There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Job, Food, Hobby, Beverage, Music-Genre, Pet.
The attributes have the following possible values:
- Job: accountant, chef, dancer, electrician, software-developer
- Food: asparagus, lemon, onion, papaya, pumpkin
- Hobby: collecting, cooking, filmmaking, fishing, reading
- Beverage: 7up, iced-tea, lemonade, milk, tea
- Music-Genre: classical, disco, dubstep, hip-hop, salsa
- Pet: chinchilla, dog, guinea-pig, hamster, mouse

Given the following premises about the line of people:
- Music-Genre:dubstep is on the left or right of Pet:mouse
- Beverage:7up == Food:asparagus or Food:asparagus == Job:dancer, but not both
- Hobby:reading is not to the right of Hobby:cooking
- Music-Genre:salsa is on the left or right of Music-Genre:dubstep
- Hobby:filmmaking is on the left or right of Beverage:tea
- Beverage:lemonade is not to the left of Job:chef
- Hobby:reading and Job:software-developer have different parity positions
- Food:papaya is somewhere between Pet:dog and Pet:mouse
- Job:dancer is not to the right of Hobby:reading
- Pet:guinea-pig and Beverage:milk have the same parity positions
- Pet:chinchilla is somewhere to the right of Music-Genre:dubstep
- Beverage:milk is not to the left of Job:electrician
- Job:electrician and Food:asparagus have the same parity positions
- Food:pumpkin == Pet:chinchilla or Hobby:filmmaking == Food:pumpkin or both
- Food:lemon is on the far left or far right
- Pet:hamster is not to the left of Hobby:reading
- Music-Genre:dubstep is on the left or right of Hobby:cooking
- Job:electrician is somewhere to the left of Music-Genre:salsa
- Music-Genre:disco == Beverage:lemonade or Food:pumpkin == Music-Genre:disco or both
- Beverage:tea and Music-Genre:classical have different parity positions
- Music-Genre:disco != Food:lemon
- Hobby:collecting is on the left or right of Hobby:filmmaking
- Hobby:cooking is on the left or right of Food:lemon
- Pet:mouse is not to the right of Hobby:filmmaking
- Beverage:tea is not to the right of Beverage:iced-tea

Answer the following question:

Question: Question: What is the Food of the person who has/is collecting??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: onion


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Food, Movie-Genre, Pet.
The attributes have the following possible values:
- Job: accountant, architect, fisherman, security-guard
- Food: apricot, blueberry, kiwi, plum
- Movie-Genre: drama, martial-arts, romance, superhero
- Pet: cat, chinchilla, dog, fish

Given the following premises about the line of people:
- Food:plum != Pet:dog
- Food:apricot is somewhere to the right of Movie-Genre:drama
- Pet:fish is on the left or right of Pet:dog
- Pet:dog is on the left of Job:architect
- Job:security-guard == Movie-Genre:romance
- Pet:cat is on the left or right of Job:fisherman
- Food:blueberry is somewhere to the right of Movie-Genre:superhero
- Food:blueberry is in an even position
- Job:security-guard is not to the right of Pet:dog
- Movie-Genre:martial-arts is on the left or right of Pet:chinchilla

Answer the following question:

Question: Question: What is the Job of the person who has/is kiwi??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: security-guard


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Beverage, Music-Genre, Sport, Transport.
The attributes have the following possible values:
- Beverage: coffee, hot-chocolate, juice, water
- Music-Genre: gospel, jazz, reggae, trance
- Sport: golf, snowboarding, surfing, swimming
- Transport: roller, taxi, trike, van

Given the following premises about the line of people:
- Transport:van and Music-Genre:gospel have the same parity positions
- Sport:surfing and Beverage:hot-chocolate have different parity positions
- Sport:swimming is not to the left of Transport:van
- Music-Genre:jazz is not to the left of Sport:surfing
- Music-Genre:gospel is not to the left of Beverage:water
- Music-Genre:reggae == Transport:taxi or Music-Genre:reggae == Beverage:water, but not both
- Beverage:water is somewhere to the left of Transport:van
- Transport:trike and Beverage:coffee have different parity positions
- Sport:golf == Transport:trike or Music-Genre:trance == Sport:golf or both
- Music-Genre:reggae is somewhere to the right of Beverage:juice
- Music-Genre:gospel and Beverage:juice have the same parity positions
- Transport:trike is somewhere to the left of Transport:taxi
- Transport:taxi is not to the right of Beverage:water

Answer the following question:

Question: Question: What is the Beverage of the person who has/is reggae??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: water


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Food, Movie-Genre, Pet, Transport.
The attributes have the following possible values:
- Job: firefighter, photographer, police-officer, security-guard
- Food: carrot, cranberry, cucumber, pineapple
- Movie-Genre: family, horror, musical, scientific
- Pet: dog, goat, mouse, pony
- Transport: bike, ship, snowmobile, trike

Given the following premises about the line of people:
- Movie-Genre:family is somewhere between Movie-Genre:musical and Job:firefighter
- Food:cranberry is somewhere between Pet:mouse and Pet:pony
- Transport:trike is on the far left or far right
- Job:security-guard is somewhere to the right of Transport:bike
- Food:pineapple is somewhere to the right of Movie-Genre:horror
- Pet:pony is somewhere between Pet:goat and Job:firefighter
- Transport:trike is on the right of Job:police-officer
- Job:photographer == Food:cucumber
- Movie-Genre:horror == Pet:pony
- Food:cranberry is on the left of Job:firefighter
- Transport:snowmobile is in an even position

Answer the following question:

Question: Question: What is the Job of the person who has/is cranberry??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: police-officer


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Job, Food, Beverage, Movie-Genre, Pet, Transport.
The attributes have the following possible values:
- Job: accountant, chef, fisherman, scientist, software-developer
- Food: blueberry, cherry, cranberry, grapefruit, tomato
- Beverage: fanta, lemonade, milk, sprite, tea
- Movie-Genre: adventure, family, romance, scientific, western
- Pet: cat, dog, goldfish, hamster, hedgehog
- Transport: airplane, quad-bike, scooter, ship, tram

Given the following premises about the line of people:
- Beverage:lemonade is somewhere to the left of Beverage:tea
- Beverage:lemonade is somewhere between Beverage:milk and Job:software-developer
- Food:tomato == Beverage:fanta or Beverage:fanta == Job:chef or both
- Pet:dog is between Beverage:fanta and Food:cherry
- Beverage:lemonade is on the left of Pet:dog
- Pet:goldfish is not to the left of Beverage:fanta
- Job:scientist is not to the left of Movie-Genre:romance
- Job:fisherman is not to the left of Movie-Genre:family
- Movie-Genre:western == Transport:tram
- Movie-Genre:western is in an even position
- Pet:hedgehog is somewhere to the left of Pet:hamster
- Transport:ship == Job:fisherman or Job:fisherman == Movie-Genre:scientific, but not both
- Beverage:sprite == Job:chef
- Beverage:sprite == Pet:hamster or Beverage:sprite == Food:cranberry or both
- Transport:airplane is somewhere between Beverage:sprite and Food:grapefruit
- Transport:quad-bike is somewhere to the right of Movie-Genre:scientific
- Pet:goldfish is not to the right of Movie-Genre:western
- Beverage:sprite is somewhere to the left of Pet:cat
- Movie-Genre:family is not to the left of Pet:cat
- Food:grapefruit is on the left of Job:accountant

Answer the following question:

Question: Question: What is the Movie-Genre of the person who is software-developer??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: western


There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes:  Job, Hobby, Music-Genre, Nationality, Transport.
The attributes have the following possible values:
- Job: architect, chef, entrepreneur, firefighter, fisherman, musician
- Hobby: card-games, chess, cooking, fishing, skydiving, traveling
- Music-Genre: blues, classical, dubstep, pop, reggae, rock
- Nationality: brazilian, egyptian, french, german, mexican, turkish
- Transport: airplane, bike, boat, bus, skateboard, train

Given the following premises about the line of people:
- Job:musician is on the left of Music-Genre:dubstep
- Nationality:brazilian is somewhere to the left of Music-Genre:rock
- Music-Genre:rock is somewhere to the right of Music-Genre:reggae
- Hobby:chess is somewhere to the left of Hobby:cooking
- Transport:bike is on the left of Job:entrepreneur
- Hobby:skydiving is on the left or right of Transport:bike
- Nationality:french is on the left of Job:fisherman
- Music-Genre:pop != Hobby:card-games
- Nationality:german is somewhere to the left of Hobby:fishing
- Transport:airplane is on the right of Transport:skateboard
- Job:fisherman is somewhere to the right of Transport:airplane
- Hobby:traveling is on the right of Transport:boat
- Job:musician is in an even position
- Music-Genre:rock == Transport:skateboard
- Music-Genre:dubstep is on the left or right of Job:chef
- Hobby:card-games is on the far right
- Music-Genre:rock is on the left or right of Music-Genre:classical
- Transport:train is on the right of Transport:boat
- Nationality:mexican is on the right of Job:firefighter
- Nationality:egyptian is on the left of Transport:bus

Answer the following question:

Question: Question: What is the Nationality of the person who is entrepreneur??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: mexican


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Nationality, Movie-Genre, Music-Genre, Beverage, Pet.
The attributes have the following possible values:
- Nationality: argentine, brazilian, british, polish, spanish
- Movie-Genre: action, animation, crime, documentary, zombie
- Music-Genre: country, electronic, funk, hip-hop, trance
- Beverage: cola, hot-chocolate, juice, lemonade, sprite
- Pet: chinchilla, horse, lizard, rabbit, rat

Given the following premises about the line of people:
- Pet:lizard is somewhere between Beverage:sprite and Beverage:lemonade
- Nationality:argentine == Pet:rabbit
- Beverage:juice is somewhere to the left of Pet:chinchilla
- Beverage:sprite is somewhere to the left of Movie-Genre:crime
- Beverage:hot-chocolate is between Music-Genre:hip-hop and Beverage:cola
- Beverage:cola is on the left of Music-Genre:country
- Beverage:juice is somewhere to the right of Nationality:argentine
- Music-Genre:electronic is on the left of Nationality:brazilian
- Music-Genre:funk == Nationality:spanish
- Movie-Genre:animation is on the far left or far right
- Music-Genre:electronic is on the left or right of Movie-Genre:zombie
- Nationality:spanish == Pet:horse
- Pet:chinchilla == Movie-Genre:action
- Nationality:argentine is on the right of Nationality:british

Answer the following question:

Question: Question: What is the Music-Genre of the person who is brazilian??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: trance


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Beverage, Food, Hobby, Movie-Genre, Sport, Transport.
The attributes have the following possible values:
- Beverage: almond-milk, fanta, hot-chocolate, tea, water
- Food: cherry, pear, pepper, potato, pumpkin
- Hobby: dancing, fishing, gardening, photography, skydiving
- Movie-Genre: family, fantasy, horror, superhero, time-travel
- Sport: biathlon, climbing, golf, lacrosse, skateboarding
- Transport: boat, helicopter, jet-ski, roller, snowmobile

Given the following premises about the line of people:
- Food:cherry == Transport:helicopter
- Sport:skateboarding is on the right of Hobby:fishing
- Hobby:dancing == Sport:climbing
- Transport:helicopter is somewhere to the right of Sport:golf
- Hobby:gardening is on the left of Transport:boat
- Food:pumpkin is on the left or right of Transport:boat
- Food:pepper == Movie-Genre:family
- Sport:skateboarding is somewhere to the left of Beverage:tea
- Beverage:almond-milk is somewhere to the right of Beverage:fanta
- Food:potato is on the left or right of Beverage:water
- Sport:lacrosse is on the left of Transport:roller
- Food:potato is on the left of Movie-Genre:time-travel
- Movie-Genre:superhero is on the far left
- Beverage:hot-chocolate is on the left or right of Sport:skateboarding
- Food:pepper is on the left or right of Beverage:water
- Hobby:photography is on the left or right of Food:potato
- Movie-Genre:fantasy is on the left or right of Hobby:dancing
- Sport:lacrosse is on the right of Transport:snowmobile
- Sport:lacrosse == Hobby:photography

Answer the following question:

Question: Question: What is the Food of the person who has/is tea??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: pear


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Nationality, Hobby, Beverage, Sport, Transport.
The attributes have the following possible values:
- Nationality: american, australian, chinese, colombian
- Hobby: collecting, fishing, hiking, traveling
- Beverage: coffee, milk, mirinda, water
- Sport: biathlon, sailing, water-polo, weightlifting
- Transport: bike, jet-ski, roller, van

Given the following premises about the line of people:
- Transport:van is on the right of Hobby:hiking
- Beverage:coffee is on the far left
- Nationality:colombian == Hobby:fishing
- Sport:sailing is not to the right of Hobby:collecting
- Beverage:water is in an odd position
- Transport:roller == Hobby:collecting
- Beverage:milk is on the far right
- Transport:van is somewhere to the right of Nationality:chinese
- Nationality:australian != Transport:roller
- Nationality:australian is somewhere to the left of Transport:bike
- Sport:biathlon is on the left of Nationality:australian
- Nationality:colombian == Sport:water-polo
- Sport:sailing is on the left or right of Transport:van

Answer the following question:

Question: Question: What is the Hobby of the person who has/is milk??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: fishing


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Job, Hobby, Beverage, Music-Genre, Nationality, Pet.
The attributes have the following possible values:
- Job: lawyer, manager, nurse, scientist, software-developer
- Hobby: baking, filmmaking, hiking, puzzles, woodworking
- Beverage: cola, iced-tea, milk, tea, water
- Music-Genre: electronic, folk, funk, soul, trance
- Nationality: argentine, australian, dutch, german, russian
- Pet: cat, dog, ferret, hedgehog, snake

Given the following premises about the line of people:
- Hobby:puzzles is somewhere to the left of Beverage:water
- Job:manager is on the far left or far right
- Nationality:dutch is somewhere to the left of Hobby:baking
- Music-Genre:electronic is on the left or right of Hobby:woodworking
- Job:nurse is between Pet:snake and Beverage:milk
- Music-Genre:folk is on the left of Nationality:argentine
- Pet:dog is somewhere to the right of Beverage:cola
- Beverage:tea == Pet:dog
- Nationality:russian is somewhere to the left of Hobby:baking
- Hobby:hiking is on the left of Music-Genre:soul
- Nationality:australian is on the left of Pet:hedgehog
- Music-Genre:trance == Job:software-developer
- Nationality:dutch is on the left or right of Beverage:tea
- Hobby:baking is on the left or right of Beverage:cola
- Job:scientist is on the left of Music-Genre:soul
- Beverage:water is somewhere to the left of Pet:cat
- Beverage:iced-tea is on the far right

Answer the following question:

Question: Question: What is the Music-Genre of the person who is scientist??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: folk


There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes:  Job, Food, Beverage, Transport.
The attributes have the following possible values:
- Job: analyst, lawyer, librarian, paramedic, pilot, videographer
- Food: grapefruit, onion, pineapple, plum, pomegranate, radish
- Beverage: 7up, hot-chocolate, milk, mirinda, soy-milk, tea
- Transport: boat, bus, jet-ski, motorbike, quad-bike, ship

Given the following premises about the line of people:
- Beverage:hot-chocolate and Food:radish have the same parity positions
- Transport:jet-ski is not to the right of Beverage:7up
- Job:videographer is somewhere to the right of Food:radish
- Food:plum is not to the left of Job:analyst
- Job:videographer and Transport:boat have the same parity positions
- Transport:bus is not to the left of Job:pilot
- Transport:quad-bike is on the far right
- Food:plum is between Food:onion and Job:librarian
- Food:plum is in an odd position
- Transport:ship is somewhere to the right of Job:videographer
- Food:grapefruit is on the left or right of Beverage:tea
- Job:paramedic is somewhere to the left of Job:librarian
- Transport:motorbike is not to the left of Transport:bus
- Beverage:mirinda is somewhere to the right of Beverage:soy-milk
- Beverage:milk is between Transport:jet-ski and Food:pomegranate
- Transport:ship is somewhere to the left of Food:pineapple
- Food:pineapple == Transport:ship or Food:pineapple == Beverage:tea, but not both
- Job:lawyer and Beverage:7up have different parity positions

Answer the following question:

Question: Question: What is the Job of the person who has/is pineapple??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: librarian


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Nationality, Music-Genre, Beverage, Pet, Sport.
The attributes have the following possible values:
- Nationality: chinese, german, indonesian, spanish
- Music-Genre: gospel, hip-hop, indie, punk
- Beverage: hot-chocolate, iced-tea, lemonade, sprite
- Pet: dog, goat, rat, turtle
- Sport: cricket, golf, soccer, surfing

Given the following premises about the line of people:
- Beverage:sprite is between Pet:dog and Nationality:indonesian
- Sport:surfing is on the left or right of Sport:cricket
- Pet:goat is somewhere to the right of Beverage:sprite
- Pet:rat is somewhere to the left of Pet:dog
- Sport:golf is on the left or right of Beverage:lemonade
- Sport:cricket is somewhere to the right of Music-Genre:punk
- Music-Genre:gospel == Nationality:indonesian
- Pet:dog is somewhere to the left of Sport:soccer
- Sport:golf is somewhere to the right of Sport:surfing
- Music-Genre:hip-hop is somewhere to the right of Beverage:iced-tea
- Nationality:spanish is on the far left or far right
- Beverage:iced-tea is not to the left of Pet:dog
- Music-Genre:hip-hop != Nationality:german

Answer the following question:

Question: Question: What is the Nationality of the person who has/is indie??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Answer: german


The base-10 integers 36, 64, and 81 can be converted into other bases so that their values are represented by the same digits $\triangle\Box\Box$, where $\triangle$ and $\Box$ are two distinct digits from 0-9. What is the value of $\triangle\Box\Box$?

Answer: 100


2. On a chessboard, four rooks are placed in such a way that they attack all the white squares.

a) Provide an example of such an arrangement.

b) Determine the number of such arrangements

Zhenodarov R.G.

Answer: 48


[ Arithmetic. Mental calculation, etc.]

When "the day after tomorrow" becomes "yesterday," then "today" will be as far from Sunday as the day that was "today" when "yesterday" was "tomorrow." What day of the week do you think it is today?

Answer: Wednesday


Given the polynomial \( p(n) = n^{3} - n^{2} - 5n + 2 \). Find all integers \( n \) such that \( p(n)^{2} \) is a perfect square of a prime number.

Answer: -3, -1, 0, 1, 3


4. If a positive integer has only the digits 0 or 1 in its decimal representation, then it is called a "simple number". Find the smallest positive integer $k$, such that for any positive integer $n$, there exists $n=a_{1} \pm a_{2} \pm \cdots \pm a_{k}$, where $a_{i}(1 \leqslant i \leqslant k)$ are simple numbers.

Answer: 9


On an $A B=8 \mathrm{~km}$ long road, buses travel in both directions at a speed of $12 \mathrm{~km}$ per hour. The first bus from each location departs at 6 o'clock, and a new bus departs every 10 minutes.

A pedestrian starts from $A$ towards $B$ at $81 / 4$ hours; their speed is $4 \mathrm{~km}$ per hour.

Determine graphically how many oncoming buses the pedestrian encounters, and when and where.

Answer: 16


4. A4 (MON) Let $\mathbb{R}$ denote the set of all real numbers and $\mathbb{R}^{+}$ the subset of all positive ones. Let $\alpha$ and $\beta$ be given elements in $\mathbb{R}$, not necessarily distinct. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}$ such that
$$ f(x) f(y)=y^{\alpha} f\left(\frac{x}{2}\right)+x^{\beta} f\left(\frac{y}{2}\right) \quad \text { for all } x \text { and } y \text { in } \mathbb{R}^{+} . $$

Answer: 0


60th Putnam 1999 Problem B1 The triangle ABC has AC = 1, angle ACB = 90 o , and angle BAC = φ. D is the point between A and B such that AD = 1. E is the point between B and C such that angle EDC = φ. The perpendicular to BC at E meets AB at F. Find lim φ→0 EF.

Answer: \dfrac{1}{3}


Given a parallelogram \( ABCD \) with point \( M \) as the midpoint of side \( BC \). On side \( AD \), point \( K \) is chosen such that \( BK = BM \) and quadrilateral \( KBMD \) is inscribed in a circle.

(a) What is the length of segment \( MD \) if \( AD = 19 \)?

(b) What is the measure of angle \( KMD \) if \( \angle BAD = 44^\circ \)?

Answer: 9.5


Square $A B C D$ has centre $O$. Points $P$ and $Q$ are on $A B, R$ and $S$ are on $B C, T$ and $U$ are on $C D$, and $V$ and $W$ are on $A D$, as shown, so that $\triangle A P W, \triangle B R Q, \triangle C T S$, and $\triangle D V U$ are isosceles and $\triangle P O W, \triangle R O Q, \triangle T O S$, and $\triangle V O U$ are equilateral. What is the ratio of the area of $\triangle P Q O$ to that of $\triangle B R Q$ ?

![](https://cdn.mathpix.com/cropped/2024_04_17_d13f9eaf1d614231018ag-4.jpg?height=393&width=401&top_left_y=389&top_left_x=924)

Answer: 1


1. In a certain sports association, each team can choose at most $t$ distinctive colors for their team uniforms. For a set $S$ of teams, if each team in $S$ can be assigned one of their distinctive colors such that no team in $S$ is assigned any distinctive color of another team in $S$, then the set $S$ is called "color-identifiable." For all positive integers $n, t$, find the maximum value of the integer $g(n, t)$ such that in any sports association where all teams' distinctive colors consist of exactly $n$ different colors, there always exists a color-identifiable set of at least $g(n, t)$ teams.

Answer: \left\lceil \frac{n}{t} \right\rceil


Answer: x(1-y)(1-z) + (1-x)y(1-z) + (1-x)(1-y)z < 1


Let \( f(x) \) be a function defined on \( \mathbf{R} \) such that \( f(0) = 0 \), \( f(x) + f(1-x) = 1 \), \( f\left(\frac{x}{3}\right) = \frac{1}{2} f(x) \), and for \( 0 \leqslant x_{1} < x_{2} \leqslant 1 \), we have \( f(x_{1}) \leqslant f(x_{2}) \). Find \( f\left(\frac{1}{2011}\right) \).

Answer: \dfrac{1}{128}


4. Let the arithmetic sequence $\left\{a_{n}\right\}$ have a common difference of $d(d \neq 0)$, and the sum of the first $n$ terms be $S_{n}$. If the sequence $\left\{\sqrt{8 S_{n}+2 n}\right\}$ is also an arithmetic sequence with a common difference of $d$, then the general term of the sequence $\left\{a_{n}\right\}$ is $a_{n}=$ $\qquad$

Answer: \dfrac{16n - 9}{4}


Answer: 44


We only know about some crates that their total weight is 10 tons, and the weight of no crate is greater than 1 ton. We want to transport the crates with 3-ton trucks. What is the minimum number of trucks that are always sufficient to transport the crates?

Answer: 5


1. A competition has $n \geqslant 2$ participants, lasting $k$ days. Each day, the participants' scores are $1,2, \cdots, n$, and all participants' total scores are 26 points. Find all possible pairs $(n, k)$.

Answer: (3, 13)


12. Grandpa Hua Luogeng was born on November 12, 1910. Arrange these numbers into an integer, and factorize it as 19101112 $=1163 \times 16424$. Are either of the numbers 1163 and 16424 prime? Please explain your reasoning.

Answer: 1163\text{ is prime, }16424\text{ is not prime}


Question 178: In the Cartesian coordinate system $x O y$, let a moving line $l$ be tangent to the parabola $\Gamma: y^{2}=4 x$, and intersect the hyperbola $\Omega: x^{2}-y^{2}=1$ at points $A$ and $B$ on the left and right branches, respectively. Find the minimum value of the area of $\triangle A O B$.

Answer: 2\sqrt{5}


10. All integers from 1 to 222222222 are written in sequence. How many times does the digit 0 appear in the recording of these numbers?

## 2. Factorization of numbers into products and their decomposition into sums

In solving some of the subsequent problems, the following notations prove useful.

The symbol $[x]$ (read as "the integer part of $x$") denotes the greatest integer not exceeding $x$. For example,

$\left[\frac{3}{2}\right]=1, \quad[10.85]=10, \quad[5]=5, \quad[-8.2]=-9$, etc.

(however, we will only need the symbol $[x]$ for positive numbers $x$).

The symbol $(x)$ (read as "the nearest integer to $x$") denotes the integer closest to the number $x$. For example,

$$
(5.4)=5, \quad(8.73)=9, \quad(6)=6, \quad(-2.8)=-3
$$

and so on.

Clearly, $(x)$ will be equal to $[x]$ or $[x]+1$ depending on whether the difference $x - [x]$ is less than or greater than half. In the case where $x - [x] = \frac{1}{2}$, $(x)$ can be taken as either $[x]$ or $[x]+1$; in this case, it is agreed to consider $(x) = [x]$ (however, we will use the symbol $(x)$ only in cases where $x - [x] \neq \frac{1}{2}$, so there will be no ambiguity as to whether to consider $(x)$ equal to $[x]$ or $[x]+1$).

In all subsequent problems where the number $n$ is mentioned in the condition, this number is considered to be a positive integer.

Answer: 175308642


Answer: (x^2 - yz)(y^2 - zx)(z^2 - xy)


1. Adil was born in 2015 . His younger sister Bav was born in 2018 . What is the minimum number of days by which Adil is older than Bav?

Answer: 732


On a table, there are 100 identical-looking coins, of which 85 are counterfeit and 15 are real. You have a miracle tester at your disposal, in which you can place two coins and get one of three results - "both coins are real," "both coins are counterfeit," and "the coins are different." Can you find all the counterfeit coins in 64 such tests? (K. Knop)

Answer: Yes


Example 5 In a local area network consisting of several computers, for any $m(m \geqslant 3)$ computers there is a unique one that is connected to all of them. How many computers at most can be connected to the same computer in this local area network?

Answer: m


Define $K(n,0)=\varnothing $ and, for all nonnegative integers m and n, $K(n,m+1)=\left\{ \left. k \right|\text{ }1\le k\le n\text{ and }K(k,m)\cap K(n-k,m)=\varnothing  \right\}$.  Find the number of elements of $K(2004,2004)$.

Answer: 127


[Isosceles, Inscribed, and Circumscribed Trapezoids]

A circle with a radius of 5 is inscribed in some angle. The chord connecting the points of tangency is 8. Two tangents parallel to the chord are drawn to the circle. Find the sides of the resulting trapezoid.

Answer: 5


In a right parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$ with bases $A B C D$ and $A_{1} B_{1} C_{1} D_{1}$, it is known that $A B=29, A D=36$, $B D=25, A A_{1}=48$. Find the area of the section $A B_{1} C_{1} D$.

Answer: 1872


Ashwin the frog is traveling on the $xy$-plane in a series of $2^{2017}-1$ steps, starting at the origin. At the $n^{\text{th}}$ step, if $n$ is odd, Ashwin jumps one unit to the right. If $n$ is even, Ashwin jumps $m$ units up, where $m$ is the greatest integer such that $2^{m}$ divides $n$. If Ashwin begins at the origin, what is the area of the polygon bounded by Ashwin's path, the line $x=2^{2016}$, and the $x$-axis?

Answer: 2^{2015} \left(2^{2017} - 2018\right)


For a real parameter $a$, solve the equation $x^4-2ax^2+x+a^2-a=0$. Find all $a$ for which all solutions are real.

Answer: [\dfrac{3}{4}, \infty)


The sequence 2, 3, 5, 6, 7, 10, 11, $\ldots$ contains all the positive integers from least to greatest that are neither squares nor cubes nor perfect fifth powers (in the form of $x^{5}$, where $x$ is an integer). What is the $1000^{\mathrm{th}}$ term of the sequence?

Answer: 1041


A sequence of positive integers with $a_1 = 1$ and $a_9+a_{10}=646$ is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all $n\ge1$, the terms $a_{2n-1}$, $a_{2n}$, and $a_{2n+1}$ are in geometric progression, and the terms $a_{2n}$, $a_{2n+1}$, and $a_{2n+2}$ are in arithmetic progression. Let $a_n$ be the greatest term in this sequence that is less than 1000.  Find $n+a_n$.

Answer: 973


Suppose that $\overline{abcd}$ is a four-digit integer with no digits equal to zero such that $\overline{ab}$, $\overline{bc}$, and $\overline{cd}$ are distinct integers that each divide into $\overline{abcd}$. Find the smallest possible value of $\overline{abcd}$.

Answer: 1155


Three circular arcs of radius 5 units bound the region shown. Arcs $AB$ and $AD$ are quarter-circles, and arc $BCD$ is a semicircle. What is the area, in square units, of the region? [asy]
/* AMC8 2000 #19 Problem */
draw((0,0)..(1,1)..(2,0));
draw((0,0)..(.7,-.3)..(1,-1));
draw((1,-1)..(1.3, -0.3)..(2,0));
label("$A$", (1,-1), SW);
label("$B$", (0,0), W);
label("$C$", (1,1),N);
label("$D$", (2,0),E);
[/asy]

Answer: 50


In the diagram, points $X$, $Y$ and $Z$ are on the sides of $\triangle UVW$, as shown.  Line segments $UY$, $VZ$ and $WX$ intersect at $P$.  Point $Y$ is on $VW$ such that $VY:YW=4:3$. If $\triangle PYW$ has an area of 30 and $\triangle PZW$ has an area of 35, determine the area of $\triangle UXP$. [asy]
size(6cm);
pair v = (0, 0); pair w = (10, 0); pair u = (3.5, 7);
pair y = 4 * w / 7;
pair x = 56 * u / 140;
pair p = IP(w--x, u--y);
pair z = IP(v--(10 * p), u--w);
draw(u--v--w--cycle);
draw(u--y);draw(x--w);draw(z--v);

label("$U$", u, N);
label("$X$", x, NW);
label("$P$", p, NE + 0.2 * W);
label("$Z$", z, NE);
label("$V$", v, SW);
label("$Y$", y, S);
label("$W$", w, SE);/*
label("$a$", centroid(p, v, y), fontsize(10));
label("$b$", centroid(p, z, u), fontsize(10));
label("$c$", centroid(p, u, x), fontsize(10));
label("$d$", centroid(p, x, v), fontsize(10));
label("$30$", centroid(p, y, w) + 0.2 * W, fontsize(10));
label("$35$", centroid(p, z, w), fontsize(10));*/
[/asy]

Answer: 84


Let $n$ be the smallest positive integer such that $mn$ is a perfect $k$th power of an integer for some $k \ge 2$, where $m=2^{1980} \cdot 3^{384} \cdot 5^{1694} \cdot 7^{343}$. What is $n+k$?

Answer: 13


A $4\times 4\times 4$ open cubical box contains 64 identical small cubes that exactly fill the box. How many of these small cubes touch the bottom or one of the four lateral sides of the box?

Answer: 52


The dartboard below has a radius of 6 inches. Each of the concentric circles has a radius two inches less than the next larger circle. If nine darts land randomly on the target, how many darts would we expect to land in a non-shaded region?

[asy]import graph;
fill(Circle((0,0),15),gray(0.7));
fill(Circle((0,0),10),white);
draw(Circle((0,0),20));
draw(Circle((0,0),15));
draw(Circle((0,0),10));
[/asy]

Answer: 6


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Carol`, `Eric`, `Bob`, `Alice`, `Peter`
 - Each person has a unique birthday month: `feb`, `mar`, `sept`, `jan`, `may`, `april`
 - Everyone has something unique for lunch: `stew`, `soup`, `grilled cheese`, `stir fry`, `spaghetti`, `pizza`
 - People have unique heights: `very short`, `average`, `super tall`, `short`, `very tall`, `tall`
 - People own unique car models: `chevrolet silverado`, `ford f150`, `bmw 3 series`, `tesla model 3`, `toyota camry`, `honda civic`

## Clues:
1. The person who owns a Honda Civic is the person who is short.
2. The person who owns a Ford F-150 is in the fifth house.
3. The person who loves stir fry is somewhere to the left of Eric.
4. The person whose birthday is in May is somewhere to the left of Carol.
5. The person who is very short is somewhere to the left of the person whose birthday is in April.
6. The person who owns a BMW 3 Series is not in the third house.
7. There are two houses between the person who loves stir fry and the person who is a pizza lover.
8. The person who loves the soup is directly left of Eric.
9. The person who loves the spaghetti eater and the person whose birthday is in May are next to each other.
10. Alice is directly left of the person who owns a BMW 3 Series.
11. The person who owns a Tesla Model 3 is somewhere to the left of the person who is tall.
12. The person who is very tall is the person who owns a Toyota Camry.
13. Peter is directly left of the person who is a pizza lover.
14. The person who loves the stew is not in the third house.
15. There is one house between the person whose birthday is in September and the person who is very short.
16. There is one house between the person whose birthday is in March and the person who is super tall.
17. The person who is tall is Bob.
18. The person whose birthday is in May is somewhere to the right of Alice.
19. The person who is very short is in the fourth house.
20. The person whose birthday is in March is the person who is short.
21. Carol is the person who owns a Tesla Model 3.
22. Eric is the person whose birthday is in January.
 What is the value of attribute CarModel for the person whose attribute Height is very tall? Please reason step by step, and put your final answer within \boxed{}

Answer: toyota camry


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Peter`, `Carol`, `Eric`
 - Each person prefers a unique type of vacation: `mountain`, `city`, `cultural`, `camping`, `beach`, `cruise`
 - Everyone has a favorite smoothie: `lime`, `desert`, `cherry`, `blueberry`, `dragonfruit`, `watermelon`
 - People have unique favorite sports: `tennis`, `volleyball`, `basketball`, `baseball`, `swimming`, `soccer`
 - Everyone has a unique favorite cigar: `blends`, `yellow monster`, `dunhill`, `pall mall`, `blue master`, `prince`
 - People have unique heights: `tall`, `very tall`, `average`, `super tall`, `very short`, `short`

## Clues:
1. Peter is the Prince smoker.
2. The person who smokes many unique blends is the person who enjoys mountain retreats.
3. The person who loves baseball is the person who is tall.
4. The person who goes on cultural tours is the person who drinks Blueberry smoothies.
5. The person who likes Cherry smoothies is not in the fourth house.
6. The person who loves soccer is the person who enjoys camping trips.
7. The person who is super tall is somewhere to the right of the person who likes going on cruises.
8. The person who drinks Blueberry smoothies is the person who is tall.
9. The person who is very tall is the Dunhill smoker.
10. The person who enjoys mountain retreats is Eric.
11. The person who is super tall is Alice.
12. Bob is the person who smokes Blue Master.
13. The person who loves volleyball is in the first house.
14. Peter and the person who drinks Lime smoothies are next to each other.
15. Carol is the person who likes Cherry smoothies.
16. Peter is the person who is short.
17. The person who smokes many unique blends is somewhere to the right of the Dragonfruit smoothie lover.
18. The person who loves swimming is Carol.
19. The person who is very short is the person who smokes Blue Master.
20. The person who loves tennis is somewhere to the left of the person who is very short.
21. The person who loves swimming is directly left of the person who enjoys camping trips.
22. The person who drinks Blueberry smoothies is not in the third house.
23. Alice is somewhere to the right of the person who drinks Lime smoothies.
24. The person who loves tennis is the person who is super tall.
25. The person who smokes Yellow Monster and the person who prefers city breaks are next to each other.
26. The Watermelon smoothie lover is the person who is very short.
27. The person who loves beach vacations is somewhere to the left of the person who is very tall.
 What is the value of attribute House for the person whose attribute FavoriteSport is volleyball? 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: `Eric`, `Arnold`, `Peter`
 - Each person has an occupation: `engineer`, `teacher`, `doctor`
 - Each person has a unique favorite drink: `water`, `milk`, `tea`
 - People use unique phone models: `google pixel 6`, `samsung galaxy s21`, `iphone 13`

## Clues:
1. The person who uses a Samsung Galaxy S21 is directly left of Arnold.
2. The person who uses a Google Pixel 6 is the one who only drinks water.
3. The person who likes milk is the person who is an engineer.
4. Eric is the person who likes milk.
5. The person who is a teacher is directly left of the tea drinker.
 What is the value of attribute Drink for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: tea


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Carol`, `Arnold`, `Eric`, `Peter`
 - Everyone has a favorite smoothie: `dragonfruit`, `watermelon`, `blueberry`, `lime`, `desert`, `cherry`
 - Each person has a unique level of education: `high school`, `doctorate`, `master`, `associate`, `trade school`, `bachelor`
 - People own unique car models: `ford f150`, `honda civic`, `chevrolet silverado`, `bmw 3 series`, `toyota camry`, `tesla model 3`

## Clues:
1. The person who owns a Chevrolet Silverado is directly left of the person who likes Cherry smoothies.
2. The person who owns a Toyota Camry is not in the first house.
3. The person who owns a BMW 3 Series is directly left of the person who owns a Ford F-150.
4. The Desert smoothie lover is the person who attended trade school.
5. The person with an associate's degree is somewhere to the left of the person who drinks Lime smoothies.
6. Alice is directly left of Eric.
7. The person with a bachelor's degree is in the sixth house.
8. Bob is the person who owns a Chevrolet Silverado.
9. The person with an associate's degree is somewhere to the right of the person who drinks Blueberry smoothies.
10. The Dragonfruit smoothie lover is somewhere to the left of the person who drinks Blueberry smoothies.
11. The person who drinks Lime smoothies is the person who owns a BMW 3 Series.
12. Eric is somewhere to the right of the person with a master's degree.
13. The person with a high school diploma is in the first house.
14. The person with a doctorate is Peter.
15. The person who drinks Blueberry smoothies is directly left of the Desert smoothie lover.
16. Carol is not in the fourth house.
17. The person who owns a Tesla Model 3 and the person with a master's degree are next to each other.
 What is the value of attribute CarModel for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}

Answer: honda civic


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 BookGenre for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: fantasy


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`, `Eric`, `Arnold`
 - Each person prefers a unique type of vacation: `mountain`, `beach`, `city`
 - Each person has a unique hobby: `gardening`, `cooking`, `photography`
 - People have unique favorite sports: `tennis`, `soccer`, `basketball`
 - They all have a unique favorite flower: `carnations`, `daffodils`, `lilies`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`

## Clues:
1. The person who loves soccer is the person who loves a bouquet of daffodils.
2. Arnold is the person who loves the boquet of lilies.
3. The person who loves beach vacations is directly left of Eric.
4. The Desert smoothie lover is the person who loves basketball.
5. The person who loves the boquet of lilies is directly left of the person who loves a carnations arrangement.
6. Eric is directly left of Peter.
7. The person who loves basketball and the person who loves a bouquet of daffodils are next to each other.
8. The person who loves a bouquet of daffodils is the person who likes Cherry smoothies.
9. The person who enjoys mountain retreats is the person who loves cooking.
10. The photography enthusiast is the person who loves basketball.
 What is the value of attribute Flower for the person whose attribute Hobby is gardening? Please reason step by step, and put your final answer within \boxed{}

Answer: lilies


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Bob`, `Eric`, `Arnold`, `Carol`
 - Each person lives in a unique style of house: `mediterranean`, `modern`, `ranch`, `craftsman`, `victorian`, `colonial`

## Clues:
1. Alice is not in the fifth house.
2. The person in a Mediterranean-style villa is directly left of the person in a Craftsman-style house.
3. The person in a Craftsman-style house is in the fifth house.
4. Carol is directly left of Alice.
5. The person in a modern-style house is Eric.
6. The person residing in a Victorian house is not in the third house.
7. Bob is in the first house.
8. Bob and Eric are next to each other.
9. Peter is not in the third house.
10. The person in a ranch-style home is somewhere to the right of the person residing in a Victorian house.
11. The person living in a colonial-style house is somewhere to the left of Carol.
 What is the value of attribute HouseStyle for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: ranch


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`, `Arnold`, `Alice`, `Eric`, `Bob`
 - People own unique car models: `honda civic`, `tesla model 3`, `bmw 3 series`, `ford f150`, `toyota camry`
 - Each person prefers a unique type of vacation: `beach`, `cruise`, `city`, `camping`, `mountain`

## Clues:
1. The person who owns a Honda Civic is in the second house.
2. Peter is the person who enjoys camping trips.
3. The person who prefers city breaks is somewhere to the left of the person who enjoys mountain retreats.
4. The person who owns a Ford F-150 is somewhere to the left of Arnold.
5. The person who owns a BMW 3 Series is Eric.
6. The person who loves beach vacations is in the first house.
7. The person who enjoys mountain retreats is Bob.
8. The person who owns a Tesla Model 3 is in the first house.
 What is the value of attribute House for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `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 Food for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: stew


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

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

Answer: watermelon


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

## Clues:
1. The fish enthusiast is in the first house.
2. Carol is somewhere to the left of the person in a Mediterranean-style villa.
3. The bird keeper is the person who loves purple.
4. The person in a ranch-style home is the one who only drinks water.
5. The coffee drinker is not in the fourth house.
6. The person living in a colonial-style house and the boba tea drinker are next to each other.
7. The cat lover is the person who is a teacher.
8. The person in a modern-style house is in the third house.
9. The person who is an engineer and Peter are next to each other.
10. Arnold is the person who is an artist.
11. The person whose favorite color is red is Arnold.
12. The coffee drinker is somewhere to the right of the person who is a teacher.
13. The bird keeper is the person in a ranch-style home.
14. The rabbit owner is in the sixth house.
15. The boba tea drinker is Peter.
16. Eric is the cat lover.
17. The person who is a nurse is directly left of the person in a modern-style house.
18. The bird keeper is Carol.
19. There are two houses between the cat lover and the boba tea drinker.
20. Bob is the tea drinker.
21. The person residing in a Victorian house is the dog owner.
22. The person whose favorite color is green is in the fourth house.
23. The person who loves blue is the root beer lover.
24. The person who loves white is not in the sixth house.
25. Peter is the person who is a doctor.
 What is the value of attribute HouseStyle for the person whose attribute House is 6? Please reason step by step, and put your final answer within \boxed{}

Answer: mediterranean


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Bob`, `Alice`, `Carol`, `Peter`
 - The people are of nationalities: `german`, `brit`, `dane`, `chinese`, `swede`, `norwegian`
 - People have unique favorite music genres: `pop`, `jazz`, `hip hop`, `classical`, `country`, `rock`
 - Each person has a unique level of education: `doctorate`, `associate`, `master`, `bachelor`, `high school`, `trade school`

## Clues:
1. The Chinese is the person who loves pop music.
2. The person with a bachelor's degree is directly left of Arnold.
3. The person with a master's degree is in the first house.
4. Carol is somewhere to the left of Peter.
5. The Dane is the person with a doctorate.
6. The person who loves classical music is the person with a master's degree.
7. The person with a bachelor's degree is in the fifth house.
8. Eric is somewhere to the right of Bob.
9. Carol is the British person.
10. The person with a doctorate is somewhere to the right of the Norwegian.
11. The person with a doctorate is the person who loves rock music.
12. The Norwegian is somewhere to the right of the German.
13. The person who attended trade school is somewhere to the left of the person with a high school diploma.
14. The person who attended trade school and the Chinese are next to each other.
15. The person who loves jazz music is Arnold.
16. The person who loves hip-hop music is directly left of Alice.
 What is the value of attribute House for the person whose attribute Nationality is dane? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Arnold`, `Bob`, `Alice`, `Peter`, `Eric`
 - People use unique phone models: `samsung galaxy s21`, `oneplus 9`, `google pixel 6`, `iphone 13`, `xiaomi mi 11`, `huawei p50`
 - Each person has a unique type of pet: `dog`, `cat`, `fish`, `rabbit`, `hamster`, `bird`
 - They all have a unique favorite flower: `daffodils`, `tulips`, `carnations`, `lilies`, `iris`, `roses`
 - People own unique car models: `toyota camry`, `chevrolet silverado`, `bmw 3 series`, `tesla model 3`, `honda civic`, `ford f150`

## Clues:
1. There is one house between the person who owns a Tesla Model 3 and the person who uses a OnePlus 9.
2. There are two houses between the person who owns a dog and the person who owns a Honda Civic.
3. There is one house between the person who uses an iPhone 13 and the person who uses a Xiaomi Mi 11.
4. The person who loves the vase of tulips is somewhere to the right of the person who loves a bouquet of daffodils.
5. The person who owns a Chevrolet Silverado is directly left of the person who loves the rose bouquet.
6. Alice is the person who owns a dog.
7. There are two houses between the person who uses a Google Pixel 6 and the person who uses a Huawei P50.
8. There are two houses between the person who owns a Ford F-150 and the person who loves the boquet of lilies.
9. The person who uses a Google Pixel 6 is somewhere to the left of the person who has a cat.
10. The person who loves a bouquet of daffodils is not in the first house.
11. There is one house between the person with a pet hamster and the person who has a cat.
12. The person who uses a OnePlus 9 is Peter.
13. The person who uses a Xiaomi Mi 11 is the person who has a cat.
14. The person who loves the boquet of lilies is Eric.
15. The person who owns a Tesla Model 3 and the person who uses a Google Pixel 6 are next to each other.
16. The person who owns a rabbit is not in the second house.
17. The person who loves a carnations arrangement is in the second house.
18. The person who loves the boquet of lilies is somewhere to the right of the person who owns a Honda Civic.
19. There are two houses between Arnold and the person who owns a Chevrolet Silverado.
20. There are two houses between Peter and Bob.
21. The person who keeps a pet bird is in the fifth house.
22. The person who owns a Toyota Camry is the person who has a cat.
 What is the value of attribute Flower for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: iris


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Eric`, `Peter`, `Carol`, `Alice`, `Arnold`
 - People have unique favorite music genres: `rock`, `classical`, `pop`, `hip hop`, `jazz`, `country`

## Clues:
1. Alice is somewhere to the right of the person who loves classical music.
2. The person who loves pop music is directly left of the person who loves classical music.
3. Bob is in the fourth house.
4. The person who loves country music is Bob.
5. The person who loves hip-hop music is Peter.
6. Carol is somewhere to the right of the person who loves jazz music.
7. Peter is directly left of Eric.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 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 Cigar for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: prince


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

Answer: jazz


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Bob`, `Eric`, `Carol`, `Alice`
 - People own unique car models: `bmw 3 series`, `ford f150`, `tesla model 3`, `toyota camry`, `honda civic`, `chevrolet silverado`
 - Each person has a unique hobby: `woodworking`, `painting`, `knitting`, `photography`, `gardening`, `cooking`

## Clues:
1. The person who loves cooking is not in the first house.
2. The person who owns a Tesla Model 3 is in the fifth house.
3. The person who owns a Honda Civic is in the third house.
4. Alice and the person who owns a Honda Civic are next to each other.
5. There is one house between the person who owns a Chevrolet Silverado and Carol.
6. The photography enthusiast is directly left of the person who paints as a hobby.
7. Peter is somewhere to the right of the person who owns a Tesla Model 3.
8. The person who owns a Honda Civic is directly left of Bob.
9. The person who enjoys knitting is somewhere to the right of the person who owns a Toyota Camry.
10. The person who enjoys knitting is somewhere to the left of Eric.
11. The woodworking hobbyist is directly left of the person who owns a BMW 3 Series.
12. The person who paints as a hobby is in the third house.
 What is the value of attribute Hobby for the person whose attribute CarModel is toyota camry? Please reason step by step, and put your final answer within \boxed{}

Answer: photography


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Alice`, `Bob`, `Arnold`, `Carol`
 - People have unique favorite book genres: `mystery`, `fantasy`, `romance`, `historical fiction`, `science fiction`, `biography`
 - Everyone has a favorite smoothie: `cherry`, `desert`, `lime`, `watermelon`, `blueberry`, `dragonfruit`
 - The people keep unique animals: `fish`, `rabbit`, `bird`, `cat`, `horse`, `dog`
 - People have unique favorite music genres: `classical`, `hip hop`, `country`, `jazz`, `rock`, `pop`
 - Everyone has a unique favorite cigar: `prince`, `dunhill`, `blends`, `pall mall`, `blue master`, `yellow monster`

## Clues:
1. The person who smokes many unique blends is the person who loves fantasy books.
2. The person who loves biography books is the Desert smoothie lover.
3. There are two houses between the person who loves pop music and the person who loves country music.
4. The rabbit owner is in the third house.
5. Arnold is the bird keeper.
6. The person who loves country music is in the fifth house.
7. The person who loves mystery books is not in the first house.
8. The person who drinks Lime smoothies is not in the third house.
9. The fish enthusiast is somewhere to the right of the person who loves rock music.
10. The Prince smoker is somewhere to the left of the person who likes Cherry smoothies.
11. Bob is the cat lover.
12. Eric is the person who likes Cherry smoothies.
13. The person who smokes Yellow Monster is in the sixth house.
14. The Prince smoker is the Dragonfruit smoothie lover.
15. Carol is directly left of Arnold.
16. The person who smokes Blue Master is directly left of the person who loves classical music.
17. The person who loves classical music is the person who loves historical fiction books.
18. The Dunhill smoker and the person who loves science fiction books are next to each other.
19. The person who keeps horses is directly left of Peter.
20. The Watermelon smoothie lover is the cat lover.
21. The person who smokes many unique blends is directly left of the person who loves jazz music.
22. The cat lover is in the first house.
23. The Desert smoothie lover is the person who loves jazz music.
24. The person who drinks Lime smoothies is directly left of the fish enthusiast.
25. The person who smokes Blue Master is not in the fifth house.
 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: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Bob`, `Arnold`, `Peter`, `Alice`, `Eric`
 - Each person has a unique favorite drink: `boba tea`, `coffee`, `root beer`, `water`, `milk`, `tea`
 - Each person lives in a unique style of house: `ranch`, `victorian`, `craftsman`, `mediterranean`, `colonial`, `modern`
 - Everyone has a unique favorite cigar: `dunhill`, `prince`, `yellow monster`, `blue master`, `blends`, `pall mall`

## Clues:
1. Eric is the Dunhill smoker.
2. Peter and the root beer lover are next to each other.
3. The Prince smoker is Bob.
4. The person in a modern-style house is somewhere to the left of Arnold.
5. The one who only drinks water is somewhere to the right of Bob.
6. The person in a modern-style house is the person who smokes Blue Master.
7. The person in a ranch-style home is directly left of the person in a modern-style house.
8. The person who smokes Yellow Monster is somewhere to the right of the coffee drinker.
9. The root beer lover is in the sixth house.
10. The person who likes milk is somewhere to the left of the tea drinker.
11. Arnold is the person living in a colonial-style house.
12. The person in a Craftsman-style house is the coffee drinker.
13. The boba tea drinker is somewhere to the left of the person living in a colonial-style house.
14. The boba tea drinker is the person partial to Pall Mall.
15. The person residing in a Victorian house is not in the fifth house.
16. The person living in a colonial-style house is in the third house.
17. Alice is not in the first house.
 What is the value of attribute Cigar for the person whose attribute Drink is water? Please reason step by step, and put your final answer within \boxed{}

Answer: yellow monster


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

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

Answer: 1


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

## Clues:
1. Bob is the photography enthusiast.
2. The person who loves eating grilled cheese is the person who is tall.
3. Peter is not in the second house.
4. The person who is tall is directly left of the person who loves stir fry.
5. The person who loves cooking is the person who has an average height.
6. Alice is directly left of the person who is a pizza lover.
7. The person who loves the spaghetti eater is not in the second house.
8. Eric is not in the fifth house.
9. The person who is short is Peter.
10. The person who has an average height and the person who enjoys gardening are next to each other.
11. The person who paints as a hobby is directly left of the person who loves eating grilled cheese.
12. The person who is very short is in the fifth house.
13. The person who is tall is in the third house.
14. Alice is somewhere to the right of the photography enthusiast.
 What is the value of attribute Hobby for the person whose attribute Height is tall? 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: `Peter`, `Eric`, `Alice`, `Arnold`
 - Each mother is accompanied by their child: `Bella`, `Fred`, `Meredith`, `Samantha`
 - People have unique hair colors: `red`, `brown`, `black`, `blonde`
 - They all have a unique favorite flower: `carnations`, `roses`, `daffodils`, `lilies`
 - The people keep unique animals: `cat`, `bird`, `fish`, `horse`

## Clues:
1. Alice is somewhere to the right of the person who has red hair.
2. The person who loves a bouquet of daffodils is directly left of Arnold.
3. The fish enthusiast is the person's child is named Samantha.
4. Alice is directly left of the bird keeper.
5. The cat lover is the person who loves the rose bouquet.
6. The person who loves a carnations arrangement is somewhere to the right of the person who has brown hair.
7. The person who loves a bouquet of daffodils is directly left of the bird keeper.
8. Eric is the person's child is named Meredith.
9. There are two houses between the person who has red hair and the person who has black hair.
10. Eric is in the fourth house.
11. The person's child is named Bella is the person who loves a bouquet of daffodils.
 What is the value of attribute House for the person whose attribute HairColor is blonde? 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`, `Eric`, `Alice`, `Bob`, `Arnold`
 - People have unique favorite book genres: `mystery`, `fantasy`, `biography`, `science fiction`, `romance`
 - Each person prefers a unique type of vacation: `mountain`, `camping`, `city`, `beach`, `cruise`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Kailyn`, `Penny`, `Aniya`

## Clues:
1. Bob is the person who loves fantasy books.
2. The person whose mother's name is Janelle and Eric are next to each other.
3. The person who enjoys camping trips is the person who loves biography books.
4. Peter is The person whose mother's name is Kailyn.
5. Peter and The person whose mother's name is Janelle are next to each other.
6. The person who loves beach vacations is Bob.
7. Peter is somewhere to the right of the person who loves romance books.
8. The person whose mother's name is Penny is the person who loves biography books.
9. Bob is directly left of the person who prefers city breaks.
10. The person who loves science fiction books is in the fourth house.
11. Arnold and Bob are next to each other.
12. The person whose mother's name is Aniya is the person who enjoys mountain retreats.
 What is the value of attribute Vacation for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: mountain


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Peter`, `Eric`, `Alice`
 - Each person has a unique birthday month: `sept`, `mar`, `april`, `feb`, `jan`
 - People have unique favorite sports: `baseball`, `basketball`, `soccer`, `tennis`, `swimming`
 - Each person has a favorite color: `white`, `red`, `yellow`, `blue`, `green`
 - Each person has an occupation: `doctor`, `teacher`, `lawyer`, `artist`, `engineer`
 - The mothers' names in different houses are unique: `Kailyn`, `Holly`, `Aniya`, `Penny`, `Janelle`

## Clues:
1. There is one house between the person who is a doctor and the person whose birthday is in March.
2. The person whose mother's name is Penny is not in the fourth house.
3. The person who loves baseball is somewhere to the right of the person who is a doctor.
4. The person who loves soccer is Peter.
5. There is one house between Bob and The person whose mother's name is Kailyn.
6. The person who loves yellow is in the third house.
7. The person who loves yellow is The person whose mother's name is Kailyn.
8. The person who loves swimming is the person who loves white.
9. Alice is in the fourth house.
10. The person whose birthday is in February is The person whose mother's name is Holly.
11. The person who loves baseball is the person whose birthday is in January.
12. The person who loves white is the person whose birthday is in April.
13. The person whose mother's name is Aniya is the person who loves swimming.
14. Alice is the person who loves baseball.
15. The person who is a doctor and the person who is a lawyer are next to each other.
16. The person whose birthday is in April is somewhere to the left of the person whose favorite color is red.
17. The person who loves tennis is somewhere to the right of the person who is a teacher.
18. The person whose favorite color is green is in the second house.
19. Arnold is the person whose birthday is in February.
20. The person whose birthday is in February is the person who is an artist.
21. The person who loves blue is not in the fifth house.
22. The person who loves tennis is not in the fifth house.
23. The person whose mother's name is Holly is the person who loves basketball.
 What is the value of attribute House for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: 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`, `Peter`, `Eric`
 - Each person prefers a unique type of vacation: `camping`, `city`, `beach`, `cruise`, `mountain`
 - People have unique heights: `very tall`, `average`, `very short`, `short`, `tall`
 - They all have a unique favorite flower: `tulips`, `roses`, `carnations`, `lilies`, `daffodils`
 - Each person has a unique level of education: `bachelor`, `master`, `associate`, `high school`, `doctorate`
 - People own unique car models: `bmw 3 series`, `tesla model 3`, `toyota camry`, `ford f150`, `honda civic`

## Clues:
1. Eric is the person who prefers city breaks.
2. Arnold is the person who is tall.
3. The person who loves beach vacations is the person who loves the boquet of lilies.
4. The person who owns a BMW 3 Series is the person who is tall.
5. The person with an associate's degree is the person who loves the boquet of lilies.
6. The person who has an average height is Eric.
7. Peter is the person who owns a Toyota Camry.
8. The person who has an average height is somewhere to the right of the person who loves a bouquet of daffodils.
9. The person who owns a Honda Civic is somewhere to the left of the person who loves the boquet of lilies.
10. Eric is the person who owns a Ford F-150.
11. The person who loves the vase of tulips is in the fourth house.
12. The person who is very short is not in the fourth house.
13. Eric is the person with a high school diploma.
14. The person who enjoys mountain retreats is somewhere to the right of the person with a master's degree.
15. The person who likes going on cruises is not in the third house.
16. Bob is the person who is short.
17. The person with an associate's degree is directly left of Arnold.
18. The person with a bachelor's degree is the person who is very tall.
19. The person who loves a carnations arrangement is directly left of the person with a doctorate.
 What is the value of attribute Vacation for the person whose attribute CarModel is tesla model 3? Please reason step by step, and put your final answer within \boxed{}

Answer: mountain


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Alice`, `Eric`, `Arnold`
 - The people keep unique animals: `cat`, `fish`, `bird`, `dog`, `horse`
 - Each person has a unique birthday month: `mar`, `jan`, `feb`, `april`, `sept`

## Clues:
1. The fish enthusiast and Arnold are next to each other.
2. The person whose birthday is in February is not in the fourth house.
3. The person who keeps horses is in the fourth house.
4. The person whose birthday is in March is somewhere to the left of the dog owner.
5. Peter is the person whose birthday is in September.
6. The dog owner is not in the fifth house.
7. The bird keeper is in the third house.
8. Peter and the person whose birthday is in February are next to each other.
9. Bob is not in the third house.
10. The dog owner is somewhere to the right of Alice.
11. The person whose birthday is in January is the bird keeper.
 What is the value of attribute House for the person whose attribute Birthday is jan? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


5. (20 points) Given a cyclic quadrilateral $A B C D$. The rays $A B$ and $D C$ intersect at point $E$, and the rays $D A$ and $C B$ intersect at point $F$. The ray $B A$ intersects the circumcircle of triangle $D E F$ at point $L$, and the ray $B C$ intersects the same circle at point $K$. The length of segment $L K$ is $5$, and $\angle E B C=15^{0}$. Find the radius of the circumcircle of triangle $E F K$.

#

Answer: 5


[ Auxiliary similar triangles ] [ Properties and characteristics of isosceles triangles. ]

From the vertex $B$ of the isosceles triangle $A B C$ to its base $A C$, a height $B D$ is dropped. Each of the lateral sides $A B$ and $B C$ of triangle $A B C$ is equal to 8. In triangle $B C D$, a median $D E$ is drawn. A circle is inscribed in triangle $B D E$, touching side $B E$ at point $K$ and side $D E$ at point $M$. The segment $KM$ is equal to 2. Find the angle $A$.

Answer: 30^\circ


As shown in the figure, let \( k>0 \) and \( k \neq 1 \). The line \( l: y = kx + 1 \) and the line \( l_{1}: y = k_{1} x + 1 \) are symmetrical about the line \( y = x + 1 \). The lines \( l \) and \( l_{1} \) intersect the ellipse \( E: \frac{x^{2}}{4} + y^{2} = 1 \) at points \( A, M \) and \( A, N \), respectively.
(1) Find the value of \( k \cdot k_{1} \);
(2) Prove that for any real number \( k \), the line \( M N \) always passes through a fixed point.

Answer: 1


Let $n \geq 3$ be a positive integer. Find the smallest positive real $k$, satisfying the following condition: if $G$ is a connected graph with $n$ vertices and $m$ edges, then it is always possible to delete at most $k(m-\lfloor \frac{n} {2} \rfloor)$ edges, so that the resulting graph has a proper vertex coloring with two colors.

Answer: \dfrac{1}{2}


4. Let

$$
n=9+99+999+\cdots+\underbrace{999 \ldots 999}_{99 \text { digits }}
$$

Determine the number and the sum of the digits of the number $n$.

Answer: 100


8. A positive real number, its fractional part, integer part, and the number itself form a geometric sequence, then the fractional part of this positive real number is $\qquad$ .

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


3Given points $A(-2 ; 1), B(2 ; 5)$ and $C(4 ;-1)$. Point $D$ lies on the extension of median $A M$ beyond point $M$, and quadrilateral $A B D C$ is a parallelogram. Find the coordinates of point $D$.

Answer: (8, 3)


484. Zero from fifty-seven. After the previous puzzle, this one will seem quite simple.

In our drawing, you see 6 cigarettes (matches would work just as well) arranged to form the number 57. The puzzle is to move two of the cigarettes without touching the others to get 0.
![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-175.jpg?height=316&width=896&top_left_y=1772&top_left_x=586)

Remember, you can only move two cigarettes. There are two completely different solutions. Can you find one or even both?

Answer: 0


5. Tom Sawyer took on the task of painting a very long fence, adhering to the condition: any two boards, between which there are exactly two, exactly three, or exactly five boards, must be painted in different colors. What is the smallest number of different colors he can manage with.

Answer: 3


The tangents at $B$ and $A$ to the circumcircle of an acute angled triangle $ABC$ meet the tangent at $C$ at $T$ and $U$ respectively.  $AT$ meets $BC$ at $P$, and $Q$ is the midpoint of $AP$; $BU$ meets $CA$ at $R$, and $S$ is the midpoint of $BR$.  Prove that $\angle ABQ=\angle BAS$.  Determine, in terms of ratios of side lengths, the triangles for which this angle is a maximum.

Answer: 1:1:1


## 14. Dromedaries ${ }^{1}$

Some of the dromedaries are sleeping, while others are awake; the number of sleeping dromedaries is seven eighths of the number of awake dromedaries plus another seven eighths of a dromedary. If half of the sleeping dromedaries woke up, the number of awake dromedaries would be between 25 and 65. And if all the sleeping dromedaries woke up, the number of awake dromedaries would be equal to...

Complete, please, the last sentence.

Answer: 59


There is a 100-digit number consisting of ones and twos. It is allowed to swap the first five digits with the next five in any sequence of ten consecutive digits. Two such numbers are called similar if one of them can be obtained from the other by several such operations. What is the maximum number of pairwise dissimilar numbers that can be selected?

Answer: 21^5


Point $Q$ has coordinates $(a+1,4 a+1)$ for some $a>1$, and lies on the line with equation $y=a x+3$. If $O$ is the origin $(0,0)$, determine the coordinates of the points $P$ and $R$ so that $O P Q R$ is a square with diagonal $O Q$.

Answer: (6, 3)


In a triangle \(ABC\), let \(M\) and \(N\) be the intersection points of the angle bisectors of \(\angle B\) and \(\angle C\) with the opposite sides, respectively. Let \(D\) be the intersection of the ray \([MN)\) with the circumcircle of \(ABC\). Show that
\[ \frac{1}{BD} = \frac{1}{AD} + \frac{1}{CD}. \]

Answer: \frac{1}{BD} = \frac{1}{AD} + \frac{1}{CD}


4. Find all five-digit numbers consisting of non-zero digits such that each time we erase the first digit, we get a divisor of the previous number.

## Solutions

Problem 1.

Answer: 91125


Exercise 4. Consider an $8 \times 8$ chessboard whose squares are alternately colored white and black. An infernal rook is a piece that can attack squares of its own color on its row, as well as squares of the opposite color on its column. What is the maximum number of infernal rooks that can be placed on the chessboard such that no two infernal rooks can ever attack each other?

Answer: 16


6. The median of a set of five positive integers is one more than the mode and one less than the mean. What is the largest possible value of the range of the five integers?

Answer: 7


1. Find all possible integer values of the expression $\frac{2}{a}+5+\sqrt{2}$, given that $a-\sqrt{2}$ is also an integer. Write the answers in any order, separated by commas or semicolons.

Example of answer format: $1 ; 2$

Answer: 3,7


12. Let $a$ be the largest positive root of the equation $x^{3}-3 x^{2}+1=0$, then among the 2006 numbers $\left[a^{n}\right](1 \leqslant n \leqslant 2006)$, there are $\qquad$ numbers that can be divided by 17.

Answer: 251


5.1. 40 people came into a room where there were 40 chairs, black and white, and sat on them. All of them said they were sitting on black chairs. Then they somehow resat, and exactly 16 claimed they were sitting on white chairs. Each of those sitting either lied both times or told the truth both times. How many of them lied the second time and were sitting on black chairs?

Answer: 8


At the New Year's school party in the city of Lzheretsark, 301 students came. Some of them always tell the truth, while the rest always lie. Each of the 200 students said: "If I leave the hall, then among the remaining students, the majority will be liars." Each of the other students stated: "If I leave the hall, then among the remaining students, there will be twice as many liars as those who tell the truth." How many liars were at the party?

Answer: 151


## Task B-4.3.

A sequence of numbers is defined by $a_{n}=n^{4}-360 n^{2}+400$. Calculate the sum of all terms of this sequence that are prime numbers.

Answer: 802


25. How many even four-digit numbers can be formed from the digits $0,1,2,3,4,5$?

Answer: 540


B2. How many natural numbers less than 100000 have a sum of digits divisible by 9 or 5?

Answer: 31109


Problem 2. Masha's classes at school end at 13:00, and her mother picks her up by car, and they drive home. One day, the classes ended at 12:00, and Masha started walking home. On the way, she met her mother, who, as usual, drove to pick up her daughter at 13:00 at the school. Then Masha and her mother drove home in the car, and they arrived 12 minutes earlier than usual. At what time did Masha meet her mother on the road? (The speeds of Masha and her mother are constant, and no time is spent on getting into the car.)

Answer: 12:54


10.4. The incircle of triangle $A B C$ touches side $A C$ at point $D$. A second circle passes through point $D$, touches ray $B A$ at point $A$, and also touches the extension of side $B C$ beyond point $C$. Find the ratio $A D: D C$.

Answer: 3:1


10. (10th US Olympiad Question) In a certain county, every two districts are connected by exactly one of three modes of transportation: car, train, or airplane. It is known that all three modes of transportation are used in the county, but no district has all three modes. Furthermore, no three districts are pairwise connected by the same mode of transportation. What is the maximum number of districts in this county?

---

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

Answer: 4


In the diagram, $P T$ and $P S$ are tangents to a circle with centre $O$. The point $Y$ lies on the circumference of the circle; and the point $Z$ is where the line $P Y$ meets the radius $O S$.

Also, $\angle S P Z=10^{\circ}$ and $\angle T O S=150^{\circ}$.

How many degrees are there in the sum of $\angle P T Y$ and $\angle P Y T$ ?

Answer: 160


$6 \cdot 2$ A function $f$ defined on the positive integers satisfies:
$$
f(n)=\left\{\begin{array}{l}
n-3 \\
f(f(n+7))
\end{array}\right.
$$
(when $n \geqslant 1000$),
(when $n<1000$).
Find $f(90)$.

Answer: 999


$d(n)$ shows the number of positive integer divisors of positive integer $n$. For which positive integers $n$ one cannot find a positive integer $k$ such that $\underbrace{d(\dots d(d}_{k\ \text{times}} (n) \dots )$ is a perfect square.

Answer: n


$\left[\begin{array}{l}\text { Combinations and Permutations } \\ {\left[\begin{array}{l}\text { Partitions into Pairs and Groups; Bijections }]\end{array}\right]}\end{array}\right]$

There are 20 people - 10 boys and 10 girls. How many ways are there to form a company in which there would be an equal number of boys and girls?

Answer: 184756


6. As shown in Figure 1, if
the three sides of $\triangle A B C$ are
$n+x, n+2 x, n$
$+3 x$, and the height $A D$ from $B C$ is $n$, where $n$
is a positive integer, and $0<x \leqslant$
1. Then the number of triangles that satisfy the above conditions is $\qquad$.

Answer: 12


8. Using weights labeled $1, 2, 3, 15, 40$ grams each, on a balance scale without graduations, if weights can be placed on both sides of the scale, then the maximum number of different gram weights (positive integer weights) that can be measured is $\qquad$ kinds.

Answer: 55


# 5. CONDITION

Vladislav Vladimirovich, taking less than 100 rubles, went for a walk. Entering any cafe and having at that moment $m$ rubles $n$ kopecks, he spent $n$ rubles $m$ kopecks ( $m$ and $n$ - natural numbers). What is the maximum number of cafes Vladislav Vladimirovich could visit?

Answer: 6


(10) There are 8 black, white, and yellow chopsticks each. Without looking, take out the chopsticks arbitrarily to ensure that there are at least two pairs of chopsticks of different colors. Then, at least $\qquad$ chopsticks need to be taken out to achieve this.

Answer: 11


Given is a non-isosceles triangle $ABC$ with $\angle ABC=60^{\circ}$, and in its interior, a point $T$ is selected such that $\angle ATC= \angle BTC=\angle BTA=120^{\circ}$. Let $M$  the intersection point of the medians in $ABC$. Let $TM$ intersect $(ATC)$ at $K$. Find $TM/MK$.

Answer: 2


Positive numbers \( x, y, \) and \( z \) satisfy the condition \( x y z \geq x y + y z + z x \). Prove the inequality:

\[
\sqrt{x y z} \geq \sqrt{x} + \sqrt{y} + \sqrt{z}
\]

(A. Khrabrov)

Answer: \sqrt{x y z} \geq \sqrt{x} + \sqrt{y} + \sqrt{z}


1 In a certain municipality, they want to set up a hydroelectric power plant, and for this purpose, they wish to utilize the energy of the water flowing at the border. Preliminary calculations are to be made to determine how many light bulbs the water's energy can power in the case of the lowest water level. The river flows in a rectangular parallelogram-shaped channel at one point, which is $5 \mathrm{~m}$ wide. Even at the lowest water level, the depth of the water is $0.9 \mathrm{~m}$. The speed of the water was determined using a floating piece of wood and it was found that it covers a distance of $70 \mathrm{~m}$ in 100 seconds. The terrain conditions are such that a drop of $3 \mathrm{~m}$ can be easily established. The turbines and associated generators convert $80 \%$ of the available energy into electrical energy, and $10 \%$ of the energy is lost in the transmission lines. A 16 standard candlepower lamp in a carbon filament operates at 110 Volts and 0.5 Amperes.[^0]

[^0]:    1* We also gladly accept solutions to starred problems from university students.

Answer: 1214


14. Given sets $A, B$ are both sets composed of positive integers, and $|A|=20,|B|=16$, set $A$ satisfies the following condition: if $a, b, m, n \in A$, and $a+b=m+n$, then it must be that $\{a, b\}=\{m, n\}$. Define $A+B=\{a+b \mid a \in A, b \in B\}$, try to determine the minimum value of $|A+B|$.

Answer: 200


12. A competition has $n \geqslant 2$ participants, and each day the scores of the participants are exactly the set $\{1,2, \cdots, n\}$. If at the end of the $k$-th day, the total score of every pair of participants is 52 points, find all pairs $(n, k)$ that make this possible.

Answer: (3, 13)


Problem 3. The set $M$ consists of all prime divisors of the number 2310. How many elements does the set of all numbers obtained as the product of exactly two elements of the set $M$ have?

Answer: 10


2. Solve the equation $\left(\frac{x}{4}\right)^{\log _{5}(50 x)}=x^{6}$.

Answer: \dfrac{1}{2}


7. Let the set $A=\{n(n+1) \mid n=1,2, \cdots\}, B=\{3 m-1 \mid m=1,2, \cdots\}$, if the elements of the set $A \cap B$ are arranged in ascending order to form a sequence $\left\{a_{k}\right\}$, then the general term formula of the sequence $\left\{a_{k}\right\}$ is $a_{k}=$ $\qquad$ .

Answer: (3k - 2)(3k - 1)


Example 3. A combination of two inverted cones sharing the same base, the lateral development radius of one cone is 15, and the central angle is $288^{\circ}$, the lateral development radius of the other cone is 13, find the volume of its inscribed sphere.

Answer: 288\pi


Problem 9.6. A white checkered $8 \times 8$ table is given. In it, 20 cells are painted black. What is the minimum number of pairs of adjacent white cells that could remain?

Answer: 34


10. (14 points) The sequence $\left\{a_{n}\right\}$ is defined as follows: $a_{1}=1$, and for $n \geqslant 2$,
$a_{n}=\left\{\begin{array}{ll}a_{\frac{n}{2}}+1, & \text { when } n \text { is even; } \\ \frac{1}{a_{n-1}}, & \text { when } n \text { is odd. }\end{array}\right.$
It is known that $a_{n}=\frac{30}{19}$. Find the positive integer $n$.

Answer: 238


Find the area enclosed by the closed curves:
1) The ellipse \( x = a \cos t, \ y = b \sin t \)
2) The loop of the Cartesian leaf \( x^{3} + y^{3} - 3 a x y = 0 \).

Figures 187 and 188.

Answer: S = \frac{3 a^2}{2}


Let \(P\) be a regular polygon with 2026 sides. A diagonal of \(P\) is said to be good if its endpoints divide the contour of \(P\) into two parts, each having an odd number of sides of \(P\). The sides of \(P\) are also said to be good. Suppose that \(P\) has been subdivided into triangles by 2023 diagonals, no two of which share a common point inside \(P\). What is the maximum number of such a subdivision that can contain isosceles triangles whose two sides are good?

Answer: 1013