8.2. During breaks, schoolchildren played table tennis. Any two schoolchildren played no more than one game with each other. By the end of the week, it turned out that Petya played half, Kolya - a third, and Vasya - a fifth of the total number of games played during the week. How many games could have been played during the week, given that Vasya did not play with Petya or Kolya?

Answer: 30


3. Given planar vectors $\boldsymbol{a}=(\sqrt{3},-1), \boldsymbol{b}=$ $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$. If there exist non-zero real numbers $k$ and angle $\alpha, \alpha \in$ $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, such that $\boldsymbol{c}=\boldsymbol{a}+\left(\tan ^{2} \alpha-3\right) \boldsymbol{b}, \boldsymbol{d}=$ $-k \boldsymbol{a}+(\tan \alpha) \boldsymbol{b}$, and $\boldsymbol{c} \perp \boldsymbol{d}$, then $k=$ $\qquad$ (expressed in terms of $\alpha$).

Answer: \dfrac{\tan^3 \alpha - 3 \tan \alpha}{4}


Let \( p(x) \) be a non-constant polynomial with integer coefficients, and let \( n(p) \) denote the number of distinct integers \( x \) that satisfy \( (p(x))^{2}=1 \). Prove that \( n(p) - \operatorname{deg} p(x) \leqslant n \).

Answer: 2


There are several stones, each with a mass not exceeding 2 kg, and the total mass of all stones is 100 kg. Some of these stones are selected such that their total mass differs from 10 kg by the smallest possible number $d$ for the given set. What is the maximum value of $d$ for all possible sets of stones?

Answer: \dfrac{10}{11}


Shapovalov A.V.

The price of a standard lunch at the "Buratino" tavern depends only on the day of the week. Anya dined for 10 consecutive days starting from July 10 and paid 70 soldo. Vanya also paid 70 soldo for 12 meals starting from July 12. Tanya paid 100 soldo for 20 meals starting from July 20. How much will Sanya pay for 24 meals starting from July 24?

Answer: 150


1. Let $A B C D E F G H$ be a rectangular cuboid. How many acute-angled triangles are formed by joining any three vertices of the cuboid?
(1 mark)
Let $A B C D E F G H$ be a rectangular cuboid. If any three vertices are joined, how many acute-angled triangles can be formed?
(1 point)

Answer: 8


A game works as follows: the player pays $2$ tokens to enter the game. Then, a fair coin is flipped. If the coin lands on heads, they receive $3$ tokens; if the coin lands on tails, they receive nothing. A player starts with $2$ tokens and keeps playing this game until they do not have enough tokens to play again. What is the expected value of the number of tokens they have left at the end?

[i]2020 CCA Math Bonanza Team Round #9[/i]

Answer: \dfrac{3 - \sqrt{5}}{2}


90. Several students went to take a math exam. One student calculated that even if none of them "fail," there could still be 59049 different exam outcomes. How many students were there?

Answer: 10


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 $4 \sqrt{6}$. Then the area of the container's inner wall that the ball can never touch is $\qquad$

Answer: 72\sqrt{3}


A grid with nine cells, as shown in the image, is filled with nine consecutive natural numbers. These numbers are arranged in ascending order from left to right and from top to bottom (i.e., the smallest number is in the top left, and the largest is in the bottom right). The thick broken line divides the grid into two parts. The sum of the numbers in the left part is 100 less than the sum of the numbers in the right part.

![](https://cdn.mathpix.com/cropped/2024_04_17_f53d56ddc45593eccca2g-2.jpg?height=377&width=376&top_left_y=477&top_left_x=840)

Which number is in the middle cell?

(L. Šimůnek)

Answer: 94


4. Alexei wrote several consecutive natural numbers on the board. It turned out that only two of the written numbers have a digit sum divisible by 8: the smallest and the largest. What is the maximum number of numbers that could have been written on the board?

Answer: 15


8. Given that $\triangle A B C$ is an inscribed triangle in the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$, and $A B$ passes through the point $P(1,0)$, then the maximum value of the area of $\triangle A B C$ is $\qquad$ .

Answer: \dfrac{16\sqrt{2}}{3}


Around a round table, 30 people are sitting - knights and liars (knights always tell the truth, while liars always lie). It is known that each of them has exactly one friend at the same table, and a knight's friend is a liar, while a liar's friend is a knight (friendship is always mutual). When asked "Is your friend sitting next to you?", those sitting every other person answered "Yes". How many of the others could have also answered "Yes"?

Answer: 0


12 Transporting goods from a riverside city $A$ to another location $B$. The shortest distance from $B$ to the riverbank $B C=b$ kilometers, and the distance along the river, $A C$, is $a$ kilometers. If the waterway transportation cost is $\frac{1}{n}$
($n$ is a real number greater than 1) of the highway transportation cost, a road is to be built from point $B$ to a point $D$ on the riverbank, to minimize the total transportation cost from $A$ to $B$, then $D C=$ $\qquad$ kilometers.

Answer: \dfrac{b}{\sqrt{n^2 - 1}}


5.1. Among all the simple fractions, where the numerator and denominator are two-digit numbers, find the smallest fraction greater than $\frac{3}{4}$. In your answer, specify its numerator.

Answer: 73


1. [3] Paul starts with the number 19. In one step, he can add 1 to his number, divide his number by 2 , or divide his number by 3 . What is the minimum number of steps Paul needs to get to 1 ?

Answer: 6


8. A sequence of numbers starts from 0 and follows the rule below, continuing up to 2020. This sequence contains $\qquad$ numbers.
$$
0 \rightarrow 1 \rightarrow 3 \rightarrow 6 \rightarrow 10 \rightarrow 11 \rightarrow 13 \rightarrow 16 \rightarrow 20 \rightarrow 21 \rightarrow 23 \rightarrow 26 \text {..... }
$$

Answer: 809


As shown in the figure, 5 identical small rectangles are combined to form a large square. It is known that the perimeter of the large square is 10 centimeters more than the perimeter of one small rectangle. What is the perimeter of one small rectangle? $\qquad$ centimeters.

Answer: 15


15. Person A and Person B start from the same point on a 300-meter circular track, running in opposite directions. Person A runs 2 meters per second, and Person B runs 4 meters per second. When they meet head-on, Person A turns around and runs back; when A and B meet again, Person B turns around and runs back. If this continues, they will meet at the starting point after ( ) seconds.

Answer: 250


A plane-parallel glass plate is normally illuminated by a thin beam of light. Behind the plate, at some distance from it, there is an ideal mirror (with a reflection coefficient of one). The plane of the mirror is parallel to the plate. It is known that the intensity of the beam passing through this system is 256 times less than the intensity of the incident beam. Assume that the reflection coefficient at the glass-air interface is constant regardless of the direction of the beam. Ignore the absorption and scattering of light in the air and glass. Find the reflection coefficient at the glass-air interface under these conditions. (10 points)

Answer: \dfrac{3}{4}


5. Choose three vertices from the eight vertices of a cube. Then the probability that these three points form a right-angled triangle is

Answer: \dfrac{6}{7}


7. Given the function $f(x)=x^{2}+2 x+a \ln x$, for any $t \geqslant 1$ it always holds that $f(2 t-1) \geqslant 2 f(t)-3$. Then the range of the real number $a$ is $\qquad$ .

Answer: (-\infty, 2]


92. On the blackboard, there are 5 consecutive two-digit numbers. Xiao Ming adds up 3 of these numbers, and the sum can be divided by 47; Xiao Jun also adds up 3 of these numbers, and the sum can be divided by 97. Then, the sum of the 5 numbers on the blackboard is $\qquad$ .

Answer: 160


13.407 Three people, A, B, and C, approached the shore of a reservoir; A set off to the opposite shore swimming at a speed of v km/h; simultaneously, B and C set off in a motorboat at a speed of $10 v$ km/h. After some time, C decided to cover the remaining distance by swimming at the same speed as A. At the same moment, B turned back to pick up A, who quickly got into the boat and continued the journey together with B. All three arrived at the opposite shore simultaneously. Determine the time of the crossing, given that the width of the reservoir is $b$ km, and the current speed is zero.

Answer: \dfrac{31b}{130v}


4. The cells of an $n \times n$ square are painted in $n$ colors, with $n$ cells of each color. For which $n$ can the square be painted in such a way that in each row and each column there are cells of exactly two colors?

Answer: 2


4. Find all positive integers $n$ and $k$, such that $(n+1)^{n}=2 n^{k}+3 n+1$.

 untranslated text remains the same as requested.

Answer: n = 3


An electric current of strength \( I \) flows from bottom to top along an infinite wire that coincides with the \( Oz \) axis. Find the vector \( \mathbf{H} \) of the magnetic field intensity created by this current at an arbitrary point \( M(x, y, z) \) in space.

Answer: \mathbf{H} = \dfrac{I}{2\pi(x^2 + y^2)}(-y\,\mathbf{i} + x\,\mathbf{j})


7.2. There are 11 kg of cereal. How can you measure out 1 kg of cereal using two weighings on a balance scale, if you have one 3 kg weight?

Answer: 1


We consider the $\lambda$-ratio inscribed triangle $H^{\prime}=A^{\prime} B^{\prime} C^{\prime}$ of a triangle $H=A B C$ (where $\lambda>0$) to be the triangle whose vertices $A^{\prime}, B^{\prime}, C^{\prime}$ lie on the segments $B C, C A, A B$ respectively, and satisfy the following condition (1). Let the perimeter of the triangle $H_{0}=A_{0} B_{0} C_{0}$ be 1 unit, and let $H_{0}^{\prime}$ be the $\lambda$-ratio inscribed triangle of $H_{0}$. Starting from these, we construct the sequence of triangles $H_{n}, H^{\prime} n$ such that $H_{n+1}$ is the $\frac{1}{\lambda}$-ratio inscribed triangle of $H_{n}^{\prime}$, and $H_{n+1}^{\prime}$ is the $\lambda$-ratio inscribed triangle of $H_{n+1}$ $(n=0,1,2, \ldots)$.

$$
\frac{A C^{\prime}}{C^{\prime} B}=\frac{B A^{\prime}}{A^{\prime} C}=\frac{C B^{\prime}}{B^{\prime} A}=\lambda
$$

a) Calculate the sum of the perimeters of the triangles $H_{n}$ $(n=0,1,2, \ldots)$.

b) Prove that the ratio of the sum of the perimeters of the triangles $H_{n}$ and $H_{n}^{\prime}$ is equal to the ratio of the perimeters of the triangles $H_{0}$ and $H_{0}^{\prime}$.

Answer: \dfrac{(\lambda + 1)^2}{3\lambda}


3. (8 points) In the figure, the radius of the large circle is 20 cm, and the radii of the 7 small circles are all 10 cm. The area of the shaded region is $\qquad$ square centimeters ( $\pi$ is taken as 3.14).

Answer: 942


## Subject III.

Lines AB and CD intersect at O. Knowing that rays [OM, [OT, [OR, are the bisectors of angles $\angle \mathrm{BOD}, \angle \mathrm{DOM}$, and respectively, $\angle \mathrm{COB}$, and $\mathrm{m}(\angle \mathrm{AOM})=130^{\circ}$, calculate:
a) $\mathrm{m}(\angle \mathrm{ROM})$;
b) $\mathrm{m}(\angle \mathrm{AOD})$;

c) the measure of the angle between the bisectors of angles $\angle \mathrm{AOC}$ and $\angle \mathrm{BOR}$.

Answer: 90^\circ


Ximena wants to number the pages of a notebook. To do this, she has a large quantity of stickers with the digits $0,1,3,4,5,6,7,8$, and 9, but she has only 100 stickers with the digit 2. Determine up to which page Ximena can number this notebook.

Answer: 244


[ Parallelepipeds (miscellaneous). ] [ Parallelism of lines and planes ]  Through the midpoints $M$ and $N$ of the edges $A D$ and $C C 1$ of the parallelepiped $A B C D A 1 B 1 C 1 D 1$, a plane is drawn parallel to the diagonal $D B 1$. Construct the section of the parallelepiped by this plane. In what ratio does it divide the edge $B B 1$?

Answer: 5:1


In the expression \(1 \square 2 \square 3 \square 4\), each \(\square\) is to be replaced by either \(+\) or \(\times\). What is the largest value of all the expressions that can be obtained in this way?

A. 10
B. 14
C. 15
D. 24
E. 25

Answer: 25


Each angle of triangle \(ABC\) is an integer degree. There is an obtuse angle at vertex \(A\). The point \(D\) is on side \(BC\) such that triangle \(ABD\) is a right triangle, and triangle \(ADC\) is isosceles. How many such triangles exist? (You do not need to list all the angles of the suitable triangles.)

Answer: 88


As shown in the figure, \(O\) is the origin, point \(F\) is the focus of the parabola \(C_{1}: x^{2}=2 p y\) \((p>0)\), and the tangent at point \(P\) on the parabola \(C_{1}\) is tangent to the circle \(C_{2}: x^{2}+y^{2}=1\) at point \(Q\).
(1) When the equation of line \(PQ\) is \(x-y-\sqrt{2}=0\), find the equation of the parabola \(C_{1}\);
(2) As the positive number \(p\) varies, let \(S_{1}\) and \(S_{2}\) be the areas of triangles \(\triangle FPQ\) and \(\triangle FOQ\) respectively. Find the minimum value of \(\frac{S_{1}}{S_{2}}\).

Answer: 2\sqrt{2}+3


The sequence \( x_{n} \) has its first two elements as \( x_{1}=1001 \) and \( x_{2}=1003 \). For \( n \geq 1 \), the recurrence relation is given by:
\[ x_{n+2}=\frac{x_{n+1}-2004}{x_{n}}. \]
What is the sum of the first 2004 terms of the sequence?

Answer: 1338004


We fill a graduated cylinder with a diameter of $2.9 \mathrm{~cm}$ with water up to a height of $4 \mathrm{~cm}$. Then, we place solid copper cubes with an edge length of $2 \mathrm{~cm}$ into the cylinder. What is the maximum number of cubes that can be submerged in the water in the cylinder?

Answer: 5


All plane angles of a trihedral angle are equal to $60^\circ$. Find the angles formed by the edges of this trihedral angle with the planes of the opposite faces.

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


Consider the sequence \(\{a_n\}\) defined by \(a_1 = 1\), \(a_2 = 3\), and \(a_n = 3a_{n-1} - a_{n-2}\) for \(n \in \mathbf{N}^*\) with \(n \geq 3\). Determine whether there exists a positive integer \(n\) such that \(2^{2016}\) divides \(a_n\) (i.e., \(2^{2016} \mid a_n\) and \(2^{2017} \nmid a_n\)). If such an \(n\) exists, find the smallest positive integer \(n\); if not, provide an explanation.

Answer: 3\times2^{2013}


In a table tennis match between players A and B, they follow the rules that: the winner of each game gets 1 point and the loser gets 0 points. The match stops when one player has 2 more points than the other or after a maximum of 6 games. Let the probability that A wins each game be $\frac{3}{4}$ and the probability that B wins each game be $\frac{1}{4}$, and assume the outcomes of each game are independent. Find the expected number of games $\xi$ played when the match stops, denoted as $\mathrm{E} \xi$.

Answer: \frac{97}{32}


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`, `Bob`, `Alice`, `Eric`, `Arnold`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `samsung galaxy s21`, `iphone 13`, `huawei p50`
 - Each person has an occupation: `lawyer`, `engineer`, `teacher`, `artist`, `doctor`

## Clues:
1. The person who uses a Samsung Galaxy S21 is directly left of Eric.
2. There are two houses between Alice and the person who is an artist.
3. The person who uses an iPhone 13 is directly left of Bob.
4. The person who is a lawyer is in the third house.
5. The person who is a doctor is Arnold.
6. The person who is an engineer is directly left of the person who is an artist.
7. The person who uses a OnePlus 9 is somewhere to the right of Eric.
8. The person who uses a Google Pixel 6 is Peter.
 What is the value of attribute PhoneModel for the person whose attribute Occupation is artist? Please reason step by step, and put your final answer within \boxed{}

Answer: google pixel 6


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Alice`, `Bob`, `Peter`
 - Each person prefers a unique type of vacation: `mountain`, `city`, `cruise`, `beach`, `camping`
 - Each person has a unique level of education: `doctorate`, `high school`, `bachelor`, `associate`, `master`
 - Each person has a favorite color: `blue`, `red`, `white`, `yellow`, `green`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `oneplus 9`, `huawei p50`, `samsung galaxy s21`
 - Everyone has something unique for lunch: `grilled cheese`, `stir fry`, `pizza`, `spaghetti`, `stew`

## Clues:
1. The person who loves the stew is not in the first house.
2. There are two houses between the person who loves stir fry and the person with an associate's degree.
3. The person who enjoys mountain retreats is the person with a bachelor's degree.
4. The person with a doctorate is somewhere to the right of Bob.
5. The person who uses a Samsung Galaxy S21 is in the third house.
6. Eric is the person with a doctorate.
7. The person with a doctorate is in the third house.
8. The person who loves stir fry is the person with a bachelor's degree.
9. The person with a doctorate is the person who is a pizza lover.
10. The person whose favorite color is green is somewhere to the right of Peter.
11. The person who enjoys camping trips is the person who uses an iPhone 13.
12. The person who likes going on cruises is Alice.
13. There is one house between the person with a high school diploma and the person who uses a Samsung Galaxy S21.
14. The person who uses a Google Pixel 6 is Arnold.
15. The person who uses a OnePlus 9 is somewhere to the right of the person who uses a Huawei P50.
16. Arnold is the person who loves eating grilled cheese.
17. The person who loves eating grilled cheese is not in the fourth house.
18. There are two houses between the person with a bachelor's degree and the person whose favorite color is red.
19. The person who loves beach vacations is somewhere to the right of the person who prefers city breaks.
20. The person whose favorite color is green is not in the second house.
21. The person who loves blue is somewhere to the right of Peter.
22. There is one house between the person who enjoys camping trips and the person who loves yellow.
 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: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Eric`, `Arnold`, `Bob`, `Carol`
 - The people are of nationalities: `dane`, `brit`, `norwegian`, `swede`, `german`, `chinese`
 - People have unique favorite book genres: `historical fiction`, `mystery`, `romance`, `fantasy`, `science fiction`, `biography`
 - The mothers' names in different houses are unique: `Holly`, `Aniya`, `Penny`, `Kailyn`, `Sarah`, `Janelle`

## Clues:
1. The person whose mother's name is Janelle is the Swedish person.
2. Alice is the person who loves science fiction books.
3. The person who loves romance books is not in the fifth house.
4. Carol and Peter are next to each other.
5. The person whose mother's name is Sarah is the Norwegian.
6. There is one house between Arnold and the British person.
7. The German and the person who loves historical fiction books are next to each other.
8. There is one house between the German and Carol.
9. The Swedish person is the person who loves fantasy books.
10. Arnold is in the second house.
11. The person whose mother's name is Sarah is in the third house.
12. The person who loves historical fiction books is somewhere to the right of The person whose mother's name is Penny.
13. The Chinese is not in the sixth house.
14. There are two houses between Alice and the person who loves mystery books.
15. Carol is somewhere to the right of Alice.
16. Arnold is the Dane.
17. The person whose mother's name is Aniya is in the fourth house.
18. The person whose mother's name is Penny is somewhere to the left of the Swedish person.
19. The person whose mother's name is Holly and Eric are next to each other.
 What is the value of attribute House for the person whose attribute BookGenre is mystery? Please reason step by step, and put your final answer within \boxed{}

Answer: 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: `Alice`, `Eric`, `Arnold`, `Peter`
 - Each person has a unique hobby: `painting`, `gardening`, `photography`, `cooking`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`, `fantasy`
 - Each person prefers a unique type of vacation: `city`, `beach`, `mountain`, `cruise`
 - People have unique favorite music genres: `rock`, `pop`, `classical`, `jazz`
 - Each person has a unique level of education: `high school`, `associate`, `bachelor`, `master`

## Clues:
1. The person who loves science fiction books is somewhere to the right of the person who loves fantasy books.
2. The person who paints as a hobby is the person who loves fantasy books.
3. Arnold is the person who loves fantasy books.
4. The person with a high school diploma is the person who loves pop music.
5. The person who loves cooking is somewhere to the right of Eric.
6. There are two houses between the photography enthusiast and the person who loves beach vacations.
7. The person who loves romance books is Peter.
8. The person who enjoys mountain retreats is not in the second house.
9. The person with a bachelor's degree is somewhere to the right of the person with a high school diploma.
10. The person who loves classical music is not in the third house.
11. The person who prefers city breaks is the person who loves science fiction books.
12. The person who loves rock music is Peter.
13. The person with a bachelor's degree is not in the fourth house.
14. The person with a master's degree is the person who loves jazz music.
15. The person who enjoys gardening is the person who loves romance books.
 What is the value of attribute Hobby for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: photography


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 Children is Alice? 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: `Carol`, `Eric`, `Peter`, `Alice`, `Bob`, `Arnold`
 - Everyone has something unique for lunch: `spaghetti`, `soup`, `grilled cheese`, `stir fry`, `stew`, `pizza`
 - Each person has a favorite color: `green`, `purple`, `blue`, `red`, `white`, `yellow`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Aniya`, `Kailyn`, `Sarah`, `Janelle`
 - Each person lives in a unique style of house: `colonial`, `craftsman`, `modern`, `ranch`, `victorian`, `mediterranean`

## Clues:
1. The person whose mother's name is Janelle is the person living in a colonial-style house.
2. Peter is the person whose favorite color is green.
3. The person who loves blue is directly left of the person who loves the spaghetti eater.
4. The person whose mother's name is Penny is directly left of the person whose favorite color is red.
5. The person who loves blue is The person whose mother's name is Penny.
6. The person whose mother's name is Aniya is directly left of the person whose favorite color is green.
7. The person who loves white is somewhere to the left of The person whose mother's name is Janelle.
8. Eric is the person who loves eating grilled cheese.
9. The person whose mother's name is Holly is directly left of Alice.
10. The person whose mother's name is Holly and the person who loves the stew are next to each other.
11. Alice is the person in a ranch-style home.
12. Carol is not in the sixth house.
13. The person who loves stir fry is the person who loves purple.
14. The person whose favorite color is red and the person living in a colonial-style house are next to each other.
15. The person who loves the soup is Peter.
16. The person who is a pizza lover is somewhere to the left of The person whose mother's name is Sarah.
17. The person residing in a Victorian house is the person who loves eating grilled cheese.
18. The person who loves the spaghetti eater is not in the fifth house.
19. There are two houses between the person who loves eating grilled cheese and The person whose mother's name is Aniya.
20. Arnold is the person whose favorite color is red.
21. The person in a modern-style house is Arnold.
22. The person who loves yellow is the person in a Mediterranean-style villa.
 What is the value of attribute Mother for the person whose attribute Color is blue? Please reason step by step, and put your final answer within \boxed{}

Answer: Penny


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`, `Alice`, `Bob`, `Eric`, `Arnold`
 - People have unique heights: `very tall`, `average`, `tall`, `very short`, `short`

## Clues:
1. The person who is short is in the second house.
2. Peter is directly left of Bob.
3. Eric is somewhere to the left of Peter.
4. The person who is very tall is directly left of Peter.
5. Alice is directly left of the person who has an average height.
6. The person who is short and the person who is very short are next to each other.
7. The person who has an average height is in the fifth house.
 What is the value of attribute Height for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: very tall


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Bob`, `Alice`
 - People have unique hair colors: `brown`, `blonde`, `black`, `gray`, `red`
 - Each person prefers a unique type of vacation: `beach`, `camping`, `city`, `cruise`, `mountain`
 - Each person has an occupation: `doctor`, `lawyer`, `engineer`, `teacher`, `artist`
 - People have unique favorite music genres: `jazz`, `pop`, `hip hop`, `classical`, `rock`

## Clues:
1. Alice is the person who has brown hair.
2. The person who is a teacher is somewhere to the right of the person who is an artist.
3. The person who has blonde hair is in the first house.
4. The person who prefers city breaks is not in the fourth house.
5. The person who enjoys mountain retreats is in the third house.
6. The person who loves beach vacations is somewhere to the right of the person who is a lawyer.
7. Eric is the person who has black hair.
8. The person who is an engineer is the person who has red hair.
9. The person who loves classical music is Bob.
10. The person who loves hip-hop music is in the fifth house.
11. The person who has blonde hair is the person who enjoys camping trips.
12. The person who has red hair is in the second house.
13. The person who is a teacher is the person who has gray hair.
14. The person who loves jazz music is the person who has black hair.
15. There is one house between the person who likes going on cruises and the person who has gray hair.
16. The person who loves rock music is the person who has red hair.
17. The person who is a lawyer is not in the third house.
18. The person who has blonde hair is Arnold.
 What is the value of attribute Occupation for the person whose attribute Vacation is camping? Please reason step by step, and put your final answer within \boxed{}

Answer: lawyer


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Peter`, `Alice`, `Eric`
 - Each person has a unique favorite drink: `milk`, `root beer`, `coffee`, `water`, `tea`
 - Each person has a unique type of pet: `dog`, `bird`, `hamster`, `cat`, `fish`
 - Each mother is accompanied by their child: `Timothy`, `Meredith`, `Fred`, `Bella`, `Samantha`
 - The mothers' names in different houses are unique: `Kailyn`, `Penny`, `Aniya`, `Janelle`, `Holly`
 - People have unique favorite sports: `tennis`, `soccer`, `swimming`, `baseball`, `basketball`

## Clues:
1. The tea drinker is the person who loves tennis.
2. The person whose mother's name is Aniya is Arnold.
3. The person whose mother's name is Janelle is somewhere to the right of the person who loves swimming.
4. The person whose mother's name is Janelle is the person's child is named Bella.
5. The person's child is named Samantha is the person who likes milk.
6. The coffee drinker is not in the fifth house.
7. Bob is the person's child is named Bella.
8. The person who loves swimming and the person who loves basketball are next to each other.
9. The person who is the mother of Timothy is somewhere to the left of the person who owns a dog.
10. The person whose mother's name is Kailyn is in the fifth house.
11. The root beer lover is Alice.
12. The person whose mother's name is Holly is directly left of the root beer lover.
13. The person who likes milk is not in the second house.
14. There is one house between the person who has a cat and the person who loves baseball.
15. The person whose mother's name is Penny is somewhere to the right of the person who loves tennis.
16. The person's child is named Fred is the person who keeps a pet bird.
17. The person who likes milk is directly left of the person with a pet hamster.
18. The person who keeps a pet bird is Alice.
19. There are two houses between Eric and The person whose mother's name is Aniya.
 What is the value of attribute House for the person whose attribute FavoriteSport is tennis? 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: `Alice`, `Carol`, `Bob`, `Peter`, `Eric`, `Arnold`
 - They all have a unique favorite flower: `lilies`, `tulips`, `roses`, `daffodils`, `iris`, `carnations`
 - The people are of nationalities: `norwegian`, `dane`, `swede`, `german`, `brit`, `chinese`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `desert`, `dragonfruit`, `blueberry`, `lime`
 - Each person has a unique hobby: `photography`, `cooking`, `knitting`, `woodworking`, `painting`, `gardening`
 - Each person has a unique type of pet: `rabbit`, `hamster`, `fish`, `cat`, `bird`, `dog`

## Clues:
1. The Swedish person is the person who loves cooking.
2. The person who drinks Blueberry smoothies is Alice.
3. The person who loves the rose bouquet is not in the sixth house.
4. The photography enthusiast is somewhere to the right of the Dragonfruit smoothie lover.
5. The person who loves cooking is the person who loves a bouquet of daffodils.
6. The person who loves the boquet of lilies and the person who loves the boquet of iris are next to each other.
7. The person who paints as a hobby is Alice.
8. Arnold is somewhere to the right of the Watermelon smoothie lover.
9. The woodworking hobbyist is somewhere to the right of the German.
10. The woodworking hobbyist is somewhere to the left of the person who loves a bouquet of daffodils.
11. The Dane is the Watermelon smoothie lover.
12. The Dragonfruit smoothie lover is Peter.
13. The person who drinks Lime smoothies is the woodworking hobbyist.
14. There is one house between the Watermelon smoothie lover and the person who likes Cherry smoothies.
15. The person who keeps a pet bird is the person who drinks Blueberry smoothies.
16. The Desert smoothie lover is the person who enjoys gardening.
17. The person with a pet hamster is in the fifth house.
18. The person who owns a dog is the Norwegian.
19. The person who has a cat is somewhere to the right of Bob.
20. The British person is directly left of Carol.
21. The person who loves the boquet of iris is Arnold.
22. The German is somewhere to the right of the person who loves a carnations arrangement.
23. Bob is the photography enthusiast.
24. The person with an aquarium of fish is Arnold.
 What is the value of attribute Hobby for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: gardening


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`, `Bob`
 - Everyone has something unique for lunch: `stir fry`, `spaghetti`, `stew`, `grilled cheese`, `pizza`
 - People own unique car models: `ford f150`, `tesla model 3`, `bmw 3 series`, `toyota camry`, `honda civic`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `oneplus 9`, `huawei p50`
 - Each person has an occupation: `teacher`, `lawyer`, `doctor`, `artist`, `engineer`
 - Each person has a unique favorite drink: `tea`, `milk`, `water`, `root beer`, `coffee`

## Clues:
1. The root beer lover is the person who owns a Honda Civic.
2. The person who likes milk is directly left of the person who loves eating grilled cheese.
3. Alice is the person who uses a Samsung Galaxy S21.
4. Alice is the person who loves stir fry.
5. The tea drinker is not in the fifth house.
6. The person who owns a BMW 3 Series is somewhere to the left of the tea drinker.
7. The person who is a doctor is Arnold.
8. The person who uses an iPhone 13 is the coffee drinker.
9. The person who is an engineer is the person who owns a BMW 3 Series.
10. The person who loves the stew is the person who uses an iPhone 13.
11. The person who is a doctor is directly left of the person who uses a OnePlus 9.
12. The person who owns a Honda Civic is directly left of the person who loves the spaghetti eater.
13. The person who uses a Google Pixel 6 is the tea drinker.
14. Alice is the person who is an artist.
15. There is one house between Alice and the person who owns a Ford F-150.
16. Arnold is the person who owns a Toyota Camry.
17. Eric is in the fourth house.
18. The person who uses a OnePlus 9 is the person who is a lawyer.
19. The person who loves eating grilled cheese is Peter.
 What is the value of attribute PhoneModel for the person whose attribute Food is pizza? Please reason step by step, and put your final answer within \boxed{}

Answer: oneplus 9


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - Each mother is accompanied by their child: `Meredith`, `Bella`, `Fred`
 - The people keep unique animals: `bird`, `cat`, `horse`
 - People own unique car models: `toyota camry`, `tesla model 3`, `ford f150`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `cherry`

## Clues:
1. The person who owns a Toyota Camry is in the first house.
2. The person's child is named Bella is in the second house.
3. The Desert smoothie lover is directly left of Peter.
4. The person who likes Cherry smoothies is the person who owns a Ford F-150.
5. The person who loves mystery books is the person's child is named Meredith.
6. The person who loves science fiction books is directly left of Eric.
7. Eric is somewhere to the right of the person who loves mystery books.
8. The cat lover is the person who loves science fiction books.
9. There is one house between the person who keeps horses and the Watermelon smoothie lover.
 What is the value of attribute BookGenre for the person whose attribute Animal is horse? Please reason step by step, and put your final answer within \boxed{}

Answer: mystery


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`, `Alice`, `Eric`, `Carol`
 - The people are of nationalities: `swede`, `dane`, `german`, `norwegian`, `chinese`, `brit`
 - Each person has a favorite color: `red`, `green`, `purple`, `blue`, `white`, `yellow`

## Clues:
1. Alice is directly left of Bob.
2. The person whose favorite color is green is the British person.
3. Alice is somewhere to the left of the German.
4. The person who loves blue is the Dane.
5. Eric is the person who loves purple.
6. Carol is in the second house.
7. The Swedish person is not in the second house.
8. The Swedish person is not in the first house.
9. The person who loves white is directly left of the Norwegian.
10. There are two houses between Alice and the person who loves yellow.
11. The Dane is in the fifth house.
12. The person who loves blue is Peter.
 What is the value of attribute House for the person whose attribute Nationality is norwegian? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


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

## Clues:
1. The person who loves swimming is directly left of the Prince smoker.
2. The person who smokes Blue Master is somewhere to the right of Eric.
3. The person who uses a OnePlus 9 is Arnold.
4. Arnold is somewhere to the right of the person with a pet hamster.
5. The person who owns a dog is Peter.
6. Eric is the Desert smoothie lover.
7. The person who loves basketball is not in the fifth house.
8. The person who has a cat is the person who uses a Samsung Galaxy S21.
9. The person who loves baseball is directly left of Peter.
10. There is one house between the Desert smoothie lover and the Dunhill smoker.
11. The person who loves baseball and the person who uses a Google Pixel 6 are next to each other.
12. The person who keeps a pet bird is in the second house.
13. The person who uses a Huawei P50 is in the fourth house.
14. The person who drinks Lime smoothies is in the fourth house.
15. The person partial to Pall Mall is in the fifth house.
16. The Dragonfruit smoothie lover is in the second house.
17. The person who loves tennis is the person with a pet hamster.
18. Bob is somewhere to the right of the person who owns a dog.
19. The Watermelon smoothie lover is Arnold.
20. The Prince smoker is somewhere to the left of the Dunhill smoker.
 What is the value of attribute House for the person whose attribute Cigar is blends? 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: `Alice`, `Bob`, `Eric`, `Arnold`, `Peter`, `Carol`
 - People have unique heights: `short`, `tall`, `very short`, `average`, `very tall`, `super tall`
 - People have unique favorite music genres: `rock`, `jazz`, `classical`, `country`, `hip hop`, `pop`
 - Each mother is accompanied by their child: `Samantha`, `Alice`, `Fred`, `Bella`, `Meredith`, `Timothy`

## Clues:
1. The person who loves hip-hop music is not in the second house.
2. The person who is very tall is directly left of the person who is very short.
3. The person's child is named Samantha is in the first house.
4. The person who has an average height is directly left of the person who loves hip-hop music.
5. The person's child is named Alice is the person who loves jazz music.
6. The person who is the mother of Timothy is somewhere to the right of the person who loves jazz music.
7. Eric is directly left of the person who is tall.
8. The person who is short is the person who loves hip-hop music.
9. The person who is very short is the person who is the mother of Timothy.
10. Peter is the person who loves rock music.
11. The person's child is named Bella is Bob.
12. The person's child is named Fred is Alice.
13. The person who loves classical music is directly left of the person who loves pop music.
14. The person who loves rock music is somewhere to the right of the person who is very tall.
15. Carol is somewhere to the left of Alice.
16. Carol is somewhere to the right of Arnold.
17. The person who loves classical music is not in the first house.
18. The person who is super tall is Eric.
 What is the value of attribute Children for the person whose attribute Height is super tall? 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: `Alice`, `Peter`, `Eric`, `Bob`, `Arnold`, `Carol`
 - Everyone has a unique favorite cigar: `pall mall`, `yellow monster`, `dunhill`, `blue master`, `prince`, `blends`
 - People have unique favorite music genres: `hip hop`, `jazz`, `country`, `pop`, `classical`, `rock`
 - Each person has a unique favorite drink: `water`, `milk`, `boba tea`, `tea`, `root beer`, `coffee`
 - The mothers' names in different houses are unique: `Kailyn`, `Penny`, `Janelle`, `Holly`, `Sarah`, `Aniya`
 - Everyone has something unique for lunch: `soup`, `pizza`, `spaghetti`, `stir fry`, `stew`, `grilled cheese`

## Clues:
1. Carol is directly left of the person who loves eating grilled cheese.
2. Eric is not in the second house.
3. The person whose mother's name is Holly is somewhere to the right of Carol.
4. The person who loves eating grilled cheese is somewhere to the right of the person who loves rock music.
5. Eric is directly left of Carol.
6. The person who loves pop music is not in the third house.
7. Eric is the person who loves country music.
8. The person who loves classical music is in the sixth house.
9. The coffee drinker is Bob.
10. The person who smokes many unique blends is Peter.
11. The person who loves the stew is not in the fifth house.
12. The root beer lover is directly left of The person whose mother's name is Janelle.
13. There are two houses between The person whose mother's name is Sarah and the person who smokes Yellow Monster.
14. Eric is the tea drinker.
15. The person partial to Pall Mall is somewhere to the right of the person who loves stir fry.
16. The person who loves the soup is Bob.
17. The person who loves hip-hop music is directly left of The person whose mother's name is Kailyn.
18. Arnold is somewhere to the right of The person whose mother's name is Kailyn.
19. The one who only drinks water is directly left of the person who smokes Blue Master.
20. The person who loves the spaghetti eater is somewhere to the left of the person who smokes many unique blends.
21. The person whose mother's name is Sarah is directly left of the person who loves jazz music.
22. The person who loves hip-hop music is directly left of the root beer lover.
23. The one who only drinks water is the person who loves the stew.
24. The Dunhill smoker is not in the second house.
25. The person who likes milk is The person whose mother's name is Janelle.
26. Eric is The person whose mother's name is Aniya.
 What is the value of attribute Mother for the person whose attribute MusicGenre is hip hop? Please reason step by step, and put your final answer within \boxed{}

Answer: Sarah


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Alice`, `Arnold`, `Bob`
 - People own unique car models: `toyota camry`, `honda civic`, `tesla model 3`, `bmw 3 series`, `ford f150`
 - Everyone has a unique favorite cigar: `dunhill`, `blends`, `prince`, `blue master`, `pall mall`

## Clues:
1. Alice is the person who owns a Honda Civic.
2. The person who owns a Ford F-150 is not in the third house.
3. Alice is the person who smokes many unique blends.
4. The person who smokes many unique blends is not in the third house.
5. Alice is not in the second house.
6. The person who owns a Tesla Model 3 is somewhere to the left of Eric.
7. Arnold is somewhere to the right of the person who smokes many unique blends.
8. Arnold is the Dunhill smoker.
9. Bob is the person who owns a BMW 3 Series.
10. The person partial to Pall Mall is directly left of the person who smokes many unique blends.
11. There is one house between the Prince smoker and the person partial to Pall Mall.
12. The person who owns a Toyota Camry is Arnold.
 What is the value of attribute CarModel for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: ford f150


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`, `Peter`, `Bob`, `Eric`, `Alice`
 - People have unique hair colors: `gray`, `brown`, `black`, `red`, `blonde`
 - Each person prefers a unique type of vacation: `city`, `mountain`, `camping`, `cruise`, `beach`
 - They all have a unique favorite flower: `roses`, `tulips`, `carnations`, `lilies`, `daffodils`
 - Everyone has something unique for lunch: `stew`, `stir fry`, `spaghetti`, `pizza`, `grilled cheese`
 - Each person has a unique hobby: `painting`, `photography`, `gardening`, `knitting`, `cooking`

## Clues:
1. The person who loves the stew is directly left of the person who loves the vase of tulips.
2. The person who enjoys camping trips is the person who loves eating grilled cheese.
3. The person who has black hair is somewhere to the left of Bob.
4. The photography enthusiast is the person who has gray hair.
5. The person who is a pizza lover is the person who likes going on cruises.
6. Alice is the person who prefers city breaks.
7. The person who loves beach vacations is directly left of the person who paints as a hobby.
8. The person who loves the spaghetti eater is Arnold.
9. Bob is in the fifth house.
10. The person who loves the boquet of lilies is in the third house.
11. The person who enjoys knitting is in the fourth house.
12. The person who has brown hair is directly left of the person who paints as a hobby.
13. There are two houses between the person who likes going on cruises and the person who enjoys knitting.
14. The person who has red hair is the person who loves the spaghetti eater.
15. The person who loves a carnations arrangement is not in the second house.
16. Eric is somewhere to the left of the person who enjoys gardening.
17. The photography enthusiast is the person who loves the stew.
18. Eric is the person who loves a bouquet of daffodils.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


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

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

Answer: romance


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`
 - People have unique favorite book genres: `science fiction`, `mystery`, `romance`
 - Each mother is accompanied by their child: `Bella`, `Fred`, `Meredith`
 - People have unique hair colors: `black`, `blonde`, `brown`
 - Each person has a unique level of education: `high school`, `associate`, `bachelor`
 - They all have a unique favorite flower: `daffodils`, `lilies`, `carnations`

## Clues:
1. The person who loves the boquet of lilies is directly left of the person who loves romance books.
2. Peter is the person with a high school diploma.
3. The person who loves the boquet of lilies is the person who has brown hair.
4. The person's child is named Bella is Arnold.
5. The person who loves science fiction books is the person's child is named Fred.
6. The person's child is named Fred is the person who loves the boquet of lilies.
7. The person who has black hair is the person with an associate's degree.
8. The person's child is named Meredith is not in the first house.
9. The person with a high school diploma is the person who loves a carnations arrangement.
10. Peter is not in the third house.
 What is the value of attribute BookGenre for the person whose attribute Flower is daffodils? Please reason step by step, and put your final answer within \boxed{}

Answer: mystery


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Eric`, `Arnold`
 - Each person has an occupation: `doctor`, `engineer`, `artist`, `teacher`
 - Each person has a unique birthday month: `sept`, `april`, `jan`, `feb`
 - Each person has a favorite color: `red`, `white`, `green`, `yellow`

## Clues:
1. Arnold is in the first house.
2. The person whose birthday is in February and the person who is a teacher are next to each other.
3. The person who is a teacher is the person whose favorite color is red.
4. The person whose favorite color is red is not in the third house.
5. The person who loves yellow is directly left of Peter.
6. The person whose favorite color is green and the person whose favorite color is red are next to each other.
7. The person who is an engineer is the person whose favorite color is green.
8. Eric is the person who is a teacher.
9. The person who loves yellow is not in the second house.
10. The person whose birthday is in September is somewhere to the left of the person whose favorite color is green.
11. The person who is a doctor is somewhere to the right of the person whose birthday is in April.
 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 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 Name for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`, `Bob`
 - Everyone has something unique for lunch: `stir fry`, `spaghetti`, `stew`, `grilled cheese`, `pizza`
 - People own unique car models: `ford f150`, `tesla model 3`, `bmw 3 series`, `toyota camry`, `honda civic`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `oneplus 9`, `huawei p50`
 - Each person has an occupation: `teacher`, `lawyer`, `doctor`, `artist`, `engineer`
 - Each person has a unique favorite drink: `tea`, `milk`, `water`, `root beer`, `coffee`

## Clues:
1. The root beer lover is the person who owns a Honda Civic.
2. The person who likes milk is directly left of the person who loves eating grilled cheese.
3. Alice is the person who uses a Samsung Galaxy S21.
4. Alice is the person who loves stir fry.
5. The tea drinker is not in the fifth house.
6. The person who owns a BMW 3 Series is somewhere to the left of the tea drinker.
7. The person who is a doctor is Arnold.
8. The person who uses an iPhone 13 is the coffee drinker.
9. The person who is an engineer is the person who owns a BMW 3 Series.
10. The person who loves the stew is the person who uses an iPhone 13.
11. The person who is a doctor is directly left of the person who uses a OnePlus 9.
12. The person who owns a Honda Civic is directly left of the person who loves the spaghetti eater.
13. The person who uses a Google Pixel 6 is the tea drinker.
14. Alice is the person who is an artist.
15. There is one house between Alice and the person who owns a Ford F-150.
16. Arnold is the person who owns a Toyota Camry.
17. Eric is in the fourth house.
18. The person who uses a OnePlus 9 is the person who is a lawyer.
19. The person who loves eating grilled cheese is Peter.
 What is the value of attribute House for the person whose attribute Occupation is doctor? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Peter`, `Eric`
 - Each mother is accompanied by their child: `Bella`, `Meredith`, `Fred`, `Samantha`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`, `stew`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`

## Clues:
1. The person who is a pizza lover is somewhere to the left of the person's child is named Samantha.
2. The person whose favorite color is red is directly left of the person who loves white.
3. Eric is somewhere to the right of the person who loves the stew.
4. The person who loves white is in the fourth house.
5. The person whose favorite color is green is not in the second house.
6. The person whose favorite color is red is the person's child is named Bella.
7. The person who loves eating grilled cheese is in the second house.
8. There is one house between the person's child is named Meredith and Alice.
9. There is one house between Peter and Alice.
10. Alice is not in the third house.
 What is the value of attribute House for the person whose attribute Children is Meredith? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Carol`, `Bob`, `Eric`
 - Each person has an occupation: `artist`, `nurse`, `engineer`, `lawyer`, `teacher`, `doctor`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Aniya`, `Janelle`, `Sarah`, `Kailyn`
 - People have unique favorite music genres: `pop`, `rock`, `classical`, `jazz`, `hip hop`, `country`

## Clues:
1. The person who is an engineer is directly left of the person who is an artist.
2. The person whose mother's name is Janelle is the person who is an artist.
3. The person who loves classical music is directly left of Peter.
4. The person whose mother's name is Janelle and the person who loves pop music are next to each other.
5. The person whose mother's name is Sarah is somewhere to the left of Arnold.
6. The person who loves country music is The person whose mother's name is Aniya.
7. The person whose mother's name is Penny is not in the second house.
8. Bob is the person who loves jazz music.
9. Carol is not in the sixth house.
10. The person who is an engineer is the person who loves classical music.
11. The person who is a nurse is somewhere to the right of the person who loves rock music.
12. There are two houses between Bob and The person whose mother's name is Penny.
13. The person who is a lawyer is directly left of the person who is a doctor.
14. The person whose mother's name is Penny is not in the third house.
15. The person who is an engineer is not in the fourth house.
16. The person who is a doctor is The person whose mother's name is Kailyn.
17. Carol is somewhere to the right of the person who loves country music.
18. The person who loves classical music is Alice.
19. The person who is a nurse is The person whose mother's name is Holly.
 What is the value of attribute MusicGenre for the person whose attribute Mother is Janelle? 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: `Carol`, `Bob`, `Eric`, `Alice`, `Peter`, `Arnold`
 - People have unique heights: `super tall`, `tall`, `very short`, `average`, `very tall`, `short`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `samsung galaxy s21`, `xiaomi mi 11`, `oneplus 9`, `huawei p50`
 - Each mother is accompanied by their child: `Fred`, `Timothy`, `Alice`, `Bella`, `Meredith`, `Samantha`

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

Answer: 4


33. How many people got lost in the mountains?

Alice and the Griffin had to wait several minutes before the Tortoise Quasi gathered his strength and could continue.

- You see,一began the Tortoise Quasi.
- I don't see anything!- the Griffin cut in.

The Tortoise Quasi did not respond, only grabbing his head with his front paws again. After remaining silent for some time, he spoke again:

- Let's say... Nine people got lost in the mountains. They only had enough food for five days. Can you imagine, just five days!

Having reached this point in his story, the Tortoise Quasi was so moved by the tragedy that he could not continue from the overflow of emotions.

- Well, that's enough, there, there! - the Griffin tried to comfort him, patting his back.

![](https://cdn.mathpix.com/cropped/2024_05_21_5afa4e43618f5800e92dg-052.jpg?height=1142&width=871&top_left_y=820&top_left_x=570)

- Can you imagine what will happen to them if they are not found! - the Tortoise Quasi sobbed. - But it happened (and this is the most beautiful part of the story!)... Yes, the most beautiful part of the story is that the next day the poor souls met another group of lost people in the mountains...
- What's so beautiful about that? - the Griffin asked.
- The most beautiful part is that the first group generously shared their supplies with the second - they divided the provisions equally among everyone, after which the food lasted for another three days.

How many people were in the second group?

Answer: 3


3. As shown in the figure, in trapezoid $A B C D$ with an area of 112, $A E: E F: F B=1: 3: 3, A D: B C=2$ : 3, then the area of triangle $C D H$ is $\qquad$.

Answer: 36


8. Given the set $A=\{1,2, \cdots, 104\}, S$ is a subset of $A$. If $x \in S$, and at the same time $x-1 \notin S$ and $x+1 \notin S$, then $x$ is called an "isolated point" of $S$. The number of all 5-element subsets of $A$ with no "isolated points" is . $\qquad$

Answer: 10000


21. Find the smallest natural number that is a multiple of 36 and in whose representation all 10 digits appear exactly once.

Answer: 1023457896


4. On a horizontal line that divides the plane into upper and lower half-planes, a segment $AB$ with a length of $72 \, \text{cm}$ is drawn. Using the endpoints of the segment, in the upper half-plane, a regular (equilateral) triangle $A M_{1} M_{2}$ and a regular pentagon $M_{5} M_{6} M_{7} M_{8} M_{9}$ are drawn, and in the lower half-plane, a regular quadrilateral (square) $M_{2} M_{3} M_{4} M_{5}$ and a regular hexagon $M_{9} M_{10} M_{11} M_{12} M_{13} B$ are drawn. Here, $M_{2}, M_{5}$, and $M_{9}$ are on the segment $AB$, point $M_{2}$ is between $A$ and $M_{5}$, and point $M_{9}$ is between $M_{5}$ and $B$. The lengths of the sides of the regular polygons are in the same ratio as the corresponding numbers of their sides. Calculate the length of the broken line

$$
L \equiv A M_{1} M_{2} M_{3} M_{4} M_{5} M_{6} M_{7} M_{8} M_{9} M_{10} M_{11} M_{12} M_{13} B
$$

Answer: 272


There are $n$ football teams participating in a league, where each team plays a match against every other team exactly once. It is stipulated that a win earns 3 points, a draw earns 1 point, and a loss earns 0 points. After the league ends, some teams may be disqualified, and thus their match results will also be canceled. The remaining teams will have their scores recalculated, and the team with the most points will become the champion of this league (if only one team is not disqualified, then that team is the champion).

Let $f_{i}(T)(i=1,2, \cdots, n)$ be the minimum number of teams that must be disqualified in this league $T$ to ensure that the $i$-th team becomes the champion, and let $F(T)=f_{1}(T)+f_{2}(T)+\cdots+f_{n}(T)$. For $n \geqslant 5$, find the maximum and minimum values of $F(T)$.

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


Task 2. Every student in the Netherlands receives a finite number of cards. Each card has a real number in the interval $[0,1]$ written on it. (The numbers on different cards do not have to be different.) Find the smallest real number $c>0$ for which the following holds, regardless of the numbers on the cards that everyone has received.

Every student whose sum of the numbers on the cards is at most 1000 can distribute the cards over 100 boxes so that the sum of the numbers in each box is at most $c$.

Answer: \dfrac{1000}{91}


58. Determine the measure of angle $\hat{A}$ of triangle $ABC$, if it is known that the bisector of this angle is perpendicular to the line passing through the point of intersection of the altitudes and the center of the circumscribed circle of this triangle.

Answer: 60^\circ


Vintik and Shpuntik built a "Push-Pull" machine that moves forward on syrup consuming 3 liters per kilometer, and backward on orange juice consuming 5 liters per kilometer. They started from home and took turns driving. Vintik drove 12 kilometers in both directions. Shpuntik drove forward half the distance that Vintik drove, and backward twice the distance after which the 75 liters of fuel they had ran out. How many kilometers will Vintik and Shpuntik have to walk back home?

Answer: 9


A midpoint plotter is an instrument which draws the exact mid point of two point previously drawn. Starting off two points  $1$  unit of distance apart and using only the midpoint plotter, you have to get two point which are strictly at a distance between  $\frac{1}{2017}$  and  $\frac{1}{2016}$  units, drawing the minimum amount of points. ¿Which is the minimum number of times you will need to use the midpoint plotter and what strategy should you follow to achieve it?

Answer: 17


1. Four elevators of a skyscraper, differing in color (red, blue, green, and yellow), are moving in different directions and at different but constant speeds. Observing the elevators, someone started a stopwatch and, looking at its readings, began to record: 36th second - the red elevator caught up with the blue one (moving in the same direction). 42nd second - the red elevator passed the green one (moving in opposite directions), 48th second - the red elevator passed the yellow one, 51st second - the yellow elevator passed the blue one, 54th second - the yellow elevator caught up with the green elevator. At what second from the start of the count will the green elevator pass the blue one, if during the observation period the elevators did not stop and did not change their direction of movement?

(12 points)

Answer: 46


1. From point A to point B, which are 12 km apart, a pedestrian and a bus set out simultaneously. Arriving at point B in less than one hour, the bus, without stopping, turned around and started moving back towards point A at a speed twice its initial speed. After 12 minutes from its departure from point B, the bus met the pedestrian. Determine the greatest possible integer value of the pedestrian's speed (in km/h), and for this value of the pedestrian's speed, determine the initial speed of the bus (in km/h). In the answer, write the sum of the found values of the pedestrian's and the bus's speeds. (5 points)

Answer: 21


Let $a, b, c, d$ be the lengths of the sides of a quadrilateral circumscribed about a circle and let $S$ be its area. Prove that $S \leq \sqrt{abcd}$ and find conditions for equality.

Answer: S \leq \sqrt{abcd}


7th Australian 1986 Problem A2 What is the smallest tower of 100s that exceeds a tower of 100 threes? In other words, let a 1 = 3, a 2 = 3 3 , and a n+1 = 3 a n . Similarly, b 1 = 100, b 2 = 100 100 etc. What is the smallest n for which b n > a 100 ?

Answer: 99


[Area of a triangle (through two sides and the angle between them)] Inscribed angle is half of the central angle ]  [Cauchy's inequality

A unit square $A B C D$ is inscribed in a circle, and a point $M$ is chosen on this circle.

What is the maximum value that the product $M A \cdot M B \cdot M C \cdot M D$ can take?

Answer: \dfrac{1}{2}


1. Observing the temperatures recorded in Cesenatico in the last months of December and January, Stefano noticed a strange coincidence: on all the days of this period (excluding the first and last day), the minimum temperature was the sum of the minimum temperature of the previous day and the next day.

Knowing that on December 3 the minimum temperature was 5 degrees, and on January 31 it was 2 degrees, determine the minimum temperature on December 25.

Answer: -3


3. Given a cube and 12 colors. Find the number of ways to paint the faces of this cube using these colors (each face in one color) such that adjacent faces are of different colors. Colorings that differ by a rotation are considered different.

Answer: 987360


44. Given an integer $n(n \geqslant 2)$. Find the largest constant $\lambda(n)$ such that: if the real number sequence $a_{0}, a_{1}, a_{2}, \cdots, a_{n}$ satisfies $0=a_{0} \leqslant a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{n}$ and $a_{i} \geqslant \frac{1}{2}\left(a_{i-1}+a_{i+1}\right)(i=$ $1,2, \cdots, n)$, then $\left(\sum_{i=1}^{n} i a_{i}\right)^{2} \geqslant \lambda(n) \sum_{i=1}^{n} a_{i}^{2}$. (2009 China National Team Selection Test for IMO)

Answer: \dfrac{n(n+1)^2}{4}


Four, in isosceles $\triangle A B C$, the base $B C$ lies on the line $x+y=0$, the vertex $\mathrm{A}(2,3)$, and it is known that one of its equal sides $\mathrm{AB}$ is parallel to the line $\mathbf{x}-4 y-6=0$. Find the equation of the other equal side $\mathrm{AC}$.

Answer: 4x - y - 5 = 0


117. Point $P(a, b)$ is in the first quadrant. A line $l$ is drawn through point $P$, intersecting the positive $x$-axis and $y$-axis at points $A$ and $B$, respectively. $O$ is the origin, and $m$ is a positive integer. Find the slope $k_{1}$ of line $l$ that minimizes $P A^{m} + P B^{m}$.

Answer: - \left( \dfrac{b}{a} \right)^{ \dfrac{m}{m + 2} }


Tomongho A.K.

Find all increasing arithmetic progressions with a finite number of terms, the sum of which is 1, and each term has the form $1 / k$, where $k$ is a natural number.

Answer: \left\{\frac{1}{6}, \frac{1}{3}, \frac{1}{2}\right\}


A strange fellow did not hesitate to write 2022 numbers in a circle, and it turned out that each number matches the absolute value of the difference between its two neighbors. Determine which numbers were written if their sum is equal to 2022.

Answer: 0


9. Given that $A B C D$ is a tetrahedron, and each edge is colored red or blue with equal probability, the probability that point $A$ can reach point $B$ through the red edges of the tetrahedron $A B C D$ is $\qquad$ .

Answer: \dfrac{3}{4}


In an acute-angled triangle \( ABC \), the largest altitude \( AH \) is equal to the median \( BM \). Prove that \( \angle B \leq 60^{\circ} \).

Answer: \angle B \leq 60^\circ


2. In a test, 25 students from eight schools participated. They correctly solved a total of 50 problems. Each student from the same school solved the same number of problems, while students from different schools solved a different number of problems. Each student correctly solved at least one problem. How many students solved only one problem?

Answer: 15


7. Given a convex quadrilateral $A B C D$. Let $P$ be the center of the circle inscribed in triangle $A B D$, and $Q$ be the center of the circle inscribed in triangle $C B D$. Ray $B P$ intersects side $D A$ at point $M$, and ray $D Q$ intersects side $B C$ at point $N$. It turns out that $A M=\frac{9}{7}, D M=\frac{12}{7}$ and $B N=\frac{20}{9}, C N=\frac{25}{9}$.

a) Find the ratio $A B: C D$.

b) Suppose it is additionally known that the circles mentioned in the condition are tangent. Find the lengths of sides $A B$ and $C D$.

Answer: \dfrac{3}{5}


In the diagram, square $P Q R S$ has side length $25, Q$ is located at $(0,7)$, and $R$ is on the $x$-axis. The square is rotated clockwise about $R$ until $S$ lies above the $x$-axis on the line with equation $x=39$. What are the new coordinates of $P$ after this rotation?

![](https://cdn.mathpix.com/cropped/2024_04_17_ab5f9723a8aff65a55ebg-2.jpg?height=464&width=439&top_left_y=1801&top_left_x=1190)

Answer: (19,35)


4. Given the functions $f(x)=x^{2}+4 x+3$ and $g(x)=x^{2}+2 x-1$. Find all integer solutions to the equation $f(g(f(x)))=g(f(g(x)))$.

Answer: -2


Question 8: Let set $A=\left\{(x, y) \mid y^{2}-x-1=0\right\}$, set $B=\left\{(x, y) \mid 4 x^{2}+2 x-2 y+5=\right.$ $0\}$, and set $\mathrm{C}=\{(\mathrm{x}, \mathrm{y}) \mid \mathrm{y}=\mathrm{kx}+\mathrm{b}\}$. Try to find all non-negative integers $\mathrm{k}$ and $\mathrm{b}$, such that $(\mathrm{A} \cup \mathrm{B}) \cap \mathrm{C}=\emptyset$.

Answer: (k, b) = (1, 2)


8. Given that $A B C D-A_{1} B_{1} C_{1} D_{1}$ is a unit cube, two ants, one black and one white, start from point $A$ and crawl along the edges. Each time they complete an edge, it is called “completing a segment”. The white ant’s crawling route is $A A_{1} \rightarrow A_{1} D_{1} \rightarrow \cdots$, and the black ant’s crawling route is $A B \rightarrow B B_{1} \rightarrow \cdots$. They both follow the rule that the $(n+2)$-th segment must be on a skew line with the $n$-th segment. Suppose both ants stop after completing 2008 segments at some vertex of the cube. At this point, the distance between the black and white ants is

Answer: \sqrt{2}


One. (20 points) Given the parabola $y=a x^{2}+b x+c$ passes through the point $(b,-3)$, and $|a| c+b|c|=0$, and the inequality $a x^{2}+b x+c+3>0$ has no solution. Find all possible values of the triplet $(a, b, c)$.

---

Please note that the formatting and line breaks have been preserved as requested.

Answer: \left(-\dfrac{1}{2}, \dfrac{1}{2}, -\dfrac{25}{8}\right)


In the plane, there are two unit-radius, tangent circles: $k$ and $k_{1}$. One of their common external tangents is the line $e$. Then we draw the circles $k_{2}, k_{3}, \ldots, k_{n}$ in sequence such that each of them is tangent to $k$, $e$, and the circle with the number one less. What is the radius of the $k_{n}$ circle?

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


Let \( a, b, c \) be the sides of a triangle, and the angles opposite to them be \(2\varphi, 3\varphi, 4\varphi\). Express \(\cos \varphi\) in terms of \( a, b, \) and \( c \) as \(\cos \varphi = \quad\).

Answer: \cos \varphi = \dfrac{a + c}{2b}


1. Let $n$ be the sum of all ten-digit numbers, which contain every decimal digit from zero to nine. Determine the remainder of $n$ when divided by 77 .

(Pavel Novotný)

Answer: 28


5. On the extension of side $A C$ of triangle $A B C$ beyond point $A$, point $T$ is marked such that $\angle B A C=2 \angle B T C$. Find the area of triangle $A B C$, given that $A B=A C, B T=70$, $A T=37$.

Answer: 420


A box contains 10 red cards and 10 blue cards, each card being numbered with $1, 3, 3^{2}, \cdots, 3^{9}$ once. The total sum of the numbers on both colored cards is denoted as $S$. For a given positive integer $n$, if it is possible to select a number of cards from the box such that the sum of their numbers is exactly $n$, it is called a "$n$ selection scheme." The number of different $n$ schemes is denoted as $f(n)$. Find the value of $P=$ $\sum_{n=1}^{1000} f(n)$.

Answer: 6423


5. A polyhedron with vertices at the midpoints of the edges of a certain cube is called a cuboctahedron. When the cuboctahedron is intersected by a plane, a regular polygon is obtained. What is the maximum number of sides this polygon can have?

(M. A. Evdokimov)

Answer: 6


10.18. Provide an example of a polynomial $P(x)$ that is divisible by $x^{2}+1$, and at the same time $P(x)-1$ is divisible by $x^{3}+1$.

Answer: -\dfrac{1}{2}x^4 - \dfrac{1}{2}x^3 - \dfrac{1}{2}x + \dfrac{1}{2}


11. There are 11 students who have signed up for the volunteer tour guide activity at the museum. The activity runs from 9 AM to 5 PM, with a public mini-lecture every hour. Each session requires 1 student to provide tour guide services to visitors. To avoid overworking the students, the museum will not schedule the same person for two consecutive sessions, and the first and last sessions must be handled by two different students. How many scheduling arrangements are possible for the museum?

Answer: 100000010


10. As shown in Figure 2, the diameter $A B=a \mathrm{~cm}$ of the large circle $\odot O$, circles $\odot O_{1}$ and $\odot O_{2}$ are drawn with $O A$ and $O B$ as diameters, respectively, and two equal circles $\odot O_{3}$ and $\odot O_{4}$ are drawn in the space between $\odot O$ and $\odot O_{1}$ and $\odot O_{2}$, such that these circles are internally or externally tangent to each other. Then the area of quadrilateral $O_{1} O_{4} O_{2} O_{3}$ is $\mathrm{cm}^{2}$.

Answer: \dfrac{a^2}{6}


In a right triangle $ABC$ ($\angle C = 90^o$) it is known that $AC = 4$ cm, $BC = 3$ cm. The points $A_1, B_1$ and $C_1$ are such that $AA_1 \parallel BC$, $BB_1\parallel A_1C$, $CC_1\parallel A_1B_1$, $A_1B_1C_1= 90^o$, $A_1B_1= 1$ cm. Find $B_1C_1$.

Answer: 12


34. In the book of records, a whole number expressing the number of meters of fabric sold was blotted out with ink. The total revenue could not be read either, but the end of this entry was visible: 7 r. 28 k. - and it is known that this amount does not exceed 500 r. The price of 1 m of fabric is 4 r. 36 k. Help the inspector restore this entry.

Answer: 98


In a $N \times N$ square table, all integers are arranged according to the following rule: 1 is placed in any cell, 2 is placed in the row whose number is equal to the column number containing 1, 3 is placed in the row whose number is equal to the column number containing 2, and so on. By how much does the sum of the numbers in the column containing $N^{2}$ differ from the sum of the numbers in the row containing 1?

Answer: N^2 - N


Exercise 12. Let $A B C$ be an isosceles triangle at $A$ such that $\widehat{C B A}=61^{\circ}$. Let $E$ be the point, other than $A$, located on the circumcircle of $A B C$ such that $E B=E C$. Let $D$ be the point other than $A$ such that $D B=D C=A B$.

Determine the value in degrees of the angle $\widehat{B E D}$.

Answer: 119


In chess, the winner receives 1 point and the loser 0 points. In the event of a draw (remis), each player receives $1 / 2$ point.
Fourteen chess players, no two of whom were the same age, participated in a tournament where each played against every other. After the tournament, a ranking list was created. In the case of two players with the same number of points, the younger one received a better ranking.
After the tournament, Jan noticed that the top three players had a total number of points exactly equal to the total number of points of the last nine players. Jörg added that the number of games that ended in a draw was maximized. Determine the number of drawn games.

Answer: 40


7.4. Does there exist a six-digit natural number that, when multiplied by 9, is written with the same digits but in reverse order?

Answer: 109989


(7) A three-digit number $\overline{x y z}, 1 \leqslant x \leqslant 9,0 \leqslant y, z \leqslant 9$, and $\overline{x y z}=x!+$ $x!+z!$. Then the value of $x+y+z$ is $\qquad$ .

Answer: 10


5. The numbers 1, 2, 3, 4, 5, 6, 7, and 8 are randomly arranged in a circle. For each such arrangement, we determine all eight sums of four consecutive numbers and denote the largest of these sums by $m$. What is the smallest possible value of the number $m$? Justify your answer.

Answer: 19


Let $E = \{1, 2, \dots , 16\}$ and let $M$ be the collection of all $4 \times 4$ matrices whose entries are distinct members of $E$. If a matrix $A = (a_{ij} )_{4\times4}$ is chosen randomly from $M$, compute the probability $p(k)$ of $\max_i \min_j a_{ij} = k$ for $k \in E$. Furthermore, determine $l \in E$ such that $p(l) = \max \{p(k) | k \in E \}.$

Answer: 5


11. If in any permutation of $1,2,3, \cdots, 18$, there always exists a sum of 6 consecutive numbers not less than $m$, then the maximum value of the real number $m$ is $\qquad$.

Answer: 57


66. Can a quadrilateral have three obtuse angles?

Answer: No


4. A right trapezoid has bases of lengths $10 \mathrm{~cm}$ and $2 \mathrm{~cm}$ and an area of $90 \mathrm{~cm}^{2}$. If we rotate it around the longer base, calculate the surface area and volume of the resulting solid.

Answer: 540\pi


Example 13 Given that the function $f(x)$ is defined on $[0,1]$. And it satisfies the following conditions: (1) $f(1)=3$; (2) $f(x) \geqslant 2$; (3) If $x_{1} \geqslant 0, x_{2} \geqslant 0, x_{1}+x_{2} \leqslant 1$, then $f\left(x_{1}+x_{2}\right) \geqslant f\left(x_{1}\right)+f\left(x_{2}\right)-2$. (1) Find the maximum and minimum values of the function $f(x)$; (2) Compare $f\left(\frac{1}{2^{*}}\right)$ with $\frac{1}{2^{*}}+2$ $\left(n \in \mathbf{N}^{+}\right)$; (3) Someone noticed that when $x=\frac{1}{2^{n}}(n \in \mathbf{N})$, $f(x)<2 x+2$. Based on this, he proposed a conjecture: For all $x \in(0,1]$, $f(x)<2 x+2$. Please determine whether the conjecture is correct and explain your reasoning.

Answer: 3


3. Find all four-digit numbers $\overline{a b c d}$ that are divisible by each of the two-digit numbers $\overline{a b}, \overline{b c}, \overline{c d}$, where the digits $a, b, c, d$ are odd and not all the same.

The school - written part of the first round in category A takes place

on Tuesday, December 5, 2000

so that it starts in the morning and the contestants have 4 hours of pure time to solve the problems. For each problem, a contestant can earn 6 points, and a successful solver is a student who scores 10 points or more. These details will be communicated to the students before the start of the competition.

Answer: 1155


85. To which number can the expression

$$
\frac{(a+b)(b+c)(a+c)}{a b c}
$$

be equal if the numbers $a, b$ and $c$ satisfy the condition of problem 84?

Answer: -1


8. An $8 \times 8$ chessboard is colored in the usual way, with 32 black squares and 32 white squares. A "path" consists of 8 white squares, one in each row, and adjacent white squares share a common vertex. The number of such paths is $\qquad$.

Answer: 296


16. (15 points) Person A and Person B start climbing a mountain from the bottom along the same path at the same time, and immediately return along the original path after reaching the top. It is known that their descending speeds are three times their respective ascending speeds. A and B meet 150 meters below the summit, and when A returns to the bottom, B is exactly halfway down the mountain. Find the distance from the bottom to the top of the mountain.

Answer: 1550


## Task 5 - 010615

How many different types of second-class passenger train tickets are needed for a route with 15 stations, if there should be a ticket for every possible connection?

How did you determine the number?

Answer: 210


Problem 12.5. A plane intersects a tetrahedron $A B C D$ and divides the medians of the triangles $D A B, D B C$ and $D C A$ through $D$ in ratios $1: 2,1: 3$ and $1: 4$ from $D$, respectively. Find the ratio of the volumes of the two parts of the tetrahedron cut by the plane.

Oleg Mushkarov

Answer: \dfrac{1}{47}


Let  $B_n$ be the set of all sequences of length $n$, consisting of zeros and ones. For every two sequences  $a,b \in B_n$ (not necessarily different) we define strings  $\varepsilon_0\varepsilon_1\varepsilon_2 \dots \varepsilon_n$ and $\delta_0\delta_1\delta_2 \dots \delta_n$ such that $\varepsilon_0=\delta_0=0$ and $$
 \varepsilon_{i+1}=(\delta_i-a_{i+1})(\delta_i-b_{i+1}), \quad \delta_{i+1}=\delta_i+(-1)^{\delta_i}\varepsilon_{i+1} \quad (0 \leq i \leq n-1).
$$. Let $w(a,b)=\varepsilon_0+\varepsilon_1+\varepsilon_2+\dots +\varepsilon_n$ . Find  $f(n)=\sum\limits_{a,b \in {B_n}} {w(a,b)} $.
.

Answer: n \cdot 4^{n-1}


78. The Walking Passenger. The train is moving at a speed of 60 km/h. A passenger is walking from the end of the train to its beginning through the passageways between the cars at a speed of 3 km/h. What is the speed of the passenger relative to the railway track?

We are not going to engage in sophistry here, like Zeno's paradox of the flying arrow, or Einstein's theory of relativity, but are talking about movement in the ordinary sense of the word relative to the railway track.

Answer: 63


Top student Vasya sums ordinary fractions without errors, while Petya sums fractions as follows: in the numerator he writes the sum of the numerators, and in the denominator - the sum of the denominators. The teacher suggested that the kids add up three irreducible fractions. Vasya got the correct answer 1. Could Petya's result be less than $\frac{1}{10}$?

Answer: \text{Yes}


27. Five friends, one of whom had a monkey, once bought a bag of nuts, which they intended to divide among themselves the next morning. However, one of the friends woke up at night and wanted nuts; he divided all the nuts in the bag into five equal parts, with one extra nut left over, which he gave to the monkey, and took one fifth for himself. Soon after, another of the nut owners woke up; not knowing that someone had already taken nuts, he divided all the remaining contents of the bag again into five parts, with one extra nut left over, which he gave to the monkey, and took one fifth for himself. Then the remaining three friends successively performed the same operation; each of them, not knowing about the actions of the others, divided all the nuts into five parts, took one fifth for himself, and each time one extra nut remained, which was given to the monkey. Finally, in the morning, all five of them together took out the bag, divided the remaining nuts into five parts, and one nut, which turned out to be extra, was again given to the monkey. It is required to determine the smallest number of nuts in the bag for which such a division is possible.

Answer: 15621


10. The center of the ellipse $\Gamma$ is at the origin $O$, with foci on the $x$-axis, and the eccentricity $e=\sqrt{\frac{2}{3}}$. The line $l$ intersects the ellipse $\Gamma$ at points $A$ and $B$, satisfying $\overrightarrow{C A}=2 \overrightarrow{B C}$, where the fixed point $C(-1,0)$. When $\triangle O A B$ attains its maximum value, find the equation of the ellipse $\Gamma$.

Answer: \dfrac{x^2}{5} + \dfrac{y^2}{\dfrac{5}{3}} = 1


Problem 2. Determine the perimeter of the scalene triangle $\mathrm{ABC}$ knowing that

$$
p^{2} \cdot \overrightarrow{G I}=(4 p-b-c) \cdot \overrightarrow{A I}+(4 p-c-a) \cdot \overrightarrow{B I}+(4 p-a-b) \cdot \overrightarrow{C I}
$$

where $\mathrm{G}$ is the centroid of $\triangle \mathrm{ABC}$, I is the incenter of $\triangle \mathrm{ABC}$, and $p$ is the semiperimeter of $\triangle \mathrm{ABC}$.

Answer: 12


3. Given that $A, B, C, D$ are points on a circle with radius 1, $\overrightarrow{A B}+2 \overrightarrow{A C}=\overrightarrow{A D}$, and $|A C|=1$. Then the area of quadrilateral $A B D C$ is $\qquad$.

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


7. Given $x, y, z \in \mathbf{R}$, then $\sum \frac{x^{2}}{(3 x-2 y-z)^{2}}$ has the minimum value of $\qquad$ ("sum" indicates cyclic sum).

Answer: \dfrac{5}{49}


A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.

Take five good haikus
Scramble their lines randomly
What are the chances
That you end up with
Five completely good haikus
(With five, seven, five)?
Your answer will be
m over n where m,n
Are numbers such that
m,n positive
Integers where gcd
Of m,n is 1.
Take this answer and
Add the numerator and
Denominator.

[i]Proposed by Jeff Lin[/i]

Answer: 3004


In a regular quadrilateral pyramid \( S A B C D \) with vertex \( S \), the side length of the base of the pyramid is \( b \), and the height of the pyramid is \( b \sqrt{2} \). A sphere inscribed in this pyramid touches the lateral face \( S A D \) at point \( K \). Find the area of the cross-section of the pyramid that passes through edge \( A B \) and point \( K \).

Answer: \dfrac{3\sqrt{17}}{16} b^2


Example 6 Rectangle $R$ is divided into 2016 small rectangles, with each small rectangle's sides parallel to the sides of rectangle $R$. The vertices of the small rectangles are called "nodes". For a line segment on the side of a small rectangle, if both endpoints are nodes and its interior does not contain any other nodes, then this line segment is called a "basic segment". Considering all possible divisions, find the maximum and minimum number of basic segments. ${ }^{[4]}$

Answer: 4122


3. (6 points) Two apprentices, A and B, are discussing who will work with the master to process a batch of parts. A says: "If I work with the master, then I will complete 20% of the entire work." B says: "That's nothing, if I work with the master, then I will complete 40% of the entire work." At this moment, the master comes and says to B: "If A joins us to help, you can process 60 fewer parts." If what they said is all correct, then the total number of parts in this batch is $\qquad$.

Answer: 1150


[ Isosceles, Inscribed, and Circumscribed Trapezoids Auxiliary Similar Triangles

An isosceles trapezoid $A B C D$ is circumscribed around a circle. The lateral sides $A B$ and $C D$ touch the circle at points $M$ and $N$, and $K$ is the midpoint of $A D$.

In what ratio does the line $B K$ divide the segment $M N$? #

Answer: 1:3


7.2. Daniil has 6 cards with letters, from which he managed to form the word WNMWNM shown in the picture. Note that this word has a remarkable property: if you rotate it 180 degrees, you get the same word. How many words with such a property can Daniil form using all 6 cards at once?

![](https://cdn.mathpix.com/cropped/2024_05_06_0f1f0fe01f8033fb92a1g-1.jpg?height=135&width=845&top_left_y=1326&top_left_x=637)

Answer: 12


Problem 1.3. Knowing the last digit of $2009^{2011}$ is very easy, but how many zeros precede that last digit?

Answer: 1


4. On the table, there are 100 weights of different masses. A weight is called successful if its mass is equal to the sum of the masses of some two other weights on the table. What is the smallest number of successful weights that would allow us to assert with certainty that the masses of some two weights differ by at least a factor of three?

Answer: 87


Jiřina has a four-digit number written on a piece of paper. If she swaps the digits in the hundreds and units places and adds this new number to the original number, she gets a result of 3332. However, if she swaps the digits in the thousands and tens places and adds this number to the original, she gets a result of 7886. Determine what number Jiřina had written on the paper.

(E. Novotná)

Answer: 1468


The figure below indicates a target fixed on a wall that cannot be rotated. It is divided into 10 parts, consisting of a central circle, a smaller ring, and a larger (outer) ring. We need to distribute the numbers from 1 to 10, one in each part, which will correspond to the scores obtained by hitting each part.

![](https://cdn.mathpix.com/cropped/2024_05_01_a50b5476db779c3986d5g-07.jpg?height=497&width=510&top_left_y=1940&top_left_x=845)
a) In how many ways can we distribute the numbers in the parts of the target?

b) In how many ways can we distribute the numbers such that numbers closer to the center cannot be smaller than numbers more distant from the center?

c) In how many ways can we distribute the numbers so that the sum of the numbers in the outer ring is equal to the sum of the numbers in the smaller ring?

Answer: 34,560


446. Two circles are given. We want to draw a common tangent - a line that touches both circles. How many such lines exist? (The answer depends on the ratio of the radii of the circles and the distance between their centers.)

Answer: 4


13. As shown in the figure, natural numbers starting from 1 are arranged according to a certain rule. What is the number in the 3rd row and 51st column?
$\qquad$
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline 3 & & 7 & 9 & 12 & & 16 & 18 & 21 & $\ldots$ \\
\hline 2 & 4 & 6 & & 11 & 13 & 15 & & 20 & $\ldots$ \\
\hline 1 & & 5 & 8 & 10 & & 14 & 17 & 19 & $\ldots$ \\
\hline
\end{tabular}

Answer: 113