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

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

Answer: white


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

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

Answer: milk


10.225. Two circles of different radii touch each other externally. Find the angle determined by the chords connecting the point of contact of the circles with the points of contact of their common external tangent.

Answer: 90^\circ


23. A person is climbing a staircase with a total of $n$ steps. If it is stipulated that he can only climb 1 step or 3 steps at a time, how many different ways are there to climb the stairs?

Answer: f(n)


5. Determine the smallest possible value of the expression $4 x^{2}+4 x y+4 y^{2}+12 x+8$. For which $x$ and $y$ will this expression have the smallest value?

Each task is scored out of 10 points.

The use of a pocket calculator or any reference materials is not allowed.

Answer: -4


8,9

In a triangular pyramid $P A B C$, the lateral edge $P B$ is perpendicular to the plane of the base $A B C$, $P B=6, A B=B C$ $=\sqrt{15}, A C=2 \sqrt{3}$. A sphere with center $O$ lying on the face $A B P$ touches the planes of the other faces of the pyramid. Find the distance from the center $O$ of the sphere to the edge $A C$.

Answer: \dfrac{8(6 - \sqrt{15})}{7}


Plane vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ satisfy: $|\vec{a}|=1, \vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{c}=1$, and $|\vec{a}-\vec{b}+\vec{c}| \leq 2 \sqrt{2}$. Then the range of values for $\vec{a} \cdot \vec{c}$ is __.

Answer: [-2\sqrt{2}, 2]


## Task 2.

A frog is located at the origin $O$ of a number line and needs to perform 2022 jumps, each of a different length from $1, 2, \ldots, 2022$. The jumps must be performed in such an order that the following rules are respected:

- if the frog is currently at point $O$ or to the left of it, it must jump to the right,
- if the frog is to the right of point $O$, it must jump to the left.

Determine the largest natural number $k$ for which the frog can jump in such a way that it never lands on any of the numbers $1, 2, \ldots, k$.

Answer: 1010


1. [5] 2021 people are sitting around a circular table. In one move, you may swap the positions of two people sitting next to each other. Determine the minimum number of moves necessary to make each person end up 1000 positions to the left of their original position.

Answer: 1021000


10. The radii of the two circles in the figure are 16 and 12, and the distance between the two centers is 24. A line passing through one of the intersection points of the two circles intercepts equal chords $Q P$ and $Q R$ on the two circles, then $P Q^{2}=$ $\qquad$ .

Answer: 520


Given \(n, m > 3\), where \(n\) trays hold \(m\) coins between them, a move consists of taking one coin from each of two trays and placing them on a third tray. Can we always transfer all the coins to a single tray by a sequence of moves?

Answer: \text{Yes}


9.3. Natural numbers $a, x$ and $y$, greater than 100, are such that $y^{2}-1=$ $=a^{2}\left(x^{2}-1\right)$. What is the smallest value that the fraction $a / x$ can take?

Answer: 2


6. Given positive numbers $a, b, c$ satisfying $a+b+c=13$. Then the minimum value of $\frac{a^{2}+b^{3}+c^{4}+2019}{10 b+123 c+26}$ is $\qquad$ .

Answer: 4


II. (50 points)
$n$ is a positive integer, let the integer closest to $\sqrt{n}$ be $a_{n}$, let $b_{n}=n+a_{n}$, remove $b_{n}$ from the set of all positive integers, and arrange the remaining positive integers in ascending order to form the sequence $\left\{c_{n}\right\}$, try to express $c_{n}$ in terms of $n$.

Answer: n^2


2. The function $f: \mathbf{R} \rightarrow \mathbf{R}$, satisfies for any real number $x$,
$$
f(1-f(x))=x \text {. }
$$
(1) Please provide a function $f$ that meets the requirement;
(2) Determine $S_{f}=\sum_{k=-2017}^{2018} f(k)$ for all possible values.

Answer: 2018


Three, (50 points) Let $A=\{1,2, \cdots, 30\}$. Find the smallest positive integer $n$, such that for any 11 subsets of $A$, if the union of any 5 of them has at least $n$ elements, then there must exist 3 of these 11 subsets whose intersection is non-empty.

Answer: 22


Given a positive number \( c \), 99 points are marked in space such that for each marked point, the distances to the two nearest marked points differ by at least \( c \) times. What is the greatest possible value of \( c \)?

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


12. (1) Let the hyperbola $C: (y+a)^{2}-(x-a)^{2}=2a$, with its asymptotes passing through the point $(3,1)$. Find the equations of the asymptotes of $C$.
(2) A certain hyperbola has the lines $y= \pm(x+1)$ as its asymptotes, and the focal distance is 4. Find its equation.

Answer: \frac{(x + 1)^2}{2} - \frac{y^2}{2} = 1


12.441 The angle between the plane of square $A B C D(A B \| C D)$ and a certain plane $P$ is $\alpha$, and the angle between side $A B$ and the same plane is $\beta$. Find the angle between side $A D$ and plane $P$.

Answer: \arcsin \sqrt{\sin^2 \alpha - \sin^2 \beta}


Example 6 Given the complex numbers $z_{1}=\frac{\cos x^{2}}{\sin y^{2}}+\mathrm{i} \cdot \frac{\cos y^{2}}{\sin x^{2}}, z_{2}=x+y i(x, y \in \mathbf{R})$. If when $x, y$ take all real numbers in the interval $\left[-\sqrt{\frac{\pi}{2}}, \sqrt{\frac{\pi}{2}}\right]$, $\left|z_{1}\right|$ is always equal to $\sqrt{2}$. Try to find the maximum and minimum values of $\left|z_{1}-z_{2}\right|$.

Answer: \sqrt{2} + \sqrt{\dfrac{\pi}{2}}


For every $n$ in the set $\mathrm{N} = \{1,2,\dots \}$ of positive integers, let $r_n$ be the minimum value of $|c-d\sqrt{3}|$ for all nonnegative integers $c$ and $d$ with $c+d=n$. Find, with proof, the smallest positive real number $g$ with $r_n \leq g$ for all $n \in \mathbb{N}$.

Answer: \dfrac{1 + \sqrt{3}}{2}


Let $n$ be a positive integer. Find the number of all polynomials $P$ with coefficients from the set $\{0,1,2,3\}$ and for which $P(2)=n$.

Answer: \left\lfloor \dfrac{n}{2} \right\rfloor + 1


In a convex quadrilateral \(ABCD\), the extensions of sides \(BA\) and \(CD\) intersect at point \(M\), and \(AD \perp BC\). A line through \(M\) intersects the lines containing the other pair of opposite sides at points \(H\) and \(L\), and intersects the line containing the diagonals at points \(H'\) and \(L'\). Prove that \(\frac{1}{MH} + \frac{1}{ML} = \frac{1}{MH'} + \frac{1}{ML'}\).

Answer: \frac{1}{MH} + \frac{1}{ML} = \frac{1}{MH'} + \frac{1}{ML'}


8. In $\triangle A B C$, the medians $A E, B F$, and $C D$ intersect at $M$. Given that $E, C, F$, and $M$ are concyclic, and $C D=n$. Find the length of segment $A B$.
(18th All-Russian Competition Problem)

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


1. Fold the triangular paper piece $A B C$ along a line parallel to $B C$, the folded part intersects $B C$ at points $F, G$. Given $B C=9 \mathrm{~cm}, F G=3 \mathrm{~cm}$, then the length of the crease $D E$ is $\qquad$ $\mathrm{cm}$.

Answer: 6


4.21. The side, bisector, and height of a triangle, all emanating from the same vertex, are equal to 5, 5, and \(2 \sqrt{6}\) cm, respectively. Find the other two sides of the triangle.

Answer: \dfrac{125}{21}


Mom prepared orange juice for Jirka's birthday party by mixing 1 liter of $100\%$ juice with $\frac{2}{3}$ liter of $30\%$ juice. Jirka poured himself a glass and tasted it. Since he prefers a weaker concentration, he topped up the prepared juice to its original amount. The resulting juice had a concentration of $61.2\%$, which suited him. How much juice did Jirka pour into his glass?

(M. Raabová)

Answer: \dfrac{1}{4}


Let $S$ be a subset of $\{0,1,2,\dots ,9\}$. Suppose there is a positive integer $N$ such that for any integer $n>N$, one can find positive integers $a,b$ so that $n=a+b$ and all the digits in the decimal representations of $a,b$ (expressed without leading zeros) are in $S$. Find the smallest possible value of $|S|$.

[i]Proposed by Sutanay Bhattacharya[/i]

[hide=Original Wording]
As pointed out by Wizard_32, the original wording is:

Let $X=\{0,1,2,\dots,9\}.$ Let $S \subset X$ be such that any positive integer $n$ can be written as $p+q$ where the non-negative integers $p, q$ have all their digits in $S.$ Find the smallest possible number of elements in $S.$
[/hide]

Answer: 5


In a line, there are 2020 white stones and one black stone. A possible operation is as follows: choose a black stone that is not at the edge and change the color of its neighboring stones. Find all possible initial positions for the black stone such that it is possible to color all the stones black with a finite number of operations.

Answer: 1011


3. Let $f(x)$ be an increasing continuous function defined on the interval $[0 ; 3]$, and $g(x)$ be its inverse function, such that $g(x) > f(x)$ for all positive $x$ for which both functions are defined. Additionally, $f(0)=0, f(3)=2$.

The area under the graph of $f(x)$ on the interval $[0 ; 3]$ is 2. Find the area of the figure bounded by the graphs of $f(x)$ and $g(x)$, as well as the segment connecting the points $(3 ; 2)$ and $(2 ; 3)$.

Answer: \dfrac{9}{2}


There are 23 students in a class. During the year, each student celebrated their birthday once, attended by some (at least one, but not all) of their classmates. Could it be possible that every pair of students in this class met the same number of times at such celebrations? (It is assumed that any two guests met at each celebration, and the birthday student met with all guests.)

(I. Bogdanov)

Answer: Yes


In the following figure, \( AD = AB \), \(\angle DAB = \angle DCB = \angle AEC = 90^{\circ} \) and \( AE = 5 \). Find the area of the quadrangle \( ABCD \).

Answer: 25


From the six digits \( 0, 1, 2, 3, 4, 5 \), select 2 odd numbers and 2 even numbers to form a 4-digit number with no repeating digits. What is the probability that the resulting 4-digit number is an even number?

Answer: \frac{8}{15}


We have 64 small wooden cubes, each with an edge of $2 \mathrm{~cm}$, that are covered with paper. Then, these small cubes are used to construct a large cube. How many $\mathrm{cm}^{2}$ of paper would we save if we first assembled the large cube and then covered it with paper, applying only one layer of paper?

Answer: 1152 \text{ cm}^2


ABC is a triangle. The points P and Q trisect BC, so BP = PQ = QC. A line parallel to AC meets the lines AB, AP, AQ at X, Y, and Z respectively. Show that YZ = 3 XY.

Answer: YZ = 3XY


There are three cars, $A$, $B$, and $C$, starting from the same location and traveling along the same road. They each catch up to a cyclist, Xiao Long, who is traveling in the same direction, at 5 minutes, 10 minutes, and 12 minutes respectively. Given that the speeds of $A$ and $B$ are 24 km/h and 20 km/h respectively, find the speed of car $C$, in km/h.

Answer: \frac{58}{3}


Let $n$ be an integer such that $1 \leq n \leq 30$. Ariane and Bérénice are playing the following game. First, Ariane chooses $n$ distinct numbers from the integers $1, 2, \ldots, 30$. Then, based on Ariane's choices, Bérénice chooses an integer $d \geq 2$. Bérénice wins if $d$ divides at least two of the numbers chosen by Ariane. Otherwise, Ariane wins.

For which values of $n$ can Ariane ensure she will win?

Answer: 1 \leq n \leq 11


On a table, there are three cones standing on their bases, touching each other. The radii of their bases are 10, 15, and 15. A truncated cone is placed on the table with its smaller base down, which has a common slant height with each of the other cones. Find the area of the smaller base of the truncated cone.

Answer: 4\pi


Given \( 0 < a < b \), lines \( l \) and \( m \) pass through two fixed points \( A(a, 0) \) and \( B(b, 0) \), respectively, intersecting the parabola \( y^2 = x \) at four distinct points. When these four points lie on a common circle, determine the locus of the intersection point \( P \) of lines \( l \) and \( m \).

Answer: x = \frac{a + b}{2}


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`
 - They all have a unique favorite flower: `carnations`, `lilies`, `daffodils`
 - People have unique hair colors: `black`, `brown`, `blonde`
 - People have unique favorite sports: `soccer`, `basketball`, `tennis`
 - Each person lives in a unique style of house: `colonial`, `ranch`, `victorian`
 - Each person has a unique type of pet: `fish`, `dog`, `cat`

## Clues:
1. The person who has a cat is the person who loves soccer.
2. The person who has blonde hair is in the second house.
3. The person who loves a bouquet of daffodils is the person who has blonde hair.
4. Peter is the person who loves basketball.
5. Arnold is directly left of the person in a ranch-style home.
6. The person who owns a dog is the person who loves basketball.
7. The person who loves a carnations arrangement is directly left of the person who has blonde hair.
8. The person who loves soccer is in the third house.
9. Arnold is somewhere to the left of the person who has black hair.
10. The person living in a colonial-style house is in the third house.
 What is the value of attribute HouseStyle for the person whose attribute Pet is fish? Please reason step by step, and put your final answer within \boxed{}

Answer: victorian


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`, `Eric`, `Bob`, `Alice`, `Arnold`
 - Each person prefers a unique type of vacation: `cruise`, `mountain`, `city`, `camping`, `cultural`, `beach`
 - Each person has a unique hobby: `knitting`, `photography`, `cooking`, `painting`, `gardening`, `woodworking`
 - Everyone has something unique for lunch: `stew`, `pizza`, `spaghetti`, `grilled cheese`, `stir fry`, `soup`
 - People own unique car models: `chevrolet silverado`, `honda civic`, `toyota camry`, `ford f150`, `tesla model 3`, `bmw 3 series`
 - They all have a unique favorite flower: `tulips`, `daffodils`, `carnations`, `lilies`, `iris`, `roses`

## Clues:
1. Arnold is the person who enjoys camping trips.
2. The photography enthusiast is the person who loves the soup.
3. Carol is the person who loves eating grilled cheese.
4. The person who owns a Honda Civic is the person who loves a carnations arrangement.
5. The person who loves beach vacations is not in the first house.
6. The person who enjoys gardening is the person who owns a Toyota Camry.
7. There is one house between the person who loves the boquet of iris and the person who loves the vase of tulips.
8. The person who prefers city breaks is the person who loves the stew.
9. The person who loves a bouquet of daffodils is Carol.
10. The person who prefers city breaks is Eric.
11. Carol is the person who loves cooking.
12. Eric is directly left of the photography enthusiast.
13. The person who loves the rose bouquet is Bob.
14. There is one house between Arnold and the person who likes going on cruises.
15. The person who owns a Tesla Model 3 and the person who enjoys mountain retreats are next to each other.
16. The person who loves the rose bouquet is the person who owns a Toyota Camry.
17. The person who owns a BMW 3 Series is somewhere to the right of the person who loves cooking.
18. The woodworking hobbyist is not in the third house.
19. The person who owns a Ford F-150 is the person who enjoys mountain retreats.
20. The person who is a pizza lover is not in the second house.
21. The person who loves the vase of tulips is not in the fourth house.
22. The person who loves the spaghetti eater is somewhere to the left of the person who enjoys knitting.
23. Peter is somewhere to the right of the photography enthusiast.
24. The person who owns a Honda Civic is the person who is a pizza lover.
25. The person who loves cooking is in the fifth house.
26. The person who likes going on cruises is the person who enjoys knitting.
 What is the value of attribute House for the person whose attribute Vacation is cultural? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Carol`, `Eric`, `Alice`, `Bob`, `Arnold`
 - People use unique phone models: `huawei p50`, `google pixel 6`, `xiaomi mi 11`, `iphone 13`, `samsung galaxy s21`, `oneplus 9`
 - Everyone has a unique favorite cigar: `dunhill`, `pall mall`, `blends`, `blue master`, `prince`, `yellow monster`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `roses`, `tulips`, `lilies`, `iris`
 - Each person has a favorite color: `yellow`, `red`, `green`, `blue`, `white`, `purple`
 - People have unique favorite sports: `soccer`, `tennis`, `basketball`, `volleyball`, `swimming`, `baseball`

## Clues:
1. The person who uses a OnePlus 9 is in the second house.
2. The person who uses a Xiaomi Mi 11 is somewhere to the left of the person who uses a Huawei P50.
3. Carol is the person who loves a carnations arrangement.
4. The person who loves purple is directly left of the person partial to Pall Mall.
5. The person whose favorite color is green is the person who smokes Blue Master.
6. The person who loves yellow and the person who loves blue are next to each other.
7. Eric is somewhere to the right of the person who uses a Samsung Galaxy S21.
8. There are two houses between Carol and the person who loves a bouquet of daffodils.
9. The Prince smoker is the person who loves basketball.
10. The Dunhill smoker is the person who loves volleyball.
11. The person who loves swimming is the person who uses a Google Pixel 6.
12. The person who uses a Huawei P50 is directly left of the person who loves white.
13. The person who uses a OnePlus 9 and the person who loves the rose bouquet are next to each other.
14. The person who loves the boquet of iris is somewhere to the left of Eric.
15. The Dunhill smoker is Peter.
16. The person who loves blue is Peter.
17. The person who loves the vase of tulips is Bob.
18. Alice is in the first house.
19. The person who loves baseball is directly left of the person who smokes Blue Master.
20. The person who uses a Google Pixel 6 is somewhere to the right of the person who smokes many unique blends.
21. The person who loves soccer is Carol.
22. The person who loves a carnations arrangement is directly left of the person who smokes many unique blends.
23. Eric is the person who smokes many unique blends.
24. The person who loves volleyball is the person who uses an iPhone 13.
 What is the value of attribute PhoneModel for the person whose attribute Color is yellow? Please reason step by step, and put your final answer within \boxed{}

Answer: xiaomi mi 11


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`
 - People use unique phone models: `iphone 13`, `samsung galaxy s21`, `google pixel 6`, `oneplus 9`, `huawei p50`
 - Each person has a unique hobby: `knitting`, `photography`, `painting`, `cooking`, `gardening`
 - Everyone has a unique favorite cigar: `pall mall`, `blends`, `blue master`, `dunhill`, `prince`
 - People have unique favorite sports: `swimming`, `baseball`, `tennis`, `basketball`, `soccer`
 - Everyone has a favorite smoothie: `cherry`, `desert`, `lime`, `watermelon`, `dragonfruit`

## Clues:
1. Peter is in the first house.
2. The person who loves soccer is the person who paints as a hobby.
3. The person partial to Pall Mall is the person who enjoys knitting.
4. The person who uses a Google Pixel 6 is not in the fifth house.
5. The person who enjoys knitting is somewhere to the left of Alice.
6. Peter is directly left of the person who smokes Blue Master.
7. The Prince smoker is the person who uses a OnePlus 9.
8. The person who loves basketball is directly left of the photography enthusiast.
9. The Dunhill smoker is somewhere to the left of the photography enthusiast.
10. There is one house between the person who drinks Lime smoothies and the person who loves basketball.
11. The person who enjoys gardening is somewhere to the right of the person who enjoys knitting.
12. The Dragonfruit smoothie lover is in the third house.
13. The person who loves baseball is the person partial to Pall Mall.
14. Eric is the person who loves tennis.
15. The person who uses a OnePlus 9 is Peter.
16. The Dunhill smoker is the person who uses a Huawei P50.
17. The Desert smoothie lover is the person who loves soccer.
18. The person who uses a Samsung Galaxy S21 is directly left of Arnold.
19. The person who loves tennis is somewhere to the right of the Watermelon smoothie lover.
 What is the value of attribute FavoriteSport for the person whose attribute Smoothie is desert? Please reason step by step, and put your final answer within \boxed{}

Answer: soccer


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`, `Alice`
 - Each person has a unique hobby: `photography`, `painting`, `gardening`, `cooking`
 - Each person prefers a unique type of vacation: `cruise`, `beach`, `mountain`, `city`
 - People own unique car models: `tesla model 3`, `honda civic`, `ford f150`, `toyota camry`

## Clues:
1. The person who loves cooking is the person who owns a Ford F-150.
2. The person who enjoys mountain retreats is the person who owns a Ford F-150.
3. Arnold is the person who prefers city breaks.
4. The person who enjoys mountain retreats is in the first house.
5. The photography enthusiast is the person who loves beach vacations.
6. The person who owns a Toyota Camry is Peter.
7. The person who enjoys gardening is somewhere to the right of the person who owns a Honda Civic.
8. Peter is the person who loves beach vacations.
9. Alice is in the first house.
10. There is one house between the person who owns a Honda Civic and Eric.
 What is the value of attribute Vacation for the person whose attribute Hobby is cooking? Please reason step by step, and put your final answer within \boxed{}

Answer: mountain


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`
 - People use unique phone models: `samsung galaxy s21`, `iphone 13`, `google pixel 6`
 - Each person has an occupation: `doctor`, `teacher`, `engineer`
 - Each person has a unique hobby: `photography`, `cooking`, `gardening`
 - The people keep unique animals: `bird`, `horse`, `cat`

## Clues:
1. The person who uses a Samsung Galaxy S21 is directly left of the person who is a teacher.
2. The person who uses a Samsung Galaxy S21 and Arnold are next to each other.
3. The person who is an engineer is the person who keeps horses.
4. The person who uses an iPhone 13 is the cat lover.
5. The bird keeper is the person who uses a Samsung Galaxy S21.
6. The person who loves cooking is not in the third house.
7. The person who enjoys gardening is directly left of the person who is a teacher.
8. There is one house between Eric and the person who uses an iPhone 13.
 What is the value of attribute Hobby for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: photography


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

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

Answer: pizza


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`, `Peter`, `Bob`, `Arnold`
 - Each person has a favorite color: `red`, `white`, `green`, `blue`, `yellow`
 - Everyone has a favorite smoothie: `watermelon`, `dragonfruit`, `cherry`, `desert`, `lime`
 - Each person has a unique birthday month: `feb`, `sept`, `april`, `mar`, `jan`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`, `fantasy`, `biography`
 - People have unique heights: `very short`, `very tall`, `tall`, `short`, `average`

## Clues:
1. The person who drinks Lime smoothies is in the second house.
2. The person who loves fantasy books is directly left of the Dragonfruit smoothie lover.
3. The person whose birthday is in September is not in the third house.
4. There is one house between the person whose birthday is in March and Peter.
5. The person whose birthday is in February is the Desert smoothie lover.
6. The person who is tall is not in the fifth house.
7. The person whose birthday is in January is somewhere to the left of the person whose birthday is in February.
8. The person who is very short is the person who loves blue.
9. The person whose favorite color is red is Arnold.
10. The person who loves science fiction books is directly left of the person who loves biography books.
11. Bob is not in the fifth house.
12. The person whose birthday is in January is the person whose favorite color is green.
13. The person who has an average height is the person who likes Cherry smoothies.
14. The person who loves fantasy books is the person whose birthday is in February.
15. The person whose favorite color is red is directly left of the person who is very short.
16. The person who loves science fiction books is the person whose birthday is in March.
17. The person who is very tall is in the second house.
18. The person who drinks Lime smoothies is somewhere to the left of the person who loves mystery books.
19. Eric is in the third house.
20. The person who loves fantasy books is the person who loves white.
 What is the value of attribute BookGenre for the person whose attribute Color is green? Please reason step by step, and put your final answer within \boxed{}

Answer: romance


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Eric`, `Arnold`
 - The people keep unique animals: `horse`, `cat`, `fish`, `bird`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `iphone 13`, `samsung galaxy s21`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`
 - Each person lives in a unique style of house: `colonial`, `victorian`, `ranch`, `craftsman`

## Clues:
1. The fish enthusiast is the person who uses an iPhone 13.
2. Alice is in the fourth house.
3. Peter is in the first house.
4. The person residing in a Victorian house is the person whose favorite color is red.
5. The person who loves white is the person in a Craftsman-style house.
6. The person whose favorite color is green is in the fourth house.
7. The person living in a colonial-style house is Eric.
8. The bird keeper is Peter.
9. The person who uses a Google Pixel 6 is in the fourth house.
10. The person who loves white is the person who uses a Samsung Galaxy S21.
11. The person living in a colonial-style house is the cat lover.
12. Eric is not in the third house.
 What is the value of attribute House for the person whose attribute Color is red? 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: `Peter`, `Eric`, `Arnold`, `Alice`
 - People have unique favorite music genres: `rock`, `jazz`, `classical`, `pop`
 - The people are of nationalities: `dane`, `brit`, `swede`, `norwegian`
 - Each person has a favorite color: `green`, `red`, `yellow`, `white`

## Clues:
1. The British person and Peter are next to each other.
2. Eric is somewhere to the left of the British person.
3. The Swedish person is directly left of the person whose favorite color is red.
4. The person who loves classical music is in the second house.
5. There is one house between the British person and the person whose favorite color is red.
6. The person who loves rock music is the Norwegian.
7. The person who loves yellow is not in the first house.
8. The Norwegian is Arnold.
9. The Dane and the person who loves white are next to each other.
10. There are two houses between the person who loves pop music and the person who loves rock music.
 What is the value of attribute House for the person whose attribute Nationality is norwegian? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Eric`, `Peter`, `Alice`
 - Each person has a unique favorite drink: `milk`, `root beer`, `coffee`, `tea`, `water`
 - Everyone has a favorite smoothie: `dragonfruit`, `lime`, `desert`, `cherry`, `watermelon`
 - Everyone has a unique favorite cigar: `prince`, `pall mall`, `dunhill`, `blends`, `blue master`
 - Each person has a favorite color: `red`, `yellow`, `green`, `blue`, `white`
 - People have unique favorite music genres: `pop`, `rock`, `jazz`, `classical`, `hip hop`

## Clues:
1. The coffee drinker is the person who likes Cherry smoothies.
2. The Watermelon smoothie lover is the person who loves rock music.
3. The person who loves yellow is the person who loves hip-hop music.
4. Eric is somewhere to the right of the person who loves classical music.
5. The one who only drinks water is the Prince smoker.
6. The person who smokes Blue Master is Eric.
7. The Dunhill smoker is Alice.
8. The root beer lover is the person whose favorite color is red.
9. The Dunhill smoker is somewhere to the left of the Prince smoker.
10. The Dragonfruit smoothie lover is directly left of the person who likes Cherry smoothies.
11. Arnold is the person whose favorite color is green.
12. Eric is the tea drinker.
13. The Dunhill smoker is the person who loves white.
14. The person who loves classical music and the person who drinks Lime smoothies are next to each other.
15. The person who drinks Lime smoothies is Eric.
16. The person who loves rock music is directly left of the root beer lover.
17. The person who loves pop music is the root beer lover.
18. The person whose favorite color is green is not in the third house.
19. The one who only drinks water is the person who loves blue.
20. The person who loves rock music is the person who loves white.
21. Peter is not in the third house.
22. The person who likes Cherry smoothies is somewhere to the right of the person partial to Pall Mall.
 What is the value of attribute Color for the person whose attribute Drink is coffee? Please reason step by step, and put your final answer within \boxed{}

Answer: green


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Bob`, `Arnold`
 - Each person has a unique type of pet: `hamster`, `fish`, `cat`, `dog`, `bird`
 - Each person has an occupation: `doctor`, `engineer`, `artist`, `lawyer`, `teacher`
 - Each person has a favorite color: `green`, `blue`, `yellow`, `red`, `white`
 - The people keep unique animals: `bird`, `cat`, `horse`, `fish`, `dog`

## Clues:
1. The person who owns a dog is Arnold.
2. The bird keeper is in the fourth house.
3. The person who keeps a pet bird is directly left of the dog owner.
4. The person who loves white is somewhere to the left of the person who is a lawyer.
5. The person who loves yellow is directly left of the person whose favorite color is green.
6. The cat lover is Bob.
7. The cat lover is somewhere to the left of Eric.
8. The person who keeps horses is in the fifth house.
9. The person who is a lawyer is directly left of the person who is a teacher.
10. The person who is a doctor is in the first house.
11. Alice is the person who loves yellow.
12. The person who loves blue is directly left of the person with an aquarium of fish.
13. The person who loves yellow is in the first house.
14. The person with a pet hamster is the person who is an artist.
15. Eric is the dog owner.
 What is the value of attribute Pet for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Bob`, `Eric`
 - People have unique heights: `very tall`, `average`, `tall`, `very short`, `short`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `cruise`, `city`, `camping`
 - Each person has a unique level of education: `master`, `doctorate`, `associate`, `high school`, `bachelor`

## Clues:
1. The person who is very tall is directly left of the person with an associate's degree.
2. The person who has an average height and the person who enjoys mountain retreats are next to each other.
3. Eric is the person who is short.
4. The person who prefers city breaks is in the fourth house.
5. The person who loves beach vacations is the person who is very tall.
6. The person with a master's degree is Bob.
7. The person who enjoys camping trips is not in the third house.
8. The person who has an average height is not in the second house.
9. The person with a doctorate is directly left of Alice.
10. The person with a master's degree is the person who enjoys camping trips.
11. The person who enjoys camping trips is not in the first house.
12. The person with a doctorate is the person who is tall.
13. The person with a high school diploma is somewhere to the right of the person who has an average height.
14. Arnold is the person with a doctorate.
 What is the value of attribute Height for the person whose attribute Education is master? Please reason step by step, and put your final answer within \boxed{}

Answer: very short


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 heights: `average`, `tall`, `very short`, `short`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `google pixel 6`, `iphone 13`
 - The mothers' names in different houses are unique: `Kailyn`, `Janelle`, `Holly`, `Aniya`

## Clues:
1. The person who uses a Samsung Galaxy S21 is The person whose mother's name is Holly.
2. The person who is tall is directly left of the person who uses a Samsung Galaxy S21.
3. Alice is the person who uses a Google Pixel 6.
4. The person who uses a OnePlus 9 is in the fourth house.
5. The person whose mother's name is Janelle is not in the third house.
6. The person whose mother's name is Janelle is somewhere to the left of Eric.
7. The person whose mother's name is Aniya is somewhere to the left of Alice.
8. Eric is the person who is very short.
9. Peter is somewhere to the left of The person whose mother's name is Kailyn.
10. The person whose mother's name is Holly and the person who has an average height are next to each other.
 What is the value of attribute House for the person whose attribute Mother is Janelle? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


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

## Clues:
1. Alice is the person who loves jazz music.
2. The person with a master's degree is somewhere to the left of The person whose mother's name is Aniya.
3. Alice is not in the second house.
4. The person with a high school diploma is the person who loves classical music.
5. The person with a master's degree is in the third house.
6. The person whose mother's name is Janelle is Eric.
7. The person with a master's degree is Arnold.
8. The person whose mother's name is Holly is in the second house.
9. Peter is the person with a high school diploma.
10. The person with an associate's degree is the person who loves rock music.
 What is the value of attribute House for the person whose attribute Education is associate? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`
 - Everyone has a unique favorite cigar: `pall mall`, `blue master`, `prince`
 - The people keep unique animals: `horse`, `cat`, `bird`
 - Each mother is accompanied by their child: `Bella`, `Fred`, `Meredith`
 - People have unique favorite book genres: `science fiction`, `romance`, `mystery`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `samsung galaxy s21`

## Clues:
1. The person who loves mystery books is the person's child is named Fred.
2. The cat lover is Eric.
3. The person partial to Pall Mall is in the second house.
4. The person who keeps horses is the person's child is named Meredith.
5. The person's child is named Bella is the Prince smoker.
6. The person who uses an iPhone 13 is directly left of the person who uses a Samsung Galaxy S21.
7. The person's child is named Fred is directly left of Arnold.
8. Peter is somewhere to the left of Eric.
9. The person who loves science fiction books is the person who uses a Samsung Galaxy S21.
10. The person who loves science fiction books is in the third house.
11. The person who loves mystery books is not in the second house.
 What is the value of attribute BookGenre for the person whose attribute Children is Meredith? Please reason step by step, and put your final answer within \boxed{}

Answer: romance


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Eric`, `Bob`, `Arnold`
 - Each person has a unique birthday month: `april`, `feb`, `mar`, `jan`, `sept`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`, `dunhill`, `blends`, `blue master`
 - Each person has a unique favorite drink: `water`, `coffee`, `tea`, `milk`, `root beer`

## Clues:
1. The root beer lover is Eric.
2. The person partial to Pall Mall is in the third house.
3. The person whose birthday is in April is Bob.
4. The Dunhill smoker is the person whose birthday is in March.
5. Peter is somewhere to the right of the root beer lover.
6. There is one house between the person whose birthday is in January and Peter.
7. The person who smokes many unique blends is the person whose birthday is in February.
8. The person whose birthday is in February is in the second house.
9. Arnold is directly left of Peter.
10. The person who likes milk is not in the fifth house.
11. The person who smokes Blue Master is the coffee drinker.
12. There is one house between the tea drinker and the coffee drinker.
13. Eric is in the third house.
 What is the value of attribute Drink for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: water


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

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

Answer: romance


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Arnold`, `Alice`, `Eric`
 - Everyone has a unique favorite cigar: `blue master`, `dunhill`, `prince`, `blends`, `pall mall`
 - People have unique favorite music genres: `rock`, `classical`, `hip hop`, `pop`, `jazz`
 - Each person has a unique birthday month: `feb`, `april`, `sept`, `jan`, `mar`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`, `biography`, `fantasy`

## Clues:
1. The person who loves pop music is the person who loves romance books.
2. The person who loves fantasy books is in the second house.
3. The person who smokes Blue Master is the person who loves science fiction books.
4. The person who loves hip-hop music is the person whose birthday is in September.
5. The person who loves biography books is Arnold.
6. Alice is the person who loves science fiction books.
7. The person who loves jazz music is somewhere to the left of Eric.
8. The person who smokes Blue Master is the person whose birthday is in April.
9. The person who smokes many unique blends is not in the third house.
10. The Dunhill smoker is the person whose birthday is in February.
11. The person who loves science fiction books is in the fifth house.
12. The person who loves classical music is Bob.
13. The person who loves pop music is Eric.
14. The person who loves mystery books is the person partial to Pall Mall.
15. Peter is the person whose birthday is in February.
16. The person whose birthday is in March is the person who smokes many unique blends.
17. The person who loves classical music is not in the first house.
 What is the value of attribute Cigar for the person whose attribute Birthday is jan? Please reason step by step, and put your final answer within \boxed{}

Answer: pall mall


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Bob`, `Alice`
 - Each person has a unique favorite drink: `milk`, `tea`, `root beer`, `water`, `coffee`
 - Each person has a favorite color: `blue`, `white`, `red`, `green`, `yellow`
 - Each mother is accompanied by their child: `Samantha`, `Meredith`, `Timothy`, `Bella`, `Fred`
 - Each person lives in a unique style of house: `modern`, `ranch`, `victorian`, `colonial`, `craftsman`
 - People use unique phone models: `samsung galaxy s21`, `iphone 13`, `huawei p50`, `oneplus 9`, `google pixel 6`

## Clues:
1. The person in a Craftsman-style house is directly left of the person living in a colonial-style house.
2. The person living in a colonial-style house is somewhere to the left of the person who uses a Samsung Galaxy S21.
3. The person who uses a Huawei P50 is the person in a Craftsman-style house.
4. The root beer lover is somewhere to the left of the person residing in a Victorian house.
5. There is one house between the person in a ranch-style home and the root beer lover.
6. The person who uses a OnePlus 9 is the person whose favorite color is red.
7. The tea drinker is not in the fourth house.
8. The person living in a colonial-style house and Peter are next to each other.
9. The person in a ranch-style home is directly left of the person whose favorite color is green.
10. The person who is the mother of Timothy and the person's child is named Meredith are next to each other.
11. The person who loves blue is the person residing in a Victorian house.
12. Bob is the person residing in a Victorian house.
13. The person's child is named Bella is the person who uses a Google Pixel 6.
14. The person who loves blue is the one who only drinks water.
15. The person who likes milk is the person who is the mother of Timothy.
16. The person who loves white is Arnold.
17. The root beer lover is directly left of the person who uses a Samsung Galaxy S21.
18. The person who likes milk is Alice.
19. The person who is the mother of Timothy is the person whose favorite color is green.
20. The person who loves white is not in the first house.
21. The person's child is named Samantha is the person whose favorite color is red.
 What is the value of attribute Color for the person whose attribute Children is Fred? Please reason step by step, and put your final answer within \boxed{}

Answer: yellow


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Eric`, `Arnold`
 - People own unique car models: `honda civic`, `toyota camry`, `ford f150`, `tesla model 3`
 - Each person has a unique birthday month: `feb`, `april`, `sept`, `jan`
 - Everyone has something unique for lunch: `pizza`, `stew`, `spaghetti`, `grilled cheese`
 - People have unique heights: `average`, `tall`, `very short`, `short`
 - The mothers' names in different houses are unique: `Janelle`, `Kailyn`, `Holly`, `Aniya`

## Clues:
1. The person who is very short is the person who loves the stew.
2. The person who is a pizza lover is Alice.
3. The person who loves the spaghetti eater is the person who owns a Honda Civic.
4. The person who owns a Toyota Camry is somewhere to the left of the person who is a pizza lover.
5. The person whose birthday is in February is not in the fourth house.
6. Eric is the person who owns a Tesla Model 3.
7. The person who owns a Toyota Camry is somewhere to the right of The person whose mother's name is Janelle.
8. The person who is tall is in the fourth house.
9. The person whose birthday is in September is Arnold.
10. The person who has an average height is somewhere to the right of Arnold.
11. The person whose mother's name is Aniya and Alice are next to each other.
12. The person whose birthday is in April and The person whose mother's name is Holly are next to each other.
13. The person who is short is in the second house.
14. The person who loves the stew and The person whose mother's name is Janelle are next to each other.
 What is the value of attribute Height for the person whose attribute Food is pizza? Please reason step by step, and put your final answer within \boxed{}

Answer: tall


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Eric`, `Bob`, `Peter`, `Arnold`, `Alice`
 - Each person prefers a unique type of vacation: `cultural`, `beach`, `mountain`, `cruise`, `camping`, `city`
 - The people are of nationalities: `chinese`, `brit`, `norwegian`, `swede`, `dane`, `german`
 - Everyone has a favorite smoothie: `lime`, `desert`, `watermelon`, `blueberry`, `dragonfruit`, `cherry`

## Clues:
1. The person who likes Cherry smoothies is the person who enjoys mountain retreats.
2. Eric is the Dane.
3. The German is Peter.
4. The Desert smoothie lover is in the sixth house.
5. The Swedish person is the Dragonfruit smoothie lover.
6. Arnold is the Dragonfruit smoothie lover.
7. The Watermelon smoothie lover is the British person.
8. The Norwegian is the person who enjoys camping trips.
9. The person who likes going on cruises is the British person.
10. There are two houses between Bob and the person who enjoys camping trips.
11. The person who drinks Blueberry smoothies is directly left of the person who goes on cultural tours.
12. Bob is directly left of the Dragonfruit smoothie lover.
13. The person who likes going on cruises is directly left of Peter.
14. The person who prefers city breaks is not in the sixth house.
15. Carol is in the fifth house.
 What is the value of attribute House for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


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`, `Eric`, `Peter`
 - Everyone has something unique for lunch: `stew`, `spaghetti`, `grilled cheese`, `pizza`, `stir fry`
 - Each person has a favorite color: `green`, `blue`, `yellow`, `red`, `white`
 - The people are of nationalities: `swede`, `brit`, `dane`, `german`, `norwegian`
 - People have unique favorite book genres: `fantasy`, `biography`, `romance`, `mystery`, `science fiction`
 - Each mother is accompanied by their child: `Timothy`, `Samantha`, `Meredith`, `Fred`, `Bella`

## Clues:
1. The Swedish person is the person's child is named Bella.
2. There is one house between the person who loves science fiction books and the person who loves the stew.
3. The person who is a pizza lover is the person's child is named Fred.
4. The person who loves white is not in the fourth house.
5. The person who loves yellow is Peter.
6. The person who loves eating grilled cheese is directly left of the person's child is named Meredith.
7. The German is the person who loves the spaghetti eater.
8. The person who loves romance books is the person who loves the stew.
9. The person's child is named Samantha is directly left of the German.
10. Alice is the Swedish person.
11. Arnold is the person whose favorite color is green.
12. Arnold is in the third house.
13. The person's child is named Meredith is the person who loves biography books.
14. The Norwegian is the person whose favorite color is red.
15. The person who loves romance books is not in the third house.
16. The person who loves the spaghetti eater is Eric.
17. The Dane is somewhere to the left of the person who loves yellow.
18. The person who loves mystery books is directly left of the person whose favorite color is red.
19. The person's child is named Fred is not in the third house.
 What is the value of attribute Food for the person whose attribute BookGenre is romance? Please reason step by step, and put your final answer within \boxed{}

Answer: stew


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`
 - People have unique favorite book genres: `romance`, `mystery`, `fantasy`, `science fiction`
 - People have unique heights: `tall`, `very short`, `short`, `average`
 - Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `desert`, `watermelon`
 - Each person has a favorite color: `red`, `green`, `yellow`, `white`

## Clues:
1. Arnold is not in the fourth house.
2. Peter is the person who loves mystery books.
3. The person who likes Cherry smoothies is somewhere to the right of the Desert smoothie lover.
4. The person who loves science fiction books is directly left of Eric.
5. The Desert smoothie lover is the person who is very short.
6. The person who loves white is the person who loves mystery books.
7. The person who loves fantasy books is the person who loves yellow.
8. The Desert smoothie lover is somewhere to the right of the person who is short.
9. The Dragonfruit smoothie lover and the person whose favorite color is green are next to each other.
10. The Dragonfruit smoothie lover is the person who loves mystery books.
11. The person who loves yellow is in the second house.
12. The person who likes Cherry smoothies is not in the third house.
13. The person who has an average height is the person who likes Cherry smoothies.
 What is the value of attribute House for the person whose attribute Smoothie is dragonfruit? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Arnold`, `Bob`, `Eric`
 - Each person has a unique hobby: `photography`, `cooking`, `knitting`, `gardening`, `painting`
 - Each person has a unique favorite drink: `root beer`, `milk`, `water`, `coffee`, `tea`

## Clues:
1. Eric is the coffee drinker.
2. The tea drinker is the person who paints as a hobby.
3. The person who enjoys knitting is not in the fourth house.
4. Peter is not in the fourth house.
5. Eric is somewhere to the right of the root beer lover.
6. Arnold is the person who loves cooking.
7. The one who only drinks water is somewhere to the right of the person who enjoys gardening.
8. There is one house between Bob and the person who paints as a hobby.
9. The person who enjoys gardening is directly left of the root beer lover.
10. The photography enthusiast is the one who only drinks water.
 What is the value of attribute Hobby for the person whose attribute Drink is coffee? Please reason step by step, and put your final answer within \boxed{}

Answer: knitting


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Arnold`, `Peter`, `Eric`
 - Each person has a unique level of education: `doctorate`, `bachelor`, `associate`, `master`, `high school`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `google pixel 6`, `iphone 13`, `huawei p50`
 - People have unique favorite music genres: `rock`, `classical`, `jazz`, `pop`, `hip hop`
 - The people keep unique animals: `bird`, `dog`, `horse`, `cat`, `fish`
 - Each person has a unique hobby: `painting`, `cooking`, `knitting`, `gardening`, `photography`

## Clues:
1. The person who loves hip-hop music is in the second house.
2. The person who loves classical music is not in the fifth house.
3. The person who loves classical music is the fish enthusiast.
4. The person who uses an iPhone 13 is the person with a doctorate.
5. The person with a master's degree is not in the second house.
6. The person who enjoys knitting is not in the first house.
7. The person with a high school diploma is somewhere to the left of the photography enthusiast.
8. The person who paints as a hobby is somewhere to the right of the person who loves rock music.
9. Arnold is the bird keeper.
10. The person with a bachelor's degree is the person who uses a Google Pixel 6.
11. The person with a bachelor's degree is somewhere to the right of the person who paints as a hobby.
12. Bob is not in the fourth house.
13. The person who loves cooking is the person who loves classical music.
14. The person who loves cooking is the person who uses a Huawei P50.
15. The person who keeps horses is somewhere to the right of Arnold.
16. The person who paints as a hobby is the person who uses a OnePlus 9.
17. The person who uses a OnePlus 9 is somewhere to the left of the person with an associate's degree.
18. The person who uses an iPhone 13 is the dog owner.
19. The person with a high school diploma is Alice.
20. Peter is the person who enjoys gardening.
21. The person who paints as a hobby is the person who loves jazz music.
 What is the value of attribute PhoneModel for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: iphone 13


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Peter`, `Eric`
 - Each person has a unique birthday month: `sept`, `feb`, `april`, `jan`
 - The mothers' names in different houses are unique: `Aniya`, `Kailyn`, `Holly`, `Janelle`
 - The people keep unique animals: `horse`, `bird`, `cat`, `fish`
 - Each person has a favorite color: `green`, `white`, `yellow`, `red`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `soccer`

## Clues:
1. The bird keeper is The person whose mother's name is Kailyn.
2. The fish enthusiast is not in the fourth house.
3. The person who loves white is Arnold.
4. The person who keeps horses is Peter.
5. The person who keeps horses is not in the first house.
6. The person whose mother's name is Kailyn is Arnold.
7. The person whose birthday is in January is in the first house.
8. The person whose birthday is in April is somewhere to the left of the person whose birthday is in September.
9. Alice is the person whose birthday is in September.
10. The person whose mother's name is Aniya and the person whose birthday is in September are next to each other.
11. Peter is directly left of the person whose favorite color is red.
12. There is one house between the person whose birthday is in April and the person who loves soccer.
13. The fish enthusiast is the person whose birthday is in September.
14. The person who loves swimming is The person whose mother's name is Holly.
15. The person whose favorite color is green is the person who loves swimming.
16. The person who loves yellow is the person who loves basketball.
 What is the value of attribute FavoriteSport for the person whose attribute Mother is Kailyn? Please reason step by step, and put your final answer within \boxed{}

Answer: soccer


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`, `Alice`, `Carol`, `Bob`
 - People have unique favorite book genres: `mystery`, `biography`, `science fiction`, `romance`, `fantasy`, `historical fiction`
 - Each person has a unique favorite drink: `water`, `milk`, `boba tea`, `coffee`, `tea`, `root beer`
 - People have unique hair colors: `blonde`, `red`, `gray`, `black`, `auburn`, `brown`
 - Each person has a unique birthday month: `may`, `mar`, `sept`, `jan`, `april`, `feb`

## Clues:
1. The person who likes milk is the person who has gray hair.
2. Alice is somewhere to the right of the person who has auburn hair.
3. The person whose birthday is in September is not in the sixth house.
4. The tea drinker is somewhere to the left of the person who has auburn hair.
5. Carol is the person who loves biography books.
6. The person who likes milk is in the fourth house.
7. The person who has auburn hair is directly left of Eric.
8. The boba tea drinker is the person who has brown hair.
9. The person whose birthday is in February is the person who has gray hair.
10. Carol is the person who has blonde hair.
11. The person who loves fantasy books is somewhere to the left of Bob.
12. The person who has blonde hair is the person whose birthday is in April.
13. Peter is the coffee drinker.
14. The person who loves historical fiction books is somewhere to the right of the person who loves science fiction books.
15. The person who loves science fiction books is Bob.
16. The root beer lover is not in the first house.
17. There is one house between the person whose birthday is in May and Peter.
18. The person who has black hair is in the second house.
19. The person whose birthday is in February is the person who loves historical fiction books.
20. Arnold is the person whose birthday is in March.
21. The root beer lover is somewhere to the left of the person whose birthday is in May.
22. The person who loves mystery books is in the sixth house.
 What is the value of attribute HairColor for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{}

Answer: red


B3. Six scouts go on a scavenger hunt. On Saturday, they go to the forest and on Sunday, they go into the mountains. On both days, they have to find their way in pairs. Their leader wants to divide them into pairs for each trip, so that no one has the same partner on the second day as on the first day. In how many ways can he do this?

Answer: 120


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


[ Combinations and Permutations ] $[$ Graph Theory (other) ]

In the parliament, there are 30 deputies. Any two of them are either friends or enemies, and each one is friends with exactly six others. Any three deputies form a committee. Find the total number of committees in which all three members are pairwise friends or all three are pairwise enemies.

Answer: 1990


G4.1 In Figure $1, A B C D$ is a parallelogram, $B E \perp C D, B F \perp A D$, $C E=2 \mathrm{~cm}, D F=1 \mathrm{~cm}$ and $\angle E B F=60^{\circ}$.
If the area of the parallelogram $A B C D$ is $R \mathrm{~cm}^{2}$, find the value of $R$.

Answer: 12\sqrt{3}


$$
\text { Three. (25 points) In } \triangle A B C \text {, } A B=8+2 \sqrt{6} \text {, }
$$
$B C=7+2 \sqrt{6}, C A=3+2 \sqrt{6}, O, I$ are the circumcenter and incenter of $\triangle A B C$, respectively. Is the length of segment $O I$ a rational number or an irrational number? Make a judgment and explain your reasoning.

Answer: \dfrac{5}{2}


4. [4] For $0 \leq y \leq 2$, let $D_{y}$ be the half-disk of diameter 2 with one vertex at $(0, y)$, the other vertex on the positive $x$-axis, and the curved boundary further from the origin than the straight boundary. Find the area of the union of $D_{y}$ for all $0 \leq y \leq 2$.

Answer: \pi


12.238. A sphere is inscribed in a truncated cone. The sum of the lengths of the diameters of the upper and lower bases of the cone is five times the length of the radius of the sphere. Find the angle between the generatrix of the cone and the plane of the base.

Answer: \arctan \frac{4}{3}


111. Find the general solution of the equation $y^{\prime \prime}=\sin 2 x$.

Answer: y = C_1 x + C_2 - \frac{1}{4} \sin(2x)


Berlov S.L.

A convex 2011-gon is drawn on the board. Petya sequentially draws diagonals in it such that each newly drawn diagonal intersects no more than one of the previously drawn diagonals at an internal point. What is the maximum number of diagonals Petya can draw?

Answer: 4016


10. (20 points) Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1, a_{n+1}=\frac{a_{n}}{2}+2^{n-1} \cdot a_{n}^{2}\left(n \in \mathbf{N}^{*}\right)$. Let $S_{n}=$ $\sum_{i=1}^{n} \frac{1}{2^{n} a_{n}+1}$, find the value of $\lim _{n \rightarrow \infty} S_{n}$.

Answer: \dfrac{1}{2}


3. Fill the numbers $1,2, \cdots, 36$ into a $6 \times 6$ grid, with each cell containing one number, such that the numbers in each row are in increasing order from left to right. The minimum value of the sum of the six numbers in the third column is $\qquad$
(2015, National Junior High School Mathematics League Competition)

Answer: 63


12. There are five cages arranged in a row, labeled from left to right as shown in Table 1.
Table 1
\begin{tabular}{|c|c|c|c|c|}
\hline \begin{tabular}{c}
Red \\
Wolf
\end{tabular} & \begin{tabular}{c}
Silver \\
Lion
\end{tabular} & \begin{tabular}{c}
Brown \\
Fox
\end{tabular} & \begin{tabular}{c}
White \\
Cow
\end{tabular} & \begin{tabular}{c}
Gray \\
Horse
\end{tabular} \\
\hline
\end{tabular}

Each cage contains exactly one animal, satisfying the following conditions:
(1) These five animals are indeed a wolf, a lion, a fox, a cow, and a horse, and their colors are indeed red, silver, brown, white, and gray;
(2) None of the labels indicate the correct color or animal inside the cage;
(3) Each animal is not in the cage labeled with its color or type, nor is it in the adjacent cage.

If the horse is not in the middle cage, then the color of the horse is . $\qquad$

Answer: White


Four numbers from the set of six numbers $\{0, 1, 2, 3, 4, 5\}$ are placed in the circles of Figure 1. The absolute difference between the numbers at the endpoints of each line segment yields 5 differences, which must be exactly $\{1, 2, 3, 4, 5\}$. Figure 2 shows one such arrangement. How many such arrangements are possible? (In the given figures, swapping 3 and 5 or 0 and 1 counts as a new arrangement.)

Answer: 32


Task 2. Two lines $p$ and $q$ are parallel to the bases of the trapezoid $ABCD$ and divide the leg $AD$ into three equal parts. Determine the lengths of the segments $x$ and $y$ of the lines $q$ and $p$ between the legs, if $\overline{AB}=13$ and $\overline{CD}=4$.

Answer: 10


1. Given $F_{1} 、 F_{2}$ are the left and right foci of the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$, and $A$ is a moving point on the ellipse. Circle $C$ is tangent to the extension of $F_{1} A$, the extension of $F_{1} F_{2}$, and the line segment $A F_{2}$. If $M(t, 0)$ is a point of tangency, then $t=$ $\qquad$ .

Answer: 2


[ Examples and counterexamples. Constructions ]  [ Partitions into pairs and groups; bijections ]  [ Decimal number system ]

A subset $X$ of the set of "two-digit" numbers $00,01, \ldots, 98,99$ is such that in any infinite sequence of digits, there are two adjacent digits that form a number in $X$. What is the smallest number of elements that $X$ can contain?

Answer: 55


5. Point $A_{1}$ lies on side $B C$ of triangle $A B C$, point $B_{1}$ lies on side $A C$. Let $K$ be the intersection point of segments $A A_{1}$ and $B B_{1}$. Find the area of quadrilateral $A_{1} C B_{1} K$, given that the area of triangle $A_{1} B K$ is 5, the area of triangle $A B_{1} K$ is 8, and the area of triangle $A B K$ is 10.

Answer: 22


2. Given the function $f(x)=a x^{2}-c$, satisfying $-4 \leqslant f(1) \leqslant-1,-1 \leqslant f(2) \leqslant 5, f(3)$'s range of values is $\qquad$ .

Answer: [-1,20]


Folklore
In Chicago, there are 36 gangsters, some of whom are at odds with each other. Each gangster is a member of several gangs, and there are no two gangs with the same membership. It turned out that gangsters who are in the same gang do not feud, but if a gangster is not a member of a certain gang, then he feuds with at least one of its members. What is the maximum number of gangs that could exist in Chicago?

Answer: 3^{12}


Two congruent equilateral triangles $A B C$ and $B D E$ are given such that the measure of angle $A B D$ is greater than $120^{\circ}$ and less than $180^{\circ}$, and points $C, E$ lie in the same half-plane defined by the line $A D$. The intersection of $C D$ and $A E$ is denoted as $F$.

Determine the measure of angle $A F D$.

(I. Jančigová)

![](https://cdn.mathpix.com/cropped/2024_04_17_e988acd5633232ee9496g-1.jpg?height=457&width=803&top_left_y=2076&top_left_x=632)

Hint. You know the sums of the interior angles in triangles or polygons.

Answer: 120^\circ


K2) Let $n$ be a natural number. A volleyball team consisting of $n$ women and $n$ men lines up for a game. Each team member occupies one of the positions $1,2, \ldots, 2 n$, whereby exactly positions 1 and $n+1$ are outside the court. During the game, all team members rotate, switching from position $i$ to position $i+1$ (or from $2 n$ to 1). How many possibilities are there for the starting line-up so that there are always at least $n-1$ women on the pitch, no matter how often the team rotates?

Note: Two starting line-ups are different if at least one team member occupies a different position.

Answer: 2^n \cdot (n!)^2


Let \( S \) be the unit circle in the complex plane (i.e., the set of complex numbers with modulus equal to 1). Let \( f \) be a mapping from \( S \) to \( S \). For any \( z \in S \), define \( f^{(1)}(z) = f(z), f^{(2)}(z) = f(f(z)), \ldots, f^{(k)}(z) = f(f^{(k-1)}(z)) \).

If \( c \in S \) and a natural number \( n \) such that \( f^{(1)}(c) \neq c, f^{(2)}(c) \neq c, \ldots, f^{(n-1)}(c) \neq c, f^{(n)}(c) = c \), then \( c \) is called an \( n \)-periodic point of \( f \).

Let \( m \) be a natural number greater than 1, and \( f \) be defined as \( f(z) = z^m \) for \( z \in S \). Calculate the number of 1989-periodic points of \( f \).

Answer: m^{1989} - m^{663} - m^{153} - m^{117} + m^{51} + m^{39} + m^{9} - m^{3}


9. (16 points) In the Cartesian coordinate system $x O y$, $\angle M O N=\frac{\pi}{3}$, and side $M N$ moves on the line $l: x=3$. Find the equation of the locus of the circumcenter of $\triangle M O N$.

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

Answer: \dfrac{(x - 4)^2}{4} - \dfrac{y^2}{12} = 1


Given a rhombus \(ABCD\) with \(\angle BCD = 60^\circ\). Points \(P\) and \(Q\) are taken on the sides \(AB\) and \(BC\) respectively such that \(AP = BQ\). Prove that triangle \(DPQ\) is equilateral.

Answer: \triangle DPQ \text{ is equilateral}


62. On a plane, there are two lines intersecting at point $O$, and two points $A$ and $B$. Denote the bases of the perpendiculars dropped from $A$ to the given lines as $M$ and $N$, and the bases of the perpendiculars dropped from $B$ as $K$ and $L$. Find the angle between the lines $M N$ and $K L$, if $\widehat{A O B}=\alpha \leqslant 90^{\circ}$.

Answer: \alpha


Exercise 16. Let $k \geqslant 1$ be an integer. What is the smallest integer $n$ such that, no matter how $n$ points are placed in the plane, it is possible to choose a subset $S$ consisting of $k$ of these points that satisfies "for every pair $P, Q$ of points in $S$, the distance between $P$ and $Q$ is less than or equal to 2" or "for every pair $\mathrm{P}, \mathrm{Q}$ of points in $\mathrm{S}$, the distance between $\mathrm{P}$ and $\mathrm{Q}$ is strictly greater than 1."

Answer: (k-1)^2 + 1


8. (10 points) As shown in the figure, a large cube with an edge length of 4 is formed by 64 smaller cubes with an edge length of 1. Then, 4 smaller cubes (the shaded part in the figure) are removed from top to bottom. If the inner and outer surfaces of the remaining solid are painted red, how many smaller cubes have exactly two faces painted red? $\qquad$

Answer: 28


Let $A = \{1, 2, 3, 4, 5, 6, 7\}$ , and let $N$ be the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Find the remainder when $N$ is divided by $1000$ .
Please give the answer directly without any intermediate steps.

Answer: 399


[ Sorting in ascending (descending) order. ]

[ Linear inequalities and systems of inequalities ] On each of four cards, a natural number is written. Two cards are randomly drawn and the numbers on them are added. With equal probability, this sum can be less than 9, equal to 9, and greater than 9. What numbers can be written on the cards?

Answer: \{1, 2, 7, 8\}


In the plane of triangle \(ABC\), a point \(M\) is given. This point starts moving linearly towards vertex \(A\), but stops halfway and starts moving towards vertex \(B\), again stops halfway and moves towards vertex \(C\), again stops halfway and continues such movement towards vertex \(A\), and so on. Prove that with unlimited continuation of such movement, the point will move infinitely close to the contour of a uniquely determined limiting triangle. Construct this triangle and calculate the ratio of its area to the area of the original triangle \(ABC\).

Answer: \dfrac{1}{7}


6. (3 points) Given the cryptarithm: ЖАЛО + ЛОЖА = ОСЕНЬ. Identical letters represent identical digits, different letters represent different digits. Find the value of the letter А.

Answer: 8


Question 2. Given non-negative real numbers $a_{1} \geq a_{2} \geq \ldots \geq a_{n}, b_{1} \leq b_{2} \leq \ldots \leq b_{n}$, satisfying
$$
a_{1} a_{n}+a_{2} a_{n-1}+\ldots+a_{n} a_{1}=b_{1} b_{n}+b_{2} b_{n-1}+\ldots+b_{n} b_{1}=1 \text {, }
$$

Find
$$
S=\sum_{1 \leq i<j \leq n} a_{i} b_{j}
$$

the minimum value.

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


*20 problems for 25 minutes.***p1.** Given that  $a + 19b = 3$  and  $a + 1019b = 5$ , what is  $a + 2019b$ ?**p2.** How many multiples of  $3$  are there between  $2019$  and  $2119$ , inclusive?**p3.** What is the maximum number of quadrilaterals a  $12$ -sided regular polygon can be quadrangulated into? Here quadrangulate means to cut the polygon along lines from vertex to vertex, none of which intersect inside the polygon, to form pieces which all have exactly  $4$  sides.**p4.** What is the value of  $|2\pi - 7| + |2\pi -  6|$ , rounded to the nearest hundredth?**p5.** In the town of EMCCxeter, there is a  $30\%$  chance that it will snow on Saturday, and independently, a  $40\%$  chance that it will snow on Sunday. What is the probability that it snows exactly once that weekend, as a percentage?**p6.** Define  $n!$  to be the product of all integers between  $1$  and  $n$  inclusive. Compute  $\frac{2019!}{2017!} \times \frac{2016!}{2018!}$  .**p7.** There are  $2019$  people standing in a row, and they are given positions  $1$ ,  $2$ ,  $3$ ,  $...$ ,  $2019$  from left to right. Next, everyone in an odd position simultaneously leaves the row, and the remaining people are assigned new positions from  $1$  to  $1009$ , again from left to right. This process is then repeated until one person remains. What was this person's original position?**p8.** The product  $1234\times 4321$  contains exactly one digit not in the set  $\{1, 2, 3, 4\}$ . What is this digit?**p9.** A quadrilateral with positive area has four integer side lengths, with shortest side  $1$  and longest side  $9$ . How many possible perimeters can this quadrilateral have?**p10.** Define  $s(n)$  to be the sum of the digits of  $n$  when expressed in base  $10$ , and let  $\gamma (n)$  be the sum of  $s(d)$  over all natural number divisors  $d$  of  $n$ . For instance,  $n = 11$  has two divisors,  $1$  and  $11$ , so  $\gamma (11) = s(1) + s(11) = 1 + (1 + 1) = 3$ . Find the value of  $\gamma (2019)$ .**p11.** How many five-digit positive integers are divisible by  $9$  and have  $3$  as the tens digit?**p12.** Adam owns a large rectangular block of cheese, that has a square base of side length  $15$  inches, and a height of  $4$  inches. He wants to remove a cylindrical cheese chunk of height  $4$ , by making a circular hole that goes through the top and bottom faces, but he wants the surface area of the leftover cheese block to be the same as before. What should the diameter of his hole be, in inches?
*Αddendum  on 1/26/19: the hole must have non-zero diameter.***p13.** Find the smallest prime that does not divide  $20! + 19! + 2019!$ .**p14.** Convex pentagon  $ABCDE$  has angles  $\angle ABC = \angle BCD = \angle DEA = \angle EAB$  and angle  $\angle CDE = 60^o$ . Given that  $BC = 3$ ,  $CD = 4$ , and  $DE = 5$ , find  $EA$ .
*Addendum on 1/26/19: ABCDE is specified to be convex.***p15.** Sophia has  $3$  pairs of red socks,  $4$  pairs of blue socks, and  $5$  pairs of green socks. She picks out two individual socks at random: what is the probability she gets a pair with matching color? **p16.** How many real roots does the function  $f(x) = 2019^x - 2019x - 2019$  have?**p17.** A  $30-60-90$  triangle is placed on a coordinate plane with its short leg of length  $6$  along the  $x$ -axis, and its long leg along the  $y$ -axis. It is then rotated  $90$  degrees counterclockwise, so that the short leg now lies along the  $y$ -axis and long leg along the  $x$ -axis. What is the total area swept out by the triangle during this rotation?**p18.** Find the number of ways to color the unit cells of a  $2\times 4$  grid in four colors such that all four colors are used and every cell shares an edge with another cell of the same color.**p19.** Triangle  $\vartriangle ABC$  has centroid  $G$ , and  $X, Y,Z$  are the centroids of triangles  $\vartriangle BCG$ ,  $\vartriangle ACG$ , and  $\vartriangle ABG$ , respectively. Furthermore, for some points  $D,E, F$ , vertices  $A,B,C$  are themselves the centroids of triangles  $\vartriangle DBC$ ,  $\vartriangle ECA$ , and  $\vartriangle FAB$ , respectively. If the area of  $\vartriangle XY Z = 7$ , what is the area of  $\vartriangle DEF$ ?**p20.** Fhomas orders three  $2$ -gallon jugs of milk from EMCCBay for his breakfast omelette. However, every jug is actually shipped with a random amount of milk (not necessarily an integer), uniformly distributed between  $0$  and  $2$  gallons. If Fhomas needs  $2$  gallons of milk for his breakfast omelette, what is the probability he will receive enough milk?

PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

Answer: 7


433. Several people have to pay 800 francs for the production of a lawsuit. But three of them have no money, so the rest have to add 60 francs to their share. How many participants are there in the payment of legal costs?

Answer: 5


The numbers from 1 to 6 are to be placed at the intersections of three circles, one number in each of the six squares. The number 6 is already placed. Which number must replace \( x \), so that the sum of the four numbers on each circle is the same?
A 1
B 2
C 3
D 4
E 5

Answer: 1


I5.1 In the figure, find \( a \).

I5.2 If the lines \( a x + b y = 1 \) and \( 10 x - 34 y = 3 \) are perpendicular to each other, find \( b \).

I5.3 If the \( b^{\text {th }} \) day of May in a year is Friday and the \( c^{\text {th }} \) day of May in the same year is Tuesday, where \( 16 < c < 24 \), find \( c \).

I5.4 \( c \) is the \( d^{\text {th }} \) prime number. Find \( d \).

Answer: 9


On a grid paper of size $60 \times 70$ cells (horizontal by vertical respectively), Lena drew a coordinate system with the origin at the center of the sheet, the $x$-axis horizontal, and the $y$-axis vertical. The axes were extended to the borders of the sheet. Lena then drew the graph of the equation $y=0.83x$ and shaded all the cells through which the graph passes, i.e., the cells that contain points of the graph. How many cells in total were shaded?

Answer: 108


Two bear cubs found a large, round piece of cheese. They wanted to split it evenly but couldn't because each one feared the other would get a larger portion. Suddenly, a fox appeared and offered to divide the cheese for them.

- "Let me divide the cheese evenly for you."
- "Great!" exclaimed the bear cubs. "Divide it!"

The fox broke the cheese into two parts, but one piece was larger than the other. The bear cubs protested:
- "This one is bigger!"

The fox reassured them:
- "I'll fix it right now."

The fox bit off a piece from the larger portion, equal in size to the smaller portion. Now the other piece became larger.
- "It's still uneven!" worried the bear cubs.

- "Don't worry," said the fox. "I know what I'm doing!"

She continued to bite off from the larger piece, an amount equal to the smaller piece. She kept dividing the cheese this way until she had eaten enough. She bit off from each portion 3 times, each time taking a piece equal to the other portion. The bear cubs, with their little black noses, watched from one piece to the other.

Finally, the pieces were equal, but there was almost no cheese left for the bear cubs: just two small pieces, each weighing 25 grams.

- "Well," said the fox, "there's not much left, but at least it's fair! Enjoy your meal, little bears!" And with that, the cunning fox ran away.

Determine the weight of the original piece of cheese found by the bear cubs.

Answer: 850


Three positive integers greater than 1000 satisfy the condition that the unit digit of the sum of any two of them is equal to the unit digit of the third number. What are the possible values for the last three digits of the product of these three numbers?

Answer: 000,250,500,750


For a polygon, define a "growth" operation as follows: as shown in Figure 1, transform one of its edges $AB$ into an outward convex broken line $ACDEB$, where $C$ and $E$ are trisection points of $AB$, and $C$, $D$, and $E$ form an equilateral triangle. Given an equilateral triangle with a side length of 9, after two "growth" operations (as shown in Figure 2), the perimeter of the resulting figure is $\qquad$ ; after four "growth" operations, the perimeter of the resulting figure is $\qquad$ .

Answer: 85\frac{1}{3}


Two regular tetrahedra are connected by two faces to form a double pyramid. The centers of the six lateral faces of this double pyramid are taken as the vertices of a right triangular prism. Calculate the volume of this prism if the edge of the tetrahedron is $a$.

Answer: \frac{\sqrt{2}^3}{54}


In a chess tournament, all players faced each other twice and scored 1 point for a win, half a point for a draw, and zero points for a loss. The winner was the one who scored the most points. Luíza, a curious mathematician, found a paper stating that the sum of the scores of all participants was 210 points.

a) How many players participated in the tournament?

b) Luíza noticed that the tournament logo depicted a knight in the center, with black squares illustrating the knight's possible moves. She wondered:

- If the board were infinite and the knight continued its pattern of movement, onto how many different squares could the knight move in two moves?

Initially, she thought “64” but quickly realized she was wrong. Now, it's your turn to correctly answer the question Luíza thought of.

c) Luíza received a message from a friend who participated in the tournament, informing her that he scored 12 points. Luíza replied: “- Don’t be sad, you will win the next championship!” How did Luíza know that her friend did not win the tournament?

Answer: 33


For a four-digit positive integer \(ABCD\), where \(A, B, C,\) and \(D\) are digits with \(A \neq 0\) and \(D \neq 0\), we define the Reverse Digit sum (RD sum) of \(ABCD\) to be the sum of \(ABCD\) and \(DCBA\). For example, the RD sum of 1205 is \(1205 + 5021 = 6226\), while the integer 2300 does not have an RD sum.

(a) Determine the RD sum of 4281.

(b) There are positive integers \(m\) and \(n\) with the property that the RD sum of the integer \(ABCD\) is always equal to \(m \times (A + D) + n \times (B + C)\). State the value of \(m\) and the value of \(n\).

(c) Determine the number of four-digit integers whose RD sum is 3883.

(d) Determine the number of four-digit integers that are equal to the RD sum of a four-digit integer.

Answer: 143


All pairwise distances between four points in space are equal to 1. Find the distance from one of these points to the plane defined by the other three.

Answer: \sqrt{\frac{2}{3}}


1. Given $A_{1}, A_{2}, \cdots, A_{n}$ are 11 stations sequentially on a straight highway, and
$$
\begin{array}{l}
A_{i} A_{i+2} \leqslant 12(i=1,2, \cdots, 9), \\
A_{i} A_{i+3} \geqslant 17(i=1,2, \cdots, 8) .
\end{array}
$$

If $A_{1} A_{11}=56 \mathrm{~km}$, then $A_{2} A_{7}=$ $\qquad$ $\mathrm{km}$.

Answer: 29


Answer: 40


Question 1: Find $\mathrm{C}_{n}^{0}+\mathrm{C}_{n}^{3}+\mathrm{C}_{n}^{6}+\cdots\left(n \in \mathbf{N}_{+}\right)$.

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

Answer: \dfrac{2^n + 2\cos\left(\dfrac{\pi n}{3}\right)}{3}


9.4. In the tournament, each of the six teams played against each other exactly once. In the end, the teams scored 12, 10, 9, 8, 7, and 6 points respectively. a) How many points were awarded for a win in a match, if 1 point was awarded for a draw, and 0 points for a loss? The answer, of course, should be a natural number. b) Find the number of wins, draws, and losses for each team and prove the uniqueness of these numbers. c) Provide an example of a corresponding tournament.

Answer: 4


8. Given that a series of non-overlapping disks with radius $R$ can cover all lattice points (in the Cartesian coordinate system, points with both integer horizontal and vertical coordinates are lattice points) on the plane. Then $R_{\max }$ $\qquad$ 4 (fill in “greater than”, “less than”, or “equal to”).

Answer: \text{less than}


4. Let there be a pentagon $A B C D E$ in which $C D=A B+A E$. Let the bisector of $\measuredangle B A E$ intersect $B E$ at point $P$, let $Q$ be a point on $C D$ such that $C Q=A B$, and let $X$ and $Y$ be the midpoints of segments $B C$ and $D E$, respectively. Let $T$ be the centroid of $\triangle A P Q$ and let $Y T$ intersect $A X$ at point $Z$. Find $\frac{A Z}{Z X}$ as a function of $A B$ and $A E$.

Answer: \dfrac{2 \cdot AE}{AB + AE}


11.102. A copper blank in the form of a rectangular parallelepiped with dimensions $80 \times 20 \times 5 \mathrm{~cm}$ is rolled into a sheet with a thickness of 1 mm. Determine the area of this sheet.

Answer: 8


12. (12 points) As shown in the figure, there are four stations numbered $1, 2, 3, 4$ at the four vertices of a square ring road: Jia, Yi, Bing, and Ding start driving at the same time from stations numbered $A, B, C, D$ respectively $(A, B, C, D$ are all different), traveling clockwise at uniform speeds. The speeds of the cars starting from stations $1, 2, 3, 4$ are $1, 2, 3, 4$ respectively, and these speeds do not change. After driving, they have the following statements:

Jia says: "I caught up with Yi for the first time exactly at station (1)".
Yi says: "I caught up with Bing for the first time exactly at station (2)".
Bing says: "I caught up with Ding for the first time exactly at station (3)".
Ding says: "I caught up with Jia for the first time exactly at station (4)".
It is known that two of them are telling the truth, and two of them are lying. What is the four-digit number $\overline{\mathrm{ABCD}}=$

Answer: 2314


For each positive integer $k$ greater than $1$, find the largest real number $t$ such that the following hold:
Given $n$ distinct points $a^{(1)}=(a^{(1)}_1,\ldots, a^{(1)}_k)$, $\ldots$, $a^{(n)}=(a^{(n)}_1,\ldots, a^{(n)}_k)$ in $\mathbb{R}^k$, we define the score of the tuple $a^{(i)}$ as
\[\prod_{j=1}^{k}\#\{1\leq i'\leq n\textup{ such that }\pi_j(a^{(i')})=\pi_j(a^{(i)})\}\]
where $\#S$ is the number of elements in set $S$, and $\pi_j$ is the projection $\mathbb{R}^k\to \mathbb{R}^{k-1}$ omitting the $j$-th coordinate. Then the $t$-th power mean of the scores of all $a^{(i)}$'s is at most $n$.

Note: The $t$-th power mean of positive real numbers $x_1,\ldots,x_n$ is defined as
\[\left(\frac{x_1^t+\cdots+x_n^t}{n}\right)^{1/t}\]
when $t\neq 0$, and it is $\sqrt[n]{x_1\cdots x_n}$ when $t=0$.

[i]Proposed by Cheng-Ying Chang and usjl[/i]

Answer: \dfrac{1}{k-1}


10. Given $f(x)=\left\{\begin{array}{cc}-2 x, & x<0 ; \\ x^{2}-1, & x \geqslant 0,\end{array}\right.$, the equation
$$
f(x)+2 \sqrt{1-x^{2}}+\left|f(x)-2 \sqrt{1-x^{2}}\right|-2 a x-4=0
$$

has three real roots $x_{1}<x_{2}<x_{3}$. If $x_{3}-x_{2}=2\left(x_{2}-x_{1}\right)$, then the real number $a=$ $\qquad$

Answer: \dfrac{\sqrt{17} - 3}{2}


5. [5 points] On a plane with a given rectangular Cartesian coordinate system, a square is drawn with vertices at points $(0 ; 0),(0 ; 59),(59 ; 59)$, and $(59 ; 0)$. Find the number of ways to choose two grid nodes inside this square (not including its boundary) such that at least one of these nodes lies on one of the lines $y=x$ or $y=59-x$, but neither of the chosen nodes lies on any line parallel to any of the coordinate axes.

Answer: 370330


Each number from the set $\{1, 2, 3, 4, 5, 6, 7\}$ must be written in each circle of the diagram, so that the sum of any three [i]aligned[/i] numbers is the same (e.g., $A+D+E = D+C+B$). What number cannot be placed on the circle $E$?

Answer: 4


$\left.\begin{array}{l}{\left[\begin{array}{l}\text { Riddles }\end{array}\right]} \\ \text { [ CaseAnalysis }\end{array}\right]$

Authors: Tokoreve S.I. Khachatryan A.B.

Find the smallest four-digit number СЕЕМ for which there is a solution to the riddle МЫ + РОЖЬ = СЕЕМ. (Identical letters correspond to identical digits, different letters correspond to different digits.)

#

Answer: 2003


43rd IMO 2002 shortlist Problem C4 T is the set of all triples (x, y, z) with x, y, z non-negative integers < 10. A chooses a member (X, Y, Z) of T. B seeks to identify it. He is allowed to name a triple (a, b, c) in T. A must then reply with |X + Y - a - b| + |Y + Z - b - c| + |Z + X - c - a|. How many triples does B need to name to be sure of determining A's triple?

Answer: 3


2. In the laboratory, there are flasks of two sizes (volume $V$ and volume $V / 2$) in a total of 100 pieces, with at least three flasks of each size. The lab assistant randomly selects three flasks in sequence, and fills the first one with an 80% salt solution, the second one with a 50% salt solution, and the third one with a 20% salt solution. Then he pours the contents of these three flasks into one dish and determines the percentage of salt in it. For what minimum number of large flasks $N$ will the event "the percentage of salt in the dish is between $45\%$ and $55\%$ inclusive" occur less frequently than the event "when two fair coins are tossed, one head and one tail appear (in any order)"? Justify your answer. (16 points)

Answer: 46


Let $ABC$ be a right-angled triangle ($\angle C = 90^\circ$) and $D$ be the midpoint of an altitude from C. The reflections of the line $AB$ about $AD$ and $BD$, respectively, meet at point $F$. Find the ratio $S_{ABF}:S_{ABC}$.
Note: $S_{\alpha}$ means the area of $\alpha$.

Answer: \dfrac{4}{3}


Problem 8.2.1. Given trapezoid $A B C D(A D \| B C)$. It turns out that $\angle A B D=\angle B C D$. Find the length of segment $B D$, if $B C=36$ and $A D=64$.

![](https://cdn.mathpix.com/cropped/2024_05_06_1ea0b100610baa73554bg-02.jpg?height=300&width=491&top_left_y=613&top_left_x=481)

Answer: 48


22. (5 points)
Jiang Liuer used 30 Go stones to form a circle, with each black stone being adjacent to exactly one white stone, and the white stones adjacent to black stones account for half of the total number of white stones. Therefore, among the 30 stones, the maximum number of black stones is $\qquad$.

Answer: 14


123. In a regular triangular pyramid $S A B C$ ( $S$ - vertex), point $E$ is the midpoint of the apothem of face $S B C$, and points $F, L$, and $M$ lie on edges $A B, A C$, and $S C$ respectively, such that $|A L|=\frac{1}{10}|A C|$. It is known that $\boldsymbol{E F L M}$ is an isosceles trapezoid and the length of its base $\boldsymbol{E} \boldsymbol{F}$ is $\sqrt{7}$. Find the volume of the pyramid.

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


We sequentially crack ten eggs into a bowl. Among the eggs, two are rotten, but this only becomes apparent when they are cracked. The yolks render all previously cracked eggs unusable. In such a case, we discard the bowl and continue the process with the remaining eggs. On average, what fraction of the good eggs will be wasted this way?

Answer: \dfrac{2}{3}


3. Given $n$ points on a plane, no three of which lie on the same line. A line is drawn through each pair of points. What is the minimum number of pairwise non-parallel lines among them?

Answer: n


9. Using the 24 -hour clock, the time $16: 11: 22$ on the date $16 / 11 / 22$ (i.e. the 16 th of November 2022) has hours, minutes and seconds equal to date, month and (two-digit) year respectively. Let $\mathrm{S}$ be the number of seconds which will elapse after that date and time until this phenomenon next occurs.
What is the value of $\sqrt{S}$ ?

Answer: 300


The altitude drawn from the vertex of the right angle of a right triangle to the hypotenuse divides the triangle into two triangles, each of which has an inscribed circle. Determine the angles and the area of the triangle formed by the legs of the original triangle and the line passing through the centers of the circles, if the altitude of the original triangle is $h$.

Answer: \frac{h^2}{2}


We need to move a pawn from the bottom left corner of a chessboard to the opposite corner, moving one step to the right or up at a time. How many such paths pass through any of the four center squares?

Answer: 2450