There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Carol`, `Peter`, `Eric`, `Bob`, `Alice`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `modern`, `craftsman`, `mediterranean`, `victorian`
 - Everyone has something unique for lunch: `pizza`, `stew`, `spaghetti`, `grilled cheese`, `stir fry`, `soup`
 - Each person prefers a unique type of vacation: `cultural`, `cruise`, `mountain`, `camping`, `city`, `beach`
 - People have unique heights: `average`, `very tall`, `very short`, `short`, `tall`, `super tall`
 - Everyone has a unique favorite cigar: `yellow monster`, `prince`, `dunhill`, `pall mall`, `blue master`, `blends`

## Clues:
1. Alice is in the fifth house.
2. The person who loves stir fry is the person living in a colonial-style house.
3. Alice is the person who loves the spaghetti eater.
4. Arnold is the person who loves the stew.
5. There is one house between the person who has an average height and Peter.
6. The person in a Craftsman-style house is not in the third house.
7. The person who has an average height is the person who loves stir fry.
8. The person who loves beach vacations is the person in a ranch-style home.
9. Eric is in the fourth house.
10. There is one house between the person living in a colonial-style house and the person who enjoys camping trips.
11. The person who enjoys mountain retreats is the person who smokes Yellow Monster.
12. The person who enjoys mountain retreats is the person who is very tall.
13. The person who enjoys mountain retreats and the Dunhill smoker are next to each other.
14. The person who loves the spaghetti eater is the person residing in a Victorian house.
15. The person who is tall is the person who loves beach vacations.
16. The person who is tall is somewhere to the left of the person residing in a Victorian house.
17. The person who loves stir fry is directly left of Bob.
18. The person in a modern-style house is somewhere to the left of Alice.
19. The person in a Craftsman-style house is somewhere to the left of the person who is short.
20. The person who loves stir fry is somewhere to the left of the Prince smoker.
21. There are two houses between the person who loves eating grilled cheese and the person who is super tall.
22. The person in a ranch-style home is the person who smokes Blue Master.
23. The person who smokes many unique blends is directly left of the person who smokes Blue Master.
24. The person who goes on cultural tours is the person who is a pizza lover.
25. The person who is a pizza lover is somewhere to the left of the person who likes going on cruises.
 What is the value of attribute Height for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: tall


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Peter`
 - Each person has a unique favorite drink: `milk`, `water`, `coffee`, `tea`
 - They all have a unique favorite flower: `lilies`, `roses`, `daffodils`, `carnations`
 - People have unique favorite sports: `tennis`, `basketball`, `soccer`, `swimming`
 - Each person has an occupation: `doctor`, `teacher`, `artist`, `engineer`
 - Each mother is accompanied by their child: `Fred`, `Samantha`, `Bella`, `Meredith`

## Clues:
1. The person who loves basketball is somewhere to the left of the person who is an engineer.
2. Eric is the person who loves a carnations arrangement.
3. The person who loves the rose bouquet is somewhere to the left of Peter.
4. The person who is a doctor is the person who loves tennis.
5. The person's child is named Bella is directly left of the person's child is named Samantha.
6. The person who is an artist is not in the third house.
7. The person who loves the rose bouquet is in the second house.
8. The person who loves a bouquet of daffodils and the person who loves tennis are next to each other.
9. The person who loves a bouquet of daffodils is the person who is a teacher.
10. The person who loves the rose bouquet is the person who likes milk.
11. The person's child is named Samantha is Alice.
12. The person who loves tennis is directly left of the one who only drinks water.
13. The person who loves a carnations arrangement is in the fourth house.
14. Arnold is somewhere to the left of the person who is an artist.
15. The person's child is named Fred is not in the fourth house.
16. The coffee drinker is the person who loves swimming.
 What is the value of attribute FavoriteSport for the person whose attribute Drink is coffee? Please reason step by step, and put your final answer within \boxed{}

Answer: swimming


Determine all rational numbers $a$ for which the matrix
$$
\left(\begin{array}{cccc}
a & -a & -1 & 0 \\
a & -a & 0 & -1 \\
1 & 0 & a & -a \\
0 & 1 & a & -a
\end{array}\right)
$$
is the square of a matrix with all rational entries. Provide the integer value of $a$ that satisfies the condition.

Answer: 0


For $R>1$ let $\mathcal{D}_{R}=\left\{(a, b) \in \mathbb{Z}^{2}: 0<a^{2}+b^{2}<R\right\}$. Compute
$$
\lim _{R \rightarrow \infty} \sum_{(a, b) \in \mathcal{D}_{R}} \frac{(-1)^{a+b}}{a^{2}+b^{2}}
$$
The original answer is in the format $m\pi \log n$. Please provide the integer value of m and n.

Answer: 1


Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_{i}=i^{k}+i$ for $i=0,1, \\ldots, p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_{2}$ upon division by $p$? Please provide the sum of all possible remainders.

Answer: 5


We throw a dice (which selects one of the numbers $1,2, \ldots, 6$ with equal probability) $n$ times. What is the probability that the sum of the values is divisible by 5? What is the value of the probability when n = 1/6?

Answer: 0


Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $(c,0)$ to $(3,3)$, divides the entire region into two regions of equal area. The original answer is in $\frac{k}{m}$ format, where $\frac{k}{m}$ is in simplest form. Please find the value of $k + m$.

Answer: 5


Points \(A\) and \(C\) lie on a circle centered at \(O\), each of \(\overline{BA}\) and \(\overline{BC}\) are tangent to the circle, and \(\triangle ABC\) is equilateral. The circle intersects \(\overline{BO}\) at \(D\). The original answer is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).

Answer: 3


Trapezoid \(ABCD\) has \(AD \parallel BC\), \(BD = 1\), \(\angle DBA = 23^\circ\), and \(\angle BDC = 46^\circ\). The ratio \(BC: AD\) is \(9: 5\). What is the value of \(CD\)? The original answer is in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a fully simplified fraction. Please provide the value of \(k + m\).

Answer: 9


Find all values of $b$ for which the equations $1988x^2 + bx + 8891 = 0$ and $8891x^2 + bx + 1988 = 0$ have a common root.

Enter all the possible values of $b,$ separated by commas.

Answer: 10879


Three circles $C_i$ are given in the plane: $C_1$ has diameter $AB$ of length $1$; $C_2$ is concentric and has diameter $k$ ($1 < k < 3$); $C_3$ has center $A$ and diameter $2k$. We regard $k$ as fixed. Consider all straight line segments $XY$ which have one endpoint $X$ on $C_2$, one endpoint $Y$ on $C_3$, and contain the point $B$. For what ratio $\frac{XB}{BY}$ will the segment $XY$ have minimal length?

Answer: 1


There are $128$ numbered seats arranged around a circle in a palaestra. The first person to enter the place would sit on seat number $1$. Since a contagious disease is infecting the people of the city, each person who enters the palaestra would sit on a seat whose distance is the longest to the nearest occupied seat. If there are several such seats, the newly entered person would sit on the seat with the smallest number. What is the number of the seat on which the $39$th person sits?

Answer: 27


Let $r$ be a nonzero real number. The values of $z$ which satisfy the equation
\[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9r^8+10r^4+1) = 0 \]
are plotted on the complex plane (i.e., using the real part of each root as the x-coordinate and the imaginary part as the y-coordinate). Show that the area of the convex quadrilateral with these points as vertices is independent of $r$, and find this area.

Answer: 8


For each positive integer $n$, let $S_n$ denote the set of positive integers $k$ such that $n^k - 1$ is divisible by $2006$. Define the function $P(n)$ by the rule:

$$P(n) = \begin{cases} \min(s)_{s \in S_n} & \text{if } S_n \neq \emptyset, \\ 0 & \text{otherwise}. \end{cases}$$

Let $d$ be the least upper bound of $\{P(1), P(2), P(3), \ldots\}$, and let $m$ be the number of integers $i$ such that $1 \leq i \leq 2006$ and $P(i) = d$. Compute the value of $d + m$.

Answer: 912


Alfred and Bonnie play a game where they take turns tossing a fair coin. The winner of the game is the first person to obtain a head. Alfred and Bonnie play this game multiple times with the rule that the loser of a game goes first in the next game. Suppose Alfred goes first in the first game, and the probability that he wins the sixth game is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What are the last three digits of $m + n$?

Answer: 93


While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$.

Answer: 132


Wu starts with exactly one coin. Each year, Wu flips every coin he has at once. For each heads he flips, Wu receives a coin, and for each tails he flips, Wu loses a coin. He continues this process each year until he has $0$ coins, at which point he stops. The probability that Wu will stop after exactly five years can be expressed as $\frac{a}{2^b}$, where $a$ and $b$ are positive integers with $a$ being odd. Find $a+b$.

Answer: 71622400


An acute triangle $ABC$ has side lengths $a$, $b$, $c$ such that $a$, $b$, $c$ form an arithmetic sequence. Given that the area of triangle $ABC$ is an integer, what is the smallest value of its perimeter?

Answer: 42


Farmer James invents a new currency such that for every positive integer $n \leq 6$, there exists an $n$-coin worth $n!$ cents. Furthermore, he has exactly $n$ copies of each $n$-coin. An integer $k$ is said to be *nice* if Farmer James can make $k$ cents using at least one copy of each type of coin. How many positive integers less than 2018 are nice?

Answer: 210


A board with dimensions $2018 \times 2018$ is divided into unit cells of size $1 \times 1$. In some cells of the board, black chips are placed, and in others, white chips are placed (each cell contains at most one chip).

First, we remove all black chips from columns that contain white chips. Then, we remove all white chips from rows that contain black chips.

Let $W$ be the number of remaining white chips, and $B$ be the number of remaining black chips on the board. Define $A = \min\{W, B\}$. Determine the maximum value of $A$.

Answer: 1018081


10. There are two sets of square paper pieces of the same size and the same number, one set black and one set white. Xiao Zhang first uses the white paper pieces to form a rectangle without any gaps in the middle, then uses the black paper pieces to surround the already formed white rectangle to create a larger rectangle, and then continues with white paper pieces. This process is repeated, and after Xiao Zhang has used the black paper pieces 5 times, the black and white paper pieces are exactly used up. Therefore, the black paper pieces are at least $\qquad$ pieces.

Answer: 350


Bakayev E.V.

A secret base is surrounded by a transparent winding fence in the shape of a non-convex polygon, with a swamp outside. A straight power line consisting of 36 poles runs through the swamp, some of which are outside the base, and some are inside. (The power line does not pass through the vertices of the fence.) A spy walks around the base along the fence from the outside, keeping the fence always to his right. Each time he reaches the power line, he counts how many poles are to his left (he can see all of them). By the time the spy has walked around the entire fence, he has counted a total of 2015 poles. How many poles are inside the base?

Answer: 1


5. There are two methods to increase the concentration of saltwater, one is to evaporate the water, and the other is to add salt. Now, 200 grams of 10% saltwater is divided into two portions. To one portion, 5 grams of salt is added; from the other portion, 5 grams of water is evaporated. As a result, the concentration of the two portions of saltwater is still equal. In the two portions of saltwater initially divided, the smaller portion originally had $\qquad$ grams.

Answer: 25


Problem 4. Let $n$ be a non-zero natural number; consider the set $M=\{1,2, \ldots, 2 n+1\}$.

Determine in how many ways the set $M$ can be partitioned into three non-empty subsets $A, B, C$ $(A \cup B \cup C=M, A \cap B=B \cap C=C \cap A=\emptyset)$ such that the following conditions are simultaneously satisfied:

(i) for any $a \in A$ and any $b \in B$, the remainder of the division of $a$ by $b$ belongs to the set $C$;

(ii) for any $c \in C$, there exist $a \in A$ and $b \in B$ such that $c$ is the remainder of the division of $a$ by $b$.

Answer: n


A semicircle with diameter $d$ is contained in a square whose sides have length 8. Given the maximum value of $d$ is $m - \sqrt{n},$ find $m+n.$
Please give the answer directly without any intermediate steps.

Answer: 544


Let \( A, B, C, \) and \( D \) be four points on a circle \(\Gamma\) such that chords \([A B]\) and \([C D]\) intersect at a point \(E\) inside \(\Gamma\), and let \(P\) be an arbitrary point on \([B E]\). The tangent \( t \) to the circumcircle of \(D E P\) at \(E\) intersects \((A C)\) at \(F\).

Demonstrate that \(\widehat{E F C} = \widehat{B D P}\).

Answer: \widehat{EFC} = \widehat{BDP}


6. Given real numbers $x, y$ satisfy $x^{2}-x y-2 y^{2}=1$. Then the minimum value of $2 x^{2}+y^{2}$ is $\qquad$ .

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


Jack has a large number of tiles, each of which is in the shape of a right-angled triangle with side lengths \(3 \text{ cm}, 4 \text{ cm}\) and \(5 \text{ cm}\). Is it possible for Jack to combine a number of these tiles, without gaps or overlap, to form a rectangle of size \(2016 \text{ cm} \times 2021 \text{ cm}\)?

Answer: \text{Yes}


In triangle \(ABC\), an inscribed circle with radius \(R\) touches side \(AC\) at point \(D\), side \(AB\) at point \(E\), and side \(BC\) at point \(F\). Given that \(AD = R\) and \(DC = a\), find the area of triangle \(BEF\).

Answer: \dfrac{R^2 (a + R)^3}{2 (a - R) (a^2 + R^2)}


Positive integers $a_0<a_1<\dots<a_n$, are to be chosen so that $a_j-a_i$ is not a prime for any $i,j$ with $0 \le i <j \le n$. For each $n \ge 1$, determine the smallest possible value of $a_n$.

Answer: 4n +1


Example 6. Let $a=\sqrt{17}, b=\mathrm{i} \sqrt{19}$. Find the maximum value of $\frac{|a-z|}{|b-z|}$, where $z$ is any complex number with modulus 1.
(2015, Harvard-MIT Mathematics Tournament)

Answer: \dfrac{4}{3}


Let \(a, b, c > 0\) and \(ab + bc + ca = 1\). Prove that \(\sqrt{a^3 + a} + \sqrt{b^3 + b} + \sqrt{c^3 + c} \geq 2\sqrt{a + b + c}\).

Answer: \sqrt{a^3 + a} + \sqrt{b^3 + b} + \sqrt{c^3 + c} \geq 2\sqrt{a + b + c}


Task 1. We have 1000 balls in 40 different colors, with exactly 25 balls of each color. Determine the smallest value of $n$ with the following property: if you randomly arrange the 1000 balls in a circle, there will always be $n$ balls next to each other where at least 20 different colors occur.

Answer: 452


4. Natural numbers $a$, $b$, and $c$, greater than 2022, are such that $a+b$ is divisible by $c-2022$, $a+c$ is divisible by $b-2022$, and $b+c$ is divisible by $a-2022$. What is the greatest value that the number $a+b+c$ can take? (S. Berlov)

Answer: 171870


1. Dima wrote a sequence of zeros and ones in his notebook. Then he noticed that a one follows a zero 16 times, a one follows 01 seven times, and a zero follows 01 eight times. What could be the last two digits on the board? (In your answer, indicate all options and prove that there are no others).

Answer: 01


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Alice`, `Eric`
 - People have unique favorite book genres: `romance`, `mystery`, `fantasy`, `science fiction`
 - People have unique heights: `tall`, `very short`, `short`, `average`
 - Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `desert`, `watermelon`
 - Each person has a favorite color: `red`, `green`, `yellow`, `white`

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

Answer: short


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Alice`, `Arnold`
 - Each mother is accompanied by their child: `Bella`, `Fred`, `Meredith`, `Samantha`
 - People have unique hair colors: `red`, `brown`, `black`, `blonde`
 - They all have a unique favorite flower: `carnations`, `roses`, `daffodils`, `lilies`
 - The people keep unique animals: `cat`, `bird`, `fish`, `horse`

## Clues:
1. Alice is somewhere to the right of the person who has red hair.
2. The person who loves a bouquet of daffodils is directly left of Arnold.
3. The fish enthusiast is the person's child is named Samantha.
4. Alice is directly left of the bird keeper.
5. The cat lover is the person who loves the rose bouquet.
6. The person who loves a carnations arrangement is somewhere to the right of the person who has brown hair.
7. The person who loves a bouquet of daffodils is directly left of the bird keeper.
8. Eric is the person's child is named Meredith.
9. There are two houses between the person who has red hair and the person who has black hair.
10. Eric is in the fourth house.
11. The person's child is named Bella is the person who loves a bouquet of daffodils.
 What is the value of attribute House for the person whose attribute Children is Meredith? 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: `Eric`, `Arnold`, `Peter`, `Alice`, `Carol`, `Bob`
 - People have unique favorite book genres: `mystery`, `biography`, `science fiction`, `romance`, `fantasy`, `historical fiction`
 - Each person has a unique favorite drink: `water`, `milk`, `boba tea`, `coffee`, `tea`, `root beer`
 - People have unique hair colors: `blonde`, `red`, `gray`, `black`, `auburn`, `brown`
 - Each person has a unique birthday month: `may`, `mar`, `sept`, `jan`, `april`, `feb`

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

Answer: mystery


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

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

Answer: Meredith


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

## Clues:
1. Arnold is the person who loves baseball.
2. The Swedish person is somewhere to the left of Arnold.
3. There are two houses between the person who loves basketball and the person who has black hair.
4. The Norwegian is in the fourth house.
5. Peter is the person who has black hair.
6. The person who has red hair is in the second house.
7. Bob is the German.
8. The person who has blonde hair is not in the third house.
9. The person who loves swimming is the person who has red hair.
10. The German is not in the first house.
11. The person who loves basketball is directly left of the Dane.
12. The person who loves soccer is directly left of Arnold.
13. Alice is the person who has gray hair.
 What is the value of attribute House for the person whose attribute FavoriteSport is basketball? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


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

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

Answer: 5


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

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

Answer: iris


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Bob`, `Alice`, `Arnold`, `Eric`, `Peter`
 - People use unique phone models: `samsung galaxy s21`, `google pixel 6`, `iphone 13`, `huawei p50`, `oneplus 9`, `xiaomi mi 11`
 - The people are of nationalities: `swede`, `chinese`, `norwegian`, `dane`, `german`, `brit`
 - Each person has a favorite color: `blue`, `red`, `yellow`, `green`, `white`, `purple`

## Clues:
1. Carol is not in the third house.
2. There is one house between the Dane and the British person.
3. Carol is the person whose favorite color is green.
4. Arnold is directly left of Alice.
5. Alice is the German.
6. The person who uses a OnePlus 9 is the person who loves purple.
7. The person who uses a Huawei P50 is not in the third house.
8. The person who uses a Samsung Galaxy S21 is in the fifth house.
9. The person who loves white is somewhere to the right of the person whose favorite color is red.
10. The person who uses a Samsung Galaxy S21 is Bob.
11. The Dane is the person who loves yellow.
12. The person who uses a Samsung Galaxy S21 is somewhere to the left of Peter.
13. The person who loves blue is Peter.
14. Peter is the British person.
15. The person who uses a Samsung Galaxy S21 is directly left of the person who uses an iPhone 13.
16. The Norwegian is the person who loves purple.
17. The person who uses a Xiaomi Mi 11 is the Chinese.
 What is the value of attribute House for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


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

## Clues:
1. The woodworking hobbyist is somewhere to the left of Bob.
2. The person who paints as a hobby is in the fifth house.
3. Arnold is directly left of the person who uses a Google Pixel 6.
4. The person who uses a Samsung Galaxy S21 is Arnold.
5. Carol is somewhere to the right of the person who uses a Samsung Galaxy S21.
6. The person who uses a Google Pixel 6 is directly left of the person who uses a OnePlus 9.
7. The person who uses a OnePlus 9 is somewhere to the right of Eric.
8. There are two houses between the person who loves cooking and Peter.
9. The person who enjoys knitting is Eric.
10. Peter is the person who paints as a hobby.
11. The person who enjoys gardening is somewhere to the right of Bob.
12. There is one house between Bob and the person who uses a Huawei P50.
13. The person who uses a Xiaomi Mi 11 is not in the first house.
14. Alice is in the sixth house.
 What is the value of attribute PhoneModel for the person whose attribute Hobby is cooking? Please reason step by step, and put your final answer within \boxed{}

Answer: samsung galaxy s21


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Alice`, `Carol`, `Bob`, `Arnold`
 - The people keep unique animals: `horse`, `fish`, `cat`, `bird`, `dog`, `rabbit`
 - Each person has a unique birthday month: `sept`, `mar`, `jan`, `feb`, `april`, `may`
 - The mothers' names in different houses are unique: `Holly`, `Sarah`, `Penny`, `Kailyn`, `Aniya`, `Janelle`
 - Each person prefers a unique type of vacation: `beach`, `cruise`, `cultural`, `camping`, `city`, `mountain`

## Clues:
1. The person whose birthday is in February is the fish enthusiast.
2. The person whose birthday is in May is somewhere to the right of The person whose mother's name is Penny.
3. The person who loves beach vacations is Peter.
4. The person whose mother's name is Janelle is Carol.
5. The rabbit owner is Bob.
6. The rabbit owner is the person whose birthday is in May.
7. The person whose mother's name is Kailyn is in the fourth house.
8. The person who enjoys camping trips is the cat lover.
9. The person who likes going on cruises is in the sixth house.
10. The person whose birthday is in April is the dog owner.
11. Alice is The person whose mother's name is Aniya.
12. The person whose birthday is in May and The person whose mother's name is Sarah are next to each other.
13. The person whose mother's name is Holly is the cat lover.
14. There are two houses between the person whose birthday is in March and the rabbit owner.
15. The dog owner is directly left of the person who keeps horses.
16. The person whose birthday is in February is in the second house.
17. The rabbit owner is somewhere to the right of the person who goes on cultural tours.
18. Carol is in the second house.
19. The person whose birthday is in September is the person who prefers city breaks.
20. Eric is not in the first house.
 What is the value of attribute House for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{}

Answer: 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: `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 MusicGenre for the person whose attribute HouseStyle is victorian? 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: `Peter`, `Arnold`, `Eric`
 - Each mother is accompanied by their child: `Bella`, `Meredith`, `Fred`
 - People use unique phone models: `samsung galaxy s21`, `iphone 13`, `google pixel 6`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`
 - Everyone has a unique favorite cigar: `blue master`, `prince`, `pall mall`
 - People have unique heights: `short`, `very short`, `average`

## Clues:
1. Eric is directly left of the Prince smoker.
2. The person who has an average height is somewhere to the right of Peter.
3. The person who uses a Samsung Galaxy S21 is in the second house.
4. The person who smokes Blue Master is in the third house.
5. The person who uses a Google Pixel 6 is Eric.
6. Arnold is somewhere to the right of the person who loves science fiction books.
7. The person who uses an iPhone 13 is the person's child is named Bella.
8. The person who is short is the Prince smoker.
9. The person who loves mystery books is the person who uses a Google Pixel 6.
10. The person's child is named Meredith and the person who is short are next to each other.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Peter`
 - Everyone has a unique favorite cigar: `blue master`, `dunhill`, `pall mall`, `prince`
 - Everyone has something unique for lunch: `stew`, `spaghetti`, `pizza`, `grilled cheese`
 - People have unique favorite book genres: `fantasy`, `mystery`, `science fiction`, `romance`

## Clues:
1. Arnold is the person who loves eating grilled cheese.
2. The person who smokes Blue Master is Alice.
3. Arnold is in the second house.
4. The person who is a pizza lover and the person who loves the stew are next to each other.
5. Arnold is the person who loves science fiction books.
6. The person partial to Pall Mall is the person who loves eating grilled cheese.
7. Eric is somewhere to the left of Alice.
8. The person who is a pizza lover is somewhere to the left of the Dunhill smoker.
9. The person who loves fantasy books is the person who loves the stew.
10. The person who loves mystery books is not in the first house.
 What is the value of attribute Food for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: grilled cheese


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Peter`
 - They all have a unique favorite flower: `lilies`, `carnations`, `roses`, `daffodils`
 - Each person has an occupation: `teacher`, `engineer`, `artist`, `doctor`
 - Everyone has a favorite smoothie: `desert`, `dragonfruit`, `watermelon`, `cherry`

## Clues:
1. The Dragonfruit smoothie lover is in the first house.
2. The person who is a doctor is somewhere to the right of Eric.
3. Eric is the person who likes Cherry smoothies.
4. Peter is directly left of the person who is an artist.
5. The Desert smoothie lover is the person who loves a carnations arrangement.
6. The person who is a teacher is directly left of the Desert smoothie lover.
7. The person who loves the boquet of lilies is directly left of the person who likes Cherry smoothies.
8. The person who is a teacher is the person who loves the rose bouquet.
9. The person who is an engineer is Alice.
 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: 2


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

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

Answer: spaghetti


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`, `Bob`, `Arnold`, `Carol`, `Eric`, `Alice`
 - Everyone has something unique for lunch: `spaghetti`, `stir fry`, `stew`, `soup`, `grilled cheese`, `pizza`
 - Everyone has a unique favorite cigar: `blends`, `pall mall`, `dunhill`, `yellow monster`, `prince`, `blue master`
 - Each person has a unique favorite drink: `boba tea`, `water`, `coffee`, `root beer`, `milk`, `tea`
 - Each person has a unique level of education: `associate`, `bachelor`, `trade school`, `doctorate`, `high school`, `master`

## Clues:
1. Bob is not in the second house.
2. The person who smokes many unique blends is somewhere to the left of the person who is a pizza lover.
3. The person partial to Pall Mall is somewhere to the right of the Dunhill smoker.
4. The person who loves the soup is not in the first house.
5. The person who likes milk is directly left of the person who loves the stew.
6. The person who loves the soup is the person with a doctorate.
7. The coffee drinker is the person with a high school diploma.
8. There are two houses between the person who loves the spaghetti eater and the person with an associate's degree.
9. The Prince smoker is not in the third house.
10. The Dunhill smoker and Bob are next to each other.
11. There are two houses between Alice and the boba tea drinker.
12. The person with a high school diploma is Eric.
13. The person who attended trade school is the Prince smoker.
14. There is one house between the Dunhill smoker and the person who loves stir fry.
15. The Dunhill smoker is directly left of the person who is a pizza lover.
16. There is one house between the person partial to Pall Mall and the person who smokes Blue Master.
17. The person who smokes Blue Master is the coffee drinker.
18. The person with a bachelor's degree is the Dunhill smoker.
19. Alice is the tea drinker.
20. Arnold is directly left of the root beer lover.
21. The person with a master's degree is Peter.
22. Arnold is somewhere to the left of the person who likes milk.
 What is the value of attribute Cigar for the person whose attribute House is 5? Please reason step by step, and put your final answer within \boxed{}

Answer: yellow monster


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`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `cherry`, `dragonfruit`
 - Each person lives in a unique style of house: `craftsman`, `colonial`, `ranch`, `victorian`
 - Each person has a unique hobby: `cooking`, `photography`, `gardening`, `painting`
 - Each person has a unique favorite drink: `coffee`, `tea`, `milk`, `water`

## Clues:
1. The person in a Craftsman-style house is the Dragonfruit smoothie lover.
2. The tea drinker is the person who loves a bouquet of daffodils.
3. The Desert smoothie lover is the person living in a colonial-style house.
4. There is one house between the Watermelon smoothie lover and the person who loves a bouquet of daffodils.
5. The person who likes Cherry smoothies is directly left of the photography enthusiast.
6. The person who likes milk is the person who loves cooking.
7. The tea drinker is Arnold.
8. The person who loves the boquet of lilies is the person living in a colonial-style house.
9. The person who loves the boquet of lilies is not in the third house.
10. The person living in a colonial-style house is directly left of the person who paints as a hobby.
11. The person in a ranch-style home is the person who likes milk.
12. The person who loves the rose bouquet is the one who only drinks water.
13. Eric is somewhere to the left of the Dragonfruit smoothie lover.
14. The person who loves the rose bouquet is somewhere to the right of the person who loves a carnations arrangement.
15. Peter is the person who loves the boquet of lilies.
 What is the value of attribute Hobby for the person whose attribute Smoothie is desert? Please reason step by step, and put your final answer within \boxed{}

Answer: gardening


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Arnold`, `Peter`, `Eric`
 - The mothers' names in different houses are unique: `Penny`, `Aniya`, `Kailyn`, `Janelle`, `Holly`
 - The people keep unique animals: `cat`, `bird`, `dog`, `fish`, `horse`
 - Everyone has a favorite smoothie: `desert`, `dragonfruit`, `cherry`, `watermelon`, `lime`
 - Each person has an occupation: `doctor`, `engineer`, `artist`, `teacher`, `lawyer`

## Clues:
1. Bob is the Desert smoothie lover.
2. The cat lover is the person who is an engineer.
3. The person whose mother's name is Penny is the bird keeper.
4. The person who is a lawyer is the fish enthusiast.
5. The person who likes Cherry smoothies is directly left of Alice.
6. The Watermelon smoothie lover and the person who is an engineer are next to each other.
7. The person whose mother's name is Holly is the dog owner.
8. The person who is a teacher is in the first house.
9. The person whose mother's name is Penny is in the second house.
10. The person who drinks Lime smoothies is the person who is a doctor.
11. The person whose mother's name is Janelle is the person who is a lawyer.
12. Arnold is not in the third house.
13. The Dragonfruit smoothie lover is directly left of Peter.
14. The person whose mother's name is Kailyn is the Dragonfruit smoothie lover.
15. Arnold is somewhere to the right of Peter.
16. The dog owner is not in the fifth house.
17. The Watermelon smoothie lover is somewhere to the left of the Desert smoothie lover.
 What is the value of attribute Animal for the person whose attribute Mother is Kailyn? Please reason step by step, and put your final answer within \boxed{}

Answer: horse


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Eric`, `Arnold`, `Peter`
 - Each mother is accompanied by their child: `Samantha`, `Meredith`, `Timothy`, `Bella`, `Fred`
 - People have unique hair colors: `red`, `blonde`, `black`, `brown`, `gray`
 - Each person has a unique hobby: `gardening`, `cooking`, `painting`, `knitting`, `photography`
 - Each person has a unique type of pet: `hamster`, `bird`, `cat`, `fish`, `dog`
 - They all have a unique favorite flower: `tulips`, `roses`, `carnations`, `lilies`, `daffodils`

## Clues:
1. The person's child is named Samantha is somewhere to the right of the person's child is named Meredith.
2. The person who loves the boquet of lilies is the person who keeps a pet bird.
3. The person who has blonde hair is the person who loves the vase of tulips.
4. Bob is the person who has brown hair.
5. The person who paints as a hobby is the person with an aquarium of fish.
6. The person's child is named Fred is Peter.
7. The person with an aquarium of fish is somewhere to the left of the person who has a cat.
8. The person with a pet hamster is the person's child is named Samantha.
9. The person's child is named Bella is Arnold.
10. The person's child is named Bella is in the third house.
11. Peter is not in the first house.
12. Alice is the person who has blonde hair.
13. The person who loves the rose bouquet is not in the third house.
14. The person who owns a dog is the person's child is named Fred.
15. The person who has red hair is directly left of the person who has black hair.
16. The photography enthusiast is directly left of Eric.
17. The person who has black hair is the person who loves the rose bouquet.
18. The person who loves a carnations arrangement is not in the third house.
19. The person who enjoys knitting is somewhere to the right of the person's child is named Bella.
20. The person who loves the boquet of lilies is the person who enjoys gardening.
21. The person who keeps a pet bird is in the fifth house.
 What is the value of attribute Flower for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: tulips


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

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

Answer: yellow


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`, `Eric`, `Arnold`, `Peter`, `Alice`
 - Everyone has a unique favorite cigar: `blends`, `prince`, `pall mall`, `blue master`, `dunhill`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `stew`, `pizza`, `stir fry`
 - The people are of nationalities: `norwegian`, `swede`, `brit`, `dane`, `german`
 - Each person has an occupation: `teacher`, `lawyer`, `doctor`, `artist`, `engineer`
 - People have unique heights: `average`, `very tall`, `short`, `very short`, `tall`

## Clues:
1. The person who smokes Blue Master is the person who is a lawyer.
2. Alice is the person partial to Pall Mall.
3. The person partial to Pall Mall is somewhere to the left of the Dunhill smoker.
4. The German is the person who loves stir fry.
5. The person who is tall is the person who smokes many unique blends.
6. The person who is an artist is the person who loves the spaghetti eater.
7. Peter is the person who is short.
8. The Dunhill smoker is not in the fourth house.
9. The person who is very short is the person who smokes Blue Master.
10. There are two houses between the person who loves the spaghetti eater and the person who loves stir fry.
11. The person who has an average height is in the second house.
12. The Prince smoker is the person who is a doctor.
13. The Swedish person is the person who is an engineer.
14. Eric is the person who loves stir fry.
15. There is one house between the British person and the person who loves the stew.
16. The person who is very tall is the Norwegian.
17. The Dane is the person who is short.
18. The person who is very tall is Arnold.
19. There are two houses between the person who is a pizza lover and the person who is very tall.
20. The person who is a teacher is not in the third house.
 What is the value of attribute Height for the person whose attribute Food is stew? Please reason step by step, and put your final answer within \boxed{}

Answer: short


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - Each person has a unique birthday month: `april`, `jan`, `sept`
 - People have unique favorite book genres: `mystery`, `romance`, `science fiction`
 - Each person lives in a unique style of house: `ranch`, `victorian`, `colonial`
 - They all have a unique favorite flower: `carnations`, `daffodils`, `lilies`

## Clues:
1. The person who loves romance books is in the third house.
2. Arnold is the person who loves a carnations arrangement.
3. The person living in a colonial-style house is the person whose birthday is in September.
4. The person who loves a carnations arrangement is in the second house.
5. The person whose birthday is in April is the person in a ranch-style home.
6. The person who loves science fiction books is Arnold.
7. The person in a ranch-style home is the person who loves a carnations arrangement.
8. The person whose birthday is in January is somewhere to the left of Peter.
9. The person whose birthday is in April is somewhere to the left of the person who loves the boquet of lilies.
 What is the value of attribute Birthday for the person whose attribute Flower is daffodils? Please reason step by step, and put your final answer within \boxed{}

Answer: jan


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

## Clues:
1. Bob is the photography enthusiast.
2. The person who loves eating grilled cheese is the person who is tall.
3. Peter is not in the second house.
4. The person who is tall is directly left of the person who loves stir fry.
5. The person who loves cooking is the person who has an average height.
6. Alice is directly left of the person who is a pizza lover.
7. The person who loves the spaghetti eater is not in the second house.
8. Eric is not in the fifth house.
9. The person who is short is Peter.
10. The person who has an average height and the person who enjoys gardening are next to each other.
11. The person who paints as a hobby is directly left of the person who loves eating grilled cheese.
12. The person who is very short is in the fifth house.
13. The person who is tall is in the third house.
14. Alice is somewhere to the right of the photography enthusiast.
 What is the value of attribute House for the person whose attribute Hobby is painting? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Peter`, `Eric`, `Alice`, `Carol`
 - Each person has a unique birthday month: `april`, `feb`, `mar`, `jan`, `may`, `sept`
 - Each person prefers a unique type of vacation: `city`, `beach`, `mountain`, `cruise`, `cultural`, `camping`
 - Each person has a favorite color: `blue`, `white`, `green`, `purple`, `red`, `yellow`

## Clues:
1. The person whose birthday is in January is the person who loves blue.
2. The person whose birthday is in January is directly left of the person who loves white.
3. The person who enjoys camping trips is in the fourth house.
4. Eric is in the sixth house.
5. There are two houses between the person who likes going on cruises and the person whose birthday is in February.
6. The person whose birthday is in May is not in the sixth house.
7. Peter is directly left of the person who loves yellow.
8. The person who goes on cultural tours is Eric.
9. The person whose birthday is in September is directly left of the person whose favorite color is green.
10. Arnold and the person who loves purple are next to each other.
11. The person who loves beach vacations is the person whose birthday is in March.
12. The person whose favorite color is green is somewhere to the left of Carol.
13. The person whose birthday is in May is somewhere to the right of Peter.
14. The person who loves blue is Bob.
15. The person who enjoys mountain retreats is in the fifth house.
16. The person who loves purple is in the third house.
17. Carol is in the fourth house.
 What is the value of attribute Color for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: purple


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`, `Carol`
 - People have unique favorite sports: `volleyball`, `soccer`, `basketball`, `baseball`, `swimming`, `tennis`
 - Each person has a unique hobby: `gardening`, `cooking`, `painting`, `woodworking`, `knitting`, `photography`
 - Each mother is accompanied by their child: `Bella`, `Meredith`, `Alice`, `Samantha`, `Fred`, `Timothy`
 - Each person has a unique birthday month: `feb`, `jan`, `mar`, `may`, `sept`, `april`
 - Everyone has something unique for lunch: `stir fry`, `grilled cheese`, `pizza`, `spaghetti`, `stew`, `soup`

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

Answer: 6


Let $p$ be a prime and $a$ and $n$ be natural numbers such that \( \frac{p^a - 1}{p-1} = 2^n \). Find the number of natural divisors of $na$.

Answer: 4


We call a number greater than $25$ a \textit{semi-prime} if it is the sum of two different prime numbers. What is the greatest number of consecutive natural numbers that can be \textit{semi-prime}?

Answer: 5


Using $600$ cards, where $200$ of them have the number $5$, $200$ have the number $2$, and the other $200$ have the number $1$, a student wants to create groups of cards such that the sum of the card numbers in each group is $9$. What is the maximum number of groups that the student can create?

Answer: 150


Let $ABC$ be a scalene triangle with inradius $1$ and exradii $r_A$, $r_B$, and $r_C$ such that
\[ 20\left(r_B^2r_C^2+r_C^2r_A^2+r_A^2r_B^2\right)=19\left(r_Ar_Br_C\right)^2. \]
If
\[ \tan\frac{A}{2}+\tan\frac{B}{2}+\tan\frac{C}{2}=2.019, \]
then the area of $\triangle{ABC}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.

Answer: 201925


A set $S$ of positive integers is \textit{sum-complete} if there are positive integers $m$ and $n$ such that an integer $a$ is the sum of the elements of some nonempty subset of $S$ if and only if $m \le a \le n$.

Let $S$ be a sum-complete set such that $\{1, 3\} \subset S$ and $|S| = 8$. Find the greatest possible value of the sum of the elements of $S$.

Answer: 223


Will has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n$ times, the probability that the next flip turns up heads is exactly \(\frac{m+1}{m+n+2}\). Suppose that the coin starts at $0$ flips. The probability that after $10$ coin flips, heads and tails have both turned up exactly $5$ times can be expressed as \(\frac{m}{n}\) for relatively prime positive integers $m$ and $n$. Compute $m+n$.

Answer: 12


A natural number is called a *prime power* if that number can be expressed as $p^n$ for some prime $p$ and natural number $n$.

Determine the largest possible $n$ such that there exists a sequence of prime powers $a_1, a_2, \dots, a_n$ such that $a_i = a_{i - 1} + a_{i - 2}$ for all $3 \le i \le n$.

Answer: 7


An $8 \times 8 \times 8$ cube is painted red on 3 faces and blue on 3 faces such that no corner is surrounded by three faces of the same color. The cube is then cut into 512 unit cubes. How many of these cubes contain both red and blue paint on at least one of their faces?

Answer: 56


A triangle is composed of circular cells arranged in $5784$ rows: the first row has one cell, the second has two cells, and so on. The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called **diagonal** if the two cells in it aren't in the same row. What is the minimum possible amount of diagonal pairs in the division?

Answer: 2892


Triangle $ABC$ has side lengths $AB=20$, $AC=14$, and $BC=22$. The median from $B$ intersects $AC$ at $M$, and the angle bisector from $C$ intersects $AB$ at $N$ and the median from $B$ at $P$. Let \( \frac{p}{q} = \frac{[AMPN]}{[ABC]} \) for positive integers \( p \) and \( q \) that are coprime. Note that \([ABC]\) denotes the area of triangle $ABC$. Find \( p+q \).

Answer: 331


Compute the remainder when $99989796\ldots 121110090807 \ldots 01$ is divided by $010203 \ldots 091011 \ldots 9798$. Note that the first sequence starts at $99$, and the second sequence ends at $98$.

Answer: 9801


Let $x_1, x_2, \ldots, x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p + x_q + x_r$ (with $1 \le p < q < r \le 5$) are equal to $0$, then $x_1 = x_2 = \cdots = x_5 = 0$.

Answer: 7


Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt{2}}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$. Determine the greatest integer less than $\prod_{n=1}^{100}\psi(3^n)$.

Answer: 5792


Every week, Ben goes to the supermarket and buys the following: $7$ apples at $\$2$ each, $4$ bottles of milk at $\$4$ each, $3$ loaves of bread at $\$3$ each, and $3$ bags of sugar at $\$6$ each. This week the store has a $25\%$ discount on all dairy products. Ben also has a coupon for $\$10$ off any order of $\$50$ or over. How much money does Ben spend on this shopping trip?

Answer: \$43


In trapezoid $EFGH$, sides $EF$ and $GH$ are equal. $GH$ is 12 units long, and the height of the trapezoid is 5 units. The top base $EF$ is 10 units long. Calculate the perimeter of trapezoid $EFGH$.

Answer: 22 + 2\sqrt{26}


Let's define two operations $\oplus$ and $\otimes$ for any number $y$, where $y \oplus = 9 - y$ and $\oplus y = y - 9$. What is the value of $\oplus(18 \oplus)$?

Answer: -18


Triangles $PQR$ and $PRS$ are right triangles with $PQ = 14$ units, $PR = 24$ units, and $PS = 26$ units. What is the area, in square units, of quadrilateral $PQRS$?

Answer: 288\text{ square units}


After a hurricane in Miami, the estimated damage was $\$$45 million in local (US) currency. Later, it was discovered that some assets were valued in Canadian dollars. The total value of these assets was $\$$15 million. At that time, 1 US dollar was worth 1.25 Canadian dollars. Determine the damage in Canadian dollars including the assets valued in Canadian dollars. Additionally, there is a recovery tax of 10% on the total damage calculated in Canadian dollars that needs to be considered.

Answer: 78,375,000 \text{ Canadian dollars}


In heptagon $GEOMETRY$, $\angle G \cong \angle E \cong \angle T \cong \angle R$ and $\angle M$ is supplementary to $\angle Y$, $\angle J$ is supplementary to $\angle O$. Find the measure of $\angle T$.

Answer: 135^\circ


The polynomial \[x^4 + bx^3 + cx^2 + 7dx + e = 0\] has integer coefficients, where $d$ is nonzero. Let $m$ be the exact number of integer roots of the polynomial, counting multiplicities. Determine all possible values for $m$.

Answer: 0, 1, 2, 4


Answer: 5


Berengere and her exchange student Emily are at a French café that accepts both euros and American dollars. They want to buy a cake which now costs 8 euros, but individually, they don't have enough money. Emily has a ten-dollar bill, and the exchange rate is 1 euro = 1.10 USD. How many euros does Berengere need to contribute so they can buy the cake?

Answer: 0 \text{ euros}


The stem-and-leaf plot shows the duration (in minutes and seconds) of songs played during a concert by a band. There are 19 songs in total. In the stem-and-leaf plot, $3 \ 15$ represents 3 minutes, 15 seconds, which is the same as 195 seconds. Determine the median duration of these songs. Express your answer in seconds.

\begin{tabular}{c|ccccc}
0&45&50&55&&\\
1&10&12&15&18&20\\
2&05&10&15&25&30\\
3&00&05&10&15&20\\
4&10&15&&&\\
\end{tabular}

Answer: 130


Four primes \(p_1\), \(p_2\), \(p_3\), and \(p_4\) form an increasing arithmetic sequence with \(p_1 > 3\) and common difference 4. What is the ones digit of \(p_1\)?

Answer: 9


In a circle with radius $6$, two chords $PQ$ and $RS$ intersect at a point $T$. Chord $RS$ bisects chord $PQ$ such that $PT = TQ = 8$ and $PQ$ is the only chord from $P$ that $RS$ bisects. It is known that the sine of the central angle subtended by minor arc $PR$ is a rational number. If expressed as a fraction $\frac{p}{q}$ in lowest terms, what is the product $pq$?

Answer: pq=20


Define an integer $N$ to be $4$-nice or $6$-nice if there exists a positive integer $a$ such that $a^4$ or $a^6$ respectively, has exactly $N$ positive divisors. Determine the number of positive integers less than or equal to $500$ that are neither $4$-nice nor $6$-nice.

Answer: 333


Tracy initially had a bag of candies. She first ate $\frac{1}{4}$ of them, then gave $\frac{1}{3}$ of the remaining candies to her friend Rachel. After that, Tracy and her mom each ate 12 candies from what was left. Lastly, her brother took between two to six candies, leaving Tracy with five candies. How many candies did Tracy have at the beginning?

Answer: 68


In the shown diagram, segment $CD$ is parallel to segment $WX$. If $CX = 30$ units, $DP = 15$ units, and $PW = 45$ units, find the length of segment $PX$. Diagram:
[asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8));
pair W = (0,0), X = (24,0), C = (0,12), D = (9,12);
draw(C--D--W--X--cycle);
label("$C$",C,W); label("$D$",D,E); label("$W$",W,W); label("$X$",X,E);
pair P = intersectionpoint(C--X,D--W);
label("$P$",P,E);
[/asy]

Answer: 22.5 \text{ units}


Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) + y) = f(x^2 - y) + 2f(x) y\]for all real numbers $x$ and $y.$ Additionally, suppose that $f(1) = 1$. Calculate the sum of all possible values of $f(3)$.

Answer: 9


5.3. Among all the irreducible fractions, where the numerator and denominator are two-digit numbers, find the smallest fraction greater than $\frac{4}{5}$. In your answer, specify its numerator.

Answer: 77


8. (10 points) Every day, Xiaoming has to pass through a flat section $AB$, an uphill section $BC$, and a downhill section $CD$ (as shown in the figure). It is known that $AB: BC: CD=1: 2: 1$, and Xiaoming's speed ratio on the flat section, uphill section, and downhill section is 3: 2: 4. What is the ratio of the time Xiaoming spends going to school to the time he spends coming back home? $\qquad$

Answer: 19:16


23. When Chief Black Cat was chasing One-Ear, he found a $4 \times 4$ number puzzle left by One-Ear. In this puzzle, the sum of the four numbers in each row, each column, and each diagonal is the same. Among them, $a+b=2018, c+d=2019$, and the sum of the 16 numbers on the puzzle is the house number of the Cat-Eating Mouse. Chief Black Cat quickly calculated the house number of the Cat-Eating Mouse and caught One-Ear and the Cat-Eating Mouse. The house number of the Cat-Eating Mouse is $\qquad$

Answer: 16148


13. A and B are two car retailers (hereinafter referred to as A and B) who ordered a batch of cars from a car manufacturer. Initially, the number of cars A ordered was 3 times the number of cars B ordered. Later, due to some reason, A transferred 6 cars from its order to B. When picking up the cars, the manufacturer provided 6 fewer cars than the total number ordered by A and B. In the end, the number of cars A purchased was twice the number of cars B purchased. Try to find: what is the maximum number of cars that A and B finally purchased in total? And what is the minimum number of cars they purchased in total?

Answer: 18


5. In the Cartesian coordinate system, grid points are numbered as follows: $(0,0)$ is No. 1, $(1,0)$ is No. 2, $(1,1)$ is No. 3, $(0,1)$ is No. 4, $(0,2)$ is No. 5, $(1,2)$ is No. 6, $(2,2)$ is No. 7, $(2,1)$ is No. 8, $(2,0)$ is No. 9, $\cdots$ (as shown in Figure 7-1). Following the order of the arrows in the figure, the coordinates of the 2006th point are . $\qquad$

Answer: (44,19)


## Task 34/65

There are four consecutive natural numbers for which the following is true:

The sum of the cubes of the two smaller numbers is equal to the difference of the cubes of the two larger numbers.

These four numbers are to be found.

Answer: 3,4,5,6


6. Given that $O$ is a point inside the equilateral $\triangle A B C$, the ratio of the angles $\angle A O B$, $\angle B O C$, and $\angle A O C$ is 6:5:4. Then, in the triangle formed by $O A$, $O B$, and $O C$, the ratio of the angles opposite these sides is

Answer: 5: 3: 7


10. 12 people gather for dinner at the same restaurant every week, each time divided into three tables, with four people at each table. It is required that any two people sit at the same table at least once. Question: What is the minimum number of weeks such dinners need to be held?

Answer: 5


On the extensions of the sides $CA$ and $CB$ of triangle $ABC$, segments $AA_1$ and $BB_1$ are laid out equal to segments $AL_3$ and $BL_3$, respectively, where $L_3$ is the foot of the angle bisector of angle $C$. Prove that the point $L_3$ is the incenter of triangle $A_1B_1C$.

Answer: L_3 \text{ is the incenter of } \triangle A_1B_1C


9.64 Two identical chessboards $(8 \times 8$ squares $)$ overlap each other. Now, one of them is rotated $45^{\circ}$ around the center. If the area of each square is 1, find the total area of the intersection of all the black squares of the two chessboards.

Answer: 32(\sqrt{2} - 1)


Due to the rainstorms, our basement filled with water. To remove the water, three pumps were set up. The first pump alone could pump out the water in 3 hours, the second in 4 hours, and the third in 6 hours. After 30 minutes of working together, the second pump broke down. The remaining water was pumped out by the first and third pumps. How long did it take in total to remove the water?

Answer: \dfrac{7}{4}


5. Given that $a, b, c$ are three real numbers in an arithmetic sequence. Then the locus of the midpoint of the chord formed by the intersection of the line $b x+a y+c=0$ and the parabola $y^{2}=-\frac{1}{2} x$ is $\qquad$ .

Answer: 4y^2 - 4y + x + 2 = 0


29. Find a function $f(x)$ such that for any real $x$, except 0 and 1, $f(1 / x) + f(1 - x) = x$.

Answer: -\dfrac{x}{2} + \dfrac{1}{2x} - \dfrac{1}{2(x - 1)}


6. (10 points) Square $ABCD$ and rectangle $BEFG$ are placed as shown in the figure, with $AG=CE=2$ cm. The area of square $ABCD$ is larger than the area of rectangle $BEFG$ by $\qquad$ square centimeters.

Answer: 4


3B. Determine the smallest natural number, the product of whose digits is equal to 75600.

Answer: 556789


Five. (20 points) Given the sequence of positive integers $\left\{T_{m, n}\right\}$ satisfies:
(1) $T_{m, m}=2^{m}$;
(2) $T_{m, n}=T_{n, m}$;
(3) $\left(T_{m, n}\right)^{m+n}=\left(T_{m, m+n}\right)^{n}$.

Here $m, n$ are any positive integers. Try to find the general term formula for the sequence $\left\{T_{m, n}\right\}$.

Answer: 2^{\operatorname{lcm}(m,n)}


Let $\omega$ be an incircle of triangle $ABC$. Let $D$ be a point on segment $BC$, which is tangent to $\omega$. Let $X$ be an intersection of $AD$ and $\omega$ against $D$. If $AX : XD : BC = 1 : 3 : 10$, a radius of $\omega$ is $1$, find the length of segment $XD$. Note that $YZ$ expresses the length of segment $YZ$.

Answer: \dfrac{3\sqrt{10}}{5}


12. Given that $F$ is the right focus of the ellipse $C: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1$, and the ratio of the distance from any point $P$ on the ellipse $C$ to the point $F$ to the distance from point $P$ to the line $l: x=m$ is $\frac{1}{2}$.
(1) Find the equation of the line $l$.
(2) Let $A$ be the left vertex of the ellipse $C$. A line passing through the point $F$ intersects the ellipse $C$ at points $D$ and $E$. The lines $A D$ and $A E$ intersect the line $l$ at points $M$ and $N$, respectively. Does the circle with diameter $M N$ always pass through a fixed point? If so, find the coordinates of the fixed point; if not, explain the reason.

Answer: x = 4


Given two points \( A \) and \( B \) on a plane, there are \( m \) lines passing through point \( A \) and \( n \) lines passing through point \( B \). What is the maximum number of regions into which these \( m+n \) lines can divide the plane? (where \( m \neq 0 \) and \( n \neq 0 \)).

Answer: mn + 2m + 2n - 1


## Task A-1.3.

Determine how many six-digit natural numbers exist such that by removing the first two, or the last two digits, we obtain two four-digit numbers that give the same remainder when divided by 99.

Answer: 8190


Let $l$ be a tangent line at the point $T\left(t,\ \frac{t^2}{2}\right)\ (t>0)$ on the curve $C: y=\frac{x^2}{2}.$ A circle $S$ touches the curve $C$ and the $x$ axis.
Denote by $(x(t),\ y(t))$ the center of the circle $S$. Find $\lim_{r\rightarrow +0}\int_r^1 x(t)y(t)\ dt.$

Answer: \dfrac{1 + \sqrt{2}}{30}


5. We have cards with numbers 5, 6, 7, ..., 55 (each card has one number). What is the maximum number of cards we can select so that the sum of the numbers on any two selected cards is not a palindrome? (A palindrome is a number that reads the same backward as forward.)

Answer: 25


Answer: \dfrac{355}{113}


$\left[\begin{array}{l}\text { Angles between lines and planes } \\ {[\underline{\text { Linear dependence of vectors }}]}\end{array}\right]$

The lateral face of a regular quadrilateral pyramid forms an angle of $45^{\circ}$ with the plane of the base. Find the angle between the apothem of the pyramid and the plane of the adjacent face.

Answer: 30^\circ


In an isosceles right triangle, one of the legs is 36 units. Starting from the right-angle vertex on one of the legs, we draw an infinite sequence of equilateral triangles, each touching the next, such that the third vertex of each inscribed triangle always lies on the hypotenuse, and the sides opposite these vertices fill the leg. Determine the sum of the areas of the equilateral triangles.

Answer: 324


[ Intersecting lines, angle between them]

In rectangle $A B C D$, the sides are given as $A B=3, B C=4$. Point $K$ is at distances $\sqrt{10}$, 2, and 3 from points $A, B$, and $C$ respectively. Find the angle between the lines $C K$ and $B D$.

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


Problem 8. For what values of the parameter a does the equation $x^{3}+6 x^{2}+a x+8=0$ have exactly three solutions?

Answer: (-\infty, -15)


p1. Given the set $H = \{(x, y)|(x -y)^2 + x^2 - 15x + 50 = 0$ where x and y are natural numbers $\}$.
Find the number of subsets of $H$.

p2. A magician claims to be an expert at guessing minds with following show. One of the viewers was initially asked to hidden write a five-digit number, then subtract it with the sum of the digits that make up the number, then name four of the five digits that make up the resulting number (in order of any). Then the magician can guess the numbers hidden. For example, if the audience mentions four numbers result: $0, 1, 2, 3$, then the magician will know that the hidden number is $3$.
a. Give an example of your own from the above process.
b. Explain mathematically the general form of the process.

p3. In a fruit basket there are $20$ apples, $18$ oranges, $16$ mangoes, $10$ pineapples and $6$ papayas. If someone wants to take $10$ pieces from the basket. After that, how many possible compositions of fruit are drawn?

p4. Inside the Equator Park, a pyramid-shaped building will be made with base of an equilateral triangle made of translucent material with a side length of the base $8\sqrt3$ m long and $8$ m high. A globe will be placed in a pyramid the. Ignoring the thickness of the pyramidal material, determine the greatest possible length of the radius of the globe that can be made.

p5. What is the remainder of $2012^{2012} + 2014^{2012}$ divided by $2013^2$?

Answer: 64


4. Given an integer $n>0$. There is a balance scale and $n$ weights with masses $2^{0}, 2^{1}, \cdots, 2^{n-1}$.

Now, through $n$ steps, all the weights are to be placed on the balance one by one, ensuring that during the process, the weight on the left side never exceeds the weight on the right side. In each step, one of the weights not yet placed on the balance is chosen and placed on either the left or the right side of the balance, until all weights are placed on the balance. Find the number of different ways to perform this process.

Answer: (2n-1)!!


A tanker of milk was delivered to the store. The seller has balance scales without weights (fляги can be placed on the pans of the scales), and three identical flasks, two of which are empty, and the third one contains 1 liter of milk. How can you pour exactly 85 liters of milk into one flask, making no more than eight weighings?

#

Answer: 85


47. There are the same number of black balls and white balls in the box. Each time, 5 black balls and 8 white balls are taken out. After several times, only 12 black balls are left in the box. The box originally had $\qquad$ balls.

Answer: 64


6. There are 100 equally divided points on a circle. The number of obtuse triangles formed by these points as vertices is $\qquad$ .

Answer: 117600


Given a sector that is one-fourth of a circle with radius $R$. Find the length of the tangent drawn from the midpoint of the arc to its intersection with the extensions of the sector's radial boundaries.

Answer: 2R


A geometric sequence has its first element as 6, the sum of the first $n$ elements is $\frac{45}{4}$, and the sum of the reciprocals of these elements is $\frac{5}{2}$. Which is this geometric sequence?

Answer: 6, 3, \frac{3}{2}, \frac{3}{4}


II. (Full marks 25 points) As shown in the figure, $\odot O_{1}$ and $\odot O_{2}$ are externally tangent at $M$, and the angle between their two external common tangents is $60^{\circ}$. The line connecting the centers intersects $\odot O_{1}$ and $\odot O_{2}$ at $A$ and $B$ (different from $M$), respectively. A line through $B$ intersects $\odot O_{1}$ at points $C$ and $D$. Find the value of $\operatorname{ctg} \angle B A C \cdot \operatorname{ctg} \angle B A D$.

Answer: 4


Let $\triangle ABC$ satisfy $AB = 17$, $AC = \frac{70}{3}$, and $BC = 19$. Let $I$ be the incenter of $\triangle ABC$ and $E$ be the excenter of $\triangle ABC$ opposite $A$. (Note: this means that the circle tangent to ray $AB$ beyond $B$, ray $AC$ beyond $C$, and side $BC$ is centered at $E$.) Suppose the circle with diameter $IE$ intersects $AB$ beyond $B$ at $D$. If $BD = \frac{a}{b}$ where $a$ and $b$ are coprime positive integers, find $a + b$.

Answer: 22


You are playing a game called "Hovse." Initially, you have the number $0$ on a blackboard. At any moment, if the number $x$ is written on the board, you can perform one of the following operations:

- Replace $x$ with $3x + 1$
- Replace $x$ with $9x + 1$
- Replace $x$ with $27x + 3$
- Replace $x$ with $\left \lfloor \frac{x}{3} \right \rfloor$

You are not allowed to write a number greater than $2017$ on the board. How many positive numbers can you make with the game of "Hovse?"

Answer: 127


Using each of the ten digits exactly once, construct two five-digit numbers such that their difference is minimized. Determine this minimal difference.

Answer: 247


How many rearrangements of the letters of "HMMTHMMT" do not contain the substring "HMMT"? For instance, one such arrangement is $HMMHMTMT$.

Answer: 361


The roots of the polynomial $f(x) = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1$ are all roots of unity. We say that a real number $r \in [0, 1)$ is nice if $e^{2i \pi r} = \cos 2\pi r + i \sin 2\pi r$ is a root of the polynomial $f$ and if $e^{2i \pi r}$ has a positive imaginary part. Let $S$ be the sum of the values of nice real numbers $r$. If $S = \frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.

Answer: 31


A bitstring of length $\ell$ is a sequence of $\ell$ $0$'s or $1$'s in a row. How many bitstrings of length $2014$ have at least $2012$ consecutive $0$'s or $1$'s?

Answer: 16


Let $A=(a_{ij})$ be the $n \times n$ matrix, where $a_{ij}$ is the remainder of the division of $i^j + j^i$ by $3$ for $i,j=1,2,\ldots,n$. Find the greatest $n$ for which $\det A \neq 0$.

Answer: 5


Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is friends with four, Elías is friends with five, and so on, with each subsequent person being friends with one more person than the previous person. This pattern continues until reaching Yvonne, the twenty-fifth person, who is friends with everyone. How many people is Zoila, the twenty-sixth person, friends with?

Clarification: If $A$ is a friend of $B$, then $B$ is a friend of $A$.

Answer: 13


Al and Bill play a game involving a fair six-sided die. The die is rolled until either a number less than $5$ is rolled on consecutive tosses, or a number greater than $4$ is rolled on consecutive tosses. Al wins if the last roll is a $5$ or $6$. Bill wins if the last roll is a $2$ or lower. Let $m$ and $n$ be relatively prime positive integers such that $\frac{m}{n}$ is the probability that Bill wins. Find the value of $m+n$.

Answer: 29


Henry rolls a fair die. If the die shows the number $k$, Henry will then roll the die $k$ more times. The probability that Henry will never roll a 3 or a 6 either on his first roll or on one of the $k$ subsequent rolls is given by $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 904


Given a board with size $25 \times 25$. Some $1 \times 1$ squares are marked, so that for each $13 \times 13$ and $4 \times 4$ sub-boards, there are at least $\frac{1}{2}$ of the squares in the sub-board that are marked. Find the least possible number of marked squares in the entire board.

Answer: 313


Jacob's analog clock has 12 equally spaced tick marks on the perimeter, but all the digits have been erased, so he doesn't know which tick mark corresponds to which hour. Jacob takes an arbitrary tick mark and measures clockwise to the hour hand and minute hand. He measures that the minute hand is 300 degrees clockwise of the tick mark, and that the hour hand is 70 degrees clockwise of the same tick mark. If it is currently morning, how many minutes past midnight is it?

Answer: 500


Define $\phi^{!}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the number of integers $2 \le n \le 50$ such that $n$ divides $\phi^{!}(n)+1$.

Answer: 30


A4. In the figure, $A Q P B$ and $A S R C$ are squares, and $A Q S$ is an equilateral triangle. If $Q S=4$ and $B C=x$, what is the value of $x$ ?

Answer: 4\sqrt{3}


In a certain logarithm system, $\log 450$ is 1.1639 more than $\log 40$. What is the base of the system and what are the logarithms of the two numbers in this system?

Answer: 8


Let \( S \) be the set of all nondegenerate triangles formed from the vertices of a regular octagon with side length 1. Find the ratio of the largest area of any triangle in \( S \) to the smallest area of any triangle in \( S \).

Answer: 3+2\sqrt{2}


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

Answer: \pi


Centipedes and three-headed dragons. In a certain herd of centipedes and three-headed dragons, there are a total of 26 heads and 298 legs. Each centipede has one head. How many legs does a three-headed dragon have?

Answer: 14


As shown in the figure, square \(ABCD\) and rectangle \(BEFG\) share a common vertex \(B\). The side length of square \(ABCD\) is \(6 \, \text{cm}\) and the length of rectangle \(BEFG\) is \(9 \, \text{cm}\). Find the width of the rectangle in centimeters.

Answer: 4


Answer: 1545


Players \( A \) and \( B \) play a game on a blackboard that initially contains 2020 copies of the number 1. In every round, player \( A \) erases two numbers \( x \) and \( y \) from the blackboard, and then player \( B \) writes one of the numbers \( x+y \) and \( |x-y| \) on the blackboard. The game terminates as soon as, at the end of some round, one of the following holds:
1. One of the numbers on the blackboard is larger than the sum of all other numbers.
2. There are only zeros on the blackboard.

Player \( B \) must then give as many cookies to player \( A \) as there are numbers on the blackboard. Player \( A \) wants to get as many cookies as possible, whereas player \( B \) wants to give as few as possible. Determine the number of cookies that \( A \) receives if both players play optimally.

Answer: 7


Let \( n \) be a positive integer. Determine the smallest number \( k \) of colors needed to color the edges of any directed simple graph with \( n \) vertices such that there is no monochromatic cycle.

Answer: 2


As shown in Figure 3, the side length of equilateral triangle \( \triangle ABC \) is 5. Extend \( BA \) to point \( P \) such that \( |AP| = 9 \). Let \( D \) be a point on segment \( BC \) (including endpoints). The line \( AD \) intersects the circumcircle of \( \triangle BPC \) at points \( E \) and \( F \), where \( |EA| < |ED| \).

(1) Let \( |BD| = x \). Express \( |EA| - |DF| \) as a function \( f(x) \).
(2) Find the minimum value of \( f(x) \).

Answer: 4\sqrt{5}


$\triangle XYZ$ is similar to $\triangle MNP$. The side $\overline{YZ}$ measures 10 cm, and the side $\overline{XZ}$ measures 7 cm. In $\triangle MNP$, the side $\overline{MN}$ measures 4.2 cm. Find the length of $\overline{NP}$, rounding your answer to the nearest tenth of a cm.

Answer: 6.0 \text{ cm}


Consider Abby the alpaca is tethered to the corner of a $4\text{ m}$ by $6\text{ m}$ barn on a $5\text{ m}$ leash. How much area (in square meters) can Abby roam if she can only walk around the barn?

Answer: 19\pi\text{ m}^2


In Mr. Petrov's 30-student history class, each student’s initials are uniquely represented by the same letter for both the first and last name. Besides the typical vowels A, E, I, O, U, the letter "X" is also considered a vowel in this peculiar setting. What is the probability of randomly selecting a student whose initials are vowels, given this modification?

Answer: \frac{1}{5}


Sandy thinks of a number. She triples it, adds 20, squares the result, and the final outcome is 2500. What was her original number?

Answer: 10


Let $x$ and $y$ be positive numbers such that $x^2 y$ is constant. Suppose $y = 8$ when $x = 3$. Find the value of $x$ when $y$ is $72$ and $x^2$ has increased by a factor of $z = 4$.

Answer: 1


Convert \(142_{10}\) to base 7. Represent any values normally represented by new symbols in decimal as \(A\) (if 10), \(B\) (if 11), and so on.

Answer: 262_7


A rectangular mirror (the shaded region) is perfectly centered within a larger frame, which itself is within an even larger frame. The outermost perimeter of the largest frame measures 100 cm by 140 cm. Each side of this frame has a width of 15 cm. Inside this, there is another frame, whose sides are also 15 cm wide. What is the area of the mirror?

Answer: 3200 \text{ cm}^2


The increasing sequence of positive integers $a_1, a_2, a_3, \dots$ follows the recursive formula
\[a_{n + 2} = a_{n + 1} + a_n\]
for all $n \geq 1$. If $a_7 = 210$, determine $a_8$.

Answer: 340


Let $Q$ be a quartic polynomial such that $Q(0) = k$, $Q(1) = 3k$, and $Q(-1) = 5k$. What is $Q(2) + Q(-2)$?

Answer: 26k


A cylindrical tower has a diameter of 25 feet and a height of 60 feet. A green stripe with a width of 2 feet wraps twice around the cylinder horizontally. Calculate the area of the stripe in square feet.

Answer: 100\pi \text{ square feet}


The time right now is precisely 6:00 a.m. What time will it be in 2700 minutes?

Answer: 3\!:\!00 \text{ a.m.}


Cutting isosceles right triangle $BFC$ out of square $ABCD$ and translating it leftwards creates the shape $ABFCDE$. The perimeter of square $ABCD$ is 40 inches. What is the perimeter, in inches, of $ABFCDE$?

[asy]
import graph;
filldraw((0,0)--(sqrt(2)*2,2)--(2+2sqrt(2),2)--(2sqrt(2),0)--(2+2sqrt(2),-2)--(sqrt(2)*2,-2)--cycle,grey,linewidth(1));
draw((sqrt(2)*2,2)--(sqrt(2)*2,-2),linewidth(1));
draw((2+2sqrt(2),-2)--(2+2sqrt(2),2)--(2sqrt(2),0)--(2+2sqrt(2),-2)--cycle,dashed);

label("A",(sqrt(2)*2,2),N);
label("B",(2+2sqrt(2),2),N);
label("C",(2+2sqrt(2),-2),S);
label("D",(sqrt(2)*2,-2),S);
label("E",(0,0),W);
label("F",(2sqrt(2),0),N);
[/asy]

Answer: 20 + 20\sqrt{2} \text{ inches}


The Gnollish language has expanded to include a fourth word, "kreeg". In a sentence, "splargh" cannot come directly before "glumph" or "kreeg". How many valid 3-word sentences are there now in this updated Gnollish language?

Answer: 48