is not in the fourth house.
15. The person who enjoys gardening is somewhere to the left of the person who loves a carnations arrangement.
16. The person whose birthday is in January is Bob.
17. Carol is the person whose birthday is in April.
18. The woodworking hobbyist is in the second house.
19. The person who likes milk is in the sixth house.
20. Peter is the person who loves a carnations arrangement.
21. The person whose birthday is in May is the woodworking hobbyist.
 What is the value of attribute Hobby for the person whose attribute Flower is lilies? Please reason step by step, and put your final answer within \boxed{}

Answer: gardening


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Eric`, `Peter`, `Arnold`
 - People use unique phone models: `iphone 13`, `samsung galaxy s21`, `huawei p50`, `google pixel 6`, `oneplus 9`
 - People have unique favorite sports: `soccer`, `swimming`, `baseball`, `basketball`, `tennis`
 - Everyone has a favorite smoothie: `cherry`, `dragonfruit`, `lime`, `desert`, `watermelon`
 - Each person prefers a unique type of vacation: `city`, `cruise`, `beach`, `mountain`, `camping`
 - The people keep unique animals: `horse`, `fish`, `cat`, `dog`, `bird`

## Clues:
1. Peter is somewhere to the right of the person who uses an iPhone 13.
2. The person who loves tennis is directly left of the person who uses a Google Pixel 6.
3. There is one house between the Watermelon smoothie lover and the person who loves beach vacations.
4. The person who loves baseball is in the fourth house.
5. The dog owner is Peter.
6. There is one house between the person who prefers city breaks and the person who keeps horses.
7. The fish enthusiast is directly left of the cat lover.
8. Arnold is the Desert smoothie lover.
9. The person who likes Cherry smoothies is the person who enjoys mountain retreats.
10. The person who loves swimming is not in the first house.
11. Bob is the person who prefers city breaks.
12. The person who uses a Huawei P50 is the fish enthusiast.
13. The person who loves tennis is the person who enjoys mountain retreats.
14. The person who loves baseball is somewhere to the left of the person who loves soccer.
15. The person who uses a OnePlus 9 is Peter.
16. There is one house between the person who prefers city breaks and Alice.
17. The person who enjoys camping trips is the Dragonfruit smoothie lover.
18. The person who loves baseball is the person who enjoys camping trips.
19. The person who loves beach vacations is directly left of the person who enjoys camping trips.
20. Bob is the fish enthusiast.
 What is the value of attribute PhoneModel for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: oneplus 9


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

Answer: 2


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Bob`, `Peter`, `Arnold`
 - People have unique favorite book genres: `fantasy`, `mystery`, `science fiction`, `biography`, `romance`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Aniya`, `Kailyn`, `Janelle`
 - People use unique phone models: `oneplus 9`, `iphone 13`, `samsung galaxy s21`, `google pixel 6`, `huawei p50`
 - The people are of nationalities: `dane`, `brit`, `german`, `swede`, `norwegian`
 - Each person lives in a unique style of house: `ranch`, `modern`, `craftsman`, `victorian`, `colonial`

## Clues:
1. The person in a ranch-style home is somewhere to the right of the person residing in a Victorian house.
2. The Dane and the person who uses a Samsung Galaxy S21 are next to each other.
3. The person who uses a Google Pixel 6 is not in the second house.
4. There are two houses between the person who loves fantasy books and the person in a modern-style house.
5. The person whose mother's name is Kailyn is the Swedish person.
6. The Swedish person is Arnold.
7. The British person is somewhere to the left of the person who uses an iPhone 13.
8. The Dane is The person whose mother's name is Penny.
9. The person who uses a Samsung Galaxy S21 is the Norwegian.
10. The Norwegian is The person whose mother's name is Holly.
11. The person who loves fantasy books is Bob.
12. Alice is directly left of the Swedish person.
13. The person who loves romance books is Arnold.
14. The person who loves science fiction books is Peter.
15. The person whose mother's name is Aniya is directly left of the person who loves science fiction books.
16. The person who loves science fiction books is the person who uses a OnePlus 9.
17. The person residing in a Victorian house is the person who loves biography books.
18. The person who loves fantasy books is somewhere to the left of the German.
19. The person who uses an iPhone 13 is the person in a Craftsman-style house.
 What is the value of attribute BookGenre for the person whose attribute PhoneModel is google pixel 6? Please reason step by step, and put your final answer within \boxed{}

Answer: fantasy


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`, `Carol`, `Eric`, `Alice`, `Arnold`
 - Each person has a unique favorite drink: `coffee`, `milk`, `water`, `root beer`, `tea`, `boba tea`
 - They all have a unique favorite flower: `tulips`, `lilies`, `daffodils`, `iris`, `roses`, `carnations`
 - Each person has an occupation: `doctor`, `artist`, `nurse`, `engineer`, `teacher`, `lawyer`
 - Each person has a unique hobby: `woodworking`, `photography`, `cooking`, `painting`, `knitting`, `gardening`

## Clues:
1. Alice is somewhere to the right of the person who loves cooking.
2. Peter is not in the second house.
3. The person who loves the rose bouquet is directly left of the person who likes milk.
4. Bob is the person who is a lawyer.
5. The person who loves a carnations arrangement is somewhere to the right of Bob.
6. The person who enjoys knitting is somewhere to the left of the coffee drinker.
7. The person who is a nurse is directly left of the person who is a doctor.
8. Eric is the photography enthusiast.
9. The person who loves the boquet of lilies is the woodworking hobbyist.
10. The person who paints as a hobby is in the first house.
11. The person who loves a carnations arrangement is the root beer lover.
12. The person who is an artist is the person who likes milk.
13. The person who loves a bouquet of daffodils is in the first house.
14. The photography enthusiast is somewhere to the right of the boba tea drinker.
15. The person who loves the rose bouquet is somewhere to the right of the person who is a doctor.
16. The tea drinker is the person who is an engineer.
17. Arnold is directly left of Carol.
18. The coffee drinker is the person who is a nurse.
19. The person who loves the vase of tulips is the person who loves cooking.
20. The person who is a teacher is somewhere to the right of the root beer lover.
21. The person who enjoys gardening is Carol.
 What is the value of attribute House for the person whose attribute Hobby is photography? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


N numbers are marked in the set $\{1,2,...,2000\}$ so that any pair of the numbers $(1,2),(2,4),...,(1000,2000)$ contains at least one marked number. Find the least possible value of $N$.

I.Gorodnin

Answer: 666


[ Dirichlet's Principle (finite number of points, lines, etc.).]  [ Central angle. Arc length and circumference length ]

On a circle of radius 1, a point $O$ is marked and a notch is made to the right with a compass of radius $l$. From the resulting point $O_{1}$, another notch is made in the same direction with the same radius, and this is done 1968 times. After this, the circle is cut at all 1968 notches, resulting in 1968 arcs. How many different arc lengths can result from this?

Answer: 3


6. A shape obtained by removing one unit square from a $2 \times 2$ grid is called an "L-shape". In an $8 \times 8$ grid, place $k$ L-shapes, each of which can be rotated, but each L-shape must cover exactly three unit squares of the grid, and any two L-shapes must not overlap. Additionally, no other L-shape can be placed besides these $k$ L-shapes. Find the minimum value of $k$.
(Zhang Yuezhu, Contributed)

Answer: 11


6. Find all prime pairs $(p, q)$ such that $p q \mid\left(5^{p}-2^{p}\right)\left(5^{q}-2^{q}\right)$.

Answer: (3, 3)


3. On the sides $BC$, $CA$, $AB$ of the equilateral $\triangle ABC$, there are internal division points $D$, $E$, $F$ dividing the sides in the ratio $3:(n-3)$ (where $n>6$). The segments $AD$, $BE$, $CF$ intersect to form $\triangle PQR$ (where $BE$ intersects $AD$ at $P$ and $FC$ at $Q$). When the area of $\triangle PQR$ is $\frac{4}{49}$ of the area of $\triangle ABC$, find the value of $n$.
(1992 Japan Mathematical Olympiad Preliminary)

Answer: 8


We built a cuboid from small cubes with an edge of $1 \mathrm{~cm}$. If we removed one column from the cuboid, the remaining structure would consist of 602 cubes. If we removed one row from the top layer instead, the remaining structure would consist of 605 cubes. What are the dimensions of the cuboid?

Answer: 4


2. For the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, in what range of the eccentricity $e$ does the ellipse always have a point $P$ such that the product of the distances from point $P$ to the two foci equals the square of the focal distance?

Answer: \left[ \dfrac{\sqrt{5}}{5}, \dfrac{1}{2} \right]


6.182. $|x|^{3}+|x-1|^{3}=9$.

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

6.182. $|x|^{3}+|x-1|^{3}=9$.

Answer: -1


5. It is known that the number $\cos 6^{0}$ is a root of the equation $32 t^{5}-40 t^{3}+10 t-\sqrt{3}=0$. Find the other four roots of this equation. (Answers in the problem should be compact expressions, not containing summation signs, ellipses, and radicals.)

#

Answer: \cos 66^\circ


1. A pedestrian left point $A$ for point $B$, and after some delay, a second pedestrian followed. When the first pedestrian had walked half the distance, the second had walked 15 km, and when the second pedestrian had walked half the distance, the first had walked 24 km. Both pedestrians arrived at point $B$ simultaneously. What is the distance between points $A$ and $B$?

Answer: 40


A wooden block floating in a cylindrical container with a base area of \( \mathrm{S} = 25 \, \mathrm{cm}^2 \), partially filled with water, has a small stone placed on it. As a result, the block remained afloat, and the water level in the container rose by \( h_1 = 1.5 \, \mathrm{cm} \). Then, the stone was removed from the block and placed in the water. The stone sank, and the water level in the container rose by \( h_2 = 0.5 \, \mathrm{cm} \). Find the average density of the stone material, assuming the density of water \( \rho = 1 \, \mathrm{g/cm^3} \).

Answer: 3


14. Given $a, b, c \in \mathbf{R}$, functions $f(x)=a x^{2}+b x+c, g(x)=a x+b$, when $|x| \leqslant$ 1, $|f(x)| \leqslant 1$.
（1）Prove: $|c|0$, when $|x| \leqslant 1$, the maximum value of $g(x)$ is 2, find $f(x)$.

Answer: 2x^2 - 1


21. The volume of tetrahedron $ZABC$ is 5. A plane is drawn through the midpoint $K$ of edge $ZA$ and the midpoint $P$ of edge $BC$, intersecting edge $ZC$ at point $M$, such that $ZM: MC = 2: 3$. If the distance from vertex $A$ to the plane is 1, find the area of the section.

Answer: 3


7. For the tetrahedron $ABCD$, $AB \perp BC$, $CD \perp BC$, $BC=2$, and the angle between the skew lines $AB$ and $CD$ is $60^{\circ}$. If the radius of the circumscribed sphere of the tetrahedron $ABCD$ is $\sqrt{5}$, then the maximum volume of the tetrahedron $ABCD$ is $\qquad$ .

Answer: 2\sqrt{3}


Problem 5.4. For a physical education class, 29 seventh-graders came, some of them brought one ball each. Sometimes during the class, one of the seventh-graders would give their ball to another seventh-grader who didn't have one.

At the end of the class, \( N \) seventh-graders said: “I received balls less frequently than I gave them away!”. Find the maximum possible value of \( N \), given that none of the students lied.

Answer: 14


4. (10 points) Professor Wang arrived at the station at 8:00 AM. When he boarded the train, the hour and minute hands of the clock on the platform were exactly symmetrical left and right. The train departed at 8:35 AM and arrived at the terminal station at 2:15 PM. When Professor Wang got off the train, the hour and minute hands of the clock on the platform were exactly symmetrical top and bottom, and he walked out of the station at exactly 3:00 PM. Therefore, the total time Professor Wang spent on the train is $\qquad$ minutes.

Answer: 360


6. Variant 1.

Petya wrote the numbers from 1 to 10 on cards and laid them out along the edge of a $3 \times 4$ rectangle. At first, one of the cards - with the number 7 - was opened (see the figure). When the other cards were opened, it turned out that the sum of the numbers in the top and bottom horizontal rows is the same and equals $A$. What is the largest possible value of the number $A$?

| $\circ$ | $\circ$ | $\circ$ | $\circ$ |
| :---: | :---: | :---: | :---: |
| 7 |  |  | $\circ$ |
| $\circ$ | $\circ$ | $\circ$ | $\circ$ |

Answer: 23


3. Papa always picked up Misha from kindergarten at the same time and in the same car. But today, after firmly establishing that there would be pea porridge for dinner, Misha resolutely left the kindergarten and set off down the road to meet Papa. Having walked 5 km, Misha met Papa, got into the car, and they arrived home 10 minutes earlier than usual. At what speed does Misha's Papa drive?

Answer: 60


5. In a box, there are 7 blue and 5 red balls. What is the minimum number of balls that need to be taken out to ensure that among them there are at least 2 blue and 1 red?

Answer: 8


6. A uniformly growing uniform grassland is divided into two unequal parts, $\mathrm{A}$ and $\mathrm{B}$. A herd of cows first eats the grass on plot A, finishing it in exactly 7 days. Then the herd continues to eat the grass on plot B at the same speed, finishing it in 4 days, at which point plot A has just recovered to the state it was in when the cows started eating. What is the simplest integer ratio of the area of plot A to plot B? $\qquad$

Answer: 105:44


2. Every 20 minutes for a week, an exact amount of liters of water (always the same amount) is transferred from a tank with 25,000 liters to another initially empty reservoir. From this second reservoir, at regular intervals of time, 1 liter is extracted first, then 2 liters, then 3 liters, and so on. At the end of the week, the last transfer and the last extraction coincide, leaving the second reservoir empty at that moment. Determine how much water has been extracted in total during the week, if the data in the problem allows it. (It is assumed that the transfers and extractions are instantaneous. The first transfer is made after the first 20 minutes and the first extraction, after the first interval of time.)

Answer: 2016


3. (8 points) The figure shown is the emblem of the Yingchun Primary School Mathematics Interest Group. Among them, $A B C D E F$ is a regular hexagon with an area of 360. What is the area of quadrilateral $A G D H$?

Answer: 160


4. Add three digits after 764 to make the resulting six-digit number divisible by 8, 9, and 11.

Answer: 764280


8. Arrange all positive integers whose sum of digits is 10 in ascending order to form the sequence $\left\{a_{n}\right\}$. If $a_{n}=2017$, then $n=$ $\qquad$ .

Answer: 120


In the drawing below, three buildings have been constructed on a plot of land divided into rectangular lots. The perimeters of buildings $A$ and $B$ are $400 \mathrm{~m}$ and $240 \mathrm{~m}$, respectively. What is the perimeter of building $C$?

|  |  |  |  |  |  |  |  |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
|  |  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |
|  | $A$ |  | $B$ |  |  | $C$ |  |
|  |  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |

Answer: 240


# Task № 6.1

## Condition:

Anya, Tanya, and Vanya had identical cardboard squares with a side of 16 cm. Each of them cut off two rectangles from their square, as shown in the figure, and all 6 rectangles are the same. The perimeter of Anya's figure is 88 cm, and the perimeter of Vanya's figure is 82 cm.

![](https://cdn.mathpix.com/cropped/2024_05_06_844ecc6fa8537977a6eag-21.jpg?height=328&width=332&top_left_y=567&top_left_x=248)

Anya

![](https://cdn.mathpix.com/cropped/2024_05_06_844ecc6fa8537977a6eag-21.jpg?height=343&width=343&top_left_y=568&top_left_x=845)

Vanya

![](https://cdn.mathpix.com/cropped/2024_05_06_844ecc6fa8537977a6eag-21.jpg?height=343&width=345&top_left_y=565&top_left_x=1438)

Tanya

Find the perimeter of Tanya's figure. Express your answer in centimeters.

Answer: 76


Example 5. Find the flux of the vector field $\mathbf{a} = \boldsymbol{y}^{2} \mathbf{j} + z \mathbf{k}$ through the part of the surface $z = x^{2} + y^{2}$, cut off by the plane $z = 2$. The normal is taken outward with respect to the region bounded by the paraboloid.

![](https://cdn.mathpix.com/cropped/2024_05_22_267d965536514099194eg-24.jpg?height=285&width=248&top_left_y=171&top_left_x=106)

Answer: -2\pi


10.6. Ali-baba came to a cave where there was gold and diamonds. Ali-baba had one large bag with him. It is known that a full bag of gold weighs 200 kg, and if the entire bag is filled with diamonds, it will weigh 40 kg (the empty bag weighs nothing). A kilogram of gold costs 20 dinars, and a kilogram of diamonds costs 60 dinars. What is the maximum amount of money Ali-baba can get if he can carry no more than 100 kg in this bag? Justify your answer.

Answer: 3000


Example 4 Plant all 3 types of crops in the 5 plots of experimental fields as shown in Figure 4, with each plot planting one type of crop and no two adjacent plots can plant the same type of crop. How many different planting methods are there?

Answer: 42


4. In a quiz, you answer 10 questions. Each question is either difficult or easy. For a difficult question, you get 5 points if you answer it correctly and -1 point if you answer it incorrectly; for an easy question, you get 3 points if you answer it correctly and -1 point if you answer it incorrectly. Moreover, if you answer a question correctly, the next one is difficult; if you answer a question incorrectly, the next one is easy. You start with a difficult question.

How many different final scores are possible after 10 questions?

Answer: 27


## Task B-2.3.

One year, January 1 and April 1 were both on a Thursday. How many months in that year have five Fridays? Justify your answer.

Answer: 5


2. As shown in Figure 1, given that $P A$ and $P B$ are tangent to $\odot O$ at points $A$ and $B$, $P C$ satisfies $A B \cdot P B - A C \cdot P C = A B \cdot P C - A C \cdot P B$, and $A P \perp P C$, $\angle P A B = 2 \angle B P C$. Then $\angle A C B =$ $\qquad$

Answer: 30^{\circ}


3. How many natural numbers have a product of digits 12, a sum of digits 12, and are divisible by 12? Determine the largest such number.

Answer: 32111112


Two brothers sold their flock of sheep. They sold each sheep for as many dollars as there were sheep in the flock. They shared the revenue, giving each other 10 dollars at a time. First, the older brother received 10 dollars, then the younger one, then again the older one, and so on. In the end, only less than 10 dollars were left for the younger brother, so the older brother gave him his pocket knife, thus they both ended up with the same amount of revenue. How much is the pocket knife worth?

Answer: 2


12. A small ball with a radius of 1 can move freely in all directions inside a regular tetrahedron container with an internal edge length of $4 \sqrt{6}$. The area of the container's inner wall that the ball can never touch is $\qquad$.

Answer: 72\sqrt{3}


3. Alice has six segments of different lengths. Five of them have lengths $6,7,8,9$ and 11. Any three of them can form a triangle. What integer values can the length of the fifth segment take? If there are multiple possible answers, write them in any order separated by a semicolon.

Answer: 10;12


Let \( E = \{1, 2, \cdots, 200\} \) and \( G = \{a_1, a_2, a_3, \cdots, a_{100}\} \subsetneq E \) satisfy the following properties:
1. For any \( 1 \leq i \leq j \leq 100 \), we have \( a_i + a_j \neq 201 \);
2. \( \sum_{i=1}^{100} a_i = 10080 \).

Prove that the number of odd numbers in \( G \) is a multiple of 4, and that the sum of the squares of all numbers in \( G \) is a constant.

Answer: 1349380


As shown in the figure, in the isosceles triangle \( \triangle ABC \), \( \angle ACB = 90^\circ \). Points \( M \) and \( N \) are on the hypotenuse \( AB \), and \( \angle MCN = 45^\circ \). Given that \( AM = 3 \) and \( BN = 5 \), find the length of \( MN \).

Answer: \sqrt{34}


Let \(\varphi(n)\) denote the number of positive integers less than or equal to \(n\) which are relatively prime to \(n\). Let \(S\) be the set of positive integers \(n\) such that \(\frac{2n}{\varphi(n)}\) is an integer. Compute the sum
\[ \sum_{n \in S} \frac{1}{n} . \]

Answer: \frac{10}{3}


A mathematics competition committee is composed of 9 professors. The finalized exam questions are stored in a safe that requires at least 6 professors to be present to open it. How many locks should be installed on the safe at a minimum, how many keys are needed, and how should the keys be distributed to the committee members?

Answer: 504


Let \( A \) be an \( m \times m \) matrix. Let \( X_{i} \) be the set of entries in the \( i \)-th row and \( Y_{j} \) the set of entries in the \( j \)-th column, \( 1 \leq i, j \leq m \). \( A \) is called cool if the sets \( X_{1}, \ldots, X_{m}, Y_{1}, \ldots, Y_{m} \) are all different. Determine the smallest value for \( n \) so that a cool \( 2005 \times 2005 \) matrix with entries from the set \( \{1,2, \ldots, n\} \) exists.

Answer: 13


At least three faces of a convex polyhedron are pentagons. What is the minimum number of faces the polyhedron can have?

Answer: 7


Let \(\left\{a_{i}\right\}_{i \geq 0}\) be a sequence of real numbers defined by
$$
a_{n+1}=a_{n}^{2}-\frac{1}{2^{2020 \cdot 2^{n}-1}}
$$
for \(n \geq 0\). Determine the largest value for \(a_{0}\) such that \(\left\{a_{i}\right\}_{i \geq 0}\) is bounded.

Answer: 1+\frac{1}{2^{2020}}


A large cube is built using 64 smaller identical cubes. Three of the faces of the large cube are painted. What is the maximum possible number of small cubes that can have exactly one face painted?

Answer: 32


Let \( n \) be the answer to this problem. Given \( n > 0 \), find the number of distinct (i.e., non-congruent), non-degenerate triangles with integer side lengths and perimeter \( n \).

Answer: 48


In the trapezoid $ABCD$ with bases $AB$ and $CD$, let $M$ be the midpoint of side $DA$. If $BC=a$, $MC=b$ and $\angle MCB=150^\circ$, what is the area of trapezoid $ABCD$ as a function of $a$ and $b$?

Answer: \dfrac{ab}{2}


Question 240, Set $S$ satisfies the following conditions: (1) The elements in $S$ are all positive integers not exceeding 100; (2) For any $a, b \in S, a \neq b$, there exists a $c \in S$ different from $a, b$, such that $(a+b, c)=1$; (3) For any $a, b \in S, a \neq b$, there exists a $c \in S$ different from $a, b$, such that $(a+b, c)>1$. Try to find the maximum value of $|S|$.

---

Note: The notation $(a, b)$ represents the greatest common divisor (gcd) of $a$ and $b$.

Answer: 50


We have a collection of identical spheres that we stack to form a tetrahedron whose edges all have $n$ spheres. Calculate, in terms of $n$, the total number of tangency points (contacts) between the spheres in the pile.

## Solution.

## The problem in the plane.

Answer: n(n^2 - 1)


4. As shown in Figure 4, in $\triangle A B C$, $B C=a, C A=b, A B=$ $\sqrt{a^{2}+b^{2}}, C D \perp$ $A B$. If the incenter of $\triangle A C D$ and $\triangle B C D$ are $I_{1}$ and $I_{2}$ respectively, then $I_{1} I_{2}=$ $\qquad$ (express the answer in terms of $a$ and $b$).

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


7. Let $\theta \in[0,2 \pi]$, if for any $x \in[0,1]$ there is always $2 x^{2} \sin \theta-4 x(1-x) \cos \theta+3(1-x)^{2}>0$, then the range of $\theta$ is $\qquad$ .

Answer: (\frac{\pi}{6}, \pi)


8. Given a positive integer $n \geqslant 3$. Find the smallest real number $k$, such that for any positive real numbers $a_{1}, a_{2}, \cdots, a_{n}$, we have
$$
\sum_{i=1}^{n-1} \frac{a_{i}}{s-a_{i}}+\frac{k a_{n}}{s-a_{n}} \geqslant \frac{n-1}{n-2},
$$

where, $s=a_{1}+a_{2}+\cdots+a_{n}$. (Proposed by Bin Zhao)

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


14. (6 points) Two cars, A and B, start from points $A$ and $B$ respectively and travel towards each other, meeting at a point 108 kilometers from $B$. After reaching each other's starting point, they immediately return to their original locations, and meet again at a point 84 kilometers from $A$. The distance between the two meeting points is $\qquad$ kilometers.

Answer: 48


Point $D$ lies on the side $AC$ of triangle $ABC$. A circle with diameter $BD$ intersects the sides $AB$ and $BC$ at points $P$ and $T$ respectively. Points $M$ and $N$ are the midpoints of segments $AD$ and $CD$ respectively. It is known that $PM \parallel TN$.

a) Find the angle $ABC$.

b) Suppose additionally that $MP = \frac{1}{2}$, $NT = 2$, and $BD = \sqrt{3}$. Find the area of triangle $ABC$.

Answer: 90^\circ


Given any 5 points on a plane, let $\lambda$ represent the ratio of the maximum distance to the minimum distance between these points. Prove that $\lambda \geq 2 \sin 54^\circ$.

Answer: \lambda \geq 2 \sin 54^\circ


Several irrational numbers are written on a blackboard. It is known that for every two numbers $ a$ and $ b$ on the blackboard, at least one of the numbers $ a\over b\plus{}1$ and $ b\over a\plus{}1$ is rational. What maximum number of irrational numbers can be on the blackboard?

[i]Author: Alexander Golovanov[/i]

Answer: 3


Example 2. Find the domain of the function $x^{2}+y^{2}+z^{2}=9$.

Answer: x^2 + y^2 \leq 9


In the diagram, the circles with centres $B$ and $D$ have radii 1 and $r$, respectively, and are tangent at $C$. The line through $A$ and $D$ passes through $B$. The line through $A$ and $S$ is tangent to the circles with centres $B$ and $D$ at $P$ and $Q$, respectively. The line through $A$ and $T$ is also tangent to both circles. Line segment $S T$ is perpendicular to $A D$ at $C$ and is tangent to both circles at $C$.

![](https://cdn.mathpix.com/cropped/2024_04_17_bceb232fd49f7f0368beg-4.jpg?height=331&width=656&top_left_y=507&top_left_x=710)

(a) There is a value of $r$ for which $A S=S T=A T$. Determine this value of $r$.

(b) There is a value of $r$ for which $D Q=Q P$. Determine this value of $r$.

(c) A third circle, with centre $O$, passes through $A, S$ and $T$, and intersects the circle with centre $D$ at points $V$ and $W$. There is a value of $r$ for which $O V$ is perpendicular to $D V$. Determine this value of $r$.

![](https://cdn.mathpix.com/cropped/2024_04_17_bceb232fd49f7f0368beg-6.jpg?height=189&width=208&top_left_y=242&top_left_x=1593)

Answer: 3


3. As shown in Figure $9, A B$ is the diameter of a semicircle, and $C$ is a point on the semicircular arc. One side $D G$ of the square $D E F G$ lies on the diameter $A B$,
and the other side $D E$ passes through the incenter $I$ of $\triangle A B C$, with point
$E$ on the semicircular arc. If the area of the square $D E F G$ is
100, find the area of $\triangle A B C$.

Answer: 100


## Task B-3.4.

A restaurant offers its guests a selection of cold meat cuts, fish pâté, and vegetable salad for appetizers. For soup, guests can choose from chicken soup, beef soup, fish soup, broccoli soup, and asparagus soup. For the main course, the restaurant offers a fish, meat, or vegetarian plate, and for dessert, a vegan cake or vegan ice cream. If a guest chooses one dish from each course, what is the probability that they will randomly select a completely vegetarian meal (a meal without meat and fish)?

Answer: \dfrac{2}{45}


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Eric`
 - Each person has an occupation: `teacher`, `doctor`, `engineer`, `artist`
 - The people are of nationalities: `swede`, `dane`, `norwegian`, `brit`
 - The people keep unique animals: `horse`, `bird`, `cat`, `fish`
 - Each person has a favorite color: `green`, `white`, `red`, `yellow`
 - People have unique favorite sports: `tennis`, `soccer`, `swimming`, `basketball`

## Clues:
1. The person who is an artist is directly left of the person who loves swimming.
2. Alice is the fish enthusiast.
3. The person who is a teacher is in the second house.
4. The Norwegian is the person who loves white.
5. Arnold is directly left of the Swedish person.
6. The person who loves white and the person who loves tennis are next to each other.
7. The person who keeps horses is Eric.
8. The person whose favorite color is green is the person who loves tennis.
9. The person who keeps horses is directly left of the person who loves tennis.
10. The person who loves yellow is Peter.
11. The Dane is in the first house.
12. Alice is somewhere to the right of Peter.
13. The person who loves white is not in the second house.
14. There is one house between the cat lover and the person who loves soccer.
15. The person who loves tennis is somewhere to the right of the person who is a doctor.
 What is the value of attribute Color for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: white


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Bob`, `Peter`, `Carol`
 - Each person has a unique hobby: `woodworking`, `knitting`, `cooking`, `gardening`, `painting`, `photography`
 - Everyone has a unique favorite cigar: `pall mall`, `blue master`, `blends`, `prince`, `dunhill`, `yellow monster`

## Clues:
1. Alice is the person who loves cooking.
2. Bob is not in the second house.
3. The person who enjoys gardening is the person who smokes Blue Master.
4. The person partial to Pall Mall is the person who enjoys knitting.
5. The person who enjoys gardening is directly left of the Dunhill smoker.
6. The person who smokes many unique blends is directly left of Carol.
7. The person partial to Pall Mall is directly left of Bob.
8. The Dunhill smoker is the person who paints as a hobby.
9. The person who smokes Yellow Monster is in the second house.
10. The woodworking hobbyist is somewhere to the right of the Dunhill smoker.
11. Eric is somewhere to the left of the person who enjoys gardening.
12. There are two houses between the photography enthusiast and Peter.
 What is the value of attribute House for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Arnold`, `Carol`, `Eric`, `Peter`
 - People use unique phone models: `xiaomi mi 11`, `google pixel 6`, `samsung galaxy s21`, `oneplus 9`, `huawei p50`, `iphone 13`
 - People have unique favorite book genres: `romance`, `historical fiction`, `fantasy`, `mystery`, `science fiction`, `biography`
 - They all have a unique favorite flower: `tulips`, `roses`, `carnations`, `daffodils`, `iris`, `lilies`
 - The people keep unique animals: `dog`, `cat`, `bird`, `fish`, `rabbit`, `horse`
 - Each person has a unique level of education: `trade school`, `high school`, `master`, `associate`, `bachelor`, `doctorate`

## Clues:
1. The bird keeper is the person who uses a Google Pixel 6.
2. The person who loves a bouquet of daffodils is the person with an associate's degree.
3. The person who uses a OnePlus 9 is not in the fourth house.
4. There are two houses between the person with a master's degree and the rabbit owner.
5. The person who loves a carnations arrangement is Arnold.
6. The person who loves the boquet of lilies is not in the sixth house.
7. The person who loves the vase of tulips is Carol.
8. The person with a high school diploma is Bob.
9. The person who attended trade school is somewhere to the left of the person who keeps horses.
10. The person who loves mystery books is Eric.
11. The fish enthusiast is directly left of Carol.
12. Bob is the fish enthusiast.
13. Eric is the dog owner.
14. The person who loves a carnations arrangement is the person who uses a Google Pixel 6.
15. The person who loves fantasy books is in the second house.
16. The person who uses a Samsung Galaxy S21 is the person who loves the boquet of iris.
17. Alice is the person with an associate's degree.
18. The person who uses a Google Pixel 6 is not in the second house.
19. The dog owner is the person with a doctorate.
20. The fish enthusiast is the person who uses a Samsung Galaxy S21.
21. The person who uses an iPhone 13 is somewhere to the right of the person who loves fantasy books.
22. The person who loves mystery books is somewhere to the right of the person who loves the boquet of iris.
23. Alice is the person who uses an iPhone 13.
24. The person who attended trade school is not in the fourth house.
25. The person who loves biography books is directly left of Eric.
26. There is one house between the rabbit owner and the person who loves historical fiction books.
27. The person who loves the boquet of iris is somewhere to the right of the person who uses a Xiaomi Mi 11.
28. The person who loves science fiction books is the person who uses a Samsung Galaxy S21.
29. The person who loves the boquet of lilies is not in the second house.
 What is the value of attribute Education for the person whose attribute BookGenre is fantasy? Please reason step by step, and put your final answer within \boxed{}

Answer: trade school


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`
 - People have unique heights: `average`, `tall`, `very short`, `short`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`, `dragonfruit`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`

## Clues:
1. Eric is the person who loves the rose bouquet.
2. The Dragonfruit smoothie lover is somewhere to the left of the person who loves the rose bouquet.
3. Arnold is the person who loves a carnations arrangement.
4. The Desert smoothie lover is in the third house.
5. The person who is very short is the Dragonfruit smoothie lover.
6. Alice is the person who likes Cherry smoothies.
7. The person who loves a bouquet of daffodils is the person who likes Cherry smoothies.
8. The person who loves a carnations arrangement is in the fourth house.
9. The person who is very short is directly left of the person who is tall.
10. The person who is short is the Desert smoothie lover.
 What is the value of attribute Height for the person whose attribute Smoothie is cherry? 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: `Arnold`, `Peter`, `Eric`
 - Each person has a unique birthday month: `sept`, `april`, `jan`
 - People have unique favorite book genres: `mystery`, `science fiction`, `romance`
 - Everyone has a unique favorite cigar: `prince`, `blue master`, `pall mall`
 - The people keep unique animals: `bird`, `cat`, `horse`
 - People own unique car models: `toyota camry`, `ford f150`, `tesla model 3`

## Clues:
1. Arnold is the Prince smoker.
2. Eric is somewhere to the right of the person who loves romance books.
3. The person who keeps horses is the Prince smoker.
4. The person who smokes Blue Master is the person who loves mystery books.
5. The person who owns a Toyota Camry is the person partial to Pall Mall.
6. The person who owns a Tesla Model 3 is the person whose birthday is in September.
7. Peter is the person partial to Pall Mall.
8. The Prince smoker is somewhere to the right of the person who loves mystery books.
9. The person who keeps horses is the person whose birthday is in January.
10. The person who owns a Toyota Camry is the bird keeper.
 What is the value of attribute House for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: 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: `Eric`, `Bob`, `Peter`, `Alice`, `Arnold`, `Carol`
 - People own unique car models: `ford f150`, `honda civic`, `toyota camry`, `tesla model 3`, `chevrolet silverado`, `bmw 3 series`
 - The mothers' names in different houses are unique: `Sarah`, `Penny`, `Holly`, `Aniya`, `Kailyn`, `Janelle`
 - Each person has a unique hobby: `photography`, `cooking`, `knitting`, `gardening`, `woodworking`, `painting`

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

Answer: Penny


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

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

Answer: jan


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
 - Each person has a unique hobby: `cooking`, `gardening`, `painting`, `photography`, `knitting`
 - People have unique favorite sports: `swimming`, `tennis`, `soccer`, `baseball`, `basketball`
 - Each person lives in a unique style of house: `ranch`, `craftsman`, `victorian`, `modern`, `colonial`
 - Each mother is accompanied by their child: `Timothy`, `Samantha`, `Bella`, `Meredith`, `Fred`
 - People have unique heights: `average`, `very tall`, `very short`, `short`, `tall`

## Clues:
1. The person who has an average height is the person's child is named Meredith.
2. The person who is tall is in the second house.
3. Peter is directly left of the person residing in a Victorian house.
4. Alice is the person who is tall.
5. The person who loves baseball is the person who is very tall.
6. The person's child is named Meredith and the person who is the mother of Timothy are next to each other.
7. Bob is the person who paints as a hobby.
8. The person who enjoys gardening is in the second house.
9. The person who is very short is somewhere to the right of Eric.
10. The person who loves tennis is the person's child is named Samantha.
11. The person who loves soccer is not in the first house.
12. The person's child is named Samantha is the person in a modern-style house.
13. The person in a Craftsman-style house is the person who has an average height.
14. The person's child is named Fred is the person residing in a Victorian house.
15. The person who is short is the person who loves basketball.
16. Peter is the person who is very tall.
17. The person in a ranch-style home is somewhere to the left of the person who loves cooking.
18. The person who enjoys knitting and the person who enjoys gardening are next to each other.
19. The person in a modern-style house is the person who loves cooking.
20. The person residing in a Victorian house is in the fifth house.
 What is the value of attribute House for the person whose attribute FavoriteSport is swimming? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


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

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

Answer: iphone 13


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

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

Answer: Fred


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`, `Carol`, `Eric`, `Alice`, `Peter`, `Bob`
 - Each person has a favorite color: `red`, `yellow`, `blue`, `white`, `green`, `purple`
 - Each person has a unique birthday month: `sept`, `jan`, `april`, `feb`, `may`, `mar`

## Clues:
1. The person whose favorite color is green is the person whose birthday is in May.
2. The person whose birthday is in September is directly left of Arnold.
3. The person who loves blue is Carol.
4. The person whose birthday is in January is somewhere to the left of Arnold.
5. Eric is not in the sixth house.
6. Eric is not in the first house.
7. The person who loves yellow is not in the third house.
8. The person whose birthday is in May and the person whose favorite color is red are next to each other.
9. Alice is the person whose favorite color is red.
10. Alice is directly left of the person who loves white.
11. The person who loves blue is not in the sixth house.
12. The person who loves purple is the person whose birthday is in April.
13. Arnold is directly left of the person whose birthday is in March.
14. Peter is in the third house.
 What is the value of attribute Name for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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

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

Answer: Janelle


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Carol`, `Peter`, `Eric`, `Bob`
 - Each person has a unique type of pet: `cat`, `bird`, `fish`, `dog`, `hamster`, `rabbit`
 - People have unique heights: `super tall`, `average`, `tall`, `very short`, `very tall`, `short`
 - People have unique favorite sports: `baseball`, `tennis`, `soccer`, `swimming`, `basketball`, `volleyball`

## Clues:
1. There is one house between the person who has an average height and the person who is short.
2. The person with an aquarium of fish is in the fifth house.
3. The person who has a cat is the person who is tall.
4. The person who is very tall is the person who loves soccer.
5. Peter is in the first house.
6. The person who loves tennis is in the sixth house.
7. Carol is somewhere to the right of the person who is very tall.
8. The person who loves volleyball is directly left of the person who loves soccer.
9. The person who loves baseball is directly left of the person who is super tall.
10. Carol and Bob are next to each other.
11. The person who is super tall is not in the sixth house.
12. The person who is short is not in the fourth house.
13. There is one house between the person who loves swimming and Bob.
14. The person who loves basketball is somewhere to the left of the person who owns a dog.
15. The person who owns a rabbit is somewhere to the left of the person who is short.
16. The person who keeps a pet bird is the person who is short.
17. Arnold is not in the second house.
18. Bob is directly left of Eric.
19. The person who owns a rabbit and the person who is very tall are next to each other.
 What is the value of attribute Pet for the person whose attribute Height is super tall? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Carol`, `Eric`, `Peter`, `Bob`, `Arnold`
 - People have unique favorite music genres: `classical`, `hip hop`, `jazz`, `pop`, `rock`, `country`
 - The mothers' names in different houses are unique: `Sarah`, `Penny`, `Aniya`, `Janelle`, `Kailyn`, `Holly`
 - Each mother is accompanied by their child: `Alice`, `Fred`, `Timothy`, `Bella`, `Samantha`, `Meredith`
 - People have unique heights: `very short`, `tall`, `short`, `very tall`, `super tall`, `average`
 - The people keep unique animals: `bird`, `dog`, `horse`, `rabbit`, `cat`, `fish`

## Clues:
1. The person who loves pop music is the cat lover.
2. The rabbit owner is directly left of The person whose mother's name is Aniya.
3. The person whose mother's name is Holly is directly left of Carol.
4. The person whose mother's name is Holly is the person's child is named Alice.
5. The person whose mother's name is Holly is the person who loves classical music.
6. The person who loves jazz music is The person whose mother's name is Sarah.
7. The person's child is named Meredith is somewhere to the right of The person whose mother's name is Aniya.
8. The person who is super tall is The person whose mother's name is Holly.
9. The person who is the mother of Timothy is Bob.
10. The person who is very short is somewhere to the left of The person whose mother's name is Aniya.
11. Eric is the fish enthusiast.
12. The person's child is named Samantha is somewhere to the right of the person who is very tall.
13. The person who loves rock music is The person whose mother's name is Janelle.
14. There is one house between the person who keeps horses and the person's child is named Meredith.
15. The person's child is named Bella is somewhere to the right of Peter.
16. The fish enthusiast is somewhere to the left of the bird keeper.
17. The fish enthusiast is somewhere to the right of the person's child is named Alice.
18. There is one house between the person's child is named Bella and the person who loves rock music.
19. The person who is short is the cat lover.
20. Alice is directly left of the person who loves classical music.
21. The person's child is named Bella is The person whose mother's name is Aniya.
22. There are two houses between The person whose mother's name is Penny and the person who is short.
23. The person who loves hip-hop music is in the first house.
24. Carol is the person who is tall.
 What is the value of attribute Children for the person whose attribute MusicGenre is jazz? Please reason step by step, and put your final answer within \boxed{}

Answer: Samantha


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Bob`, `Peter`, `Carol`, `Alice`
 - Each mother is accompanied by their child: `Fred`, `Bella`, `Timothy`, `Meredith`, `Samantha`, `Alice`
 - Everyone has something unique for lunch: `pizza`, `stir fry`, `grilled cheese`, `spaghetti`, `soup`, `stew`
 - Each person has a unique type of pet: `rabbit`, `fish`, `cat`, `dog`, `bird`, `hamster`
 - Each person has a unique birthday month: `may`, `sept`, `mar`, `jan`, `april`, `feb`
 - Each person has a unique favorite drink: `water`, `milk`, `coffee`, `tea`, `boba tea`, `root beer`

## Clues:
1. The person whose birthday is in May is the person who loves the spaghetti eater.
2. The person's child is named Fred is the person who loves the stew.
3. The person who likes milk is the person with a pet hamster.
4. There is one house between the person who loves the spaghetti eater and the person who has a cat.
5. The person who is the mother of Timothy is Bob.
6. The person whose birthday is in January is directly left of the one who only drinks water.
7. Carol is the coffee drinker.
8. Peter is not in the fifth house.
9. The person whose birthday is in April is the person's child is named Alice.
10. The person who loves eating grilled cheese is somewhere to the right of the person whose birthday is in September.
11. The person's child is named Bella and Bob are next to each other.
12. The person who owns a rabbit is Peter.
13. The person who keeps a pet bird is the person who loves the soup.
14. The person who owns a rabbit is somewhere to the right of the person who loves stir fry.
15. The person who loves eating grilled cheese is somewhere to the left of the person who owns a rabbit.
16. The person whose birthday is in March is the tea drinker.
17. Peter is the person's child is named Bella.
18. The person who is the mother of Timothy is somewhere to the right of the person's child is named Bella.
19. The person with an aquarium of fish and the person's child is named Bella are next to each other.
20. The one who only drinks water is the person's child is named Fred.
21. The person whose birthday is in May is somewhere to the right of the person with an aquarium of fish.
22. The person with a pet hamster is the person's child is named Meredith.
23. The root beer lover is the person who loves the spaghetti eater.
24. Carol is directly left of Alice.
25. Eric is in the third house.
26. The person who is a pizza lover is Bob.
 What is the value of attribute House for the person whose attribute Birthday is may? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


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`
 - Everyone has a unique favorite cigar: `dunhill`, `pall mall`, `prince`, `blue master`
 - Each person has a unique level of education: `associate`, `high school`, `master`, `bachelor`
 - They all have a unique favorite flower: `lilies`, `daffodils`, `carnations`, `roses`
 - Each person has a unique hobby: `cooking`, `photography`, `gardening`, `painting`

## Clues:
1. The person with a bachelor's degree is the person who enjoys gardening.
2. Arnold is somewhere to the left of the person who loves a carnations arrangement.
3. Eric is not in the second house.
4. The person who loves a bouquet of daffodils is the person who paints as a hobby.
5. The person who loves the rose bouquet is the person who smokes Blue Master.
6. The person partial to Pall Mall is Peter.
7. Arnold is in the third house.
8. Peter is directly left of the person who enjoys gardening.
9. Arnold is the person with a high school diploma.
10. The person who smokes Blue Master is not in the second house.
11. The Prince smoker is the photography enthusiast.
12. The person with a master's degree is somewhere to the left of the person who smokes Blue Master.
 What is the value of attribute Cigar for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: prince


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 person has a unique type of pet: `dog`, `fish`, `cat`
 - People have unique favorite book genres: `mystery`, `romance`, `science fiction`
 - People own unique car models: `ford f150`, `tesla model 3`, `toyota camry`
 - Each person has a unique birthday month: `april`, `jan`, `sept`
 - People have unique favorite music genres: `rock`, `classical`, `pop`

## Clues:
1. The person who loves romance books is directly left of the person who owns a Ford F-150.
2. The person whose birthday is in September is the person who owns a Toyota Camry.
3. The person whose birthday is in September is Peter.
4. The person who owns a Tesla Model 3 is the person whose birthday is in January.
5. The person with an aquarium of fish is not in the second house.
6. Arnold is the person who has a cat.
7. The person who owns a Tesla Model 3 is the person who loves science fiction books.
8. The person who loves science fiction books is not in the third house.
9. The person who owns a Ford F-150 is the person who loves pop music.
10. The person who loves rock music is directly left of Eric.
 What is the value of attribute House for the person whose attribute Pet is fish? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Bob`, `Eric`, `Carol`
 - Each person has a unique hobby: `cooking`, `gardening`, `painting`, `knitting`, `photography`, `woodworking`
 - Each person has a unique birthday month: `april`, `sept`, `mar`, `jan`, `feb`, `may`
 - Everyone has a unique favorite cigar: `blends`, `prince`, `pall mall`, `dunhill`, `yellow monster`, `blue master`

## Clues:
1. The person whose birthday is in April is Eric.
2. Arnold is in the first house.
3. The person whose birthday is in January is the person who smokes many unique blends.
4. The Dunhill smoker is Alice.
5. Bob is the person whose birthday is in May.
6. The person whose birthday is in September and the photography enthusiast are next to each other.
7. Bob is the person who smokes Blue Master.
8. The person who loves cooking is the person whose birthday is in May.
9. The woodworking hobbyist and the person whose birthday is in March are next to each other.
10. The person who enjoys knitting is directly left of the person who smokes Yellow Monster.
11. Peter and the person who paints as a hobby are next to each other.
12. The person whose birthday is in March is somewhere to the left of the Prince smoker.
13. The person whose birthday is in February is somewhere to the left of the person who smokes many unique blends.
14. The person whose birthday is in March is somewhere to the right of the person whose birthday is in September.
15. The person who smokes many unique blends is directly left of the person whose birthday is in May.
16. The photography enthusiast and Peter are next to each other.
 What is the value of attribute Name for the person whose attribute Birthday is feb? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Alice`
 - People have unique favorite music genres: `rock`, `jazz`, `classical`, `pop`
 - The people are of nationalities: `dane`, `brit`, `swede`, `norwegian`
 - Each person has a favorite color: `green`, `red`, `yellow`, `white`

## Clues:
1. The British person and Peter are next to each other.
2. Eric is somewhere to the left of the British person.
3. The Swedish person is directly left of the person whose favorite color is red.
4. The person who loves classical music is in the second house.
5. There is one house between the British person and the person whose favorite color is red.
6. The person who loves rock music is the Norwegian.
7. The person who loves yellow is not in the first house.
8. The Norwegian is Arnold.
9. The Dane and the person who loves white are next to each other.
10. There are two houses between the person who loves pop music and the person who loves rock music.
 What is the value of attribute House for the person whose attribute Color is green? 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: `Eric`, `Bob`, `Alice`, `Peter`, `Arnold`, `Carol`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `roses`, `tulips`, `iris`, `lilies`
 - People have unique heights: `very tall`, `super tall`, `average`, `short`, `very short`, `tall`
 - Each person has a favorite color: `blue`, `yellow`, `purple`, `red`, `green`, `white`
 - Everyone has a unique favorite cigar: `yellow monster`, `pall mall`, `prince`, `blue master`, `blends`, `dunhill`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `xiaomi mi 11`, `samsung galaxy s21`, `huawei p50`, `oneplus 9`

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

Answer: prince


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`
 - The people keep unique animals: `horse`, `bird`, `cat`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `desert`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`
 - People have unique favorite music genres: `pop`, `classical`, `rock`
 - Each person has a favorite color: `yellow`, `white`, `red`

## Clues:
1. The cat lover is in the third house.
2. There is one house between the person who loves yellow and the cat lover.
3. The person who loves classical music is directly left of Eric.
4. Arnold is the Desert smoothie lover.
5. The bird keeper is the Watermelon smoothie lover.
6. The person whose mother's name is Janelle is somewhere to the left of the person who loves white.
7. The person who loves yellow is the person who loves rock music.
8. The person whose mother's name is Holly is directly left of Peter.
 What is the value of attribute Name for the person whose attribute MusicGenre is classical? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Alice`, `Eric`, `Arnold`
 - The mothers' names in different houses are unique: `Penny`, `Kailyn`, `Aniya`, `Janelle`, `Holly`
 - Everyone has a unique favorite cigar: `blue master`, `pall mall`, `prince`, `dunhill`, `blends`
 - The people keep unique animals: `bird`, `cat`, `horse`, `fish`, `dog`
 - Everyone has something unique for lunch: `grilled cheese`, `stir fry`, `pizza`, `spaghetti`, `stew`

## Clues:
1. The person who loves the stew is somewhere to the left of the fish enthusiast.
2. The person who is a pizza lover is somewhere to the right of the Dunhill smoker.
3. Peter is the cat lover.
4. The dog owner is directly left of the person who loves the spaghetti eater.
5. The person who loves stir fry is The person whose mother's name is Aniya.
6. There are two houses between the fish enthusiast and Eric.
7. The person who smokes many unique blends is The person whose mother's name is Kailyn.
8. The fish enthusiast is the person who loves the spaghetti eater.
9. The fish enthusiast is in the second house.
10. The person whose mother's name is Kailyn is the person who loves the stew.
11. The person whose mother's name is Janelle is not in the fifth house.
12. The person whose mother's name is Holly is the person who smokes Blue Master.
13. Arnold is The person whose mother's name is Aniya.
14. The person who loves the spaghetti eater and the person partial to Pall Mall are next to each other.
15. Peter is not in the third house.
16. Arnold is the person who keeps horses.
17. The person whose mother's name is Holly is Bob.
 What is the value of attribute Animal for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: horse


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`, `Carol`, `Peter`, `Eric`, `Alice`
 - People have unique favorite music genres: `classical`, `hip hop`, `jazz`, `pop`, `rock`, `country`
 - Everyone has something unique for lunch: `soup`, `stir fry`, `grilled cheese`, `spaghetti`, `stew`, `pizza`
 - Each person lives in a unique style of house: `craftsman`, `ranch`, `mediterranean`, `modern`, `colonial`, `victorian`
 - Each person has a unique birthday month: `feb`, `sept`, `april`, `may`, `jan`, `mar`
 - People have unique hair colors: `red`, `black`, `blonde`, `brown`, `gray`, `auburn`

## Clues:
1. The person whose birthday is in April is Bob.
2. The person residing in a Victorian house is the person who loves stir fry.
3. There are two houses between the person who loves jazz music and the person who has auburn hair.
4. The person who loves the stew is in the second house.
5. The person who loves stir fry is the person whose birthday is in February.
6. The person who loves classical music is in the fifth house.
7. The person who loves jazz music is Alice.
8. The person living in a colonial-style house is in the fifth house.
9. The person who has gray hair is the person who loves the stew.
10. The person whose birthday is in March is the person who has red hair.
11. The person who loves country music is directly left of the person whose birthday is in September.
12. The person whose birthday is in May is somewhere to the left of the person who loves country music.
13. The person in a ranch-style home is the person who loves the soup.
14. The person whose birthday is in January is the person who loves the stew.
15. The person who has blonde hair is the person in a Mediterranean-style villa.
16. Arnold is somewhere to the right of the person whose birthday is in April.
17. The person whose birthday is in February is the person who loves hip-hop music.
18. The person who has blonde hair is somewhere to the left of Eric.
19. The person residing in a Victorian house is Carol.
20. The person who has black hair is the person in a ranch-style home.
21. The person in a modern-style house is not in the second house.
22. The person who loves the spaghetti eater is somewhere to the left of the person who loves country music.
23. The person who is a pizza lover is Bob.
24. The person who loves pop music is somewhere to the left of the person whose birthday is in September.
 What is the value of attribute MusicGenre for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: rock


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`, `Carol`, `Eric`
 - People have unique hair colors: `red`, `brown`, `blonde`, `black`, `gray`, `auburn`
 - Each person has an occupation: `engineer`, `lawyer`, `nurse`, `doctor`, `artist`, `teacher`

## Clues:
1. The person who has brown hair is not in the second house.
2. Alice is not in the fourth house.
3. The person who is an artist is Bob.
4. The person who has gray hair is the person who is a doctor.
5. Bob is the person who has auburn hair.
6. The person who is a doctor and Bob are next to each other.
7. The person who is an engineer is directly left of Alice.
8. Arnold is directly left of the person who is a lawyer.
9. Bob is directly left of the person who is a teacher.
10. Peter is somewhere to the right of the person who has brown hair.
11. Arnold is not in the second house.
12. The person who has black hair is Arnold.
13. The person who is a doctor is somewhere to the right of the person who has red hair.
14. The person who has brown hair and Carol are next to each other.
15. Arnold is somewhere to the left of the person who is a doctor.
 What is the value of attribute HairColor for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: red


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`, `Peter`, `Carol`, `Alice`, `Eric`
 - Everyone has something unique for lunch: `stir fry`, `stew`, `soup`, `pizza`, `grilled cheese`, `spaghetti`
 - Each person has an occupation: `doctor`, `artist`, `engineer`, `teacher`, `lawyer`, `nurse`
 - Each person has a unique type of pet: `dog`, `bird`, `hamster`, `cat`, `fish`, `rabbit`
 - People have unique favorite music genres: `pop`, `country`, `jazz`, `classical`, `rock`, `hip hop`
 - Everyone has a favorite smoothie: `blueberry`, `desert`, `watermelon`, `cherry`, `lime`, `dragonfruit`

## Clues:
1. The person who loves the spaghetti eater is the Dragonfruit smoothie lover.
2. The person who drinks Lime smoothies is in the second house.
3. Carol is somewhere to the left of the Desert smoothie lover.
4. The person who is an engineer is not in the second house.
5. The person who loves the soup is in the second house.
6. The person who owns a dog is not in the second house.
7. The person who is a pizza lover is Alice.
8. Arnold is somewhere to the left of Eric.
9. The person who loves classical music is the Dragonfruit smoothie lover.
10. The person who owns a rabbit is somewhere to the left of the Desert smoothie lover.
11. The person who is a nurse is not in the sixth house.
12. There are two houses between the person who is a nurse and the person who loves the spaghetti eater.
13. The person who likes Cherry smoothies is the person with an aquarium of fish.
14. The person who has a cat is Alice.
15. The person who likes Cherry smoothies is somewhere to the right of the Desert smoothie lover.
16. There is one house between the person who drinks Blueberry smoothies and the person who loves eating grilled cheese.
17. The person who loves classical music is somewhere to the left of the person who loves the stew.
18. The person who is a lawyer is the person who loves classical music.
19. The person who loves jazz music is Bob.
20. Bob is the person who drinks Lime smoothies.
21. The person who loves pop music is the person who is a teacher.
22. The person who is an engineer is directly left of the person who loves rock music.
23. The person who loves hip-hop music is not in the first house.
24. The person with an aquarium of fish is Peter.
25. The person who drinks Blueberry smoothies is the person who keeps a pet bird.
26. The person who has a cat is in the first house.
27. The person who is a doctor is not in the sixth house.
28. The person who owns a rabbit is not in the second house.
29. Carol is not in the fourth house.
 What is the value of attribute House for the person whose attribute MusicGenre is country? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Bob`, `Arnold`, `Alice`
 - Each mother is accompanied by their child: `Timothy`, `Bella`, `Meredith`, `Samantha`, `Fred`

## Clues:
1. Arnold is not in the first house.
2. Alice is directly left of the person's child is named Samantha.
3. Peter is directly left of Alice.
4. Arnold is the person's child is named Bella.
5. Eric is not in the third house.
6. The person's child is named Meredith is in the fifth house.
7. The person who is the mother of Timothy is not in the first house.
 What is the value of attribute Name for the person whose attribute Children is Fred? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


10.4. In a $100 \times 100$ grid, each cell contains a positive integer. If the sum of the numbers in the cells of a rectangle (composed of several cells) is a multiple of 17, then the rectangle is called "good". We can color the cells of some good rectangles in the grid, with each cell being colored at most once. It is known that for any such number grid, at least $d$ cells can be colored according to the above rule. Find the maximum value of $d$.

Answer: 9744


A pyramid's base is a square with unit side length. One of its side edges is also of unit length and coincides with the height of the pyramid. What is the largest dihedral angle?

Answer: 120


. A chess knight has injured his leg and is limping. He alternates between a normal move and a short move where he moves to any diagonally neighbouring cell.

![](https://cdn.mathpix.com/cropped/2024_04_17_6a5543a8a260575a56cdg-05.jpg?height=315&width=317&top_left_y=385&top_left_x=732)

Normal move

![](https://cdn.mathpix.com/cropped/2024_04_17_6a5543a8a260575a56cdg-05.jpg?height=199&width=195&top_left_y=503&top_left_x=1136)

Short move

The limping knight moves on a $5 \times 6$ cell chessboard starting with a normal move. What is the largest number of moves he can make if he is starting from a cell of his own choice and is not allowed to visit any cell (including the initial cell) more than once?

#

Answer: 25


Let \(f(x)\) be a real-valued function satisfying \(af(x) + bf(-x) = px^2 + qx + r\). Here, \(a\) and \(b\) are distinct real numbers, and \(p\), \(q\), \(r\) are non-zero real numbers. Determine the condition under which \(f(x) = 0\) will have real solutions. The original answer is in the form \(\left(\frac{k(a+b)}{m(a-b)}\right) \leq \frac{q^2}{4pr}\). Please provide the value of \(k + m\).

Answer: 2


In triangle $ABC$, $AB = AC$. If there is a point $P$ strictly between $A$ and $B$ such that $AP = PC = CB$, then if the answer is $x^\circ$, what is the value of $x$?

Answer: 36


If the (convex) area bounded by the x-axis and the lines $y=mx+4$, $x=1$, and $x=4$ is $7$, find the value of $m$. The original answer is in \frac{k}{m} format, please give the value of k + m.

Answer: 1


A mass is attached to an ideal spring. At time \(t = 0\) the spring is at its natural length and the mass is given an initial velocity; the period of the ensuing (one-dimensional) simple harmonic motion is \(T\). At what time \(t\) is the power delivered to the mass by the spring first a maximum, in terms of \(T\)? The answer is in the form \(\frac{kT}{m}\). Please find the value of \(k + m\).

Answer: 9


Dave's Amazing Hotel has $3$ floors. If you press the up button on the elevator from the $3$rd floor, you are immediately transported to the $1$st floor. Similarly, if you press the down button from the $1$st floor, you are immediately transported to the $3$rd floor. Dave gets in the elevator at the $1$st floor and randomly presses up or down at each floor. After doing this $482$ times, the probability that Dave is on the first floor can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is the remainder when $m+n$ is divided by $1000$?

Answer: 803


A square $ABCD$ with sides of length 1 is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points $E, F, G$ where $E$ is the midpoint of $BC$, $F$ and $G$ are on $AB$ and $CD$, respectively. They are positioned such that $AF < FB$, $DG < GC$, and $F$ is directly opposite $G$. If $FB = x$, the length of the longer parallel side of each trapezoid, find the value of $x$. The original answer is in the form of \frac{k}{m}, please give the value of k + m.

Answer: 11


Circles with centers \(O\) and \(P\) have radii 2 and 4, respectively, and are externally tangent. Points \(A\) and \(B\) are on the circle centered at \(O\), and points \(C\) and \(D\) are on the circle centered at \(P\), such that \(\overline{AD}\) and \(\overline{BC}\) are common external tangents to the circles. Find the area of hexagon \(AOBCPD\). The original answer is in the form \(k\sqrt{m}\). Please give the value of \(k + m\).

Answer: 26


Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ($\angle C=90^\circ$). It is known that the circumcircles of triangles $AEM$ and $CDM$ are tangent. Find the angle $\angle BMC.$

Answer: 90^\circ


Let the sequence $u_n$ be defined by $u_0=0$ and $u_{2n}=u_n$, $u_{2n+1}=1-u_n$ for each $n\in\mathbb N_0$.
(a) Calculate $u_{1990}$.
(b) Find the number of indices $n\le1990$ for which $u_n=0$.
(c) Let $p$ be a natural number and $N=(2^p-1)^2$. Find $u_N$.

Answer: 0


5. Let the complex number $z=\cos \frac{4 \pi}{7}+\mathrm{i} \sin \frac{4 \pi}{7}$. Then the value of $\left|\frac{z}{1+z^{2}}+\frac{z^{2}}{1+z^{4}}+\frac{z^{3}}{1+z^{6}}\right|$
is $\qquad$ (answer with a number).

Answer: 2


II. (25 points) As shown in Figure 5, in $\triangle A B C$, $\angle C=$ $90^{\circ}$, $D$ is a point on side $B C$, and $\angle A D C=45^{\circ}$. Draw $D E \perp A B$ at point $E$, and $A E: B E=10: 3$. If $D E=\sqrt{2}$, try to find the length of the angle bisector $C F$ of $\angle C$.

Answer: \dfrac{6\sqrt{13}}{5}


One, (50 points) As shown in Figure 4, in the acute triangle $\triangle ABC$, the altitude $CE$ from $A B$ intersects with the altitude $BD$ from $A C$ at point $H$. The circle with diameter $DE$ intersects $AB$ and $AC$ at points $F$ and $G$, respectively. $FG$ intersects $AH$ at point $K$. Given that $BC=25, BD=20, BE=7$. Find the length of $AK$.

Answer: \dfrac{216}{25}


## 276. Math Puzzle $5 / 88$

This time, those who have a broad, application-oriented general knowledge in addition to some mathematical knowledge will have an advantage:

The following word fragments are to be preceded by the number word for a natural number so that meaningful terms are formed. Let $a$ be the sum of the numbers inserted under A, and $b$ the sum of the numbers inserted under B, then $a$ minus $b$ is the year of Carl Friedrich Gauss's death.

A ...schönchen, ...tagsfliege, ...erbahn, ...topfesser, ...füßler

B ...schaft, ...meter, ...auge, ...meilenstiefel, ...waldstätter See, ...zack, ...gestirn, ...baum, ...käsehoch, ...schläfer, ...groschenoper

Answer: 1855


## Task 4 - 191234

Investigate whether among all tetrahedra $ABCD$ with a given volume $V$ and right angles $\angle BDC, \angle CDA, \angle ADB$, there exists one with the smallest possible sum $AB + AC + AD + BC + BD + CD$.

If this is the case, determine (in dependence on $V$) this smallest possible sum.

Answer: 3(1 + \sqrt{2}) \sqrt[3]{6V}


7、A river flows at a uniform speed, with docks A and B located upstream and downstream, respectively, 200 kilometers apart. Boats A and B depart from docks A and B simultaneously and head towards each other. After meeting, they continue to their respective destinations, immediately turn around, and meet again on their return journey. If the interval between the two meetings is 4 hours, and the still water speeds of boats A and B are $36 \mathrm{~km} / \mathrm{h}$ and $64 \mathrm{~km} / \mathrm{h}$, respectively, then the speed of the current is $\qquad$ $\mathrm{km} / \mathrm{h}$

Answer: 14


The probability of twins being born in Shvambria is $p$, and triplets are not born in Shvambria.

a) Estimate the probability that a Shvambrian met on the street is one of a pair of twins?

b) In a certain Shvambrian family, there are three children. What is the probability that there is a pair of twins among them?

c) In Shvambrian schools, twins are always enrolled in the same class. There are a total of $N$ students in Shvambria.

What is the expected number of pairs of twins among them?

Answer: \dfrac{2p}{1 + p}


From four identical squares, an ornament was created as shown in the picture. The sides of the squares are $4 \mathrm{~cm}$ long, and they are either parallel or perpendicular to each other, intersecting either at their quarters or halves. Libor wanted to color the ornament and found that the cost of the paint for $1 \mathrm{~cm}^{2}$ of each contiguous area would be as many crowns as the number of squares that area is shared by.

How many crowns will it cost to paint the ornament?

(K. Pazourek)

![](https://cdn.mathpix.com/cropped/2024_04_17_96587d6cc949a3c812fag-4.jpg?height=429&width=734&top_left_y=2007&top_left_x=664)

Answer: 64


Opposite sides of the quadrilateral \(ABCD\) intersect at points \(E\) and \(F\) respectively. Lines drawn through the vertices of the quadrilateral parallel to a given direction intersect line \(EF\) at points \(A_1, B_1, C_1, D_1\) respectively. Prove that

\[
\frac{1}{AA_1} + \frac{1}{CC_1} = \frac{1}{BB_1} + \frac{1}{DD_1}
\]

2) Triangle \(A_1B_1C_1\) is inscribed in triangle \(ABC\) and is perspective with it. Lines are drawn through the vertices of both triangles parallel to a given direction and intersect the Desargues line of these triangles at points \(A', B', C', A_1', B_1', C_1'\) respectively. Prove that

\[
\frac{1}{AA'} + \frac{1}{BB'} + \frac{1}{CC'} = \frac{1}{A_1A_1'} + \frac{1}{B_1B_1'} + \frac{1}{C_1C_1'}
\]

(The line segments are assumed to be oriented).

Answer: \frac{1}{AA_1} + \frac{1}{CC_1} = \frac{1}{BB_1} + \frac{1}{DD_1}


36. [25] The Marathon. Let $\omega$ denote the incircle of triangle $A B C$. The segments $B C, C A$, and $A B$ are tangent to $\omega$ at $D, E$, and $F$, respectively. Point $P$ lies on $E F$ such that segment $P D$ is perpendicular to $B C$. The line $A P$ intersects $B C$ at $Q$. The circles $\omega_{1}$ and $\omega_{2}$ pass through $B$ and $C$, respectively, and are tangent to $A Q$ at $Q$; the former meets $A B$ again at $X$, and the latter meets $A C$ again at $Y$. The line $X Y$ intersects $B C$ at $Z$. Given that $A B=15, B C=14$, and $C A=13$, find $\lfloor X Z \cdot Y Z\rfloor$.
Answer: 101. Construct $D^{\prime}$ diametrically opposed to $D$, so that $\angle D F D^{\prime}$ and $\angle D E D^{\prime}$ are right, and note that $P$ lies on $D D^{\prime}$. By standard angle chasing, $m \angle F D D^{\prime}=\beta$ (half angle $B$ ) and $m \angle D^{\prime} D E=\gamma$. Thus, $m \angle D D^{\prime} F=90^{\circ}-\beta$ and $m \angle E D^{\prime} D=90^{\circ}-\gamma$. Then by the law of sines, $D E: E D^{\prime}: D^{\prime} F$ : $F D=\cos (\gamma): \sin (\gamma): \sin (\beta): \sin (\gamma)$. Now using $\triangle D E P \sim \triangle F D^{\prime} P$ and $\triangle D F P \sim \triangle E D^{\prime} P$, we have
$$
\frac{E P}{P F}=\frac{E D \cdot E D^{\prime}}{F D \cdot F D^{\prime}}=\frac{\sin (\gamma) \cos (\gamma)}{\sin (\beta) \sin (\beta)}=\frac{c}{b} .
$$

Let the dilation centered at $A$ sending $E$ to $C$ map $P$ and $F$ to $P^{\prime}$ and $F^{\prime}$, respectively. Note that $A F^{\prime}=A C$ as $A E$ and $A F$ are equal tangents, and $C P^{\prime}: P^{\prime} F^{\prime}=E P: P F=c: b$ by similarity. Then by Menelaus' theorem,
$$
1=\frac{B Q}{Q C} \frac{C P^{\prime}}{P^{\prime} F^{\prime}} \frac{F^{\prime} A}{A B}=\frac{B Q}{Q C} \frac{c}{b} \frac{b}{c},
$$
so that $B Q=Q C$ and $A Q$ is actually a median. So, $A Q^{2}=\frac{1}{4}\left(2 b^{2}+2 c^{2}-a^{2}\right)=148$. Now by Power of a Point, $A B \cdot A X=A Q^{2}=A C \cdot A Y$, so $A X=148 / 15$ and $A Y=148 / 13$. Moreover, $B X C Y$ is cyclic as $\triangle A B C \sim \triangle A Y X$. Thus, $X Z \cdot Y Z=B Z \cdot C Z$, and it suffices to compute $B Z / C Z$. Menelaus once more gives
$$
1=\frac{B Z}{Z C} \frac{C Y}{Y A} \frac{A X}{X B},
$$

Answer: 101


3. As shown in Figure 1, in a regular hexagon $A B C D E F$ with side length 10, $H$ is the midpoint of side $D E$, and $G$ is a point on side $B C$ such that $\angle A G B = \angle C G H$. Then the area of pentagon $A F E H G$ is $\qquad$

Answer: \dfrac{205\sqrt{3}}{2}


Three numbers $a,b,c$ , none zero, form an arithmetic progression. Increasing $a$ by $1$ or increasing $c$ by $2$ results
in a geometric progression. Then $b$ equals:
$\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 8$

Answer: 12


3. Given the equation about $x$: $x^{3}+(1-a) x^{2}-2 a x+a^{2}$ $=0$ has only one real root. Then the range of the real number $a$ is $\qquad$.

Answer: (-\infty, -\dfrac{1}{4})


At first 272, from the eight points consisting of the vertices and midpoints of the sides of a square, how many isosceles triangles can be formed by selecting three points?

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

Answer: 20


5. If the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ contains a point $P(1,-1), F$ is the right focus, and there is a point $M$ on the ellipse such that $|M P|+$ $2|M F|$ is minimized, then the point $M$ is $\qquad$.

Answer: \left( \dfrac{2\sqrt{6}}{3}, -1 \right)


Three. (50 points)
There are 21 points on a circle. Among all the arcs with these 21 points as endpoints, what is the minimum number of arcs that do not exceed $120^{\circ}$?

Answer: 100


Standa and Jana received two three-digit numbers. Standa added a decimal point after the first digit in the first number, and after the second digit in the second number, then added the resulting decimal numbers and got the result 50.13. Jana added a decimal point after the second digit in the first number, and after the first digit in the second number, then added the resulting decimal numbers and got the result 34.02.

Determine the sum of the original three-digit numbers.

(K. Pazourek)

Answer: 765


Example 1: In a 10000-meter race on a 400-meter circular track at a school sports meet, two athletes, A and B, start running at the same time. B runs faster than A. At the 15th minute, A speeds up. At the 18th minute, A catches up with B and begins to overtake B. At the 23rd minute, A catches up with B again, and at 23 minutes and 50 seconds, A reaches the finish line. Therefore, the time B takes to complete the entire course is $\qquad$ minutes.
(Ninth Hope Cup National Mathematics Invitational Competition)

Answer: 25


## Task A-2.3.

Two quadratic functions $f_{1}(x)$ and $f_{2}(x)$ are given.

Function $f_{1}(x)$ achieves its minimum value at $x=-1$, and one of its roots is $x=3$. Function $f_{2}(x)$ achieves its maximum value at $x=3$, and one of its roots is $x=-1$.

Determine all values of $x$ for which the product $f_{1}(x) f_{2}(x)$ achieves its maximum value.

Answer: 1 + 2\sqrt{5}