Given an equilateral triangle \( \triangle ABC \) with an incircle that touches sides \( AB \), \( BC \), and \( CA \) at points \( D \), \( E \), and \( F \), respectively. Let \( P \) be any point on the arc \( DF \) of the incircle. The distances from point \( P \) to the sides \( AC \), \( AB \), and \( BC \) are denoted as \( d_1 \), \( d_2 \), and \( d_3 \), respectively. Prove that \( \sqrt{d_1} + \sqrt{d_2} = \sqrt{d_3} \).

Answer: \sqrt{d_1} + \sqrt{d_2} = \sqrt{d_3}


19. (15 points) As shown in Figure 4, there is a pointer on a disk, initially pointing to the top of the disk. The pointer rotates clockwise around the center of the disk by an angle $\alpha$ each time, and $3.6^{\circ}<\alpha<180^{\circ}$. After 2,004 rotations, it returns to its initial position for the first time, pointing to the top of the disk again. How many different possible values can $\alpha$ have?

Answer: 325


## Task 2 - 040722

In a 7th grade class, at the end of the last school year in mathematics, no student received the grade "5", every ninth student received the grade "1", every third received the grade "2", and every sixth received the grade "4".

The number of students $\mathrm{n}$ is known: $20<n<40$.

How many students received the grade "3"?

Answer: 14


Let \( S = \{1, 2, \ldots, 100\} \). Find the largest integer \( k \) such that \( S \) has \( k \) distinct non-empty subsets with the following property: For any two different subsets among these \( k \) subsets, if their intersection is non-empty, then the smallest element in their intersection is not the same as the largest element in either of the two subsets.

Answer: 2^{99} - 1


Eva thought of two natural numbers. She first correctly added the numbers, then subtracted them. In both cases, she got a two-digit result. The product of the two-digit numbers thus created was 645.

Which numbers did Eva think of?

(E. Novotná)

Hint. Every natural number has a finite number of divisors.

Answer: 29


The area of a cross-section of a regular tetrahedron, which is in the shape of a square, is \(m^2\). Find the surface area of the tetrahedron.

Answer: 4\sqrt{3}m^2


11. (This question is worth 20 points) In the plane rectangular coordinate system $x O y$, let $A B$ be a chord of the parabola $y^{2}=4 x$ passing through the point $F(1,0)$, and the circumcircle of $\triangle A O B$ intersects the parabola at point $P$ (different from points $O, A, B$). If $P F$ bisects $\angle A P B$, find all possible values of $|P F|$.

Answer: \sqrt{13} - 1


5. First-grader (6th grade. 2 points). Maksim will go to the first grade on September 1, 2014, at the age of 6 years, and this day will not be his birthday. What is the probability that he was born in 2008?

Answer: \dfrac{244}{365}


Adva van egy ferde helyzetű sík és egy az első képsíkra merőleges egyenes. Keressük ezen egyenes hajlásszögét a megadott síkkal.

Answer: 90^\circ - \alpha


Three. (20 points) A certain mathematics competition had a total of 15 questions. The table below shows the statistics for the number of people who got $n (n=0,1,2, \cdots, 15)$ questions correct.
\begin{tabular}{c|c|c|c|c|c|c|c|c|c}
\hline$n$ & 0 & 1 & 2 & 3 & $\cdots$ & 12 & 13 & 14 & 15 \\
\hline Number of people who got $n$ questions correct & 7 & 8 & 10 & 21 & $\cdots$ & 15 & 6 & 3 & 1 \\
\hline
\end{tabular}

If it is also known that the students who got 4 questions and more than 4 questions correct each averaged 6 questions correct, and the students who got 10 questions and 10 questions or fewer correct each averaged 4 questions correct. How many people does this table at least include?

Answer: 200


\section*{Problem 4 - 111034}

A straight circular cone with radius \(R=6\) and height \(h\) is drilled through cylindrically such that the axis of the cone coincides with that of the drill hole.

How large must the radius \(r\) ( \(R, h, r\) measured in cm) of the drill hole be chosen so that the volume of the remaining body is half as large as that of the cone?

Answer: 3


Example 7. In $\triangle A B C$, $D, E$ are points on $B C$, $C A$ respectively, and $B D: D C=m: 1, C E: E A$ $=n: 1, A D$ intersects $B E$ at $F$. Then the area of $\triangle A B F$ is $\qquad$ times the area of $\triangle A B C$.

Answer: \dfrac{m}{m(n+1) + 1}


1. Does the number 1... (1000 ones) have a ten-digit divisor, all digits of which are different?

Answer: No


Example 2. Find the equation of the circle that passes through the intersection points of the circles $\mathrm{x}^{2}+\mathrm{y}^{2}+6 \mathrm{y}-4=0$ and $x^{2}+y^{2}+6 y-28=0$, and whose center lies on the line $\mathrm{x}-\mathrm{y}-4=0$.

Answer: x^2 + y^2 - x + 7y - 32 = 0


10. (1991, 2nd "Hope" Math Competition) Let $\{x\}$ denote the smallest integer not less than the real number $x$. Then the value of $\left\{\log _{2} 1\right\}+\left\{\log _{2} 2\right\}+\left\{\log _{2} 3\right\}+\cdots+\left\{\log _{2} 1991\right\}$ is what?

Answer: 19854


75. In quadrilateral $A B C D$, it is given that $\widehat{D A B}=150^{\circ}$, $\widehat{D A C}+\widehat{A B D}=120^{\circ}, \widehat{D B C}-\widehat{A B \bar{D}}=60^{\circ}$. Find $\widehat{B D C}$.

Answer: 30^\circ


4. Given a circle (c) with center $O$ and radius $R$. The point $O_{1}$, which lies on the circle (c), is the center of another circle with radius $\frac{R}{2}$.

If $A$ and $B$ are the intersection points of these two circles, determine the approximate value of $\measuredangle A O B$ (in radians), and then the volume of the intersection of the spheres obtained by rotating the two circles around the line $O O_{1}$.

Answer: 1.01


2. Circle $k$ is inscribed in trapezoid $A B C D, A B \| C D$, touching side $A B$ at point $E$. If $A E=15, B E=10$ and $C D=8$, determine the radius of circle $k$.

Answer: 4\sqrt{3}


9. (16 points) As shown in Figure 1, $A_{1}, A_{2}, A_{3}, A_{4}$ are four distinct points on the parabola $y^{2}=4 x$, and $A_{1}, F, A_{3}$ and $A_{2}, F, A_{4}$ are collinear respectively. Let the lines $A_{1} A_{2}$ and $A_{3} A_{4}$, $A_{1} A_{4}$ and $A_{2} A_{3}$ intersect at points $M$ and $N$, respectively, where $F(1,0)$. Find the minimum value of the area of $\triangle F M N$.

Answer: 4


Shirov V.

On a circle, $2 N$ points are marked ($N$ is a natural number). It is known that through any point inside the circle, no more than two chords with endpoints at the marked points pass. We will call a matching a set of $N$ chords with endpoints at the marked points such that each marked point is the endpoint of exactly one of these chords. We will call a matching even if the number of points where its chords intersect is even, and odd otherwise. Find the difference between the number of even and odd matchings.

Answer: 1


Pentagon $ABCDE$ consists of a square $ACDE$ and an equilateral triangle $ABC$ that share the side $\overline{AC}$. A circle centered at $C$ has area 24. The intersection of the circle and the pentagon has half the area of the pentagon. Find the area of the pentagon.
[asy]/* File unicodetex not found. */

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pen dotstyle = black; /* point style */
real xmin = -1.52, xmax = 2.74, ymin = -2.18, ymax = 6.72;  /* image dimensions */
draw((0,1)--(2,1)--(2,3)--(0,3)--cycle);
draw((0,3)--(2,3)--(1,4.73)--cycle);
 /* draw figures */
draw((0,1)--(2,1));
draw((2,1)--(2,3));
draw((2,3)--(0,3));
draw((0,3)--(0,1));
draw((0,3)--(2,3));
draw((2,3)--(1,4.73));
draw((1,4.73)--(0,3));
draw(circle((0,3), 1.44));
label("$C$",(-0.4,3.14),SE*labelscalefactor);
label("$A$",(2.1,3.1),SE*labelscalefactor);
label("$B$",(0.86,5.18),SE*labelscalefactor);
label("$D$",(-0.28,0.88),SE*labelscalefactor);
label("$E$",(2.1,0.8),SE*labelscalefactor);
 /* dots and labels */
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
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Answer: 20


We measured a distance at a temperature of $t=26^{\circ} \mathrm{C}$ using a copper rod that is exactly $a=3 \mathrm{~m}$ long at $0^{\circ} \mathrm{C}$. Thus, we found $n=7856.728 \mathrm{~m}$. What is the actual distance if the linear expansion coefficient of copper is $\lambda=0.000017$?

Answer: 7860.201


2. The old man has a certain number of apples and has said the following about them.

- If two of my grandchildren come, I will not be able to divide the apples equally between them.
- If three of my grandchildren come, I will be able to divide the apples equally among all of them.
- If four of my grandchildren come, I will be able to divide the apples equally among all of them.
- If five of my grandchildren come, I will be able to divide the apples equally among all of them.
- If six of my grandchildren come, I will not be able to divide the apples equally among all of them.

However, the old man is a bit forgetful, so exactly one of these statements is not true. Under these conditions, what is the minimum number of apples the old man can have?

Answer: 15


Šárka enjoys celebrating, so in addition to birthdays, she came up with antibirthdays: the date of the antibirthday is created by swapping the day and month numbers in the birth date. She was born on 8.11., so her antibirthday is on 11.8. Her mother cannot celebrate her antibirthday: she was born on 23.7., her antibirthday would be on 7.23., which is not a valid date. Her brother can celebrate his antibirthday, but it falls on the same day as his birthday: he was born on 3.3.

How many days in a year are such that a person born on that day can celebrate their antibirthday, and it is on a different day than their birthday?

(V. Hucíková)

Hint. What day of the month must a person be born on to be able to celebrate their antibirthday?

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

---

Šárka enjoys celebrating, so in addition to birthdays, she came up with antibirthdays: the date of the antibirthday is created by swapping the day and month numbers in the birth date. She was born on 8.11., so her antibirthday is on 11.8. Her mother cannot celebrate her antibirthday: she was born on 23.7., her antibirthday would be on 7.23., which is not a valid date. Her brother can celebrate his antibirthday, but it falls on the same day as his birthday: he was born on 3.3.

How many days in a year are such that a person born on that day can celebrate their antibirthday, and it is on a different day than their birthday?

(V. Hucíková)

Hint. What day of the month must a person be born on to be able to celebrate their antibirthday?

Answer: 132


4. Pete was given a new electric jigsaw for his birthday, with a function to count the length of the cuts made. To try out the gift, Pete took a square piece of plywood with a side of $50 \mathrm{cm}$ and cut it into squares with sides of 10 cm and squares with sides of 20 cm. How many squares in total were obtained, if the electric jigsaw shows a total length of cuts of 2 m $80 \mathrm{cm}$?

Answer: 16


# 7. Problem 7.1*

Misha thought of a five-digit number, all digits of which are different, and Igor is trying to guess it. In one move, Igor can choose several digits of the number, and Misha, in any order, tells the digits that stand in these places. The order in which to tell the digits is chosen by Misha. For example, if the number thought of is 67890, and Igor asked about the digits in the 1st and 5th places, then Misha can answer either "6 and 0" or "0 and 6". In what minimum number of moves can Igor guarantee to find out the number? Points for the problem: 13.

#

Answer: 3


The nineteenth question: Find the smallest real number $\lambda$ such that $\sum_{i=1}^{100}\left(a_{i}-a_{i+1}\right)^{2} \leq \lambda\left(100-\left(\sum_{i=1}^{100} a_{i}\right)\right)$ holds for any real numbers $a_{1} 、 a_{2} 、 \ldots 、 a_{100}$ satisfying $\sum_{i=1}^{100} a_{i}^{2}=100$. Here $a_{101}=a_{1}$.

Answer: 8


How many pairs $(n, m)$ exist such that $n \wedge m = 50$ and $n \vee m = 75$?

And such that $n \wedge m = 50$ and $n \vee m = 600$?

Answer: 0


8.1. Large sandglasses measure an hour, and small ones measure 11 minutes. How can you use these sandglasses to measure a minute?

Answer: 1


## Problem 2.

Determine the largest natural number $n$ for which the following statement is true:

There exist $n$ distinct non-zero natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ with the property that for any numbers $a_{1}, a_{2}, \ldots, a_{n} \in\{-1,0,1\}$, not all zero, the number $n^{3}$ does not divide the number $a_{1} x_{1}+a_{2} x_{2}+\ldots+a_{n} x_{n}$.

Answer: 9


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Carol`, `Bob`, `Peter`, `Arnold`, `Alice`
 - Everyone has something unique for lunch: `stew`, `stir fry`, `soup`, `grilled cheese`, `pizza`, `spaghetti`
 - Each person has a unique favorite drink: `root beer`, `milk`, `boba tea`, `water`, `coffee`, `tea`
 - People have unique heights: `super tall`, `short`, `tall`, `very short`, `very tall`, `average`
 - Each person has a unique birthday month: `sept`, `feb`, `may`, `mar`, `jan`, `april`

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

Answer: Peter


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Eric`, `Peter`
 - People have unique favorite music genres: `pop`, `classical`, `rock`, `jazz`
 - Each person has a unique type of pet: `fish`, `dog`, `bird`, `cat`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`, `honda civic`
 - Each person has a unique favorite drink: `water`, `milk`, `tea`, `coffee`

## Clues:
1. The one who only drinks water is the person who owns a dog.
2. The one who only drinks water is Eric.
3. The tea drinker is somewhere to the right of the person who loves rock music.
4. The person who likes milk is the person who owns a Tesla Model 3.
5. The person who loves rock music and the person who loves classical music are next to each other.
6. There is one house between the person who likes milk and the person who loves classical music.
7. The person who owns a Honda Civic is Arnold.
8. The person who has a cat and the coffee drinker are next to each other.
9. Alice is not in the second house.
10. The person who loves classical music is not in the second house.
11. The person with an aquarium of fish is the tea drinker.
12. The person who owns a Ford F-150 is somewhere to the right of the person with an aquarium of fish.
13. The person who owns a Toyota Camry is not in the third house.
14. The tea drinker and the person who loves jazz music are next to each other.
 What is the value of attribute MusicGenre for the person whose attribute Drink is water? Please reason step by step, and put your final answer within \boxed{}

Answer: jazz


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 Drink for the person whose attribute House is 5? Please reason step by step, and put your final answer within \boxed{}

Answer: water


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

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

Answer: ranch


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`, `Carol`, `Alice`, `Bob`
 - People have unique favorite music genres: `country`, `classical`, `pop`, `hip hop`, `jazz`, `rock`
 - Each person lives in a unique style of house: `modern`, `mediterranean`, `victorian`, `craftsman`, `ranch`, `colonial`
 - People use unique phone models: `huawei p50`, `iphone 13`, `google pixel 6`, `oneplus 9`, `xiaomi mi 11`, `samsung galaxy s21`
 - Each person has a unique type of pet: `hamster`, `dog`, `cat`, `rabbit`, `fish`, `bird`

## Clues:
1. The person who has a cat is the person in a Mediterranean-style villa.
2. Peter is in the second house.
3. The person in a modern-style house is the person who uses a Samsung Galaxy S21.
4. The person who keeps a pet bird is the person who uses a Google Pixel 6.
5. The person who keeps a pet bird is directly left of the person who uses a Xiaomi Mi 11.
6. Arnold is the person residing in a Victorian house.
7. The person who loves rock music is the person living in a colonial-style house.
8. The person in a Craftsman-style house is Carol.
9. The person who owns a rabbit is in the sixth house.
10. The person who loves pop music is somewhere to the left of the person living in a colonial-style house.
11. The person who uses a Huawei P50 and the person who loves country music are next to each other.
12. The person who loves hip-hop music is the person who uses a Samsung Galaxy S21.
13. Carol and Peter are next to each other.
14. The person who uses a OnePlus 9 is Peter.
15. The person with a pet hamster is not in the first house.
16. The person in a ranch-style home is somewhere to the left of the person who loves classical music.
17. There are two houses between Carol and Alice.
18. The person who owns a dog is in the fifth house.
19. The person in a modern-style house is Eric.
20. Peter is the person living in a colonial-style house.
21. The person who has a cat is not in the fourth house.
 What is the value of attribute House for the person whose attribute MusicGenre is country? Please reason step by step, and put your final answer within \boxed{}

Answer: 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: `Arnold`, `Alice`, `Carol`, `Peter`, `Bob`, `Eric`
 - The people keep unique animals: `fish`, `bird`, `rabbit`, `cat`, `horse`, `dog`
 - Each person has a unique favorite drink: `tea`, `root beer`, `water`, `milk`, `boba tea`, `coffee`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `huawei p50`, `xiaomi mi 11`, `oneplus 9`
 - Each mother is accompanied by their child: `Alice`, `Fred`, `Meredith`, `Samantha`, `Timothy`, `Bella`

## Clues:
1. The tea drinker is the person who uses an iPhone 13.
2. There is one house between the dog owner and the tea drinker.
3. The fish enthusiast is somewhere to the left of the tea drinker.
4. Eric and the person's child is named Alice are next to each other.
5. The one who only drinks water is not in the third house.
6. The root beer lover is directly left of the person's child is named Samantha.
7. The person who is the mother of Timothy is the coffee drinker.
8. The fish enthusiast is directly left of the person who likes milk.
9. Peter is the person who likes milk.
10. The person who uses a OnePlus 9 is directly left of the person's child is named Alice.
11. The person who keeps horses is not in the third house.
12. The person's child is named Meredith is directly left of Bob.
13. The person's child is named Bella is the person who uses a Google Pixel 6.
14. The bird keeper is directly left of the person's child is named Meredith.
15. The person who uses a Samsung Galaxy S21 is directly left of the person's child is named Bella.
16. The one who only drinks water is somewhere to the left of the person's child is named Fred.
17. Arnold is the tea drinker.
18. The person who keeps horses is somewhere to the left of the cat lover.
19. The person who uses a Huawei P50 and Alice are next to each other.
20. Bob is directly left of the root beer lover.
21. The person who is the mother of Timothy is Eric.
 What is the value of attribute Drink for the person whose attribute Animal is dog? Please reason step by step, and put your final answer within \boxed{}

Answer: water


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`
 - Each person lives in a unique style of house: `victorian`, `ranch`, `craftsman`, `colonial`
 - Everyone has a unique favorite cigar: `pall mall`, `dunhill`, `blue master`, `prince`
 - Each person has an occupation: `doctor`, `teacher`, `engineer`, `artist`
 - They all have a unique favorite flower: `lilies`, `daffodils`, `roses`, `carnations`
 - The people keep unique animals: `horse`, `bird`, `fish`, `cat`

## Clues:
1. The fish enthusiast is somewhere to the right of Eric.
2. The person who is an engineer is Arnold.
3. The person who loves a carnations arrangement is the person in a ranch-style home.
4. Alice is the person in a Craftsman-style house.
5. The person partial to Pall Mall is the person who loves a bouquet of daffodils.
6. The cat lover is Arnold.
7. The Dunhill smoker is Peter.
8. The person living in a colonial-style house is not in the fourth house.
9. The person who is a teacher is the bird keeper.
10. The person residing in a Victorian house is Peter.
11. The person who loves the boquet of lilies is the person who smokes Blue Master.
12. Peter is somewhere to the right of the person who is an artist.
13. The person who is an engineer and the person who is a teacher are next to each other.
14. The person who is a doctor is the person who loves a carnations arrangement.
15. The person partial to Pall Mall is the person living in a colonial-style house.
 What is the value of attribute House for the person whose attribute Animal is cat? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Alice`, `Arnold`
 - People have unique hair colors: `brown`, `black`, `blonde`, `red`
 - People have unique favorite sports: `tennis`, `basketball`, `soccer`, `swimming`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `dragonfruit`, `desert`
 - Each person has a unique birthday month: `april`, `sept`, `feb`, `jan`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `iphone 13`, `google pixel 6`

## Clues:
1. The person whose birthday is in January is somewhere to the left of the person who loves basketball.
2. The person whose birthday is in February is directly left of the person who has red hair.
3. Eric is the person who uses a OnePlus 9.
4. The person whose birthday is in April is Arnold.
5. The person who uses a Samsung Galaxy S21 and the person who likes Cherry smoothies are next to each other.
6. The person who has blonde hair and Peter are next to each other.
7. The person who has black hair is somewhere to the right of the person who has brown hair.
8. The person who has black hair is the person whose birthday is in February.
9. The Dragonfruit smoothie lover is in the second house.
10. The person who loves soccer is somewhere to the right of the person who uses a OnePlus 9.
11. There are two houses between the person who loves tennis and the person who loves basketball.
12. The person who has black hair is the person who uses a Samsung Galaxy S21.
13. Alice is the person who likes Cherry smoothies.
14. The Desert smoothie lover is the person who uses a Google Pixel 6.
 What is the value of attribute PhoneModel for the person whose attribute Birthday is feb? Please reason step by step, and put your final answer within \boxed{}

Answer: samsung galaxy s21


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Bob`, `Peter`, `Alice`, `Carol`
 - People have unique heights: `very tall`, `tall`, `average`, `short`, `super tall`, `very short`
 - Each person has a unique birthday month: `sept`, `may`, `jan`, `april`, `feb`, `mar`
 - People have unique hair colors: `blonde`, `red`, `black`, `auburn`, `gray`, `brown`
 - People have unique favorite book genres: `science fiction`, `romance`, `biography`, `mystery`, `fantasy`, `historical fiction`
 - Each person has a unique type of pet: `bird`, `hamster`, `fish`, `dog`, `cat`, `rabbit`

## Clues:
1. The person with an aquarium of fish is the person who loves mystery books.
2. The person who has black hair is the person who owns a dog.
3. The person who has gray hair is the person who loves science fiction books.
4. The person who loves historical fiction books is somewhere to the left of the person who has black hair.
5. The person who loves historical fiction books is directly left of the person who owns a rabbit.
6. The person whose birthday is in February is the person who owns a dog.
7. Bob is the person who keeps a pet bird.
8. The person who keeps a pet bird is the person who loves romance books.
9. Alice is directly left of the person who loves fantasy books.
10. The person who owns a rabbit is directly left of the person who is very short.
11. The person who is short is somewhere to the left of the person who is super tall.
12. The person who has an average height is in the sixth house.
13. The person who has red hair is not in the first house.
14. The person who is very tall is somewhere to the right of the person who is super tall.
15. There are two houses between the person who loves romance books and Eric.
16. The person with an aquarium of fish is in the sixth house.
17. The person who has blonde hair is the person who loves historical fiction books.
18. The person who is tall is somewhere to the right of the person whose birthday is in February.
19. The person who has auburn hair is Carol.
20. The person with an aquarium of fish is the person whose birthday is in March.
21. The person who loves science fiction books is directly left of the person with an aquarium of fish.
22. The person whose birthday is in September is not in the first house.
23. The person who loves biography books is directly left of the person who has a cat.
24. The person whose birthday is in January is the person who loves science fiction books.
25. Arnold is the person whose birthday is in May.
 What is the value of attribute Birthday for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`
 - Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `desert`, `watermelon`
 - People have unique favorite sports: `soccer`, `tennis`, `basketball`, `swimming`
 - People own unique car models: `tesla model 3`, `toyota camry`, `honda civic`, `ford f150`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`

## Clues:
1. The person who owns a Tesla Model 3 is the person who loves the rose bouquet.
2. Peter is the Dragonfruit smoothie lover.
3. The Desert smoothie lover is the person who owns a Toyota Camry.
4. The person who loves tennis is in the first house.
5. The person who owns a Toyota Camry and the person who loves basketball are next to each other.
6. Arnold is the person who loves basketball.
7. The person who owns a Honda Civic is the person who loves a bouquet of daffodils.
8. Eric is the person who loves the rose bouquet.
9. The Watermelon smoothie lover is not in the first house.
10. The person who owns a Honda Civic is somewhere to the right of the Desert smoothie lover.
11. The person who loves basketball is the person who loves the boquet of lilies.
12. The person who loves tennis and the person who loves soccer are next to each other.
 What is the value of attribute House for the person whose attribute CarModel is ford f150? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


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

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

Answer: short


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

## Clues:
1. The person's child is named Samantha is somewhere to the left of the person who loves biography books.
2. Arnold is the person who loves biography books.
3. The person who is the mother of Timothy is the person who loves science fiction books.
4. The person's child is named Bella is Peter.
5. There is one house between the person who loves biography books and Bob.
6. Alice is somewhere to the right of Eric.
7. The person who loves mystery books is in the fourth house.
8. The person who loves biography books is directly left of the person who is the mother of Timothy.
9. The person who loves fantasy books is directly left of the person's child is named Meredith.
 What is the value of attribute BookGenre for the person whose attribute Children is Timothy? Please reason step by step, and put your final answer within \boxed{}

Answer: science fiction


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Alice`, `Eric`, `Arnold`
 - Each mother is accompanied by their child: `Meredith`, `Bella`, `Samantha`, `Fred`, `Timothy`
 - People have unique favorite music genres: `classical`, `hip hop`, `pop`, `jazz`, `rock`
 - Everyone has a favorite smoothie: `watermelon`, `lime`, `dragonfruit`, `cherry`, `desert`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `city`, `cruise`, `camping`

## Clues:
1. The person who loves rock music is somewhere to the left of the person who loves classical music.
2. The person who loves beach vacations and the person who loves rock music are next to each other.
3. Eric is the person who is the mother of Timothy.
4. The person who drinks Lime smoothies is not in the second house.
5. The Watermelon smoothie lover is somewhere to the left of the person who loves rock music.
6. Alice is the person who loves beach vacations.
7. The Dragonfruit smoothie lover is somewhere to the right of Peter.
8. The person who loves jazz music is the person's child is named Samantha.
9. Bob is the person's child is named Samantha.
10. Alice is somewhere to the left of Eric.
11. The person who loves pop music is the person who likes going on cruises.
12. The person's child is named Fred is in the fifth house.
13. The person who prefers city breaks is the person who loves jazz music.
14. The Dragonfruit smoothie lover is the person's child is named Samantha.
15. The person's child is named Meredith is the person who likes going on cruises.
16. The Desert smoothie lover is somewhere to the left of the person who likes Cherry smoothies.
17. The person who enjoys mountain retreats is not in the fifth house.
18. The person who drinks Lime smoothies is directly left of the Desert smoothie lover.
 What is the value of attribute MusicGenre for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: rock


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 BookGenre is mystery? 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`, `Bob`, `Alice`, `Eric`, `Peter`, `Carol`
 - People have unique favorite sports: `volleyball`, `soccer`, `basketball`, `baseball`, `swimming`, `tennis`
 - Each person has a unique hobby: `gardening`, `cooking`, `painting`, `woodworking`, `knitting`, `photography`
 - Each mother is accompanied by their child: `Bella`, `Meredith`, `Alice`, `Samantha`, `Fred`, `Timothy`
 - Each person has a unique birthday month: `feb`, `jan`, `mar`, `may`, `sept`, `april`
 - Everyone has something unique for lunch: `stir fry`, `grilled cheese`, `pizza`, `spaghetti`, `stew`, `soup`

## Clues:
1. The person who loves soccer is somewhere to the right of Eric.
2. The person who paints as a hobby is somewhere to the left of the woodworking hobbyist.
3. The person whose birthday is in February is not in the third house.
4. The person who is the mother of Timothy is directly left of Arnold.
5. The person who loves cooking is somewhere to the left of the person's child is named Fred.
6. Alice is the person who paints as a hobby.
7. The person's child is named Alice is the person who loves cooking.
8. The person whose birthday is in March is the person's child is named Samantha.
9. The person who loves the soup is the person whose birthday is in May.
10. Carol is somewhere to the right of the person whose birthday is in May.
11. The person whose birthday is in March and the person who is a pizza lover are next to each other.
12. The person whose birthday is in February is not in the fourth house.
13. There are two houses between the person who loves baseball and the person who loves the stew.
14. The person whose birthday is in May is somewhere to the left of the person who loves volleyball.
15. The person whose birthday is in January is somewhere to the right of Arnold.
16. The person who loves cooking is not in the third house.
17. The person who enjoys knitting is the person whose birthday is in September.
18. The person who enjoys gardening is the person's child is named Meredith.
19. The person who is a pizza lover is the person who loves swimming.
20. There is one house between the person who loves basketball and the person whose birthday is in May.
21. The person who loves the soup is directly left of the person who loves the spaghetti eater.
22. The person who loves the stew is the person's child is named Bella.
23. The person who enjoys knitting is somewhere to the left of the person who loves the soup.
24. Bob is somewhere to the left of the person who loves the stew.
25. Arnold is the person who loves stir fry.
26. The woodworking hobbyist is directly left of the person who loves the soup.
 What is the value of attribute Children for the person whose attribute Hobby is gardening? Please reason step by step, and put your final answer within \boxed{}

Answer: Meredith


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Carol`, `Bob`, `Peter`, `Alice`
 - Each person lives in a unique style of house: `ranch`, `victorian`, `modern`, `colonial`, `craftsman`, `mediterranean`
 - Each person has a unique favorite drink: `coffee`, `water`, `root beer`, `tea`, `milk`, `boba tea`
 - Each person has an occupation: `engineer`, `nurse`, `doctor`, `artist`, `lawyer`, `teacher`
 - The people keep unique animals: `horse`, `fish`, `dog`, `cat`, `rabbit`, `bird`
 - Each person has a favorite color: `red`, `purple`, `white`, `green`, `yellow`, `blue`

## Clues:
1. The fish enthusiast is in the first house.
2. Carol is somewhere to the left of the person in a Mediterranean-style villa.
3. The bird keeper is the person who loves purple.
4. The person in a ranch-style home is the one who only drinks water.
5. The coffee drinker is not in the fourth house.
6. The person living in a colonial-style house and the boba tea drinker are next to each other.
7. The cat lover is the person who is a teacher.
8. The person in a modern-style house is in the third house.
9. The person who is an engineer and Peter are next to each other.
10. Arnold is the person who is an artist.
11. The person whose favorite color is red is Arnold.
12. The coffee drinker is somewhere to the right of the person who is a teacher.
13. The bird keeper is the person in a ranch-style home.
14. The rabbit owner is in the sixth house.
15. The boba tea drinker is Peter.
16. Eric is the cat lover.
17. The person who is a nurse is directly left of the person in a modern-style house.
18. The bird keeper is Carol.
19. There are two houses between the cat lover and the boba tea drinker.
20. Bob is the tea drinker.
21. The person residing in a Victorian house is the dog owner.
22. The person whose favorite color is green is in the fourth house.
23. The person who loves blue is the root beer lover.
24. The person who loves white is not in the sixth house.
25. Peter is the person who is a doctor.
 What is the value of attribute House for the person whose attribute HouseStyle is victorian? 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: `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`, `honda civic`, `bmw 3 series`
 - The people are of nationalities: `brit`, `norwegian`, `german`, `swede`, `dane`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `lilies`, `tulips`, `roses`

## Clues:
1. The person who owns a Tesla Model 3 is the person who loves a carnations arrangement.
2. The person who owns a Toyota Camry is Alice.
3. The person who owns a BMW 3 Series is somewhere to the right of Arnold.
4. Arnold is not in the third house.
5. The Swedish person is in the first house.
6. The German is the person who loves a bouquet of daffodils.
7. Arnold is not in the second house.
8. The Norwegian is the person who owns a Tesla Model 3.
9. The Dane is somewhere to the left of the person who owns a Ford F-150.
10. The person who owns a BMW 3 Series is somewhere to the right of the person who loves a carnations arrangement.
11. Alice is directly left of the person who loves the rose bouquet.
12. Peter and the person who loves the vase of tulips are next to each other.
13. The British person is somewhere to the right of the Norwegian.
14. The person who loves the vase of tulips is Eric.
15. The person who owns a Ford F-150 is Eric.
 What is the value of attribute Nationality for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: norwegian


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Alice`, `Eric`
 - They all have a unique favorite flower: `roses`, `daffodils`, `carnations`, `lilies`
 - Each person has a unique hobby: `photography`, `painting`, `cooking`, `gardening`
 - Each person has a unique type of pet: `dog`, `fish`, `bird`, `cat`
 - Each person has a favorite color: `red`, `yellow`, `green`, `white`
 - Each person lives in a unique style of house: `craftsman`, `colonial`, `ranch`, `victorian`

## Clues:
1. The person in a Craftsman-style house is Arnold.
2. The person who loves the rose bouquet is somewhere to the right of Peter.
3. The photography enthusiast is the person who owns a dog.
4. The person who loves a bouquet of daffodils is not in the fourth house.
5. The person who loves the rose bouquet is the person whose favorite color is red.
6. The person in a Craftsman-style house is in the second house.
7. Eric is the person residing in a Victorian house.
8. The person with an aquarium of fish is the person who loves white.
9. The person who loves cooking is somewhere to the right of the person whose favorite color is red.
10. The person who loves white is the person who loves a carnations arrangement.
11. The person who loves white is somewhere to the right of the person who enjoys gardening.
12. The person who loves a bouquet of daffodils is the person who loves yellow.
13. The person living in a colonial-style house is the person whose favorite color is red.
14. The person who has a cat is Eric.
 What is the value of attribute House for the person whose attribute Pet is cat? 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: `Peter`, `Alice`, `Eric`, `Arnold`
 - People own unique car models: `ford f150`, `tesla model 3`, `toyota camry`, `honda civic`
 - People have unique favorite music genres: `pop`, `jazz`, `rock`, `classical`
 - People have unique favorite sports: `basketball`, `tennis`, `soccer`, `swimming`
 - Each person prefers a unique type of vacation: `mountain`, `beach`, `cruise`, `city`

## Clues:
1. The person who loves beach vacations is the person who loves soccer.
2. The person who likes going on cruises and Arnold are next to each other.
3. Eric is the person who owns a Toyota Camry.
4. The person who owns a Toyota Camry is the person who loves swimming.
5. The person who loves beach vacations is in the fourth house.
6. The person who loves jazz music is somewhere to the right of the person who loves rock music.
7. The person who owns a Toyota Camry is the person who loves pop music.
8. The person who prefers city breaks is the person who loves jazz music.
9. The person who loves basketball is directly left of the person who owns a Tesla Model 3.
10. The person who likes going on cruises is somewhere to the left of the person who owns a Ford F-150.
11. The person who loves jazz music is directly left of the person who enjoys mountain retreats.
12. Peter is somewhere to the right of the person who loves rock music.
 What is the value of attribute CarModel for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: tesla model 3


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Eric`, `Bob`, `Carol`, `Arnold`
 - People have unique favorite sports: `baseball`, `basketball`, `swimming`, `tennis`, `soccer`, `volleyball`
 - Each person has a favorite color: `purple`, `green`, `white`, `red`, `yellow`, `blue`
 - People have unique hair colors: `red`, `auburn`, `blonde`, `black`, `gray`, `brown`
 - Everyone has something unique for lunch: `stew`, `spaghetti`, `stir fry`, `grilled cheese`, `soup`, `pizza`

## Clues:
1. Arnold is somewhere to the left of the person who loves blue.
2. There are two houses between the person whose favorite color is green and Bob.
3. The person who loves white is in the fifth house.
4. The person who loves baseball is directly left of the person who has brown hair.
5. The person who loves tennis is not in the fifth house.
6. The person who has auburn hair is the person who loves the stew.
7. The person whose favorite color is red and the person who loves baseball are next to each other.
8. The person who is a pizza lover is Eric.
9. The person who loves purple is Carol.
10. Peter is in the third house.
11. Carol is the person who loves soccer.
12. The person who loves volleyball is somewhere to the left of the person who has auburn hair.
13. Eric is in the first house.
14. There is one house between the person who has gray hair and the person who loves blue.
15. The person who loves volleyball is directly left of the person who loves baseball.
16. The person who loves the spaghetti eater is in the fifth house.
17. The person who loves eating grilled cheese is the person who has brown hair.
18. The person who loves the soup is in the third house.
19. The person who has blonde hair is somewhere to the left of the person who has gray hair.
20. The person who has red hair is the person who loves white.
21. The person who loves blue is somewhere to the right of the person who loves swimming.
 What is the value of attribute House for the person whose attribute FavoriteSport is basketball? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Peter`, `Arnold`, `Bob`, `Eric`, `Alice`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `oneplus 9`, `huawei p50`, `samsung galaxy s21`, `xiaomi mi 11`
 - People have unique hair colors: `gray`, `auburn`, `red`, `brown`, `black`, `blonde`
 - Each person has a unique favorite drink: `coffee`, `water`, `root beer`, `tea`, `milk`, `boba tea`
 - People have unique favorite music genres: `classical`, `jazz`, `rock`, `country`, `hip hop`, `pop`

## Clues:
1. Alice is the person who uses a Huawei P50.
2. The person who uses a Xiaomi Mi 11 is somewhere to the left of the one who only drinks water.
3. The one who only drinks water is somewhere to the left of Bob.
4. Carol is the person who has gray hair.
5. Eric is the tea drinker.
6. The person who has black hair is not in the sixth house.
7. The person who uses a Samsung Galaxy S21 is directly left of the person who loves classical music.
8. The person who uses a OnePlus 9 is not in the third house.
9. Alice is somewhere to the right of Carol.
10. The person who loves jazz music is Alice.
11. The person who loves jazz music is the boba tea drinker.
12. There is one house between the person who uses a OnePlus 9 and Eric.
13. The person who has brown hair is the person who loves hip-hop music.
14. The person who loves pop music is not in the first house.
15. The root beer lover is in the second house.
16. Eric is the person who uses an iPhone 13.
17. The person who has red hair is the person who uses an iPhone 13.
18. The person who has red hair and the coffee drinker are next to each other.
19. Peter is in the first house.
20. Arnold is somewhere to the right of the person who has auburn hair.
21. The person who has gray hair is not in the second house.
22. The person who loves country music is in the fifth house.
 What is the value of attribute House for the person whose attribute PhoneModel is google pixel 6? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Carol`, `Peter`, `Alice`, `Eric`, `Arnold`
 - They all have a unique favorite flower: `daffodils`, `iris`, `lilies`, `roses`, `tulips`, `carnations`
 - People have unique hair colors: `gray`, `red`, `blonde`, `auburn`, `brown`, `black`
 - Each person has a favorite color: `white`, `blue`, `green`, `purple`, `red`, `yellow`
 - People own unique car models: `tesla model 3`, `ford f150`, `honda civic`, `toyota camry`, `chevrolet silverado`, `bmw 3 series`
 - The mothers' names in different houses are unique: `Janelle`, `Holly`, `Kailyn`, `Sarah`, `Penny`, `Aniya`

## Clues:
1. There is one house between Arnold and the person who loves blue.
2. The person whose mother's name is Janelle is directly left of the person who has auburn hair.
3. The person who owns a Chevrolet Silverado and the person who owns a BMW 3 Series are next to each other.
4. Carol is the person who loves a carnations arrangement.
5. The person who loves the boquet of iris is the person who has gray hair.
6. The person who loves the rose bouquet is The person whose mother's name is Kailyn.
7. Bob is the person who loves the boquet of lilies.
8. The person who loves a bouquet of daffodils is somewhere to the left of the person who owns a Chevrolet Silverado.
9. Bob is in the third house.
10. The person who owns a Honda Civic is in the sixth house.
11. The person who loves the vase of tulips is somewhere to the left of The person whose mother's name is Penny.
12. The person who has brown hair is in the fourth house.
13. The person who has brown hair is somewhere to the left of the person who has blonde hair.
14. The person whose mother's name is Holly is Bob.
15. The person who owns a Ford F-150 is in the third house.
16. Eric is directly left of the person who has auburn hair.
17. The person who owns a Tesla Model 3 is the person who loves yellow.
18. There are two houses between the person who loves purple and The person whose mother's name is Aniya.
19. The person who loves purple is in the first house.
20. The person whose favorite color is green is not in the fourth house.
21. The person who has gray hair is the person who owns a BMW 3 Series.
22. The person who has gray hair is somewhere to the left of Alice.
23. The person whose mother's name is Holly is somewhere to the left of the person who loves the vase of tulips.
24. The person who has red hair is the person who loves white.
 What is the value of attribute Flower for the person whose attribute HairColor is red? Please reason step by step, and put your final answer within \boxed{}

Answer: lilies


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Carol`, `Bob`, `Arnold`, `Peter`
 - People have unique favorite music genres: `hip hop`, `jazz`, `country`, `classical`, `rock`, `pop`
 - People use unique phone models: `iphone 13`, `oneplus 9`, `xiaomi mi 11`, `huawei p50`, `samsung galaxy s21`, `google pixel 6`
 - People have unique favorite book genres: `biography`, `historical fiction`, `fantasy`, `romance`, `science fiction`, `mystery`

## Clues:
1. Arnold is not in the sixth house.
2. There are two houses between the person who loves country music and the person who loves hip-hop music.
3. The person who loves romance books is Carol.
4. The person who loves biography books is somewhere to the left of the person who loves rock music.
5. The person who loves mystery books is somewhere to the right of Peter.
6. The person who loves romance books is the person who loves classical music.
7. The person who loves mystery books is directly left of the person who loves fantasy books.
8. Bob is the person who loves pop music.
9. The person who loves jazz music is somewhere to the left of the person who loves pop music.
10. Arnold is somewhere to the right of the person who loves hip-hop music.
11. The person who loves hip-hop music and the person who uses a OnePlus 9 are next to each other.
12. The person who loves science fiction books is the person who uses a Xiaomi Mi 11.
13. The person who loves classical music is in the second house.
14. The person who uses a Xiaomi Mi 11 is directly left of Eric.
15. The person who loves historical fiction books is somewhere to the left of the person who loves classical music.
16. The person who uses a Huawei P50 is Peter.
17. The person who uses a Samsung Galaxy S21 is in the fourth house.
18. The person who uses a Samsung Galaxy S21 is somewhere to the right of the person who uses an iPhone 13.
 What is the value of attribute House for the person whose attribute MusicGenre is country? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Bob`, `Peter`, `Arnold`
 - Each person has a unique birthday month: `mar`, `april`, `sept`, `feb`, `jan`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Kailyn`, `Penny`, `Aniya`
 - Each person has an occupation: `engineer`, `doctor`, `lawyer`, `artist`, `teacher`
 - People have unique hair colors: `red`, `blonde`, `black`, `gray`, `brown`

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

Answer: 1


Let $a$ and $b$ be positive integers. When $a^{2}+b^{2}$ is divided by $a+b,$ the quotient is $q$ and the remainder is $r.$ Find all pairs $(a,b)$ such that $q^{2}+r=1977$.

Answer: (7, 50)


Richard has an infinite row of empty boxes labeled $1, 2, 3, \ldots$ and an infinite supply of balls. Each minute, Richard finds the smallest positive integer $k$ such that box $k$ is empty. Then, Richard puts a ball into box $k$, and if $k \geq 3$, he removes one ball from each of boxes $1,2,\ldots,k-2$. Find the smallest positive integer $n$ such that after $n$ minutes, both boxes $9$ and $10$ have at least one ball in them.

[i]Proposed by [b]vvluo[/b] & [b]richy[/b][/i]

Answer: 89


Task B-4.5. On the chord $\overline{A B}$ of the circle $k(S, r)$, two isosceles triangles are constructed on the same side of the chord, with the chord $\overline{A B}$ being their common base. One has its third vertex at the center $S$, and the other at a point $C$ on the circle. If the ratio of their areas is $3:(2 \sqrt{3}+3)$, calculate the measure of the angle between the legs of the triangle $\triangle A B S$.

Answer: 60^\circ


Problem 9.8. From the set $\{1,2,3, \ldots, n\}$, 9 distinct numbers are chosen and written in the cells of a $3 \times 3$ table such that the products of the numbers in each row, column, and diagonal are equal. Determine the smallest value of $n$ for which such a table exists.

Answer: 36


1. In a certain triangle, the sum of the tangents of the angles turned out to be 2016. Estimate (at least to the nearest degree) the magnitude of the largest of its angles.

Answer: 90


16. Given the even function
$$
\begin{aligned}
f(x)= & 5 \cos \theta \cdot \sin x-5 \sin (x-\theta)+ \\
& (4 \tan \theta-3) \sin x-5 \sin \theta
\end{aligned}
$$

has a minimum value of -6.
(1) Find the maximum value of $f(x)$ and the set of $x$ at which this maximum occurs.
(2) Let the function
$$
g(x)=\lambda f(\omega x)-f\left(\omega x+\frac{\pi}{2}\right)(\lambda>0, \omega>0) .
$$

It is known that $y=g(x)$ attains its minimum value at $x=\frac{\pi}{6}$ and the point $\left(\frac{2 \pi}{3}, 3-3 \lambda\right)$ is a center of symmetry of its graph. Try to find the minimum value of $\lambda+\omega$.

Answer: 0


80. One day, Xiao Ben told a joke. Except for Xiao Ben himself, four-fifths of the classmates in the classroom heard it, but only three-quarters of the classmates laughed. It is known that one-sixth of the classmates who heard the joke did not laugh. Then, what fraction of the classmates who did not hear the joke laughed?

Answer: \dfrac{5}{12}


Let \( P(x) \) be a polynomial with integer coefficients, and let \( a \) and \( b \) be integers such that \( a > b \). Prove that the number \( P(a) - P(b) \) is divisible by \( a - b \).

Answer: P(a) - P(b) \text{ is divisible by } a - b


[ $\underline{\text { Cone }}$

Through the edge $B C$ of the triangular pyramid $P A B C$ and the point $M$, the midpoint of the edge $P A$, a section $B C M$ is drawn. The vertex of the cone coincides with the vertex $P$ of the pyramid, and the circle of the base is inscribed in the triangle $B C M$, touching the side $B C$ at its midpoint. The points of tangency of the circle with the segments $B M$ and $C M$ are the points of intersection of the medians of the faces $A P B$ and $A P C$. The height of the cone is twice the radius of the base. Find the ratio of the lateral surface area of the pyramid to the area of the base of the pyramid.

#

Answer: 2


For real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$ , and define $\{x\} = x - \lfloor x \rfloor$ to be the fractional part of $x$ . For example, $\{3\} = 0$ and $\{4.56\} = 0.56$ . Define $f(x)=x\{x\}$ , and let $N$ be the number of real-valued solutions to the equation $f(f(f(x)))=17$ for $0\leq x\leq 2020$ . Find the remainder when $N$ is divided by $1000$ .
Please give the answer directly without any intermediate steps.

Answer: 10


Two unit-radius spheres intersect in such a way that the resulting 3 parts have equal volume. How far apart are the centers of the spheres?

Answer: 4 \cos \left( \dfrac{4\pi}{9} \right)


Gorskiy E.A.

Two natural numbers $x$ and $y$ ($x \leq y$) are written on the board in ascending order. Petya writes down $x^2$ (the square of the first number) on a piece of paper, and then replaces the numbers on the board with $x$ and $y-x$, writing them in ascending order. With the new numbers on the board, he performs the same operation, and so on, until one of the numbers on the board becomes zero. What will be the sum of the numbers on Petya's paper at this moment?

Answer: xy


4. (10 points) A two-digit number multiplied by 109 results in a four-digit number, which can be divided by 23 with a quotient being a single digit. The largest two-digit number equals $\qquad$ .

Answer: 69


The initial terms of a sequence are: $c_{0}=2, c_{1}=3$, and subsequent terms for $k=2,3,4, \ldots$ can be calculated using the relation $c_{k}=3 c_{k-1}-2 c_{k-2}$. Write $c_{k}$ as a function of $k$ alone. What is the sum $S_{n}$ of the first $n$ terms of this sequence? Express $S_{n}$ in terms of $S_{n-1}$ and $S_{n-2}$.

Answer: 2^k + 1


A set $S$ of natural numbers is called good, if for each element $x \in S, x$ does not divide the sum of the remaining numbers in $S$. Find the maximal possible number of elements of a good set which is a subset of the set $A=\{1,2,3, \ldots, 63\}$.

Answer: 61


fantasticbobob is proctoring a room for the SiSiEyMB with $841$ seats arranged in $29$ rows and $29$ columns. The contestants sit down, take part $1$ of the contest, go outside for a break, and come back to take part $2$ of the contest. fantasticbobob sits among the contestants during part $1$, also goes outside during break, but when he returns, he finds that his seat has been taken. Furthermore, each of the $840$ contestants now sit in a chair horizontally or vertically adjacent to their original chair. How many seats could fantasticbobob have started in?

[i]2019 CCA Math Bonanza Team Round #8[/i]

Answer: 421


The sequence 1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, ... is formed as follows: First, we take one odd number, then two even numbers, then three odd numbers, then four even numbers, and so on. Find the number in the sequence which is closest to 1994.

Answer: 1995


In triangle \(ABC\), \(AB = 32\), \(AC = 35\), and \(BC = x\). What is the smallest positive integer \(x\) such that \(1 + \cos^2 A\), \(\cos^2 B\), and \(\cos^2 C\) form the sides of a non-degenerate triangle?

Answer: 48


Given 3 skew lines. Prove that they will be common perpendiculars to their common perpendiculars.

Answer: 3


The number 2019 is expressed as a sum of different odd natural numbers. What is the maximum possible number of terms in this sum?

Answer: 43


Pedrinho is playing with three triangular pieces with sides $(5,8,10),(5,10,12)$, and $(5,8,12)$ as shown in the figures below. He can join two pieces by sticking exactly the sides of the same length together. For example, he can join the side 10 of the first piece with the side 10 of the second piece, but he cannot join the side 10 of the first piece with the side 8 of the third piece because they do not have the same length. What is the greatest perimeter Pedrinho can obtain by joining the three pieces?

(Note: The figures provided in the original problem have been omitted for text-based translation purposes.)

Answer: 49


The Fibonacci numbers are defined recursively by \( F_{0}=0 \), \( F_{1}=1 \), and \( F_{i}=F_{i-1}+F_{i-2} \) for \( i \geq 2 \). Given 15 wooden blocks of weights \( F_{2}, F_{3}, \ldots, F_{16} \), compute the number of ways to paint each block either red or blue such that the total weight of the red blocks equals the total weight of the blue blocks.

Answer: 32


A trapezoid with its smaller base equal to 4 is inscribed in a circle with radius \(\sqrt{7}\). Through a point on this circle, a tangent parallel to one of the trapezoid's non-parallel sides is drawn. A chord of length 5, parallel to the bases of the trapezoid, is also drawn through this point. Find the length of the trapezoid's diagonal and the area of the trapezoid.

Answer: \frac{975\sqrt{3}}{196}


Given the parabola \( y^{2} = 4ax \) where \( 0 < a < 1 \) with a focus at \( F \). Taking \( A(a + 4, 0) \) as the center of a circle, and \(|AF|\) as the radius, draw a circle above the x-axis that intersects the parabola at two distinct points \( M \) and \( N \). Let \( P \) be the midpoint of \( MN \).

(1) Find the value of \( |MF| + |NF| \).

(2) Does a value of \( a \) exist such that \( |MF| \), \( |PF| \), and \( |NF| \) form an arithmetic sequence? If such a value exists, find \( a \); if it does not, provide an explanation.

Answer: 8


Let \( \mathbb{N}_{>1} \) denote the set of positive integers greater than 1. Let \( f: \mathbb{N}_{>1} \rightarrow \mathbb{N}_{>1} \) be a function such that \( f(m n) = f(m) f(n) \) for all \( m, n \in \mathbb{N}_{>1} \). If \( f(101!) = 101! \), compute the number of possible values of \( f(2020 \cdot 2021) \).

Answer: 66


On the segment \( A C \), point \( B \) is marked. Semicircles are constructed on the segments \( A B \), \( B C \), and \( A C \) as diameters in the same half-plane. A circle with center at point \( O \) is tangent to all these semicircles (see figure). Find the radius of this circle, given that \( A B = 4 \) and \( B C = 2 \). If necessary, round your answer to the nearest hundredth.

Answer: 0.86


In $\triangle ABC$, let \( D, E, F \) be the midpoints of \( BC, AC, AB \) respectively, and let \( G \) be the centroid of the triangle. For each value of \(\angle BAC\), how many non-similar triangles are there in which \( AEGF \) is a cyclic quadrilateral?

Answer: 1


Three. (25 points) Find all integers $n$ such that $2 n^{2}+1$ divides $n^{3}-5 n+5$.

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

---

Three. (25 points) Find all integers $n$ such that $2 n^{2}+1$ divides $n^{3}-5 n+5$.

Answer: -1


8,9

On a circle, a point $A$ is taken, on the diameter $B C$ - points $D$ and $E$, and on its extension beyond point $B$ - point $F$. Find $B C$, if $\angle B A D=\angle A C D, \angle B A F=\angle C A E, B D=2, B E=5$ and $B F=4$.

Answer: 11


We have selected the vertices, the centroid, and the points that trisect the sides of an equilateral triangle. How many points can we retain from among them so that no three form an equilateral triangle?

Answer: 6


7. Leshа has 10 million rubles. Into what minimum number of banks should he deposit them to receive the full amount through ACB insurance payouts in case the banks cease operations?

Answer: 8


3.4. The angle at the base of an acute isosceles triangle \(ABC (AB = BC)\) is \(\alpha\). In what ratio, counting from vertex \(A\), does the height \(BD\) divide the height \(AE\) ?

Answer: \dfrac{1}{2 \sin^2 \alpha - 1}


Answer: 1000001


Task B-2.7. Calculate the area of that part of the complex plane in which all complex numbers $z$ for which $\operatorname{Im}\left(z^{2}\right) \leq \operatorname{Re}(z) \text{ and } |z| \leq 1$ are located.

Answer: \dfrac{\pi}{2}


$ m$ and $ n$ are positive integers. In a $ 8 \times 8$ chessboard, $ (m,n)$ denotes the number of grids a Horse can jump in a chessboard ($ m$ horizontal $ n$ vertical or $ n$ horizontal $ m$ vertical ). If a $ (m,n) \textbf{Horse}$ starts from one grid, passes every grid once and only once, then we call this kind of Horse jump route a $ \textbf{H Route}$. For example, the $ (1,2) \textbf{Horse}$ has its $ \textbf{H Route}$. Find the smallest positive integer $ t$, such that from any grid of the chessboard, the $ (t,t\plus{}1) \textbf{Horse}$ does not has any $ \textbf{H Route}$.

Answer: 2


8. Let $p$ be a prime number, and $A$ be a set of positive integers, satisfying the following conditions:
(1) The set of prime factors of the elements in $A$ contains $p-1$ elements;
(2) For any non-empty subset of $A$, the product of its elements is not a $p$-th power of an integer.
Find the maximum number of elements in $A$.
(Iran provided)

Answer: (p-1)^2


10. (5 points) The New Year's party has a total of 8 programs, including 3 non-singing programs. When arranging the program list, it is stipulated that non-singing programs should not be adjacent, and the first and last programs must be singing programs. Therefore, there are $\qquad$ different ways to arrange the program list.

Answer: 2880


5. A cube with a side of 5 is made up of 125 smaller cubes with a side of 1. How many small cubes does a plane perpendicular to one of the cube's diagonals and passing through its midpoint intersect?

Answer: 55


Vasya and Petya, who are participating in a school sports-entertainment game, need to cover a distance of 3 km as quickly as possible, having only one pair of roller skates between the two of them. They start simultaneously, one running and the other running on roller skates. At any moment, the one on roller skates can leave them for the other to use and continue running without them. This exchange can be done as many times as necessary. Find the minimum time needed to cover the distance, determined by the last one to finish, if Vasya's running and skating speeds are 4 km/h and 8 km/h respectively, and Petya's are 5 km/h and 10 km/h. Assume no time is lost during the transitions.

Answer: \dfrac{1}{2}


6. Given $F_{1} 、 F_{2}$ are the foci of the ellipse $\Gamma: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>$ $b>0)$, $P$ is a point on $\Gamma$, the perimeter of $\triangle P F_{1} F_{2}$ is 6, and the minimum value of $\frac{4}{a}+\frac{1}{c}$ is 3. A line passing through the point $Q(-4,0)$ intersects $\Gamma$ at two distinct points $A 、 B$. Then the range of $|Q A||Q B|$ is

Answer: \left( \dfrac{45}{4}, 12 \right]


Pythagorean Theorem (direct and inverse). Isosceles, inscribed, and circumscribed trapezoids.

A circle with radius 6 is circumscribed around trapezoid $ABCD$. The center of this circle lies on the base $AD$, and $BC=4$. Find the area of the trapezoid.

Answer: 32\sqrt{2}


Tokarev S.i.

Among 2000 indistinguishable balls, half are aluminum with a mass of 10 g, and the rest are duralumin with a mass of 9.9 g. It is required to separate the balls into two piles such that the masses of the piles are different, but the number of balls in them is the same. What is the smallest number of weighings on a balance scale without weights that can achieve this?

#

Answer: 1


7.5. At a round table, 10 people are sitting, each of whom is either a knight, who always tells the truth, or a liar, who always lies. Each of them was given a token. Then each of them passed their token to one of their two neighbors. After that, 5 people said: "I have one token," while the other 5 said: "I have no tokens." What is the maximum number of knights that could have been among these 10 people?

Answer: 7


5. Parallelogram $A B C D$ can be divided into four equilateral triangles with a side length

(8) of $2 \mathrm{~cm}$. What is the length of the longer diagonal of the parallelogram?

Answer: 2\sqrt{7}


Three, (15 points) Given the linear function $f(x)=a x+b$
for any $x, y \in[0,1]$ it satisfies
$|f(x)+f(y)-x y| \leqslant \frac{1}{4}$.
Determine such $f(x)$.

Answer: f(x) = \dfrac{1}{2}x - \dfrac{1}{8}


Given in $\triangle A B C$, $A B=A C, \angle A=20^{\circ}$, $D$ is on $A C$ and $E$ is on $A B$. If $\angle A B D=10^{\circ}, \angle B D E$ $=20^{\circ}$, find the degree measure of $\angle A C E$.

Answer: 20


Some of $100$ towns of a kingdom are connected by roads.It is known that for each two towns $A$ and $B$ connected by a road there is a town $C$ which is not connected by a road with at least one of the towns $A$ and $B$. Determine the maximum possible number of roads in the kingdom.

Answer: 4900


Let \( a \) and \( b \) be positive numbers. Prove that the inequality \( \sqrt{a} + 1 > b \) holds if and only if for any real number \( x > 1 \), we have \( a x + \frac{x}{x-1} > b \).

Answer: \sqrt{a} + 1 > b


Example 4. In a group of 21 students, including 5 excellent students, 10 well-performing students, and 6 poorly performing students. On the upcoming exam, excellent students can only receive excellent grades. Well-performing students can receive good or excellent grades with equal probability. Poorly performing students can receive good, satisfactory, or unsatisfactory grades with equal probability. One student is randomly invited to take the exam. Find the probability that he will receive a good or excellent grade (event $A$).

Answer: \dfrac{17}{21}


69. As shown in the figure, in $\triangle A B C$, $B C>A C, \angle A=60^{\circ}, D$ and $E$ are the midpoints of $A B$ and $A C$ respectively. If $P C$ bisects $\angle A C B$ and $P D$ bisects $\angle A D E$, then $\angle D P C=$ $\qquad$ ـ.

Answer: 120^{\circ}


Answer: 4


Example 3 Let $A A_{1} B_{1} B$ be the axial section of a cylinder, $C$ be a point on the circumference of the base circle, $A A_{1}=1, A B=4, \angle B A C=$ $60^{\circ}$. Find the dihedral angle formed by the plane $A_{1} C B_{1}$ and the base $A B$ of the cylinder.

Answer: 30^\circ


5. It is known that positive numbers $x, y, z$ satisfy the system:

$$
\left\{\begin{array}{c}
x^{2}+y^{2}+x y=529 \\
x^{2}+z^{2}+\sqrt{3} x z=441 \\
z^{2}+y^{2}=144
\end{array}\right.
$$

Find the value of the expression $\sqrt{3} x y+2 y z+x z$.

Answer: 224\sqrt{5}


13.223. Point $C$ is located 12 km downstream from point $B$. A fisherman set out from point $A$, located upstream from point $B$, to point $C$. After 4 hours, he arrived at $C$, and the return trip took 6 hours. On another occasion, the fisherman used a motorboat, thereby tripling his own speed relative to the water, and reached from $A$ to $B$ in 45 minutes. It is required to determine the speed of the current, assuming it is constant.

Answer: 1


Let \( f \) be a real-valued function defined on the nonnegative real numbers. Assume that \( f(x) f(y) \leq y^{2} f\left(\frac{x}{2}\right) + x^{2} f\left(\frac{y}{2}\right) \) holds for all nonnegative \( x, y \), and that \( |f(x)| \leq 1997 \) for all \( 0 \leq x \leq 1 \). Prove that \( f(x) \leq \frac{x^{2}}{2} \) for all nonnegative real \( x \).

Answer: f(x) \leq \frac{x^2}{2}


10. In the complex plane, region $A$ consists of points $Z$ corresponding to complex numbers $z$, if the real and imaginary parts of $\frac{z}{40}$ and $\frac{40}{z}$ are both between 0 and 1, find the area of $A$.

Answer: 1200 - 200\pi


Four identical pieces, in the shape of a right triangle, are arranged in two different ways as shown in the given figures. The squares $ABCD$ and $EFGH$ have sides measuring $3 \text{ cm}$ and $9 \text{ cm}$ respectively. Determine the side length of the square $IJKL$.

Answer: 3\sqrt{5}


Problem 8. For what values of the parameter $a$ does the equation

$$
\log _{2}\left(2 x^{2}-x-2 a-4 a^{2}\right)+3 \log _{1 / 8}\left(x^{2}-a x-2 a^{2}\right)=0
$$

have two distinct roots, the sum of whose squares lies in the interval $(4 ; 8)$?

Answer: \left( \dfrac{3}{5}, 1 \right)


1. If the graph of the function $f(x)=\left(x^{2}-a x+2 a\right) \ln (x+1)$ passes through four quadrants, then the range of the real number $a$ is $\qquad$ .

Answer: \left( -\dfrac{1}{3}, 0 \right)


A-6. Take any four points on a sphere to form a tetrahedron. What is the probability that the center of the sphere is inside the tetrahedron? (It can be assumed that each point is independently chosen and is uniformly distributed on the sphere)

Answer: \dfrac{1}{8}


Three. (50 points) Let $M=\{1,2, \cdots, 65\}, A \subseteq M$ be a subset. If $|A|=33$, and there exist $x, y \in A, x<y$, $x \mid y$, then $A$ is called a "good set". Find the largest $a \in M$ such that any 33-element subset containing $a$ is a good set.

Answer: 21


7. (10 points) Five contestants $A, B, C, D, E$ participate in the "The Voice" competition, and the five of them stand in a row for a group appearance. They each have a contestant number on their chest, and the sum of the five numbers is 35. It is known that the sum of the numbers of the contestants standing to the right of $E$ is 13; the sum of the numbers of the contestants standing to the right of $D$ is 31; the sum of the numbers of the contestants standing to the right of $A$ is 21; the sum of the numbers of the contestants standing to the right of $C$ is 7. What is the sum of the numbers of the contestants standing at the far left and the far right? $\qquad$

Answer: 11


2. Let the incircle $\odot O$ of $\triangle A B C$ touch $B C$ at point $D$, and draw the diameter $D E$ through $D$. Connect $A E$ and extend it to intersect $B C$ at point $F$. If $B F+C D=1998$, then $B F+2 C D=$

Answer: 2997


2. In $\triangle A B C$, it is known that the three angles $A, B, C$ form an arithmetic sequence. Suppose the sides opposite to these angles are $a, b, c$, respectively, and $c-a$ equals the altitude $h$ from vertex $A$ to side $AC$, then $\sin \frac{C-A}{2}=$ $\qquad$.

Answer: \dfrac{1}{2}


11. (20 points) For any 2016 complex numbers $z_{1}, z_{2}$, $\cdots, z_{2016}$, we have
$$
\sum_{k=1}^{2016}\left|z_{k}\right|^{2} \geqslant \lambda \min _{1 \leqslant k \leqslant 2016}\left\{\left|z_{k+1}-z_{k}\right|^{2}\right\},
$$

where, $z_{2017}=z_{1}$. Find the maximum value of $\lambda$.

Answer: 504


Problem 8'2. Let $M$ be the midpoint of the side $B C$ of $\triangle A B C$ and $\angle C A B=45^{\circ} ; \angle A B C=30^{\circ}$.

a) Find $\angle A M C$.

b) Prove that $A M=\frac{A B \cdot B C}{2 A C}$.

Chavdar Lozanov

Answer: 45^\circ


15. (15 points) 100 people participate in a quick calculation test, with a total of 10 questions. The number of people who answered each question correctly is shown in the table below:
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline Question Number & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline \begin{tabular}{l}
Correct \\
Answers
\end{tabular} & 93 & 90 & 86 & 91 & 80 & 83 & 72 & 75 & 78 & 59 \\
\hline
\end{tabular}

Rule: Answering 6 or more questions correctly is considered passing. Based on the table above, calculate the minimum number of people who passed.

Answer: 62


5. A point $P$ in space is at distances of $2$ and $3$ from the vertices $A$ and $B$ of a regular tetrahedron $ABCD$. When the edge length of the tetrahedron changes, the maximum distance from point $P$ to the line $CD$ is $\qquad$

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


Find all real numbers $c$ such that there exists a function $f: \mathbb{R}_{ \ge 0} \rightarrow \mathbb{R}$ which satisfies the following.

For all nonnegative reals $x, y$, $f(x+y^2) \ge cf(x)+y$.

Here $\mathbb{R}_{\ge 0}$ is the set of all nonnegative reals.

Answer: (-\infty, 1)


4 Place $r$ chess pieces on an $m \times n (m>1, n>1)$ chessboard, with at most one piece per square. If the $r$ pieces have the following property $p$: each row and each column contains at least one piece. However, removing any one of the pieces will cause them to no longer have the above property $p$. Find the maximum value of $r$, denoted as $r(m, n)$.

Answer: m + n - 2


In triangle $A B C \angle C A B=75^{\circ}, \angle A B C=45^{\circ}$. On side $C A$ a point $K$ is taken, and on side $C B$ a point $M$, $C K: A K=3: 1$.

Find $K M: A B$, if this ratio is less than $3 / 4$, and line $M K$ cuts off from triangle $A B C$ a triangle similar to it.

Answer: \dfrac{3(\sqrt{3} - 1)}{4}


3. Let $F_{1}$ and $F_{2}$ be the two foci of the hyperbola $x^{2}-y^{2}=4$, and $P$ be any point on the hyperbola. A perpendicular line is drawn from $F_{1}$ to the angle bisector of $\angle F_{1} P F_{2}$, with the foot of the perpendicular being $M$. Then the equation of the locus of point $M$ is $\qquad$

Answer: x^2 + y^2 = 4


6. The first circle with center at point $O$ is inscribed in triangle $A B C$. Points $A$ and $B$ lie on the second circle with the same center $O$. Line $A C$ intersects the second circle at point $D$ $(D \neq A)$, and line $B C$ intersects the second circle at point $E(B \neq E)$. It is known that angle $A B C$ is equal to angle $C A E$. Find the cosine of angle $B A C$.

Answer: \dfrac{1 + \sqrt{5}}{4}


# Task 3.

Maximum 15 points

In an equilateral triangle with area $S_{1}$, a circle is inscribed, and in this circle, an equilateral triangle with area $S_{2}$ is inscribed. In the resulting new triangle, another circle is inscribed, and in this circle, another equilateral triangle with area $S_{3}$ is inscribed. The procedure is repeated $n$ times. The area of the triangle $S_{n}$ turned out to be 1. Find the function $S_{1}(n)$.

Answer: 4^{n-1}


24. The width of a rectangle is called the length of its shortest side *). In how many different ways can a rectangle of width 3 be cut out from a square sheet of paper consisting of 100 cells? (Cuts must be made only along the cell boundaries.)

Answer: 512


Given a triangle \(ABC\) with pairwise distinct sides. On its sides, externally constructed equilateral triangles \(ABC_1\), \(BCA_1\), and \(CAB_1\). Prove that the triangle \(A_1B_1C_1\) cannot be equilateral.

Answer: \text{The triangle } A_1B_1C_1 \text{ cannot be equilateral.}


The entries of a $7 \times 5$ table are filled with numbers such that in each $2 \times 3$ rectangle (either vertical or horizontal), the sum of the numbers is $0$. For $100$ dollars, Peter may choose any single entry and learn the number in it. What is the least amount of dollars he should spend to determine the total sum of numbers in the table for sure?

Answer: 0


The following diagram shows an equilateral triangle and two squares that share common edges. The area of each square is $75$. Find the distance from point $A$ to point $B$.

```latex
\begin{asy}
size(175);
defaultpen(linewidth(0.8));
pair A=(-3,0), B=(3,0), C=rotate(60,A)*B, D=rotate(270,B)*C, E=rotate(90,C)*B, F=rotate(270,C)*A, G=rotate(90,A)*C;
draw(A--G--F--C--A--B--C--E--D--B);
label("$A$", F, N);
label("$B$", E, N);
\end{asy}
```

Answer: 15


Let $ABCD$ be a convex quadrilateral with $AB = AD$, $m\angle A = 40^{\circ}$, $m\angle C = 130^{\circ}$, and $m\angle ADC - m\angle ABC = 20^{\circ}$. Find the measure of the non-reflex angle $\angle CDB$ in degrees.

Answer: 35


Let $\triangle ABC$ have $AB = 14$, $BC = 30$, $AC = 40$ and $\triangle AB'C'$ with $AB' = 7\sqrt{6}$, $B'C' = 15\sqrt{6}$, $AC' = 20\sqrt{6}$ such that $\angle BAB' = \frac{5\pi}{12}$. The lines $BB'$ and $CC'$ intersect at point $D$. Let $O$ be the circumcenter of $\triangle BCD$, and let $O'$ be the circumcenter of $\triangle B'C'D$. Then the length of segment $OO'$ can be expressed as $\frac{a+b \sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.

Answer: 55


A $10$-digit number is called interesting if its digits are distinct and the number is divisible by $11111$. Determine the total number of interesting numbers.

Answer: 3456


We have four registers, $R_1, R_2, R_3, R_4$, such that $R_i$ initially contains the number $i$ for $1 \leq i \leq 4$. We are allowed two operations:

1. Simultaneously swap the contents of $R_1$ and $R_3$ as well as $R_2$ and $R_4$.

2. Simultaneously transfer the contents of $R_2$ to $R_3$, the contents of $R_3$ to $R_4$, and the contents of $R_4$ to $R_2$. For example, if we do this once, then $(R_1, R_2, R_3, R_4) = (1, 4, 2, 3)$.

Using these two operations as many times as desired and in whatever order, what is the total number of possible outcomes?

Answer: 12


Esmeralda has created a special knight to play on quadrilateral boards that are identical to chessboards. If a knight is in a square, it can move to another square by moving 1 square in one direction and 3 squares in a perpendicular direction (which is a diagonal of a $2 \times 4$ rectangle instead of $2 \times 3$ like in chess). In this movement, it doesn't land on the squares between the beginning square and the final square it lands on.

A trip of the length $n$ of the knight is a sequence of $n$ squares $C_1, C_2, \ldots, C_n$ which are all distinct such that the knight starts at the $C_1$ square and for each $i$ from $1$ to $n-1$, it can use the movement described before to go from the $C_i$ square to the $C_{i+1}$.

Determine the greatest $N \in \mathbb{N}$ such that there exists a path of the knight with length $N$ on a $5 \times 5$ board.

Answer: 12


For an olympiad geometry problem, Tina wants to draw an acute triangle whose angles each measure a multiple of $10^\circ$. She doesn't want her triangle to have any special properties, so none of the angles can measure $30^\circ$ or $60^\circ$, and the triangle should definitely not be isosceles.

How many different triangles can Tina draw? (Similar triangles are considered the same.)

Answer: 0


Each triangle is a 30-60-90 triangle, and the hypotenuse of one triangle is the longer leg of an adjacent triangle. The hypotenuse of the largest triangle is 8 centimeters. What is the number of centimeters in the length of the longer leg of the smallest triangle? Express your answer as a common fraction.

[asy] pair O; for(int i = 0; i < 5; ++i){
draw(O--((2/sqrt(3))^i)*dir(30*i));
}
for(int g = 0; g < 4; ++g){
draw( ((2/sqrt(3))^g)*dir(30*g)-- ((2/sqrt(3))^(g+1))*dir(30*g+30));
}
label("8 cm", O--(16/9)*dir(120), W);
label("$30^{\circ}$",.4*dir(0),dir(90));
label("$30^{\circ}$",.4*dir(25),dir(115));
label("$30^{\circ}$",.4*dir(50),dir(140));
label("$30^{\circ}$",.4*dir(85),dir(175));
real t = (2/(sqrt(3)));
draw(rightanglemark((1,.1),(1,0),(.9,0),s=3));
draw(rightanglemark(rotate(30)*(0,t**4),rotate(0)*(0,t**3),O,s=3));
draw(rightanglemark(rotate(0)*(0,t**3),rotate(-30)*(0,t**2),O,s=3));
draw(rightanglemark(rotate(-30)*(0,t**2),rotate(-60)*(0,t**1),O,s=3));
[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 11


Ryan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, then turns 3 random lamps on. What is the probability that the leftmost lamp on the shelf is red, and the leftmost lamp which is turned on is also red?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 27


Each of five, standard, six-sided dice is rolled once. What is the probability that there is at least one pair but not a three-of-a-kind (that is, there are two dice showing the same value, but no three dice show the same value)?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 61


Find the volume of the region given by the inequality
\[|x + y + z| + |x + y - z| + |x - y + z| + |-x + y + z| \le 4.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 23


For $1 \leq i \leq 215$ let $a_i = \dfrac{1}{2^{i}}$ and $a_{216} = \dfrac{1}{2^{215}}$. Let $x_1, x_2, \dots, x_{216}$ be positive real numbers such that $\sum_{i=1}^{216} x_i=1$ and
\[\sum_{1 \leq i < j \leq 216} x_ix_j = \dfrac{107}{215} + \sum_{i=1}^{216} \dfrac{a_i x_i^{2}}{2(1-a_i)}.\]Find the maximum possible value of $x_2.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 863


There are $5$ yellow pegs, $4$ red pegs, $3$ green pegs, $2$ blue pegs, and $1$ orange peg to be placed on a triangular peg board. Find the number of ways the pegs can be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color.

Answer: 1


Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is \(m\) times the area of the square. The original answer is in \(\frac{k}{nm}\) format, where \(m\) is given in the problem. Please find the value of \(k+n\).

Answer: 5


Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is \(m\) times the area of the square. The original answer is in the form \(\frac{k}{nm}\). Please find the value of \(k + n\), where \(k\) and \(n\) are integers.

Answer: 5


A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white? Express your answer in the form \(\frac{k}{m}\) and find the value of \(k + m\).

Answer: 8


46. In a convex quadrilateral $A B C D$, point $M$ is the midpoint of side $A D$, $C M \| A B$, $A D=B D$ and $3 \angle B A C=\angle A C D$. Find the angle $A C B$.

(S. Berlov)

Answer: 90^\circ


Elsa uses ice blocks to make ice sculptures. One ice block can make a small sculpture, and three ice blocks can make a large sculpture. The leftover fragments from making 2 small sculptures or 1 large sculpture can perfectly form 1 new ice block. Given 30 ice blocks, and ensuring that the number of small sculptures made is greater than the number of large sculptures made, what is the maximum number of large sculptures that can be made?

Answer: 11


11. As shown in the figure, in the Cartesian coordinate system $x O y$, $F$ is a moving point on the positive half-axis of the $x$-axis. A parabola $C$ is constructed with $F$ as the focus and $O$ as the vertex. Let $P$ be a point on $C$ in the first quadrant, and $Q$ be a point on the negative half-axis of the $x$-axis such that $P Q$ is a tangent to $C$, and $P Q=2$. Circles $C_{1}$ and $C_{2}$ are both tangent to the line $O P$ at point $P$, and both are tangent to the $x$-axis. Find the coordinates of point $F$ such that the sum of the areas of circles $C_{1}$ and $C_{2}$ is minimized.

Answer: \left( \sqrt{\dfrac{3 + \sqrt{3}}{6}}, 0 \right)


7. In a lottery with 100000000 tickets, each ticket number consists of eight digits. A ticket number is called "lucky" if and only if the sum of its first four digits equals the sum of its last four digits. Then, the sum of all lucky ticket numbers, when divided by 101, leaves a remainder of

Answer: 0


3. The set of positive integers coincides with the union of the sets

$$
\{f(1), f(2), \ldots, f(n), \ldots\} \quad \text { and }\{g(1), g(2), \ldots, g(n), \ldots\}
$$

which are disjoint, where

$$
\begin{gathered}
f(1)<f(2)<\cdots<f(n)<\ldots \\
g(1)<g(2)<\cdots<g(n)<\ldots
\end{gathered}
$$

and $g(n)=f(f(n))+1$ for all $n \geqq 1$.

Determine the value of $f(240)$! (United Kingdom, 8 points)

Answer: 388


Given a finite set $S \subset \mathbb{R}^3$, define $f(S)$ to be the mininum integer $k$ such that there exist $k$ planes that divide $\mathbb{R}^3$ into a set of regions, where no region contains more than one point in $S$. Suppose that
\[M(n) = \max\{f(S) : |S| = n\} \text{ and } m(n) = \min\{f(S) : |S| = n\}.\]
Evaluate $M(200) \cdot m(200)$.

Answer: 2189