B3. On a billiard table in the shape of a regular hexagon $A B C D E F$ with side 4, a ball is struck from $P$ towards $Q$. Calculate the length of the path $P Q R S P$, neglecting the dimensions of the ball. $P$ is the midpoint of $D E$ and $R$ is the midpoint of $A B, Q$ lies on $A F$ and $S$ lies on $B C$.

Answer: 4\sqrt{21}


A2 From each of the eight vertices of a cube, a piece is sawn off so that instead of a vertex, a triangular face with three vertices appears. Each pair of the 24 vertices of the resulting body is connected by a line segment.

How many of these line segments lie, except for their endpoints, entirely within the body?

![](https://cdn.mathpix.com/cropped/2024_04_17_3181bece6cd13f041828g-1.jpg?height=294&width=279&top_left_y=1646&top_left_x=1531)

Answer: 120


Let $f(n)$ denote the number of ways a positive integer $n$ can be expressed as a sum of numbers of the form $2^{a} 5^{b}$ (where the order of summands does not matter) such that none of the summands is a divisor of another (with $a$ and $b$ being non-negative integers). Prove that $f(n)$ is unbounded.

Answer: f(n) \text{ is unbounded.}


All empty white triangles in figure are to be filled with integers such that for each gray triangle the three numbers in the white neighboring triangles sum to a multiple of $5$. The lower left and the lower right white triangle are already filled with the numbers $12$ and $3$, respectively.
Find all integers that can occur in the uppermost white triangle.

(G. Woeginger, Eindhoven, The Netherlands)
[img]https://cdn.artofproblemsolving.com/attachments/8/a/764732f5debbd58a147e9067e83ba8d31f7ee9.png[/img]

Answer: 5k


3. On the number line, points with coordinates $1, 2, \cdots, 2006$ are called "marked points". A frog starts from point 1, and after 2006 jumps, it visits all the marked points and returns to the starting point. What is the maximum length of the path the frog can jump? Explain your reasoning.
(2006, Shandong Province Junior High School Mathematics Competition)

Answer: 2012018


7. As shown in Figure $2, \triangle A B C$ has an area of 1, points $D, G, E,$ and $F$ are on sides $A B, A C, B C$ respectively, $B D < D A$, $D G \parallel B C, D E \parallel A C, G F \parallel A B$. Then the maximum possible area of trapezoid $D E F G$ is $\qquad$

Answer: \dfrac{1}{3}


Define the two sequences $a_0, a_1, a_2, \ldots$ and $b_0, b_1, b_2, \ldots$ by $a_0 = 3$ and $b_0 = 1$ with the recurrence relations:
- $a_{n+1} = 3a_n + b_n$
- $b_{n+1} = 3b_n - a_n$

for all nonnegative integers $n$. Let $r$ and $s$ be the remainders when $a_{32}$ and $b_{32}$ are divided by $31$, respectively. Compute $100r + s$.

Answer: 3010


The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$, the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$. Given that $P(3) = 89$, determine the value of $P(10)$.

Answer: 859


For a natural number $n$, a string $s$ of $n$ binary digits, and a natural number $k \leq n$, define an $n,s,k$-block as a string of $k$ consecutive elements from $s$. We say that two $n,s,k$-blocks, namely, $a_1a_2\ldots a_k$ and $b_1b_2\ldots b_k$, are incompatible if there exists an $i \in \{1,2,\ldots ,k\}$ such that $a_i \neq b_i$. Additionally, for two natural numbers $r \leq n$ and $l$, we say that $s$ is $r,l$-typed if there are at most $l$ pairwise incompatible $n,s,r$-blocks.

Let $t$ be a $3,7$-typed string consisting of $10,000$ binary digits. Determine the maximum number $M$ such that $t$ is $10,M$-typed.

Answer: 128


Hong and Song each have a shuffled deck of eight cards, consisting of four red cards and four black cards. On each turn, both players reveal the top two cards of their decks. The possible pairs that can be played are:

- Two black cards
- Two red cards
- One black and one red card

Determine the probability that Hong and Song play exactly the same pair of cards in each of the four turns. This probability is given as $\frac{m}{n}$, where $m$ and $n$ are positive integers such that $\gcd(m, n) = 1$. Calculate $100m + n$.

Answer: 25150


A set $D$ of positive integers is called indifferent if there are at least two integers in the set, and for any two distinct elements $x, y \in D$, their positive difference $|x-y|$ is also in $D$. Let $M(x)$ be the smallest size of an indifferent set whose largest element is $x$. Compute the sum $M(2) + M(3) + \dots + M(100)$.

Answer: 1257


Let $\zeta = \cos \frac{2\pi}{13} + i \sin \frac{2\pi}{13}$. Suppose $a > b > c > d$ are positive integers satisfying:

$$|\zeta^a + \zeta^b + \zeta^c + \zeta^d| = \sqrt{3}.$$

Compute the smallest possible value of $1000a + 100b + 10c + d$.

Answer: 7521


Let $\mathbb{N}_0$ be the set of non-negative integers. Consider a triple $(f, a, b)$, where $f$ is a function from $\mathbb{N}_0$ to $\mathbb{N}_0$, and $a, b \in \mathbb{N}_0$ that satisfies the following conditions:

- $f(1) = 2$
- $f(a) + f(b) \leq 2\sqrt{f(a)}$
- For all $n > 0$, $f(n) = f(n-1)f(b) + 2n - f(b)$

Find the sum of all possible values of $f(b + 100)$.

Answer: 10201


In the triangle $ABC$, $\angle ABC = \angle ACB = 78^\circ$. On the sides $AB$ and $AC$, respectively, the points $D$ and $E$ are chosen such that $\angle BCD = 24^\circ$, $\angle CBE = 51^\circ$. Find the measure of angle $\angle BED$.

Answer: 12


Compute the number of sequences of integers $(a_1, \ldots, a_{200})$ such that the following conditions hold:

1. $0 \leq a_1 < a_2 < \cdots < a_{200} \leq 202$.
2. There exists a positive integer $N$ with the following property: for every index $i \in \{1, \ldots, 200\}$, there exists an index $j \in \{1, \ldots, 200\}$ such that $a_i + a_j - N$ is divisible by $203$.

Answer: 20503


Several players try out for the USAMTS basketball team, and they all have integer heights and weights when measured in centimeters and pounds, respectively. In addition, they all weigh less in pounds than they are tall in centimeters. All of the players weigh at least $190$ pounds and are at most $197$ centimeters tall, and there is exactly one player with every possible height-weight combination.

The USAMTS wants to field a competitive team, so there are some strict requirements:

- If person $P$ is on the team, then anyone who is at least as tall and at most as heavy as $P$ must also be on the team.
- If person $P$ is on the team, then no one whose weight is the same as $P$’s height can also be on the team.

Assuming the USAMTS team can have any number of members (including zero), how many different basketball teams can be constructed?

Answer: 128


Let $n$ be the answer to this problem. Suppose square $ABCD$ has side-length $3$. Then, congruent non-overlapping squares $EHGF$ and $IHJK$ of side-length $\frac{n}{6}$ are drawn such that $A$, $C$, and $H$ are collinear, $E$ lies on $BC$ and $I$ lies on $CD$. Given that $\triangle AJG$ is an equilateral triangle, the area of $\triangle AJG$ is $a + b\sqrt{c}$, where $a$, $b$, $c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a + b + c$.

Answer: 48


An $n \times n$ table is filled with real numbers such that the elements decrease monotonically in every row and every column. We want to decide whether a number $a$ is in the table. For this, we can compare $a$ with any element of the table. How many comparisons are needed to solve the problem for sure?

Answer: 2n - 1


4. Let $P$ be any point on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, with foci at $F_{1}(-c, 0), F_{2}(c, 0)$. $P F_{1}$ and $P F_{2}$ intersect the ellipse at points $A$ and $B$, respectively. If $a^{2}$, $b^{2}$, and $c^{2}$ form an arithmetic sequence, then $\frac{\left|P F_{1}\right|}{\left|A F_{1}\right|}+\frac{\left|P F_{2}\right|}{\left|B F_{2}\right|}=$ $\qquad$.

Answer: 4


12th Putnam 1952 Problem A4 A map represents the polar cap from latitudes -45 o to 90 o . The pole (latitude 90 o ) is at the center of the map and lines of latitude on the globe are represented as concentric circles with radii proportional to (90 o - latitude). How are east-west distances exaggerated compared to north-south distances on the map at a latitude of -30 o ? Solution

Answer: \dfrac{4\pi\sqrt{3}}{9}


1. A complete number is a 9 digit number that contains each of the digits 1 to 9 exactly once. The difference number of a number $N$ is the number you get by taking the differences of consecutive digits in $N$ and then stringing these digits together. For instance, the difference number of 25143 is equal to 3431 . The complete number 124356879 has the additional property that its difference number, 12121212, consists of digits alternating between 1 and 2 .

Determine all $a$ with $3 \leqslant a \leqslant 9$ for which there exists a complete number $N$ with the additional property that the digits of its difference number alternate between 1 and $a$.

Answer: 4


There are $169$ lamps, each equipped with an on/off switch. You have a remote control that allows you to change exactly $19$ switches at once. (Every time you use this remote control, you can choose which $19$ switches are to be changed.)

(a) Given that at the beginning some lamps are on, can you turn all the lamps off, using the remote control?
(b) Given that at the beginning all lamps are on, how many times do you need to use the remote control to turn all lamps off?

Answer: Yes


## Task 4 - 330524

Rita calculates the three numbers

$$
1+9-9+3=a, \quad 1 \cdot 9+9-3=b, \quad 1 \cdot 9 \cdot 9 \cdot 3=c
$$

She considers further possibilities for filling in the boxes in the row

$$
1 \square 9 \square 9 \square 3=
$$

with signs that are either + or - or $\cdot$. She is looking for all those insertions where the number to be calculated is greater than 30 but less than 100.

Find all such insertions; show that you have found all of them!

Add the numbers that result from these insertions!

To the sum thus found, add the product of the two smallest numbers found between 30 and 100! Finally, add the numbers calculated above as $a, b$, and $c$!

Answer: 1993


10. As shown in the figure, two circles are placed side by side on a straight line, touching each other. The area of each circle is 1680 square centimeters. The shaded area is the region between the two circles and the straight line. If a circle of the largest possible size is to be placed inside the shaded area, what is the area of this circle in square centimeters?

Answer: 105


332. In parallelogram $ABCD$, which is not a rhombus, the ratio of the lengths of the diagonals is given as $AC: BD = k$. Let ray $AM$ be symmetric to ray $AD$ with respect to line $AC$, and ray $BM$ be symmetric to ray $BC$ with respect to line $BD$, with $M$ being the common point of rays $AM$ and $BM$. Find the ratio $AM: BM$.

333*. On a circle, $3k$ points are marked, dividing it into $3k$ arcs, of which $k$ arcs have length 1, another $k$ arcs have length 2, and the remaining $k$ arcs have length 3. Prove that among the marked points, there are two diametrically opposite points.

Answer: k


Let $a,b,c>0$ the sides of a right triangle. Find all real $x$ for which $a^x>b^x+c^x$, with $a$ is the longest side.

Answer: (2, \infty)


3. A large cube is formed by $8 \times 8 \times 8$ unit cubes. How many lines can pass through the centers of 8 unit cubes?

Answer: 244


4. Given that the function $f(x)$ is defined on $\mathbf{R}$ as an even function, and for any real number $x$, $f(x+1)=f(1-x)$ holds. When $1 \leqslant x \leqslant 2$, $f(x)=\ln x$. If the equation $f(x)+a x-1=0$ has two distinct real roots for $x \in[3,5]$, then the range of values for $a$ is

Answer: \left( \dfrac{1 - \ln 2}{4}, \dfrac{1}{5} \right)


11.095. A sphere is circumscribed around a regular triangular prism, the height of which is twice the side of the base. How does its volume relate to the volume of the prism?

Answer: \dfrac{64\pi}{27}


4. Consider triangle $ABC$, where $AC = BC$, $m(ACB) = 90^{\circ}$, and triangle $DAB$, where $DA = DB$, located in perpendicular planes. Let $\quad M \in (BC), \quad BM = 2CM, \quad N \in (AC)$, $AC = 3AN, P \in MN \cap AB$, $T$ be the midpoint of segment $[AB]$, and $G$ be the centroid of triangle $DAB$. Calculate the tangent of the plane angle corresponding to the dihedral angle determined by the planes $(ABC)$ and $(DBC)$, given that $3SD = 5CT$, where $S \in PG \cap AD$.

Narcis Gabriel Turcu, Brăila

Answer: \sqrt{6}


## 7. At the Dentist's Office

In the waiting room, two women and ten men are waiting for their turn. At their disposal are eight copies of the latest magazine and four copies of the morning newspaper. In how many ways can they distribute the newspapers and magazines among themselves, if both women insist on reading the same thing?

Answer: 255


16. Let $\alpha, \beta$ be real numbers. If for any real numbers $x, y, z$, we have $\alpha(x y+y z+z x) \leqslant M \leqslant \beta\left(x^{2}+y^{2}+z^{2}\right)$ always holds, where $M=\sqrt{x^{2}+x y+y^{2}} \cdot \sqrt{y^{2}+y z+z^{2}}+\sqrt{y^{2}+y z+z^{2}} \cdot \sqrt{z^{2}+z x+x^{2}}+\sqrt{z^{2}+z x+x^{2}} \cdot \sqrt{x^{2}+x y+y^{2}}$. Find the maximum value of $\alpha$ and the minimum value of $\beta$.

Answer: 3


## Inscribed Quadrilateral with Perpendicular Diagonals Pythagorean Theorem (direct and inverse) Ratio of Areas of Similar Triangles ] Area of a Quadrilateral ] Complex ]  [ Ratio of Areas of Triangles with a Common Base or Height]

Diagonals $A C$ and $B D$ of an inscribed quadrilateral intersect at point $Q$ at a right angle. Lines $A B$ and $C D$ intersect at point $P$. It is known that $B C=5, A D=10, B Q=3$. Find $A P$.

#

Answer: \dfrac{20\sqrt{5}}{3}


Suppose that $a$ and $b$ are natural numbers such that \[p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}\] is a prime number. What is the maximum possible value of $p$?

Answer: 5


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


Example 3. Find the equation of the line passing through $\mathrm{A}$ $(-6,3)$ and tangent to the parabola $2 y^{2}=9 x$.

 untranslated text remains the same as requested.

Answer: x - 4y + 18 = 0


A tetrahedron inscribed in a parallelepiped cuts off four tetrahedra from the parallelepiped with its face planes. What fraction of the volume of the parallelepiped is the volume of the tetrahedron whose vertices are the centroids of the cut-off tetrahedra?

Answer: \dfrac{1}{24}


3. Solve the equation $\log _{x}(5 \cdot \sqrt{5})-\frac{5}{4}=\left(\log _{x} \sqrt{5}\right)^{2}$.

Answer: 5


## 32. Xenon, Zephyr, and Enofa

Two friends at the races were speculating about the outcome of the next race. One of them bet on Xenon and Enofa, while the other bet on Zephyr.

- I bet,- said the second,- that if my horse finishes in the top three, then Xenon will also be in the top three.
- And I,- replied the first,- bet that if at least one of my horses finishes in the top three, then you will lose your bet.

Suppose the last bet was not lost. Which of the three horses, Xenon, Zephyr, or Enofa, in this case, has the highest chance of finishing in the top three?

Answer: Zephyr


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Eric`
 - People have unique favorite music genres: `rock`, `jazz`, `classical`, `pop`
 - People have unique favorite book genres: `mystery`, `science fiction`, `fantasy`, `romance`
 - Everyone has a unique favorite cigar: `prince`, `blue master`, `pall mall`, `dunhill`
 - Each person has a unique birthday month: `april`, `jan`, `feb`, `sept`
 - Each person has a unique favorite drink: `water`, `tea`, `milk`, `coffee`

## Clues:
1. The person whose birthday is in September is the one who only drinks water.
2. The person who loves rock music and the person who loves jazz music are next to each other.
3. The person who loves mystery books is the person who smokes Blue Master.
4. Alice is the person who loves classical music.
5. The person who loves jazz music is the person who likes milk.
6. The person who loves jazz music is directly left of the person who loves romance books.
7. The person who loves rock music is not in the fourth house.
8. The person whose birthday is in February is Arnold.
9. The Prince smoker and the person who loves classical music are next to each other.
10. The coffee drinker is the person who loves science fiction books.
11. The person who loves mystery books is the person whose birthday is in September.
12. Peter is the person who loves jazz music.
13. The Dunhill smoker is the person whose birthday is in April.
14. The person who loves mystery books is in the second house.
 What is the value of attribute MusicGenre for the person whose attribute Birthday is jan? Please reason step by step, and put your final answer within \boxed{}

Answer: jazz


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`
 - Everyone has something unique for lunch: `spaghetti`, `grilled cheese`, `pizza`
 - People have unique favorite sports: `basketball`, `tennis`, `soccer`
 - People have unique hair colors: `black`, `blonde`, `brown`
 - People have unique favorite music genres: `classical`, `rock`, `pop`
 - People have unique favorite book genres: `science fiction`, `mystery`, `romance`

## Clues:
1. The person who loves basketball is somewhere to the left of the person who loves science fiction books.
2. The person who has brown hair is Eric.
3. The person who has brown hair is the person who is a pizza lover.
4. The person who loves romance books is the person who is a pizza lover.
5. Peter is the person who has blonde hair.
6. The person who loves science fiction books is the person who loves soccer.
7. The person who loves eating grilled cheese is the person who loves basketball.
8. The person who loves pop music is the person who has blonde hair.
9. The person who has brown hair is somewhere to the left of the person who loves rock music.
10. The person who loves classical music is not in the first house.
 What is the value of attribute FavoriteSport for the person whose attribute MusicGenre is classical? Please reason step by step, and put your final answer within \boxed{}

Answer: tennis


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

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

Answer: swimming


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`, `Bob`, `Alice`
 - Everyone has a favorite smoothie: `watermelon`, `lime`, `cherry`, `dragonfruit`, `desert`
 - The people keep unique animals: `fish`, `bird`, `horse`, `cat`, `dog`
 - Each person prefers a unique type of vacation: `camping`, `beach`, `mountain`, `city`, `cruise`
 - Each person has a unique favorite drink: `root beer`, `water`, `milk`, `tea`, `coffee`

## Clues:
1. The person who keeps horses is the person who drinks Lime smoothies.
2. There is one house between the Watermelon smoothie lover and the person who likes Cherry smoothies.
3. Bob and the coffee drinker are next to each other.
4. The fish enthusiast is the one who only drinks water.
5. The bird keeper is Bob.
6. The person who enjoys mountain retreats is directly left of the root beer lover.
7. The person who loves beach vacations is somewhere to the left of the dog owner.
8. Eric is the person who likes Cherry smoothies.
9. The tea drinker is Alice.
10. The person who prefers city breaks is somewhere to the right of the person who enjoys mountain retreats.
11. Peter is somewhere to the left of the coffee drinker.
12. Eric is the person who loves beach vacations.
13. The person who enjoys mountain retreats is Alice.
14. There are two houses between the Watermelon smoothie lover and Alice.
15. The person who enjoys camping trips is not in the first house.
16. The dog owner is not in the fifth house.
17. The fish enthusiast and the Dragonfruit smoothie lover are next to each other.
 What is the value of attribute Animal for the person whose attribute Vacation is cruise? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Peter`
 - Each person has a unique type of pet: `fish`, `cat`, `dog`, `bird`
 - Each person prefers a unique type of vacation: `cruise`, `mountain`, `city`, `beach`

## Clues:
1. The person who likes going on cruises is the person who owns a dog.
2. The person who keeps a pet bird is the person who prefers city breaks.
3. The person who enjoys mountain retreats is directly left of the person who has a cat.
4. There are two houses between the person with an aquarium of fish and the person who prefers city breaks.
5. Peter is somewhere to the right of the person who loves beach vacations.
6. The person with an aquarium of fish is Arnold.
7. Alice is in the third house.
 What is the value of attribute Name for the person whose attribute Pet is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Eric`
 - People have unique heights: `tall`, `short`, `very short`, `average`
 - Each person has a unique hobby: `photography`, `painting`, `cooking`, `gardening`

## Clues:
1. Alice is somewhere to the right of Peter.
2. There are two houses between the person who paints as a hobby and the person who has an average height.
3. Peter is somewhere to the left of the person who loves cooking.
4. Arnold is the person who is very short.
5. The person who is tall and Alice are next to each other.
6. The person who loves cooking is not in the third house.
7. The photography enthusiast is Peter.
8. The photography enthusiast and the person who enjoys gardening are next to each other.
9. The person who loves cooking is Eric.
 What is the value of attribute Height for the person whose attribute Hobby is photography? Please reason step by step, and put your final answer within \boxed{}

Answer: tall


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Peter`, `Eric`, `Bob`, `Alice`, `Arnold`
 - Each person prefers a unique type of vacation: `cruise`, `mountain`, `city`, `camping`, `cultural`, `beach`
 - Each person has a unique hobby: `knitting`, `photography`, `cooking`, `painting`, `gardening`, `woodworking`
 - Everyone has something unique for lunch: `stew`, `pizza`, `spaghetti`, `grilled cheese`, `stir fry`, `soup`
 - People own unique car models: `chevrolet silverado`, `honda civic`, `toyota camry`, `ford f150`, `tesla model 3`, `bmw 3 series`
 - They all have a unique favorite flower: `tulips`, `daffodils`, `carnations`, `lilies`, `iris`, `roses`

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

Answer: 5


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Peter`, `Eric`
 - People own unique car models: `ford f150`, `tesla model 3`, `toyota camry`, `honda civic`
 - Each person has a unique birthday month: `april`, `sept`, `feb`, `jan`
 - Each person has an occupation: `teacher`, `artist`, `engineer`, `doctor`
 - Everyone has something unique for lunch: `pizza`, `stew`, `spaghetti`, `grilled cheese`

## Clues:
1. The person who owns a Toyota Camry is in the first house.
2. Eric and the person who loves the spaghetti eater are next to each other.
3. The person who is an engineer is the person who loves eating grilled cheese.
4. Peter is the person who owns a Toyota Camry.
5. The person whose birthday is in April is directly left of the person whose birthday is in February.
6. The person who is a doctor is somewhere to the right of the person who is an artist.
7. The person who owns a Ford F-150 is the person who is a doctor.
8. The person whose birthday is in September is somewhere to the left of Alice.
9. Arnold is the person who is an artist.
10. The person who loves the spaghetti eater is the person who owns a Ford F-150.
11. The person who is an engineer is the person whose birthday is in April.
12. The person whose birthday is in February and the person who is a pizza lover are next to each other.
13. The person who owns a Honda Civic is Eric.
 What is the value of attribute House for the person whose attribute CarModel is honda civic? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


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

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

Answer: hip hop


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

## Clues:
1. There is one house between Alice and the fish enthusiast.
2. The person in a ranch-style home is not in the fifth house.
3. The person whose birthday is in March is Alice.
4. The person in a ranch-style home is not in the fourth house.
5. Arnold is somewhere to the right of the person whose birthday is in April.
6. The person who has a cat is the person who keeps horses.
7. There are two houses between the cat lover and the person living in a colonial-style house.
8. Eric is somewhere to the left of the person whose birthday is in September.
9. The person whose birthday is in April is directly left of the person who keeps horses.
10. Alice is the person residing in a Victorian house.
11. The person whose birthday is in March and the cat lover are next to each other.
12. Peter is the person whose birthday is in February.
13. The cat lover is the person who owns a dog.
14. The bird keeper is somewhere to the left of the person who keeps a pet bird.
15. Bob is the person who keeps a pet bird.
16. Eric is in the fourth house.
17. The person who has a cat is somewhere to the left of the person with a pet hamster.
18. The person who keeps a pet bird is the person in a modern-style house.
 What is the value of attribute House for the person whose attribute HouseStyle is colonial? 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: `Arnold`, `Peter`, `Alice`, `Eric`, `Bob`
 - Each person has a unique favorite drink: `root beer`, `tea`, `water`, `coffee`, `milk`
 - People have unique favorite music genres: `pop`, `hip hop`, `classical`, `jazz`, `rock`
 - The people are of nationalities: `dane`, `norwegian`, `swede`, `german`, `brit`
 - Everyone has a unique favorite cigar: `dunhill`, `prince`, `blue master`, `pall mall`, `blends`

## Clues:
1. The Dunhill smoker is not in the second house.
2. The person who loves rock music is directly left of the person who loves classical music.
3. Peter is the person who loves jazz music.
4. Arnold is directly left of the Dane.
5. The root beer lover is the German.
6. The tea drinker is the Swedish person.
7. The Swedish person is the person who loves classical music.
8. The Dunhill smoker is not in the fourth house.
9. The person who loves jazz music is the Prince smoker.
10. Eric and the person who loves hip-hop music are next to each other.
11. There are two houses between the person partial to Pall Mall and the Swedish person.
12. Arnold is somewhere to the right of Eric.
13. The coffee drinker is somewhere to the right of the Dunhill smoker.
14. The Norwegian is somewhere to the right of the person who smokes Blue Master.
15. There is one house between the one who only drinks water and the person who smokes many unique blends.
16. Alice is the tea drinker.
 What is the value of attribute Nationality for the person whose attribute Cigar is pall mall? Please reason step by step, and put your final answer within \boxed{}

Answer: german


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Carol`, `Eric`, `Peter`, `Arnold`, `Bob`
 - Everyone has a unique favorite cigar: `pall mall`, `yellow monster`, `dunhill`, `blends`, `blue master`, `prince`
 - Everyone has a favorite smoothie: `watermelon`, `dragonfruit`, `lime`, `desert`, `blueberry`, `cherry`

## Clues:
1. The person who drinks Lime smoothies is in the third house.
2. The person partial to Pall Mall is the Watermelon smoothie lover.
3. The Desert smoothie lover is Arnold.
4. The Watermelon smoothie lover is somewhere to the left of Eric.
5. There is one house between Peter and Carol.
6. The person who drinks Blueberry smoothies is somewhere to the left of the person who drinks Lime smoothies.
7. The Watermelon smoothie lover is somewhere to the left of the person who smokes many unique blends.
8. The Dragonfruit smoothie lover and Carol are next to each other.
9. Bob is directly left of Carol.
10. The person who likes Cherry smoothies is the Dunhill smoker.
11. The Dragonfruit smoothie lover is the Prince smoker.
12. The person who smokes Blue Master is Peter.
13. The Watermelon smoothie lover and the Desert smoothie lover are next to each other.
 What is the value of attribute Cigar for the person whose attribute Smoothie is watermelon? Please reason step by step, and put your final answer within \boxed{}

Answer: pall mall


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Bob`, `Alice`, `Eric`, `Arnold`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `samsung galaxy s21`, `iphone 13`, `huawei p50`
 - Each person has an occupation: `lawyer`, `engineer`, `teacher`, `artist`, `doctor`

## Clues:
1. The person who uses a Samsung Galaxy S21 is directly left of Eric.
2. There are two houses between Alice and the person who is an artist.
3. The person who uses an iPhone 13 is directly left of Bob.
4. The person who is a lawyer is in the third house.
5. The person who is a doctor is Arnold.
6. The person who is an engineer is directly left of the person who is an artist.
7. The person who uses a OnePlus 9 is somewhere to the right of Eric.
8. The person who uses a Google Pixel 6 is Peter.
 What is the value of attribute House for the person whose attribute PhoneModel is huawei p50? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Eric`, `Arnold`, `Bob`
 - Everyone has a favorite smoothie: `lime`, `cherry`, `desert`, `dragonfruit`, `watermelon`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `pizza`, `stew`, `stir fry`
 - People have unique hair colors: `gray`, `black`, `red`, `brown`, `blonde`

## Clues:
1. The person who has blonde hair is the person who loves eating grilled cheese.
2. The Desert smoothie lover is somewhere to the left of the person who has red hair.
3. The person who loves the spaghetti eater is the person who has red hair.
4. Bob and the person who has blonde hair are next to each other.
5. The person who likes Cherry smoothies is in the fourth house.
6. Arnold is the person who has black hair.
7. The Desert smoothie lover is in the third house.
8. The Desert smoothie lover is the person who loves eating grilled cheese.
9. Eric is the person who has blonde hair.
10. The person who has gray hair is somewhere to the left of the person who has black hair.
11. The person who has black hair is the Dragonfruit smoothie lover.
12. Bob is the Watermelon smoothie lover.
13. The person who loves stir fry is the person who has black hair.
14. The person who is a pizza lover is the person who has brown hair.
15. Peter is the person who has gray hair.
 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: 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 has a unique hobby: `cooking`, `gardening`, `painting`, `photography`, `knitting`
 - People have unique favorite sports: `swimming`, `tennis`, `soccer`, `baseball`, `basketball`
 - Each person lives in a unique style of house: `ranch`, `craftsman`, `victorian`, `modern`, `colonial`
 - Each mother is accompanied by their child: `Timothy`, `Samantha`, `Bella`, `Meredith`, `Fred`
 - People have unique heights: `average`, `very tall`, `very short`, `short`, `tall`

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

Answer: short


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

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

Answer: Samantha


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

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

Answer: romance


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Bob`, `Arnold`, `Eric`, `Alice`
 - Each person has a favorite color: `red`, `white`, `green`, `blue`, `yellow`
 - Each person has a unique hobby: `gardening`, `cooking`, `painting`, `photography`, `knitting`
 - People have unique heights: `very tall`, `tall`, `average`, `short`, `very short`
 - People have unique hair colors: `black`, `blonde`, `brown`, `gray`, `red`

## Clues:
1. The person whose favorite color is green is somewhere to the right of the person who has black hair.
2. The person who has an average height is somewhere to the right of the person who loves blue.
3. The person who has red hair is Alice.
4. The person who loves cooking is not in the first house.
5. The person who loves yellow is the person who has gray hair.
6. The person who is short is somewhere to the left of the person who loves yellow.
7. The person who enjoys gardening is the person who loves white.
8. Arnold is somewhere to the left of the person who has black hair.
9. The person who is short is somewhere to the right of the person who has blonde hair.
10. Eric and the person who loves cooking are next to each other.
11. The person who has black hair is directly left of the person who loves blue.
12. The person who is very short is the person who has black hair.
13. Bob is the person who loves blue.
14. The person who is tall is directly left of the photography enthusiast.
15. The person who loves white is not in the first house.
16. The person who loves yellow is the person who enjoys knitting.
17. The person who loves blue is the person who has blonde hair.
 What is the value of attribute Hobby for the person whose attribute Height is very tall? Please reason step by step, and put your final answer within \boxed{}

Answer: painting


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Bob`, `Eric`, `Alice`, `Peter`, `Arnold`
 - People have unique heights: `super tall`, `tall`, `very short`, `average`, `very tall`, `short`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `samsung galaxy s21`, `xiaomi mi 11`, `oneplus 9`, `huawei p50`
 - Each mother is accompanied by their child: `Fred`, `Timothy`, `Alice`, `Bella`, `Meredith`, `Samantha`

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

Answer: 1


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`
 - The people are of nationalities: `dane`, `swede`, `german`, `brit`, `norwegian`
 - Each person has a unique hobby: `gardening`, `painting`, `knitting`, `photography`, `cooking`
 - Everyone has a favorite smoothie: `dragonfruit`, `lime`, `watermelon`, `desert`, `cherry`
 - People have unique hair colors: `brown`, `gray`, `black`, `blonde`, `red`
 - People have unique favorite sports: `baseball`, `soccer`, `swimming`, `tennis`, `basketball`

## Clues:
1. The person who loves cooking is not in the second house.
2. The Watermelon smoothie lover is not in the fourth house.
3. The person who loves baseball is somewhere to the right of the Desert smoothie lover.
4. The person who drinks Lime smoothies is the person who has brown hair.
5. The person who has gray hair is somewhere to the right of the Swedish person.
6. The photography enthusiast is not in the fifth house.
7. The Swedish person is not in the first house.
8. The German is the person who has blonde hair.
9. There is one house between the Watermelon smoothie lover and the photography enthusiast.
10. Alice and the person who drinks Lime smoothies are next to each other.
11. The British person and the person who enjoys knitting are next to each other.
12. The person who has black hair is the Swedish person.
13. The British person is the person who enjoys gardening.
14. The person who loves swimming is the person who loves cooking.
15. The Dragonfruit smoothie lover is not in the third house.
16. The Watermelon smoothie lover is not in the second house.
17. Eric is somewhere to the left of the person who drinks Lime smoothies.
18. The person who loves soccer is Alice.
19. Peter is the photography enthusiast.
20. The person who enjoys gardening is directly left of the person who loves tennis.
21. The Norwegian and the photography enthusiast are next to each other.
22. The British person is Alice.
23. Bob is the person who has blonde hair.
 What is the value of attribute Hobby for the person whose attribute Smoothie is dragonfruit? Please reason step by step, and put your final answer within \boxed{}

Answer: gardening


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Carol`, `Peter`, `Bob`
 - Each person has a unique level of education: `high school`, `trade school`, `bachelor`, `doctorate`, `master`, `associate`
 - Each person has a favorite color: `purple`, `blue`, `white`, `red`, `green`, `yellow`
 - Each person prefers a unique type of vacation: `cultural`, `cruise`, `beach`, `mountain`, `camping`, `city`
 - People have unique favorite book genres: `science fiction`, `biography`, `historical fiction`, `fantasy`, `romance`, `mystery`

## Clues:
1. Alice is the person with a master's degree.
2. There is one house between the person with an associate's degree and Peter.
3. The person who loves historical fiction books is somewhere to the left of the person who loves biography books.
4. The person who loves historical fiction books is the person with a high school diploma.
5. Bob is the person with a bachelor's degree.
6. Carol is somewhere to the right of Eric.
7. The person with a master's degree is somewhere to the right of the person who enjoys camping trips.
8. The person who loves science fiction books is somewhere to the left of the person who loves yellow.
9. The person who loves historical fiction books is somewhere to the left of the person whose favorite color is green.
10. The person with a high school diploma is not in the second house.
11. The person who loves beach vacations is the person with a doctorate.
12. The person who loves mystery books is somewhere to the left of the person with a bachelor's degree.
13. The person with a doctorate is in the first house.
14. The person with a high school diploma is directly left of the person who enjoys camping trips.
15. The person who loves romance books is not in the third house.
16. There is one house between the person who loves purple and the person with a bachelor's degree.
17. The person who loves biography books is the person whose favorite color is red.
18. There is one house between the person who loves fantasy books and the person who loves blue.
19. The person who goes on cultural tours is the person who loves blue.
20. Carol is not in the fifth house.
21. The person who loves purple is the person with a doctorate.
22. The person who enjoys mountain retreats is somewhere to the right of the person who loves science fiction books.
23. The person who prefers city breaks is the person whose favorite color is green.
 What is the value of attribute Education for the person whose attribute House is 6? Please reason step by step, and put your final answer within \boxed{}

Answer: master


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Eric`, `Peter`
 - Each person has a unique level of education: `master`, `bachelor`, `high school`, `associate`
 - People have unique heights: `short`, `average`, `tall`, `very short`
 - Each person has a unique favorite drink: `water`, `coffee`, `tea`, `milk`

## Clues:
1. The person who likes milk is somewhere to the left of the person with a bachelor's degree.
2. The person who likes milk and the person with a bachelor's degree are next to each other.
3. Alice is the person who is tall.
4. Peter is not in the second house.
5. Arnold is in the first house.
6. The person who has an average height is not in the second house.
7. Arnold is directly left of the person with a master's degree.
8. The coffee drinker is the person who is short.
9. There is one house between the person with a master's degree and the person with a high school diploma.
10. The one who only drinks water is directly left of Alice.
11. The person who is short is somewhere to the left of the person who is tall.
 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: 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: `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
 - People own unique car models: `toyota camry`, `bmw 3 series`, `ford f150`, `honda civic`, `tesla model 3`
 - Each person has a unique favorite drink: `tea`, `milk`, `water`, `root beer`, `coffee`
 - The people are of nationalities: `dane`, `brit`, `german`, `norwegian`, `swede`
 - People use unique phone models: `oneplus 9`, `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `huawei p50`

## Clues:
1. The person who uses a Google Pixel 6 is somewhere to the right of the Dane.
2. The person who uses a Google Pixel 6 is somewhere to the right of the root beer lover.
3. The person who owns a Tesla Model 3 is the British person.
4. The person who owns a Ford F-150 is the Dane.
5. Eric is the tea drinker.
6. The British person is directly left of the person who owns a Honda Civic.
7. Peter is somewhere to the right of the person who likes milk.
8. Bob is directly left of the Swedish person.
9. The person who uses an iPhone 13 is somewhere to the right of the person who uses a Huawei P50.
10. Arnold is the person who uses a Google Pixel 6.
11. Eric is not in the third house.
12. The person who owns a BMW 3 Series is somewhere to the right of the person who uses a Samsung Galaxy S21.
13. The person who owns a Tesla Model 3 is not in the first house.
14. Alice is somewhere to the left of the person who owns a Ford F-150.
15. The person who uses a OnePlus 9 is somewhere to the right of the person who uses a Samsung Galaxy S21.
16. The German is somewhere to the right of the person who owns a Tesla Model 3.
17. Eric is not in the fourth house.
18. Peter is directly left of the person who uses a Samsung Galaxy S21.
19. The coffee drinker is Bob.
 What is the value of attribute Name for the person whose attribute Drink is root beer? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Bob`, `Eric`, `Alice`, `Peter`, `Arnold`
 - People have unique heights: `super tall`, `tall`, `very short`, `average`, `very tall`, `short`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `samsung galaxy s21`, `xiaomi mi 11`, `oneplus 9`, `huawei p50`
 - Each mother is accompanied by their child: `Fred`, `Timothy`, `Alice`, `Bella`, `Meredith`, `Samantha`

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

Answer: oneplus 9


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Peter`, `Eric`
 - People own unique car models: `ford f150`, `tesla model 3`, `toyota camry`, `honda civic`
 - Each person has a unique birthday month: `april`, `sept`, `feb`, `jan`
 - Each person has an occupation: `teacher`, `artist`, `engineer`, `doctor`
 - Everyone has something unique for lunch: `pizza`, `stew`, `spaghetti`, `grilled cheese`

## Clues:
1. The person who owns a Toyota Camry is in the first house.
2. Eric and the person who loves the spaghetti eater are next to each other.
3. The person who is an engineer is the person who loves eating grilled cheese.
4. Peter is the person who owns a Toyota Camry.
5. The person whose birthday is in April is directly left of the person whose birthday is in February.
6. The person who is a doctor is somewhere to the right of the person who is an artist.
7. The person who owns a Ford F-150 is the person who is a doctor.
8. The person whose birthday is in September is somewhere to the left of Alice.
9. Arnold is the person who is an artist.
10. The person who loves the spaghetti eater is the person who owns a Ford F-150.
11. The person who is an engineer is the person whose birthday is in April.
12. The person whose birthday is in February and the person who is a pizza lover are next to each other.
13. The person who owns a Honda Civic is Eric.
 What is the value of attribute CarModel for the person whose attribute Food is spaghetti? Please reason step by step, and put your final answer within \boxed{}

Answer: ford f150


Given a triangle \(ABC\), let \(O\) be the center of its circumcircle, \(G\) be its centroid (i.e., the point of intersection of its medians), and \(H\) be its orthocenter (i.e., the point of intersection of its altitudes). Let \(A'\), \(B'\), and \(C'\) be the midpoints of \([BC]\), \([CA]\), and \([AB]\) respectively, and \(D\), \(E\), and \(F\) be the feet of the altitudes from \(A\), \(B\), and \(C\) respectively.

- Show that \(A'B'C'\) is the image of \(ABC\) under a homothety centered at \(G\).
- Deduce that \(O\), \(G\), and \(H\) are collinear.

Answer: O, G, H \text{ are collinear}


X is a subset of the rationals which is closed under addition and multiplication. 0 ∉ X. For any rational \( x \neq 0 \), exactly one of \( x \) or \( -x \) belongs to X. Show that X is the set of all positive rationals.

Answer: X \text{ is the set of all positive rational numbers}


6. Given $A_{i}\left(t_{i}+\frac{1}{t_{i}}, t_{i}-\frac{1}{t_{i}}\right)(i=1,2,3$,
4) are four distinct points on a plane. If these four points are concyclic, then $t_{1} 、 t_{2} 、 t_{3} 、 t_{4}$ should satisfy $\qquad$ .

Answer: t_1 t_2 t_3 t_4 = 1


5.4. Find the angle $C$ of triangle $A B C$, if vertex ' $A$ is equidistant from the centers of the excircles *) tangent to sides $A B$ and $B C$.

*) Excircle - a circle that is tangent to one side of the triangle and the extensions of the other two sides.

Answer: 90^\circ


The first $9$ positive integers are placed into the squares of a $3\times 3$ chessboard. We are taking the smallest number in a column. Let $a$ be the largest of these three smallest number. Similarly, we are taking the largest number in a row. Let $b$ be the smallest of these three largest number. How many ways can we distribute the numbers into the chessboard such that $a=b=4$?

Answer: 25920


10. (15 points) Given that $P$ is a point on the hyperbola $E: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ $(a>0, b>0)$, and points $A$ and $B$ are on the two asymptotes of the hyperbola, with $A$ in the first quadrant and $B$ in the fourth quadrant. When $P$, $A$, and $B$ are collinear, find the range of the area of $\triangle A O B$.

Answer: [ab, +\infty)


7. As shown in the figure, 10 identical regular pentagons are joined edge-to-edge, forming a regular decagon in the middle. 3 identical regular $n$-sided polygons are joined edge-to-edge, forming a regular triangle in the middle, then the value of $n$ is $\qquad$ .

Answer: 12


Two altitudes of a triangle are equal to 10 and 6. Prove that the third altitude is less than 15.

Answer: h_c < 15


In each cell of a 15 x 15 square, there is a natural number not exceeding 4, and the sum of the numbers in each 2 x 2 square is 7. What is the maximum possible sum of all the numbers in the table?

Answer: 417


Example 3 The condition for three line segments to form a triangle is: the sum of the lengths of any two line segments is greater than the length of the third line segment. There is a wire of length $144 \mathrm{~cm}$, which is to be cut into $n$ $(n>2)$ small segments, with each segment being no less than $1 \mathrm{~cm}$ in length. If any three of these segments cannot form a triangle, then the maximum value of $n$ is $\qquad$

Answer: 10


1. When purchasing goods for an amount of no less than 1000 rubles, the store provides a discount on subsequent purchases of $50 \%$. Having 1200 rubles in her pocket, Dasha wanted to buy 4 kg of strawberries and 6 kg of sugar. In the store, strawberries were sold at a price of 300 rubles per kg, and sugar - at a price of 30 rubles per kg. Realizing that she didn't have enough money for the purchase, Dasha still managed to buy what she intended. How did she do it?

Answer: 1200


11. String 6 red balls, 1 white ball, and 8 yellow balls into a necklace, then the number of possible arrangements is $\qquad$ (balls of the same color are indistinguishable).

Answer: 1519


11. The sequence $\left\{a_{n}\right\}$ is: $1,1,2,1,1,2,3,1,1,2,1,1,2,3,4, \cdots$, that is, first take $a_{1}=1$, then copy this item and paste it behind as $a_{2}$, and add the successor number 2 as $a_{3}$, then copy all items $1,1,2$ and paste them behind as $a_{4}, a_{5}, a_{6}$, and add the successor number 3 as $a_{7}, \cdots$ and continue in this manner. Then $a_{2021}=$ $\qquad$

Answer: 1


12 Let $E$ be the intersection of the diagonals $AC$ and $BD$ of the cyclic quadrilateral $ABCD$. Given: $AC=BC, AD=5, BE=12, DE=3$, find $\angle BCD$.

Answer: 90^\circ


Let's inscribe a regular hexagon and rotate it around its axes of symmetry. How do the surface areas of the resulting solids of revolution compare?

Answer: \frac{7}{4\sqrt{3}}


7. A regular tetrahedron frame with edge length 3 contains a solid sphere with radius 1. Then the maximum volume of the common part of the tetrahedron and the sphere is $\qquad$

Answer: \dfrac{(21\sqrt{6} - 32)\pi}{24}


a) Given a rectangular table of size \(4 \times 10\) (cells). What is the maximum number of crosses that can be placed in the cells of this table such that the following condition is met: each row and each column of the table must contain an odd number of crosses?

b) Is it possible to place several crosses in a table of size \(5 \times 10\) such that the indicated condition is met?

Answer: 30


Given the equation \( k x^2 + \frac{|x|}{x+6} = 0 \) has four distinct real solutions for \( x \). Determine the range of values for the real number \( k \).

Answer: (-\infty, -\dfrac{1}{9})


The solid shown has a square base of side length $s$ . The upper edge is parallel to the base and has length $2s$ . All other edges have length $s$ . Given that $s=6\sqrt{2}$ , what is the volume of the solid?

Answer: 288


A cat broke into a wine store at night. It jumped onto a shelf where bottles of wine were lined up in a long row - the first third of the bottles at the edge cost 160 Kč each, the next third cost 130 Kč each, and the last third cost 100 Kč each. First, the cat knocked over a 160 Kč bottle at the very beginning of the row, and then continued to knock over one bottle after another without skipping any. Before it got bored, it had knocked over 25 bottles, all of which broke. In the morning, the owner regretted that the cat had not started its mischief at the other end of the shelf. Even if it had broken the same number of bottles, the loss would have been 660 Kč less. How many bottles were originally on the shelf?

(L. Šimůnek)

Answer: 36


5. Let $a$ and $n$ be natural numbers, and it is known that $a^{n}-2014$ is a number with $2014$ digits. Find the smallest natural number $k$ such that $a$ cannot be a $k$-digit number.

Answer: 49


2. Let $n \in \mathbb{N}^{*}$. Find $x \in \mathbb{Z}$ for which $\frac{x^{5 n}+2 x^{2 n}-x^{n}+1}{x^{5 n}+x^{n}-1} \in \mathbb{Z}$.

Carmen and Viorel Botea, Brăila

Answer: -1


A computer science professor is writing a series of publications on theoretical concepts. He plans to write a total of 12 publications this year. Each publication takes him 3 weeks of dedicated work. Additionally, he allocates 2 hours each week to review and edit previous work. If the professor works 5 days a week and dedicates 6 hours each day to writing and reviewing his publications, how many hours in total will he spend on writing and reviewing all 12 publications by the end of the year?

Answer: 1080


Alex is a detail-oriented digital project manager overseeing a web development project. The project requires coordinating a team of developers to complete 5 different tasks. Each task takes a different amount of time to finish: Task A requires 3 days, Task B takes 2 days, Task C needs 4 days, Task D takes 5 days, and Task E requires 1 day. Alex needs to plan the project timeline efficiently by assigning tasks in parallel when possible, but no more than two tasks can be worked on simultaneously due to resource limitations. What is the minimum number of days required to complete all tasks if Alex schedules them optimally?

Answer: 8


A loyal fan of CD Mirandés has 3 different CD Mirandés jerseys and wants to wear a different one each time the team plays at home this month. If CD Mirandés has 5 home games this month, how many different combinations of jerseys can the fan wear if they choose a jersey at random for each game?

Answer: 243


A film producer who transitioned from theater is planning a new project that involves both film and live theater performances. For the film, they need to schedule 12 shooting days, each costing $5,000. For the theater, they plan to have 10 performances, each generating $2,000 in ticket sales. However, they need to share 20% of the theater revenue with the theater house. The producer’s friend, who understands the modern entertainment industry, suggests investing an additional $3,000 in marketing to boost ticket sales by 25%. Calculate the total profit or loss the producer will make if they proceed with this plan, considering the costs and revenues from both the film and theater projects.

Answer: -43000


Jamie is an investment analyst who specializes in drone tech companies. She is analyzing two different companies, SkyHigh Drones and AeroTech Innovations. SkyHigh Drones has reported a quarterly profit increase of $1,200,000, while AeroTech Innovations has reported a quarterly profit increase of $850,000. Jamie predicts that SkyHigh Drones' profits will grow by an additional 5% next quarter, and AeroTech Innovations' profits will grow by an additional 10%.

Calculate the projected profit increases for the next quarter for both SkyHigh Drones and AeroTech Innovations, and find out the total projected profit increase for both companies combined.

Answer: 2195000


Alex, a Marvel fan who absolutely cannot stand spoilers, is planning a Marvel movie marathon. He wants to watch all the movies in the correct order, so he avoids any spoilers about character developments and plot twists. Alex has a list of 23 Marvel movies to watch. He plans to watch 3 movies each day during the weekend (Saturday and Sunday) and 2 movies each day from Monday to Friday. However, he had to skip watching any movies on Wednesday because he was busy avoiding spoiler-heavy social media comments. How many days in total will it take Alex to finish watching all the Marvel movies?

Answer: 11


It's not difficult to verify that the power of $3^{m}$, where $m=3^{2}$, starts with the digit sequence 1968. - On the other hand, someone claims that in the sequence of powers of $2^{n}$ - where $n$ is a natural number - the first one that starts with the digit sequence 1968 has more than $2^{k}$ digits, where $k=2^{3}$. Let's find the exponent of this power.

Answer: 868


In a mathematical student competition, three problems were posed: $A$, $B$, and $C$. 25 students in total solved at least one of the problems. Among the students who did not solve $A$, there were twice as many who solved $B$ as those who solved $C$. One more student solved only $A$ than those who solved $A$ along with other problems. Half of the students who solved only one problem did not solve $A$. - How many students solved only $B$?

Answer: 6


3. In the Cartesian coordinate system, the set of points $(x, y)$ that satisfy $x<y,|x|<3,|y|<3$, and make the equation $\left(x^{3}-y^{3}\right) t^{4}+$ $(3 x+y) t^{2}+\frac{1}{x-y}=0$ have no real roots is denoted as $N$. Then the area of the region formed by the point set $N$ is $\qquad$

Answer: \dfrac{81}{5}


Example 1 Find the value of $\xi_{0}=\langle-1,1,4, \overline{3,1,1,1,3,7}\rangle$.

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

Answer: \dfrac{3 - \sqrt{57}}{24}


7.1. The sides of a rectangle are in the ratio $3: 4$, and its area is numerically equal to its perimeter. Find the sides of the rectangle.

Answer: \dfrac{7}{2}


Define the sequence $a_1, a_2, a_3, \ldots$ by $a_n = \sum\limits_{k=1}^n \sin{k}$ , where $k$ represents radian measure. Find the index of the 100th term for which $a_n < 0$ .
Please give the answer directly without any intermediate steps.

Answer: 628


21. A sequence of numbers is arranged according to the rule:
$$
1^{1}, 2^{2}, 3^{3}, 4^{4}, 5^{5}, 1^{6}, 2^{7}, 3^{8}, 4^{9}, 5^{10}, \cdots \cdots
$$

The bases cycle through $1 \sim 5$, and the exponents start from 1 and increase sequentially. Let the sum of the first $n$ numbers be denoted as $P(n)$. When $P(n)$ is a multiple of 10 for the first time, $n=$ $\qquad$ .

Answer: 37


5. If for any positive divisor $a$ of a positive integer $n$ we have $(a+1) \mid (n+1)$, then $n$ is called a "good number". Find all good numbers.

Answer: 1


14. (10 points) Person A departs from location $A$ at a constant speed to location $B$. When A departs, person B departs from location $B$ at a constant speed to location $A$. They meet at location $C$ on the way. After meeting, A continues for another 150 meters and then turns around to chase B. A catches up with B 540 meters from $C$. When A catches B, A immediately turns around to head to $B$. As a result, when A arrives at $B$, B also arrives at $A$. The distance between $A$ and $B$ is $\qquad$ meters.

Answer: 2484


Let $p(x)$ be a polynomial with all roots real and distinct. Prove that $\left(p^{\prime}(x)\right)^{2} \geqslant p(x) p^{\prime \prime}(x)$.

Answer: \left(p^{\prime}(x)\right)^{2} \geqslant p(x) p^{\prime \prime}(x)


Three. (20 points) Let real numbers $a, m$ satisfy
$$
a \leqslant 1, 0 < m \leqslant 2 \sqrt{3} \text{, }
$$

The function $f(x)=\frac{a m x - m x^{2}}{a + a(1-a)^{2} m^{2}}(x \in (0, a))$. If there exist $a, m, x$ such that $f(x) \geqslant \frac{\sqrt{3}}{2}$, find all real values of $x$.

Answer: \dfrac{1}{2}


3. An electronic flea can jump from one circle to an adjacent circle with each step. Now, a red flea starts from the circle marked with the number "0" and jumps 1991 steps in a clockwise direction, landing in one circle. A black flea also starts from the circle marked with the number "0", but it jumps 1949 steps in a counterclockwise direction, landing in another circle. What is the product of the numbers in these two circles?

Answer: 77


a) Two players are playing a game. The first player writes two rows of 10 numbers each, one under the other, such that the following rule holds: if the number \( b \) is written below the number \( a \), and the number \( d \) is written below the number \( c \), then \( a+d = b+c \). The second player, knowing this rule, wants to determine all the written numbers. He is allowed to ask the first player questions like "What number is in the first row in the third position?" or "What number is in the second row in the ninth position?", etc. What is the minimum number of such questions the second player needs to ask to determine all the numbers?

b) In an \( m \times n \) table, numbers are written such that for any two rows and any two columns, the sum of the numbers at two opposite corners of the rectangle formed by them is equal to the sum of the numbers at the other two corners. Some of the numbers have been erased, but the erased numbers can be recovered from the remaining ones. Prove that at least \( m + n - 1 \) numbers remain.

Answer: 11


8. There is a square on the plane, whose sides are parallel to the horizontal and vertical directions. Draw several line segments parallel to the sides of the square such that: no two line segments belong to the same straight line, and any two intersecting line segments intersect at an endpoint of one of the line segments. It is known that the drawn line segments divide the square into several rectangles, and any vertical line intersecting the square but not containing any of the drawn line segments intersects exactly $k$ rectangles, and any horizontal line intersecting the square but not containing any of the drawn line segments intersects exactly $l$ rectangles. Find all possible values of the number of rectangles in the square.

Answer: k \times l


19. $\overline{P Q Q P Q Q}, \overline{P Q P Q P Q}, \overline{Q P Q P Q P}, \overline{P P P P P P}, \overline{P P P Q Q Q}$ are five six-digit numbers, where the same letters represent the same digits, and different letters represent different digits. Regardless of what $P$ and $Q$ are, among these five six-digit numbers, $\qquad$ of them are definitely multiples of 7.

Answer: 4


7. Let set $S$ contain $n$ elements, $A_{1}, A_{2}, \cdots, A_{k}$ are different subsets of $S$, they all have non-empty intersections with each other, and no other subset of $S$ can intersect with all of $A_{1}, A_{2}, \cdots, A_{k}$. Express this in terms of $n$:

Answer: 2^{n-1}


Given a triangle  $OAB$  with the vetices  $O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0)$  in the  $xyz$  space.
Let  $V$  be the cone obtained by rotating the triangle around the  $x$ -axis.
Find the volume of the solid obtained by rotating the cone  $V$  around the  $y$ -axis.

Answer: \frac{8\pi}{3}


We are given a map divided into  $13\times 13$  fields. It is also known that at one of the fields a tank of the enemy is stationed, which we must destroy. To achieve this we need to hit it twice with shots aimed at the centre of some field. When the tank gets hit it gets moved to a neighbouring field out of precaution. At least how many shots must we fire, so that the tank gets destroyed certainly?
*We can neither see the tank, nor get any other feedback regarding its position.*

Answer: 254


An engineer is designing an engine. Each cycle, it ignites a negligible amount of fuel, releasing  $ 2000 \, \text{J} $  of energy into the cubic decimeter of air, which we assume here is gaseous nitrogen at  $ 20^\circ \, \text{C} $  at  $ 1 \, \text{atm} $  in the engine in a process which we can regard as instantaneous and isochoric. It then expands adiabatically until its pressure once again reaches  $ 1 \, \text{atm} $ , and shrinks isobarically until it reaches its initial state. What is the efficiency of this engine?

*Problem proposed by B. Dejean*

Answer: 0.33


The diagram below shows an  $8$ x $7$  rectangle with a 3-4-5 right triangle drawn in each corner. The lower two triangles have their sides of length 4 along the bottom edge of the rectangle, while the upper two
triangles have their sides of length 3 along the top edge of the rectangle. A circle is tangent to the hypotenuse of each triangle. The diameter of the circle is  $\frac{m}{n}$  , where m and n are relatively prime positive integers. Find m + n.
For diagram go to http://www.purplecomet.org/welcome/practice, go to the 2015 middle school contest questions, and then go to #20

Answer: 47


Let the circles  $k_1$  and  $k_2$  intersect at two points  $A$  and  $B$ , and let  $t$  be a common tangent of  $k_1$  and  $k_2$  that touches  $k_1$  and  $k_2$  at  $M$  and  $N$  respectively. If  $t\perp AM$  and  $MN=2AM$ , evaluate the angle  $NMB$ .

Answer: 45


Let  $T = \{9^k : k \ \text{is an integer}, 0 \le k \le 4000\}$ .  Given that  $9^{4000}$  has 3817 digits and that its first (leftmost) digit is 9, how many elements of  $T$  have 9 as their leftmost digit?

Answer: 184


Five identical circles are placed in a line inside a larger one as shown. If the shown chord has length  $16,$  find the radius of the large circle.

Answer: 8


Let  $S$  be a real number. It is known that however we choose several numbers from the interval  $(0, 1]$  with sum equal to  $S$ , these numbers can be separated into two subsets with the following property: The sum of the numbers in one of the subsets doesn’t exceed 1 and the sum of the numbers in the other subset doesn’t exceed 5.

Find the greatest possible value of  $S$ .

Answer: 5.5


A set  $\mathcal{S}$  of distinct positive integers has the following property: for every integer  $x$  in  $\mathcal{S},$  the arithmetic mean of the set of values obtained by deleting  $x$  from  $\mathcal{S}$  is an integer.  Given that 1 belongs to  $\mathcal{S}$  and that 2002 is the largest element of  $\mathcal{S},$  what is the greatet number of elements that  $\mathcal{S}$  can have?

Answer: 30


There is a very popular race course where runners frequently go for a daily run. Assume that all runners randomly select a start time, a starting position on the course, and a direction to run. Also assume that all runners make exactly one complete circuit of the race course, all runners run at the same speed, and all runners complete the circuit in one hour. Suppose that one afternoon you go for a run on this race course, and you count  $300$  runners which you pass going in the opposite direction, although some of those runners you count twice since you pass them twice. What is the expected value of the number of different runners that you pass not counting duplicates?

Answer: 225


A block  $Z$  is formed by gluing one face of a solid cube with side length 6 onto one of the circular faces of a right circular cylinder with radius  $10$  and height  $3$  so that the centers of the square and circle coincide. If  $V$  is the smallest convex region that contains Z, calculate  $\lfloor\operatorname{vol}V\rfloor$  (the greatest integer less than or equal to the volume of  $V$ ).

Answer: 2827


Laurie loves multiplying numbers in her head. One day she decides to multiply two  $2$ -digit numbers  $x$  and  $y$  such that  $x\leq y$  and the two numbers collectively have at least three distinct digits. Unfortunately, she accidentally remembers the digits of each number in the opposite order (for example, instead of remembering  $51$  she remembers  $15$ ). Surprisingly, the product of the two numbers after flipping the digits is the same as the product of the two original numbers. How many possible pairs of numbers could Laurie have tried to multiply?

Answer: 28


 The deputies in a parliament were split into  $10$  fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation.
Find the smallest possible number of independent deputies after the vacation.

Answer: 50


Let  $\triangle ABC$  be a right triangle with hypotenuse  $AC$ . A square is inscribed in the triangle such that points  $D,E$  are on  $AC$ ,  $F$  is on  $BC$ , and  $G$  is on  $AB$ . Given that  $AG=2$  and  $CF=5$ , what is the area of  $\triangle BFG$ ?

Answer: \frac{500}{841}


We draw a unit-radius quarter circle around each vertex of a unit-sided square, which connects the adjacent vertices. Calculate the area of the common part of the four circle segments.

Answer: 1 + \dfrac{\pi}{3} - \sqrt{3}


An alphabet consists of $n$ letters. What is the maximal length of a word if we know that any two consecutive letters $a,b$ of the word are different and that the word cannot be reduced to a word of the kind $abab$ with $a\neq b$ by removing letters.

Answer: 2n-1


Let \( X \) be a set of \( n \) elements and \( Y \) be a subset of \( X \) with \( k \) elements. Prove that the largest family \( S \) of subsets of \( X \) that contain exactly \( r \) elements of \( Y \) consists of \( C_{k} C_{n}^{[\frac{n-k}{2}]} \) subsets.

Answer: \binom{k}{r} \binom{n-k}{\left\lfloor \frac{n-k}{2} \right\rfloor}


Inside a convex polygon \( M \), a circle with the maximum possible radius \( R \) is inscribed (meaning no larger circle can fit inside \( M \)). It is known that a segment of length 1 can be rotated at any angle within the polygon (i.e., we can move the unit segment as a rigid rod across the plane so that it does not protrude outside the polygon \( M \) and can be rotated to any given angle). Prove that \( R \geq 1 / 3 \).

Answer: R \geq \frac{1}{3}


4. Pjotr has a bathtub with a plug at the bottom that he can fill with two identical taps. The water flows from both taps at the same constant speed, and the bathtub drains (if the plug is not in) at a constant speed.

On Monday, Pjotr lets the bathtub fill to the brim by turning on one tap fully, after which he pulls the plug and waits until the bathtub is empty again. Only when the bathtub is empty does he turn the tap off. On Tuesday, he lets the bathtub fill to the brim by turning on both taps fully, after which he pulls the plug and waits until the bathtub is empty again. Only when the bathtub is empty does he turn the taps off. On both days, Pjotr was busy for exactly 45 minutes from turning on the tap/taps to turning them off again.

How many minutes does it take for a full bathtub to drain completely if both taps are off?

Answer: 10


The two numbers $0$ and $1$ are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the number $a+b$ between all pairs of consecutive numbers $a$, $b$ on the board. How many odd numbers will be on the board after $10$ such operations?

[i]Proposed by Michael Kural[/i]

Answer: 683


11. As shown in Figure 1, given
$\odot C:(x+1)^{2}+y^{2}=$
8, a fixed point $A(1,0)$, and $M$ is
a moving point on the circle. Points $P$ and $N$
are on $A M$ and $M C$, respectively, and
satisfy $\overrightarrow{A M}=2 \overrightarrow{A P}, \overrightarrow{N P}$.
$\overline{A M}=0$, the trajectory of point $N$ is curve $E$. If a line passing through the fixed point $F$
$(2,0)$ intersects curve $E$ at two different points $G$ and $H$
(point $G$ is between points $F$ and $H$), and satisfies $\overrightarrow{F G}=\lambda \overrightarrow{F H}$, then the range of $\lambda$ is

Answer: [3 - 2\sqrt{2}, 1)


3.- Let $p \geq 3$ be a prime number. Each side of a triangle is divided into $p$ equal parts, and each of the division points is connected to the opposite vertex. Calculate the maximum number of regions, disjoint from each other, into which the triangle is divided.

## SOLUTION:

## SOLUTION:

Answer: 3p^2 - 3p + 1


A set $S$ is constructed as follows. To begin, $S = \{0,10\}$ . Repeatedly, as long as possible, if $x$ is an integer root of some polynomial $a_{n}x^n + a_{n-1}x^{n-1} + \dots + a_{1}x + a_0$ for some $n\geq{1}$ , all of whose coefficients $a_i$ are elements of $S$ , then $x$ is put into $S$ . When no more elements can be added to $S$ , how many elements does $S$ have?

Answer: 9


The secant passing through the intersection points of the circle with center $O_{1}$ and radius $4 \mathrm{~cm}$ and the circle with center $O_{2}$ and radius $6 \mathrm{~cm}$ intersects the segment $O_{1} O_{2}$ at point $T$, and we know that the length of this segment is not less than $6 \mathrm{~cm}$. The larger circle intersects the segment $O_{1} O_{2}$ at point $A$, and the smaller circle intersects it at point $B$, and $A T: B T=1: 2$. Calculate the length of the segment $O_{1} O_{2}$.

Answer: 6


9.5. What is the smallest number of colors needed to color all the cells of a 6 by 6 square so that in each row, column, and diagonal of the square, all cells have different colors? Explanation: a diagonal of the square is understood to mean all rows of at least two cells running diagonally from one edge of the square to the other at an angle of $45^{\circ}$ or $135^{\circ}$ to the horizontal.

Answer: 7


Answer: 4 + 4\sqrt{10}


In the right triangle $ABC$, the foot of the altitude from $C$ to the hypotenuse $AB$ is $D$. The angle bisectors of $\angle ACD$ and $\angle BCD$ intersect the hypotenuse $AB$ at points $E$ and $F$, respectively. Determine the ratio of the radius of the incircle of triangle $ABC$ to the radius of the circumcircle of triangle $CEF$.

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


283 Find the unit digit of $\left(\frac{5+\sqrt{21}}{2}\right)^{2010}$.

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

Answer: 7


A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n = (x_n, y_n),$ the frog jumps to $P_{n+1},$ which may be any of the points $(x_n + 7, y_n + 2),$ $(x_n + 2, y_n + 7),$ $(x_n - 5, y_n - 10),$ or $(x_n - 10, y_n - 5).$ There are $M$ points $(x, y)$ with $|x| + |y| \le 100$ that can be reached by a sequence of such jumps. Find the remainder when $M$ is divided by $1000.$
Please give the answer directly without any intermediate steps.

Answer: 373


Through the ends of three edges emerging from the vertices \( B \), \( D \), \( A_{1} \), and \( C_{1} \) of a cube \( ABCD A_{1} B_{1} C_{1} D_{1} \) with edge length \( a \), planes are drawn. Prove that the resulting figure is a regular tetrahedron, and calculate its total surface area and volume.

Answer: 2\sqrt{3}a^2


$f(n)$ is the least number that there exist a $f(n)-$mino that contains every $n-$mino.
Prove that $10000\leq f(1384)\leq960000$.
Find some bound for $f(n)$

Answer: 10000 \leq f(1384) \leq 960000


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Eric`, `Bob`, `Carol`, `Arnold`
 - Everyone has a unique favorite cigar: `dunhill`, `blends`, `blue master`, `yellow monster`, `pall mall`, `prince`
 - People have unique favorite book genres: `fantasy`, `romance`, `science fiction`, `mystery`, `historical fiction`, `biography`

## Clues:
1. The person partial to Pall Mall is Alice.
2. The Prince smoker is not in the sixth house.
3. Carol is directly left of the person who loves fantasy books.
4. The Prince smoker is not in the fifth house.
5. Peter is in the first house.
6. The person partial to Pall Mall is directly left of the Dunhill smoker.
7. The person partial to Pall Mall is in the second house.
8. Carol is the person who loves mystery books.
9. Bob is not in the fourth house.
10. The person who smokes Yellow Monster is not in the sixth house.
11. The person who loves biography books is not in the third house.
12. The person who loves historical fiction books is in the first house.
13. The person who smokes many unique blends is not in the fifth house.
14. The person who loves romance books is somewhere to the right of Eric.
15. There is one house between Arnold and the person who smokes Yellow Monster.
16. There is one house between the person who smokes many unique blends and Bob.
 What is the value of attribute House for the person whose attribute Cigar is blends? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Bob`, `Carol`, `Arnold`, `Eric`
 - Each person has an occupation: `engineer`, `doctor`, `lawyer`, `artist`, `nurse`, `teacher`
 - Everyone has something unique for lunch: `soup`, `stew`, `grilled cheese`, `spaghetti`, `pizza`, `stir fry`
 - Everyone has a favorite smoothie: `cherry`, `dragonfruit`, `watermelon`, `lime`, `blueberry`, `desert`
 - Each person prefers a unique type of vacation: `camping`, `cultural`, `city`, `cruise`, `beach`, `mountain`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `oneplus 9`, `samsung galaxy s21`, `xiaomi mi 11`, `huawei p50`

## Clues:
1. The person who likes Cherry smoothies is directly left of the person who enjoys camping trips.
2. The person who is a lawyer is the person who uses a Xiaomi Mi 11.
3. The person who is a doctor is Eric.
4. The person who is a doctor and the Desert smoothie lover are next to each other.
5. The person who is an artist is the person who prefers city breaks.
6. The person who loves beach vacations is Alice.
7. There are two houses between the person who is a pizza lover and the person who uses a Huawei P50.
8. The person who uses a Xiaomi Mi 11 is the person who enjoys camping trips.
9. The person who is an artist is directly left of the person who loves eating grilled cheese.
10. The person who uses a Google Pixel 6 is in the first house.
11. There is one house between the person who is a teacher and the person who loves stir fry.
12. There are two houses between the person who is a pizza lover and the person who is an engineer.
13. The person who is a pizza lover is somewhere to the left of the person who loves eating grilled cheese.
14. The Watermelon smoothie lover is the person who uses a Xiaomi Mi 11.
15. The person who loves eating grilled cheese is Carol.
16. Alice is the person who drinks Blueberry smoothies.
17. The person who likes Cherry smoothies is the person who loves the stew.
18. The person who loves the soup is the person who uses a Samsung Galaxy S21.
19. The person who uses an iPhone 13 is directly left of the Dragonfruit smoothie lover.
20. The person who loves the stew is somewhere to the left of Peter.
21. Arnold is somewhere to the right of the Desert smoothie lover.
22. The person who likes going on cruises is the person who uses a Google Pixel 6.
23. The person who goes on cultural tours is not in the fifth house.
24. The person who is a teacher is Alice.
 What is the value of attribute PhoneModel for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: xiaomi mi 11


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Carol`, `Peter`, `Bob`, `Eric`
 - People use unique phone models: `huawei p50`, `iphone 13`, `xiaomi mi 11`, `oneplus 9`, `samsung galaxy s21`, `google pixel 6`

## Clues:
1. The person who uses an iPhone 13 is Alice.
2. The person who uses a Huawei P50 is in the first house.
3. The person who uses a OnePlus 9 is in the sixth house.
4. The person who uses a Google Pixel 6 is not in the second house.
5. The person who uses an iPhone 13 is not in the second house.
6. There is one house between Bob and Carol.
7. The person who uses a Huawei P50 is Eric.
8. The person who uses a Xiaomi Mi 11 is in the third house.
9. Alice is somewhere to the left of Carol.
10. Arnold is the person who uses a OnePlus 9.
 What is the value of attribute PhoneModel for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}

Answer: google pixel 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: `Eric`, `Alice`, `Arnold`, `Carol`, `Peter`, `Bob`
 - Each person lives in a unique style of house: `mediterranean`, `modern`, `craftsman`, `ranch`, `colonial`, `victorian`
 - People have unique favorite music genres: `country`, `hip hop`, `pop`, `jazz`, `classical`, `rock`
 - Each person has a unique hobby: `cooking`, `painting`, `photography`, `woodworking`, `gardening`, `knitting`

## Clues:
1. The person who loves rock music is in the fifth house.
2. The person who loves classical music and the woodworking hobbyist are next to each other.
3. The person in a Mediterranean-style villa is the person who loves hip-hop music.
4. There are two houses between Arnold and the person residing in a Victorian house.
5. The person who loves jazz music is directly left of Eric.
6. The person who loves hip-hop music is somewhere to the left of the person who enjoys knitting.
7. Carol is the person who loves hip-hop music.
8. The person in a Craftsman-style house is Arnold.
9. The person in a ranch-style home is Eric.
10. The woodworking hobbyist is the person residing in a Victorian house.
11. The person who loves country music is in the first house.
12. There is one house between the person who paints as a hobby and the person living in a colonial-style house.
13. Alice is the photography enthusiast.
14. The person who enjoys gardening is Eric.
15. Bob is in the third house.
 What is the value of attribute House for the person whose attribute Hobby is painting? Please reason step by step, and put your final answer within \boxed{}

Answer: 1