Subsets $S$ of the first 3$5$ positive integers $\{1, 2, 3, ..., 35\}$ are called [i]contrived [/i] if $S$ has size $4$ and the sum of the squares of the elements of $S$ is divisible by $7$. Find the number of contrived sets.

Answer: 8605


A triangle $\vartriangle A_0A_1A_2$ in the plane has sidelengths $A_0A_1 = 7$,$A_1A_2 = 8$,$A_2A_0 = 9$. For $i \ge 0$, given $\vartriangle A_iA_{i+1}A_{i+2}$, let $A_{i+3}$ be the midpoint of $A_iA_{i+1}$ and let Gi be the centroid of $\vartriangle A_iA_{i+1}A_{i+2}$. Let point $G$ be the limit of the sequence of points $\{G_i\}^{\infty}_{i=0}$. If the distance between $G$ and $G_0$ can be written as $\frac{a\sqrt{b}}{c}$ , where $a, b, c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime, find $a^2 + b^2 + c^2$.

Answer: 422


In space, there are 1989 points, none of which are collinear. These points are divided into 30 groups, each containing a different number of points. From any three different groups, one point is taken to form the vertices of a triangle. To maximize the total number of such triangles, how many points should each group have?

Answer: 51, 52, 53, 54, 55, 56, 58, 59, \ldots, 81


A terrain ( $ABCD$ ) has a rectangular trapezoidal shape.  The angle in $A$ measures $90^o$. $AB$ measures $30$ m, $AD$ measures $20$ m and $DC$ measures 45 m. This land must be divided into two areas of the same area, drawing a parallel to the $AD$ side . At what distance from $D$ do we have to draw the parallel?
[img]https://1.bp.blogspot.com/-DnyNY3x4XKE/XNYvRUrLVTI/AAAAAAAAKLE/gohd7_S9OeIi-CVUVw-iM63uXE5u-WmGwCK4BGAYYCw/s400/image002.gif[/img]

Answer: \dfrac{75}{4}


[ Proof by contradiction ] [ Examples and counterexamples. Constructions ]

There are several stones, the mass of each of which does not exceed 2 kg, and the total mass is 100 kg. From them, several stones are selected, the total mass of which differs from 10 kg by the smallest possible number $d$ for this set. What is the maximum value that the number $d$ can take for all possible sets of stones?

Answer: \dfrac{10}{11}


Let $O$ and $I$ be the circumcenter and incenter of triangle $ABC$. The perpendicular from $I$ to $OI$ meets $AB$ and the external bisector of angle $C$ at points $X$ and $Y$ respectively. In what ratio does $I$ divide the segment $XY$?

Answer: \dfrac{1}{2}


32nd Putnam 1971 Problem B2 Let X be the set of all reals except 0 and 1. Find all real valued functions f(x) on X which satisfy f(x) + f(1 - 1/x) = 1 + x for all x in X.

Answer: \frac{1}{2} \left( x + \frac{1}{x} + \frac{1}{1 - x} \right)


89. A circle of radius $r$ touches internally a circle of radius $R$. $A$ is the point of tangency. A line perpendicular to the line of centers intersects one circle at point $B$, and the other at point $C$. Find the radius of the circle circumscribed about triangle $A B C$.

Answer: \sqrt{Rr}


The sum $\frac{2}{3\cdot 6} +\frac{2\cdot 5}{3\cdot 6\cdot 9} +\ldots +\frac{2\cdot5\cdot \ldots \cdot 2015}{3\cdot 6\cdot 9\cdot \ldots \cdot 2019}$ is written as a decimal number. Find the first digit after the decimal point.

Answer: 6


4. Determine the smallest natural number $n$ with the following property: If we choose $n$ different natural numbers less than 2006, there are two among them such that the ratio of the sum and the difference of their squares is greater than three.

Answer: 21


10.5. Let $M$ be a finite set of numbers. It is known that from any 3 elements of it, two numbers can be found whose sum belongs to $M$. How many elements can $M$ have at most?

Answer: 7


Example 3 Let the midpoint of side $AB$ of $\triangle ABC$ be $N$, $\angle A > \angle B$, and $D$ be a point on ray $AC$ such that $CD = BC$. $P$ is a point on ray $DN$ and is on the same side of line $BC$ as point $A$, satisfying $\angle PBC = \angle A$. $PC$ intersects $AB$ at point $E$, and $BC$ intersects $DP$ at point $T$. Find the value of the expression $\frac{BC}{TC} - \frac{EA}{EB}$.

Answer: 2


7. Let $R S T U V$ be a regular pentagon. Construct an equilateral triangle $P R S$ with point $P$ inside the pentagon. Find the measure (in degrees) of angle $P T V$.

Answer: 6


Let \( n \) be an integer greater than 1. If \( n \) is greater than 1200 times any one of its prime factors, find the smallest possible value of \( n \).

Answer: 3888


7. Given $z \in \mathrm{C}$, if the equation $x^{2}-2 z x+\frac{3}{4}+\mathrm{i}=0$ (i being the imaginary unit) has real roots, then the minimum value of the modulus of the complex number $z$ $|z|$ is $\qquad$ .

Answer: 1


5. A little squirrel collected 15 nuts in the forest weighing $50, 51, \ldots, 64$ grams. He knows the weight of each nut. Using a balance scale, the little squirrel tries to prove to his friends that the first nut weighs 50 g, the second 51 g, the third 52 g, and so on (initially, the friends know nothing about the weights of the nuts). What is the minimum number of weights the little squirrel will need, if both the weights and the nuts can be placed on either pan of the balance, and the number of weighings is unlimited? (The weights of the weights are known to both the little squirrel and his friends. There is an unlimited supply of weights weighing $1, 2, \ldots, 1000$ g.)

Answer: 1


4. Given the equation
$$
x^{2}+(b+2) x y+b y^{2}=0(b \in \mathbf{R})
$$

represents two lines. Then the range of the angle $\theta$ between them is

Answer: \left[ \arctan\left( \frac{2\sqrt{5}}{5} \right), \frac{\pi}{2} \right]


On the side \( BC \) of triangle \( ABC \) a point \( D \) is marked. On the side \( AB \) a point \( P \) is chosen. The segments \( PC \) and \( AD \) intersect at point \( Q \). The point \( R \) is the midpoint of segment \( AP \). Prove that there exists a fixed point \( X \), through which the line \( RQ \) passes for any choice of point \( P \). (A. Kuznetsov)

Answer: X


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

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

Answer: watermelon


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`
 - People have unique favorite music genres: `classical`, `pop`, `rock`
 - Each person has a unique birthday month: `sept`, `jan`, `april`
 - Everyone has a favorite smoothie: `watermelon`, `desert`, `cherry`
 - The people are of nationalities: `swede`, `dane`, `brit`
 - Each person has a favorite color: `white`, `red`, `yellow`

## Clues:
1. The person whose birthday is in January is Eric.
2. The person whose birthday is in September is Peter.
3. The person whose birthday is in April is in the third house.
4. The Desert smoothie lover is directly left of the Swedish person.
5. The person who loves classical music is the person whose favorite color is red.
6. The British person is the person who loves rock music.
7. The person who loves white is directly left of the Desert smoothie lover.
8. The person whose birthday is in September is the person who loves classical music.
9. The Desert smoothie lover is directly left of the Watermelon smoothie lover.
 What is the value of attribute Birthday for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: jan


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`, `Peter`, `Arnold`
 - Each person has a unique favorite drink: `milk`, `water`, `tea`
 - Each person prefers a unique type of vacation: `mountain`, `city`, `beach`
 - Each person lives in a unique style of house: `colonial`, `victorian`, `ranch`
 - The people keep unique animals: `cat`, `bird`, `horse`
 - Each person has a unique birthday month: `jan`, `sept`, `april`

## Clues:
1. The person living in a colonial-style house is somewhere to the left of the person who likes milk.
2. The person who prefers city breaks is directly left of the person residing in a Victorian house.
3. The person whose birthday is in January is directly left of the cat lover.
4. The one who only drinks water is the person who enjoys mountain retreats.
5. The person who keeps horses is Peter.
6. The person residing in a Victorian house is somewhere to the right of the person who loves beach vacations.
7. Peter is the person who prefers city breaks.
8. The person who enjoys mountain retreats is the person whose birthday is in April.
9. Eric is the one who only drinks water.
 What is the value of attribute Drink for the person whose attribute Animal is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: tea


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`
 - People own unique car models: `toyota camry`, `ford f150`, `tesla model 3`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `victorian`
 - Each person has a unique type of pet: `cat`, `dog`, `fish`
 - Each person has an occupation: `engineer`, `doctor`, `teacher`
 - Each person prefers a unique type of vacation: `city`, `mountain`, `beach`

## Clues:
1. The person with an aquarium of fish is in the first house.
2. The person who owns a Toyota Camry is in the second house.
3. The person who enjoys mountain retreats is not in the second house.
4. The person who prefers city breaks is not in the second house.
5. The person in a ranch-style home is somewhere to the left of Peter.
6. The person who owns a Toyota Camry is directly left of the person living in a colonial-style house.
7. Arnold is the person who has a cat.
8. Eric is somewhere to the left of the person who enjoys mountain retreats.
9. The person who is an engineer is not in the third house.
10. The person who owns a Tesla Model 3 is somewhere to the left of the person who is a teacher.
11. The person who owns a dog is the person who is an engineer.
 What is the value of attribute House for the person whose attribute Vacation is mountain? 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: `Arnold`, `Alice`, `Eric`, `Peter`, `Bob`, `Carol`
 - They all have a unique favorite flower: `daffodils`, `roses`, `tulips`, `iris`, `carnations`, `lilies`
 - People have unique favorite sports: `basketball`, `swimming`, `soccer`, `volleyball`, `tennis`, `baseball`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `lime`, `blueberry`, `dragonfruit`, `desert`
 - Each person has an occupation: `lawyer`, `doctor`, `teacher`, `artist`, `nurse`, `engineer`

## Clues:
1. The person who loves a bouquet of daffodils is the person who is an artist.
2. The person who loves tennis is not in the second house.
3. The person who is an artist is somewhere to the right of the person who likes Cherry smoothies.
4. The person who loves soccer is somewhere to the right of the person who is a teacher.
5. The person who loves basketball is directly left of the person who loves the vase of tulips.
6. The person who is a nurse is in the sixth house.
7. The person who loves a bouquet of daffodils and the Dragonfruit smoothie lover are next to each other.
8. The person who loves tennis is not in the third house.
9. The person who loves the boquet of iris is not in the second house.
10. The person who is a teacher is not in the second house.
11. The person who loves a carnations arrangement is the Watermelon smoothie lover.
12. There are two houses between the person who loves a bouquet of daffodils and the person who is a nurse.
13. The person who loves volleyball is the Dragonfruit smoothie lover.
14. Peter is the person who is a nurse.
15. There are two houses between the person who is a doctor and the person who is a lawyer.
16. The person who loves baseball is somewhere to the left of the person who drinks Blueberry smoothies.
17. The person who drinks Lime smoothies is somewhere to the left of Carol.
18. The person who loves the vase of tulips is the Desert smoothie lover.
19. The person who drinks Blueberry smoothies is Alice.
20. There is one house between the person who is an artist and Arnold.
21. Eric is directly left of Arnold.
22. There is one house between the person who is a doctor and Alice.
23. The person who loves the boquet of lilies is in the first house.
24. The person who is a doctor is not in the fifth house.
 What is the value of attribute Smoothie for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: lime


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 HairColor for the person whose attribute Flower is carnations? Please reason step by step, and put your final answer within \boxed{}

Answer: gray


Given any triangle  $ABC$  and any positive integer  $n$ , we say that  $n$  is a *decomposable* number for triangle  $ABC$  if there exists a decomposition of the triangle  $ABC$  into  $n$  subtriangles with each subtriangle similar to   $\triangle ABC$ . Determine the positive integers that are decomposable numbers for every triangle.

Answer: 1


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

Answer: 2


Let  $(x_n)$ ,  $n=1,2, \ldots $  be a sequence defined by  $x_1=2008$  and
\begin{align*} x_1 +x_2 + \ldots + x_{n-1} = \left( n^2-1 \right) x_n \qquad ~ ~ ~ \forall n \geq 2 \end{align*}
Let the sequence  $a_n=x_n + \frac{1}{n} S_n$ ,  $n=1,2,3, \ldots $  where  $S_n$   $=$   $x_1+x_2 +\ldots +x_n$ . Determine the values of  $n$  for which the terms of the sequence  $a_n$  are perfect squares of an integer.

Answer: 251,1004,4016


Let  $\triangle A_1B_1C_1$  be an equilateral triangle of area  $60$ .  Chloe constructs a new triangle  $\triangle A_2B_2C_2$  as follows.  First, she flips a coin.  If it comes up heads, she constructs point  $A_2$  such that  $B_1$  is the midpoint of  $\overline{A_2C_1}$ .  If it comes up tails, she instead constructs  $A_2$  such that  $C_1$  is the midpoint of  $\overline{A_2B_1}$ .  She performs analogous operations on  $B_2$  and  $C_2$ .  What is the expected value of the area of  $\triangle A_2B_2C_2$ ?

Answer: 195


Let  $d$  be any positive integer not equal to  $2, 5$  or  $13$ . Show that one can find distinct  $a,b$  in the set  $\{2,5,13,d\}$  such that  $ab-1$  is not a perfect square.

Answer: 1


You are playing a game called "Hovse."
Initially you have the number  $0$  on a blackboard.
If at any moment the number  $x$  is written on the board, you can either: $\bullet$  replace  $x$  with  $3x + 1$  $\bullet$  replace  $x$  with  $9x + 1$  $\bullet$  replace  $x$  with  $27x + 3$  $\bullet$  or replace  $x$  with  $\left \lfloor \frac{x}{3} \right \rfloor $ .
However, 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


A wooden cube, whose edges are one centimeter long, rests on a horizontal surface.  Illuminated by a point source of light that is  $x$  centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface.  The area of the shadow, which does not inclued the area beneath the cube is 48 square centimeters.  Find the greatest integer that does not exceed  $1000x.$

Answer: 166


In a convex quadrilateral $A B C D A B=B C$. The rays $B A$ and $C D$ intersect at point $E$, and the rays $A D$ and $B C$ intersect at point $F$. It is also known that $B E=B F$ and

$\angle D E F=25^{\circ}$. Find the angle $E F D$.

Answer: 25


2. At the front of the cycling race column are cyclists Aco and Boro, riding side by side. They have so far outpaced the other cyclists that no one can catch them before the finish line. At $20 \mathrm{~km}$ before the finish, Aco's tire goes flat. Boro continues to the finish line at an average speed of $40 \mathrm{~km} / \mathrm{h}$. Aco lost 3 minutes fixing the tire, and then continued riding at an average speed of $45 \mathrm{~km} / \mathrm{h}$. Who won the race and by what distance?

Answer: \dfrac{2}{9}\ \text{km}


[b]p16.[/b] When not writing power rounds, Eric likes to climb trees. The strength in his arms as a function of time is $s(t) = t^3 - 3t^2$. His climbing velocity as a function of the strength in his arms is $v(s) = s^5 + 9s^4 + 19s^3 - 9s^2 - 20s$. At how many (possibly negative) points in time is Eric stationary?

[b]p17[/b]. Consider a triangle $\vartriangle ABC$ with angles $\angle ACB = 60^o$, $\angle ABC = 45^o$. The circumcircle around $\vartriangle ABH$, where $H$ is the orthocenter of $\vartriangle ABC$, intersects $BC$ for a second time in point $P$, and the center of that circumcircle is $O_c$. The line $PH$ intersects $AC$ in point $Q$, and $N$ is center of the circumcircle around $\vartriangle AQP$. Find $\angle NO_cP$.

[b]p18.[/b] If $x, y$ are positive real numbers and $xy^3 = \frac{16}{9}$ , what is the minimum possible value of $3x + y$?

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

Answer: 8


In the game Pogémon, Pogé Balls contain Pichagus with a $50 \%$ chance and otherwise contain nothing. The new Pogémon universe started with one Pichagu and has existed for five hours. Every hour, every Pichagu at the time opened a new Pogé Ball.

a) What is the expected number of Pichagus created in the last hour?

b) Every two currently existing Pichagus will now battle each other. What is the expected number of battles?

Answer: \dfrac{81}{32}


Call a day a [i]perfect[/i] day if the sum of the digits of the month plus sum of the digits of the day equals the sum of digits of the year. For example, February $28$th, $2028$ is a perfect day because $2+2+8=2+0+2+8$. Find the number of perfect days in $2018$.

[i]2018 CCA Math Bonanza Team Round #5[/i]

Answer: 36


[ Properties and signs of tangents ] [ Rectangles and squares. Properties and signs ]

Two circles are given. Their common internal tangents are perpendicular to each other. The chords connecting the points of tangency are 3 and 5. Find the distance between the centers of the circles.

Answer: 8


While out for a stroll, you encounter a vicious velociraptor. You start running away to the northeast at $ 10 \text{m/s}$, and you manage a three-second head start over the raptor. If the raptor runs at $ 15\sqrt{2} \text{m/s}$, but only runs either north or east at any given time, how many seconds do you have until it devours you?

Answer: 6


We know that we currently have six denominations of banknotes in circulation: 500, 1000, 2000, 5000, 10000, and 20000 forints. How many different amounts can be paid with three banknotes?

Answer: 49


Exercise 10. Find all triplets ( $\mathbf{p , q , r} \mathbf{)}$ ) of prime numbers such that the 3 differences

$$
|p-q|,|q-r|,|r-p|
$$

are also prime numbers.

Answer: (2, 5, 7)


## Task 5 - 300735

Let $A B C D$ be a rectangle with side lengths $A B=a$ and $B C=b$, and let $a>b$. On $A B$, points $C_{1}$ and $D_{1}$, and on $C D$, points $A_{1}$ and $B_{1}$ are marked such that the segments $A A_{1}, B B_{1}, C C_{1}, D D_{1}$ are the angle bisectors of an interior angle of $A B C D$.

The intersection points $E, F, G, H$ of these angle bisectors are labeled as shown in the figure.

Determine the area $I$ of the quadrilateral $E F G H$, given that $a=8 \mathrm{~cm}$ and $b=5 \mathrm{~cm}$!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0527.jpg?height=346&width=443&top_left_y=358&top_left_x=1292)

Answer: \dfrac{9}{2}


10.374 The bases of an isosceles trapezoid are 4 and 8 cm, and its area is 21 cm². Which side does the bisector of the angle at the larger base intersect: the smaller base or the lateral side of the trapezoid?

Answer: \text{the lateral side}


From a triangle, three triangles were cut off, and each of the three cuts touched the inscribed circle of the triangle. It is known that the perimeters of the cut-off triangles are $P_{1}, P_{2}, P_{3}$. Find the perimeter of the original triangle.

Answer: P_1 + P_2 + P_3


2. Find all such triples of prime numbers $p, q$ and $r$, that $\frac{p}{q}=\frac{8}{r-1}+1$.

Answer: (3, 2, 17)


2. Find the smallest possible value of $\left|2015 m^{5}-2014 n^{4}\right|$, given that $m, n$ are natural numbers.

Answer: 0


A fighting game club has $2024$ members. One day, a game of Smash is played between some pairs of members such that every member has played against exactly $3$ other members. Each match has a winner and a loser. A member will be happy if they won in at least $2$ of the matches. What is the maximum number of happy members over all possible match-ups and all possible outcomes?

Answer: 1012


At a party, there were some couples attending. As they arrive, each person gets to talk with all the other persons who are already in the room. During the party, after all the guests arrive, groups of persons form such that:
1. No two persons forming a couple belong to the same group.
2. For each two persons that do not form a couple, there is one and only one group to which both belong.

Find the number of couples attending the party, knowing that there are fewer groups than persons at the party.

Answer: 4


A sequence $a_n$ of real numbers satisfies the following conditions:

- $a_1 = 1$
- $a_2 = 0$
- $a_n = (S_{n-1} + 1) S_{n-2}$ for all integers $n \geq 3$

where $S_k = a_1 + a_2 + \dots + a_k$ for positive integers $k$.

Determine the smallest integer $m > 2$ such that $127$ divides $a_m$.

Answer: 10


For some positive integer $n$, consider the usual prime factorization \[ n = \prod_{i=1}^{k} p_{i}^{e_{i}} = p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}, \] where $k$ is the number of prime factors of $n$ and $p_{i}$ are the prime factors of $n$. Define $Q(n), R(n)$ by \[ Q(n) = \prod_{i=1}^{k} p_{i}^{p_{i}} \text{ and } R(n) = \prod_{i=1}^{k} e_{i}^{e_{i}}. \] For how many $1 \leq n \leq 70$ does $R(n)$ divide $Q(n)$?

Answer: 56


Compute $\left\lceil\sum_{k=2018}^{\infty}\frac{2019!-2018!}{k!}\right\rceil$. The notation $\left\lceil x\right\rceil$ denotes the least integer $n$ such that $n\geq x$.

Answer: 2019


Triangle $\triangle ABC$ has circumcenter $O$ and incircle $\gamma$. Suppose that $\angle BAC = 60^\circ$ and $O$ lies on $\gamma$. If \[ \tan B \tan C = a + \sqrt{b} \] for positive integers $a$ and $b$, compute $100a+b$.

Answer: 408


On a $10 \times 10$ chessboard, several knights are placed. In any $2 \times 2$ square, there is at least one knight. What is the smallest number of cells these knights can threaten? (Note: A knight does not threaten the square on which it stands, but it does threaten the squares on which other knights are standing.)

Answer: 50


For real numbers $x_1, x_2, x_3, \dots, x_{333} \in [-1, \infty)$, let $S_k = \sum_{i = 1}^{333} x_i^k$ for each $k$. If $S_2 = 777$, compute the least possible value of $S_3$.

Answer: 999


A square is inscribed in a circle with a radius of $6$. A quarter circle is inscribed in the square, as depicted in the diagram below. Given that the area of the region inside the circle but outside the quarter circle is $n\pi$ for some positive integer $n$, determine the value of $n$.

[asy]
size(5 cm);
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw(circle((1,1),1.41));
draw(arc((0,0),2,0,90));[/asy]

Answer: 18


Let $a, b, c, d$ be four positive integers such that $a > b > c > d$. Given that $ab + bc + ca + d^2 \mid (a + b)(b + c)(c + a)$. Find the minimal value of $\Omega(ab + bc + ca + d^2)$. Here $\Omega(n)$ denotes the number of prime factors $n$ has. For example, $\Omega(12) = 3$.

Answer: 3


Kelvin the Frog likes numbers whose digits strictly decrease, but numbers that violate this condition in at most one place are good enough. In other words, if $d_i$ denotes the $i$th digit, then $d_i \le d_{i+1}$ for at most one value of $i$. For example, Kelvin likes the numbers $43210$, $132$, and $3$, but not the numbers $1337$ and $123$. How many 5-digit numbers does Kelvin like?

Answer: 14034


How many ways are there to write $657$ as a sum of powers of two, where each power of two is used at most twice in the sum? For example, $256 + 256 + 128 + 16 + 1$ is a valid sum.

Answer: 41


For each integer $1 \le j \le 2017$, let $S_j$ denote the set of integers $0 \le i \le 2^{2017} - 1$ such that $\left\lfloor \frac{i}{2^{j-1}} \right\rfloor$ is an odd integer. Let $P$ be a polynomial such that
\[P\left(x_0, x_1, \ldots, x_{2^{2017} - 1}\right) = \prod_{1 \le j \le 2017} \left(1 - \prod_{i \in S_j} x_i\right).\]
Compute the remainder when
\[ \sum_{\left(x_0, \ldots, x_{2^{2017} - 1}\right) \in \{0, 1\}^{2^{2017}}} P\left(x_0, \ldots, x_{2^{2017} - 1}\right)\]
is divided by $2017$.

Answer: 1840


In triangle \(ABC\), the altitude from vertex \(A\) is \(AA^{\prime}\) and \(AD\) bisects the angle \(\alpha\). Let \(B^{\prime}\) be the projection of vertex \(B\) onto \(AD\), and \(C^{\prime}\) be the projection of vertex \(C\) onto \(AD\). Let \(M\) be the midpoint of side \(BC\).

1. Prove that \(MB^{\prime} = MC^{\prime}\).
2. Show that quadrilateral \(A^{\prime}B^{\prime}MC^{\prime}\) is a cyclic quadrilateral.

Answer: MB' = MC'


A cube-shaped cake is completely covered with chocolate, including the bottom. We want to share the cake among $K$ people so that everyone gets the same amount of dough and the same amount of chocolate coating. We want to achieve this by dividing the cake into $N \times N \times N$ identical small cubes, and giving each person the same number of pieces, ensuring that the total area of the chocolate-coated sides of the pieces received by each person is the same. Is this possible for any $K$? What is the minimum number of pieces the cake must be divided into if $K=1997$?

Answer: 1997^3


3. A $4 \mathrm{~cm} \times 4 \mathrm{~cm}$ square is split into four rectangular regions using two line segments parallel to the sides.
How many ways are there to do this so that each region has an area equal to an integer number of square centimetres?

Answer: 17


3. (physics) A wire is made into a right-angled triangle with legs of length $l=1$ m. From vertices A and B, two bugs start crawling along the legs simultaneously with speeds $v_{A}=5$ cm/s and $v_{B}=2 v_{A}=10$ cm/s (see figure). What is the minimum time after the start of their movement when the distance between the bugs reaches a minimum?

![](https://cdn.mathpix.com/cropped/2024_05_06_25c5d6100f66704b4a85g-2.jpg?height=354&width=354&top_left_y=1770&top_left_x=1545)

Answer: 4


I1.3 Given that $p$ and $q$ are real numbers with $p q=b$ and $p^{2} q+q^{2} p+p+q=70$. If $c=p^{2}+q^{2}$, find the value of $c$.

Answer: 31


2. Given that point $E$ is the midpoint of side $B C$ of rhombus $A B C D$, $\angle A B C=30^{\circ}$, and $P$ is a point on diagonal $B D$ such that $P C + P E = \sqrt{13}$. Then the maximum area of rhombus $A B C D$ is $\qquad$

Answer: 10 + 4\sqrt{3}


19. A riot broke out in the Kingdom of Natural Numbers. Among the digits from $0 \sim 9$, the larger digits felt superior and refused to be placed behind smaller digits when appearing together. Thus, natural numbers like 36, 121, and 1234 disappeared, leaving only numbers like 2, 55, 433, and 4321. So, among the original natural numbers from 1 to 2022, there are now $\qquad$ numbers left.

Answer: 287


$[$ Properties and characteristics of the tangent $]$

In a circle with radius $R=4$, a chord $A B$ and a diameter $A K$ are drawn, forming an angle of $\frac{\pi}{8}$ with the chord. A tangent is drawn at point $B$, intersecting the extension of the diameter $A K$ at point $C$. Find the median $A M$ of triangle $A B C$.

Answer: 2\sqrt{3} + 2\sqrt{6}


Example 6.21. On average, $85 \%$ of the items coming off the conveyor are of the first grade. How many items need to be taken so that with a probability of 0.997, the deviation of the frequency of first-grade items from 0.85 in absolute value does not exceed 0.01?

Answer: 11475


Example 3. A region is divided into five administrative areas as shown in the figure. There are 4 colors available to color the map, with the requirement that adjacent areas must not use the same color. How many different coloring methods are there?

Answer: 72


Example 5. For an isosceles trapezoid $\triangle B C D$ with a base angle of $67.5^{\circ}$, a circle is drawn with the leg $B C$ as the diameter, intersecting the lower base $A B$ at $E$. 11. It is exactly tangent to the leg $A D$ at $M$. Find $B E: A E$. (1981, Beijing High School Competition)

Answer: \sqrt{2} : 2


Let \(ABCD\) be a trapezoid with \( (AB) \parallel (CD) \). Let \(M\) be the midpoint of \([AB]\) and \(P\) a point on \((BC)\). Define the points \(X = (PD) \cap (AB)\), \(Q = (PM) \cap (BD)\), and \(Y = (PQ) \cap (AB)\). Show that \(M\) is the midpoint of \([XY]\).

Answer: M \text{ is the midpoint of } [XY]


3. (50 points) Given that $p$ is a prime number, the fractional part of $\sqrt{p}$ is $x$, and the fractional part of $\frac{1}{x}$ is $\frac{\sqrt{p}-31}{75}$. Find all prime numbers $p$ that satisfy the condition.

Answer: 2011


2. 10 people go to the bookstore to buy books, it is known that: (1) each person bought three types of books; (2) any two people have at least one book in common.

How many people at most could have bought the book that was purchased by the fewest people?

Answer: 5


8. The sequence $\left\{a_{n}\right\}$ satisfies $a_{n+1}=\frac{1}{2-a_{n}}$. If for any positive integer $n$, we have $a_{n+1}>a_{n}$, then the range of $a_{1}$ is $\qquad$

Answer: (-\infty, 1)


Mitranov I.V.

The King decided to reward a group of $n$ sages. They will be placed in a row, all facing the same direction, and a black or white hat will be placed on each of them. Each sage will see the hats of all those in front of them. The sages will take turns, from the last to the first, naming a color (white or black) and a natural number of their choice. At the end, the number of sages who named the color matching their own hat will be counted: the group will receive a bonus for exactly that many days. The sages are allowed to agree in advance on how to answer. However, the sages know that exactly $k$ of them are insane (who exactly is unknown to them). An insane sage will name a white or black color and a number regardless of the agreement. What is the maximum number of days with a bonus that the sages can guarantee for the group, regardless of the position of the insane in the queue?

Answer: n - k - 1


Let \( S \) be the set of all non-negative integers with at most \( n \) digits in decimal, and \( S_{k} \) be the subset of \( S \) consisting of elements whose digit sum is less than \( k \). For which \( n \), does there exist \( k \) such that \( |S| = 2|S_{k}| \)?

Answer: n \text{ is odd}


## Properties and characteristics of a parallelogram [Trigonometric ratios in a right triangle]

On the side $C D$ of parallelogram $A B C D$ with an obtuse angle at vertex $D$, an equilateral triangle $C D E$ is constructed such that points $A$ and $E$ lie on opposite sides of line $C D$. It is known that the distances from points $D$ and $E$ to line $B C$ are 3 and 8, respectively, and the distance from point $E$ to line $A B$ is 13. Find the area of parallelogram $A B C D$.

Answer: 28\sqrt{3}


2. A little goat is tied to a stake with a 4.7 m long rope, located on a flat meadow next to an old tree. The trunk of the tree has the shape of a circular cylinder with a radius of 0.5 m, and the shortest distance from the stake to the surface of the tree is $1 \mathrm{m}$.

Can the goat walk around the tree and approach the stake so closely that the end of the rope coincides with its beginning? Justify your answer.

Assume that the rope, when taut, lies in a horizontal plane.

Answer: \text{No}


Example 9 Find all positive integers $a, b, c$, where $1<a<b<c$, such that $(a-1)(b-1)(c-1)$ is a divisor of $abc-1$.

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

Answer: (2, 4, 8)


A merchant sold goods for 39 K. How much did the goods cost, if the profit margin was the same percentage as the cost of the goods?

Translate the above text back into Hungarian, please keep the original text's line breaks and format, and output the translation result directly.
preneurial context.
A kereskedő 39 K-ért adott el árut. Mennyiért vette az árút, ha ugyanannyi százalékot nyert, mint a mennyibe az áru került?

Answer: 30


Sámson writes the number 123456789 on a piece of paper. Then he can insert multiplication signs between any two adjacent digits, at different places, or not at all. The digits between the multiplication signs, when read together, form a number, resulting in an expression that is a product of numbers, for example, 1234$\cdot$56$\cdot$789. What is the maximum possible value of the resulting number?

Answer: 123456789


2. Each card has a number on one side and a letter on the other side (two cards that have the same letter on one side do not necessarily have the same number on the other side, and vice versa). On the table, there are cards that show the following on their visible side:

$$
M, A, T, E, M, A, T, И, K, A, 2,0,1,8
$$

Vlada claims the following: "If a card has a vowel, then the other side has an even number." Miljan wants to check Vlada's claim. How many and which cards does he need to turn over to verify the correctness of the statement?

Answer: 6


Two identical equilateral triangles $A B C$ and $C D E$ with side length 1 are positioned such that they share only one common point $C$ and the angle $B C D$ is less than $60^{\circ}$. Point $K$ is the midpoint of $A C$, point $L$ is the midpoint of $C E$, and point $M$ is the midpoint of $B D$. The area of triangle $K L M$ is $\frac{\sqrt{3}}{5}$. Find $B D$.

Answer: \dfrac{2\sqrt{5} - \sqrt{15}}{5}


Let \( n \geq 2 \) be an integer. Let nonzero real numbers \( x_{1}, x_{2}, \ldots, x_{n} \) satisfy \( \sum_{i=1}^{n} x_{i} = 0 \), and for any positive real number \( t \), there are at most \(\frac{1}{t} \) pairs \((i, j)\) such that \(\left|x_{i}-x_{j}\right| \geq t \). Prove that:
\[
\sum_{i=1}^{n} x_{i}^{2} < \frac{1}{n}\left(\max_{1 \leq i \leq n} x_{i} - \min_{1 \leq i \leq n} x_{i} \right).
\]

Answer: \sum_{i=1}^{n} x_{i}^{2} < \frac{1}{n}\left(\max_{1 \leq i \leq n} x_{i} - \min_{1 \leq i \leq n} x_{i} \right)


15. Arrange 5 teachers $A, B, C, D, E$ and 25 identical seats in a row, after which 25 students will sit in the seats to take photos with the teachers. Requirements: $A, B, C, D, E$ must appear in alphabetical order from left to right in this row, and there must be at least two seats between each adjacent pair of teachers. Then there are $\qquad$ different arrangement methods (Note: The arrangement refers to the placement of teachers and empty seats, not the subsequent seating of students).

Answer: 26334


9. Given that $M N$ is a moving chord of the circumcircle of equilateral $\triangle A B C$ with side length $2 \sqrt{6}$, $M N=4$, and $P$ is a moving point on the sides of $\triangle A B C$. Then the maximum value of $\overrightarrow{M P} \cdot \overrightarrow{P N}$ is

Answer: 4


【Question 7】As shown in the figure, on a square piece of paper with a side length of 15 cm, cut along a $45^{\circ}$ angle from 4 cm away from each vertex, forming a small square in the middle. The area of this small square is $\qquad$ (square centimeters).

Answer: 32


Putnam 1998 Problem B2 Let P be the point (a, b) with 0 < b < a. Find Q on the x-axis and R on y = x, so that PQ + QR + RP is as small as possible.

Answer: \sqrt{2(a^2 + b^2)}


Answer: (-1, -6)


You are given a positive integer $n$. What is the largest possible number of numbers that can be chosen from the set
$\{1, 2, \ldots, 2n\}$ so that there are no two chosen numbers $x > y$ for which $x - y = (x, y)$?

Here $(x, y)$ denotes the greatest common divisor of $x, y$.

[i]Proposed by Anton Trygub[/i]

Answer: n


7. Peggy picks three positive integers between $1$ and $25$, inclusive, and tells us the following information about those numbers:
[list]
[*] Exactly one of them is a multiple of $2$;
[*] Exactly one of them is a multiple of $3$;
[*] Exactly one of them is a multiple of $5$;
[*] Exactly one of them is a multiple of $7$;
[*] Exactly one of them is a multiple of $11$.
[/list]
What is the maximum possible sum of the integers that Peggy picked?

8. What is the largest positive integer $k$ such that $2^k$ divides $2^{4^8}+8^{2^4}+4^{8^2}$?

9. Find the smallest integer $n$ such that $n$ is the sum of $7$ consecutive positive integers and the sum of $12$ consecutive positive integers.

Answer: 48


In a triangle $ABC$ with $ \angle A = 36^o$ and $AB = AC$, the bisector of the angle at $C$ meets the oposite side at $D$. Compute the angles of $\triangle BCD$. Express the length of side $BC$ in terms of the length $b$ of side $AC$ without using trigonometric functions.

Answer: 36^\circ


7.4. Several knights and liars are standing in a circle, all of different heights. Each of them said the phrase: "I am taller than exactly one of my neighbors." Could there have been exactly 2023 liars among those standing in the circle? (Knights always tell the truth, while liars always lie).

Answer: No


3. On the sides $B C$ and $A D$ of the inscribed quadrilateral $A B C D$, points $K$ and $M$ are marked respectively, such that $B K: K C = A M: M D$. On the segment $K M$, a point $L$ is chosen such that $K L: L M = B C: A D$. Find the ratio of the areas of triangles $A C L$ and $B D L$, given that $A C = p$ and $B D = q$.

Answer: \dfrac{p}{q}


There are 2 houses, numbered 1 to 2 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`
 - The people keep unique animals: `horse`, `cat`
 - Each person has a favorite color: `yellow`, `red`
 - Everyone has a favorite smoothie: `cherry`, `desert`
 - People have unique favorite book genres: `science fiction`, `mystery`
 - Everyone has something unique for lunch: `pizza`, `grilled cheese`

## Clues:
1. Eric is the person whose favorite color is red.
2. The Desert smoothie lover is in the second house.
3. The cat lover is the person who loves eating grilled cheese.
4. Arnold is in the second house.
5. The person who keeps horses is somewhere to the left of the person who loves mystery books.
 What is the value of attribute Color for the person whose attribute Food is grilled cheese? Please reason step by step, and put your final answer within \boxed{}

Answer: yellow


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`, `Alice`
 - Everyone has something unique for lunch: `spaghetti`, `pizza`, `stew`, `grilled cheese`
 - People own unique car models: `tesla model 3`, `honda civic`, `toyota camry`, `ford f150`
 - The people keep unique animals: `fish`, `cat`, `bird`, `horse`

## Clues:
1. The cat lover is somewhere to the right of the fish enthusiast.
2. Eric is the person who owns a Ford F-150.
3. The person who owns a Honda Civic is in the fourth house.
4. The person who is a pizza lover is the fish enthusiast.
5. The person who keeps horses is somewhere to the left of Arnold.
6. Arnold is the person who loves the spaghetti eater.
7. The person who owns a Toyota Camry is the person who loves the stew.
8. Alice and Arnold are next to each other.
9. Alice is in the first house.
 What is the value of attribute CarModel for the person whose attribute Animal is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: tesla model 3


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

## Clues:
1. Alice is the person who uses a Huawei P50.
2. The person who uses a Xiaomi Mi 11 is somewhere to the left of the one who only drinks water.
3. The one who only drinks water is somewhere to the left of Bob.
4. Carol is the person who has gray hair.
5. Eric is the tea drinker.
6. The person who has black hair is not in the sixth house.
7. The person who uses a Samsung Galaxy S21 is directly left of the person who loves classical music.
8. The person who uses a OnePlus 9 is not in the third house.
9. Alice is somewhere to the right of Carol.
10. The person who loves jazz music is Alice.
11. The person who loves jazz music is the boba tea drinker.
12. There is one house between the person who uses a OnePlus 9 and Eric.
13. The person who has brown hair is the person who loves hip-hop music.
14. The person who loves pop music is not in the first house.
15. The root beer lover is in the second house.
16. Eric is the person who uses an iPhone 13.
17. The person who has red hair is the person who uses an iPhone 13.
18. The person who has red hair and the coffee drinker are next to each other.
19. Peter is in the first house.
20. Arnold is somewhere to the right of the person who has auburn hair.
21. The person who has gray hair is not in the second house.
22. The person who loves country music is in the fifth house.
 What is the value of attribute House for the person whose attribute Name is Arnold? 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: `Alice`, `Peter`, `Eric`, `Bob`, `Arnold`, `Carol`
 - Everyone has a unique favorite cigar: `pall mall`, `yellow monster`, `dunhill`, `blue master`, `prince`, `blends`
 - People have unique favorite music genres: `hip hop`, `jazz`, `country`, `pop`, `classical`, `rock`
 - Each person has a unique favorite drink: `water`, `milk`, `boba tea`, `tea`, `root beer`, `coffee`
 - The mothers' names in different houses are unique: `Kailyn`, `Penny`, `Janelle`, `Holly`, `Sarah`, `Aniya`
 - Everyone has something unique for lunch: `soup`, `pizza`, `spaghetti`, `stir fry`, `stew`, `grilled cheese`

## Clues:
1. Carol is directly left of the person who loves eating grilled cheese.
2. Eric is not in the second house.
3. The person whose mother's name is Holly is somewhere to the right of Carol.
4. The person who loves eating grilled cheese is somewhere to the right of the person who loves rock music.
5. Eric is directly left of Carol.
6. The person who loves pop music is not in the third house.
7. Eric is the person who loves country music.
8. The person who loves classical music is in the sixth house.
9. The coffee drinker is Bob.
10. The person who smokes many unique blends is Peter.
11. The person who loves the stew is not in the fifth house.
12. The root beer lover is directly left of The person whose mother's name is Janelle.
13. There are two houses between The person whose mother's name is Sarah and the person who smokes Yellow Monster.
14. Eric is the tea drinker.
15. The person partial to Pall Mall is somewhere to the right of the person who loves stir fry.
16. The person who loves the soup is Bob.
17. The person who loves hip-hop music is directly left of The person whose mother's name is Kailyn.
18. Arnold is somewhere to the right of The person whose mother's name is Kailyn.
19. The one who only drinks water is directly left of the person who smokes Blue Master.
20. The person who loves the spaghetti eater is somewhere to the left of the person who smokes many unique blends.
21. The person whose mother's name is Sarah is directly left of the person who loves jazz music.
22. The person who loves hip-hop music is directly left of the root beer lover.
23. The one who only drinks water is the person who loves the stew.
24. The Dunhill smoker is not in the second house.
25. The person who likes milk is The person whose mother's name is Janelle.
26. Eric is The person whose mother's name is Aniya.
 What is the value of attribute Food for the person whose attribute MusicGenre is classical? Please reason step by step, and put your final answer within \boxed{}

Answer: soup


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Bob`, `Eric`, `Arnold`
 - They all have a unique favorite flower: `tulips`, `daffodils`, `lilies`, `roses`, `carnations`
 - Each mother is accompanied by their child: `Fred`, `Bella`, `Timothy`, `Meredith`, `Samantha`
 - Each person has a unique level of education: `high school`, `bachelor`, `doctorate`, `master`, `associate`
 - The people keep unique animals: `cat`, `fish`, `dog`, `horse`, `bird`

## Clues:
1. Peter is somewhere to the right of the dog owner.
2. There is one house between the person who loves a carnations arrangement and the person with a bachelor's degree.
3. There is one house between the person with a doctorate and the cat lover.
4. The person with an associate's degree is directly left of the person who loves the vase of tulips.
5. There is one house between the cat lover and the person who keeps horses.
6. The person's child is named Fred is somewhere to the left of the person with a bachelor's degree.
7. The person's child is named Meredith is not in the third house.
8. The person with a master's degree is the person's child is named Bella.
9. The person with a bachelor's degree is Bob.
10. The person who is the mother of Timothy is the person who loves a bouquet of daffodils.
11. The person's child is named Samantha is Bob.
12. Arnold is directly left of Peter.
13. The bird keeper is directly left of the person who loves a bouquet of daffodils.
14. The person who loves the rose bouquet is the bird keeper.
15. There is one house between the dog owner and the fish enthusiast.
16. The person who loves the vase of tulips is Eric.
 What is the value of attribute Name for the person whose attribute Children is Fred? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Eric`, `Bob`
 - Each person has a unique favorite drink: `root beer`, `tea`, `water`, `coffee`, `milk`
 - People have unique favorite music genres: `pop`, `hip hop`, `classical`, `jazz`, `rock`
 - The people are of nationalities: `dane`, `norwegian`, `swede`, `german`, `brit`
 - Everyone has a unique favorite cigar: `dunhill`, `prince`, `blue master`, `pall mall`, `blends`

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

Answer: pall mall


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Bob`, `Peter`, `Arnold`
 - People have unique favorite book genres: `fantasy`, `mystery`, `science fiction`, `biography`, `romance`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Aniya`, `Kailyn`, `Janelle`
 - People use unique phone models: `oneplus 9`, `iphone 13`, `samsung galaxy s21`, `google pixel 6`, `huawei p50`
 - The people are of nationalities: `dane`, `brit`, `german`, `swede`, `norwegian`
 - Each person lives in a unique style of house: `ranch`, `modern`, `craftsman`, `victorian`, `colonial`

## Clues:
1. The person in a ranch-style home is somewhere to the right of the person residing in a Victorian house.
2. The Dane and the person who uses a Samsung Galaxy S21 are next to each other.
3. The person who uses a Google Pixel 6 is not in the second house.
4. There are two houses between the person who loves fantasy books and the person in a modern-style house.
5. The person whose mother's name is Kailyn is the Swedish person.
6. The Swedish person is Arnold.
7. The British person is somewhere to the left of the person who uses an iPhone 13.
8. The Dane is The person whose mother's name is Penny.
9. The person who uses a Samsung Galaxy S21 is the Norwegian.
10. The Norwegian is The person whose mother's name is Holly.
11. The person who loves fantasy books is Bob.
12. Alice is directly left of the Swedish person.
13. The person who loves romance books is Arnold.
14. The person who loves science fiction books is Peter.
15. The person whose mother's name is Aniya is directly left of the person who loves science fiction books.
16. The person who loves science fiction books is the person who uses a OnePlus 9.
17. The person residing in a Victorian house is the person who loves biography books.
18. The person who loves fantasy books is somewhere to the left of the German.
19. The person who uses an iPhone 13 is the person in a Craftsman-style house.
 What is the value of attribute Mother for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: Kailyn


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 Mother is Aniya? Please reason step by step, and put your final answer within \boxed{}

Answer: 6


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`, `Alice`
 - Each person has a unique level of education: `master`, `high school`, `associate`, `bachelor`
 - Everyone has something unique for lunch: `stew`, `grilled cheese`, `pizza`, `spaghetti`
 - People have unique favorite sports: `soccer`, `swimming`, `tennis`, `basketball`
 - People have unique hair colors: `blonde`, `black`, `brown`, `red`
 - People have unique favorite book genres: `mystery`, `fantasy`, `science fiction`, `romance`

## Clues:
1. The person who loves the stew is in the first house.
2. The person who loves science fiction books is the person who has black hair.
3. The person who has blonde hair is not in the second house.
4. The person with a master's degree is in the first house.
5. Eric is the person who loves soccer.
6. The person with a master's degree and the person who loves mystery books are next to each other.
7. The person who loves tennis is the person with a high school diploma.
8. The person who loves the spaghetti eater is the person who loves science fiction books.
9. The person who is a pizza lover is the person with an associate's degree.
10. The person who has blonde hair is not in the fourth house.
11. The person with a master's degree is directly left of Eric.
12. The person with a bachelor's degree is directly left of the person who loves fantasy books.
13. The person who has red hair is Peter.
14. The person who loves swimming is directly left of the person who is a pizza lover.
15. Alice is not in the first house.
 What is the value of attribute House for the person whose attribute Food is grilled cheese? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


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

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

Answer: very short


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Eric`, `Arnold`, `Peter`, `Alice`
 - People have unique favorite sports: `tennis`, `baseball`, `basketball`, `soccer`, `swimming`
 - Each mother is accompanied by their child: `Fred`, `Bella`, `Meredith`, `Timothy`, `Samantha`
 - The mothers' names in different houses are unique: `Kailyn`, `Holly`, `Penny`, `Janelle`, `Aniya`
 - The people keep unique animals: `horse`, `cat`, `dog`, `bird`, `fish`

## Clues:
1. The person whose mother's name is Penny is in the fourth house.
2. Eric is The person whose mother's name is Penny.
3. The person who keeps horses is the person who loves swimming.
4. The person who loves baseball is the person's child is named Meredith.
5. The dog owner is directly left of the person who loves basketball.
6. The person who loves soccer is the bird keeper.
7. The person who loves swimming is directly left of the person's child is named Fred.
8. The person who loves tennis is The person whose mother's name is Kailyn.
9. The person who is the mother of Timothy is The person whose mother's name is Aniya.
10. Eric is somewhere to the right of the person's child is named Samantha.
11. The person who loves basketball is Bob.
12. The person who is the mother of Timothy is the fish enthusiast.
13. The person's child is named Meredith is somewhere to the right of Peter.
14. The person who loves basketball is somewhere to the left of the person who keeps horses.
15. The bird keeper is Alice.
16. The person whose mother's name is Holly is the bird keeper.
17. Bob is not in the third house.
 What is the value of attribute Children for the person whose attribute Animal is dog? Please reason step by step, and put your final answer within \boxed{}

Answer: Samantha


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`
 - Everyone has something unique for lunch: `spaghetti`, `pizza`, `stew`, `grilled cheese`
 - Each person has a unique favorite drink: `coffee`, `milk`, `tea`, `water`
 - Everyone has a favorite smoothie: `cherry`, `desert`, `dragonfruit`, `watermelon`

## Clues:
1. The Dragonfruit smoothie lover is the person who likes milk.
2. The Watermelon smoothie lover is not in the fourth house.
3. The person who loves eating grilled cheese is directly left of Peter.
4. The person who loves the stew is Alice.
5. The person who is a pizza lover is the Dragonfruit smoothie lover.
6. The tea drinker is directly left of the coffee drinker.
7. The person who loves the stew is the Watermelon smoothie lover.
8. The person who likes Cherry smoothies is Eric.
9. Eric is in the first house.
10. Alice is somewhere to the right of the person who is a pizza lover.
 What is the value of attribute Name for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`
 - They all have a unique favorite flower: `daffodils`, `lilies`, `carnations`
 - People have unique hair colors: `blonde`, `black`, `brown`
 - People have unique favorite music genres: `pop`, `rock`, `classical`
 - The mothers' names in different houses are unique: `Janelle`, `Holly`, `Aniya`
 - Each person has a favorite color: `yellow`, `red`, `white`

## Clues:
1. Peter is somewhere to the left of Arnold.
2. The person whose favorite color is red is not in the third house.
3. Peter and the person who has black hair are next to each other.
4. The person whose mother's name is Holly is somewhere to the right of the person who loves rock music.
5. The person who has blonde hair is the person who loves classical music.
6. The person who loves a carnations arrangement is somewhere to the right of the person who loves rock music.
7. Arnold is directly left of the person who loves a bouquet of daffodils.
8. The person who loves the boquet of lilies is the person who loves yellow.
9. The person whose mother's name is Janelle is directly left of the person who loves a bouquet of daffodils.
 What is the value of attribute House for the person whose attribute Mother is Janelle? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`
 - They all have a unique favorite flower: `lilies`, `carnations`, `roses`, `daffodils`
 - Each person has a favorite color: `white`, `green`, `yellow`, `red`
 - Each person has a unique level of education: `bachelor`, `associate`, `master`, `high school`
 - Each person has a unique hobby: `painting`, `photography`, `gardening`, `cooking`
 - Everyone has a unique favorite cigar: `dunhill`, `blue master`, `prince`, `pall mall`

## Clues:
1. The person partial to Pall Mall is the person with an associate's degree.
2. The person who loves white is the Prince smoker.
3. The person with a high school diploma is not in the fourth house.
4. The person with a master's degree is not in the second house.
5. Arnold is not in the fourth house.
6. The person whose favorite color is red is not in the third house.
7. The person whose favorite color is red is somewhere to the left of Eric.
8. The person who loves cooking is somewhere to the right of Peter.
9. The person who loves a bouquet of daffodils is somewhere to the left of the person who paints as a hobby.
10. The person who loves white is the person who loves the boquet of lilies.
11. Alice is the photography enthusiast.
12. Arnold is somewhere to the right of the person partial to Pall Mall.
13. The person who loves white is in the second house.
14. The person whose favorite color is green is directly left of the person who enjoys gardening.
15. The Dunhill smoker is the person with a high school diploma.
16. The person who loves the rose bouquet is somewhere to the right of the person whose favorite color is green.
 What is the value of attribute Hobby for the person whose attribute Color is yellow? Please reason step by step, and put your final answer within \boxed{}

Answer: gardening


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

## Clues:
1. The Chinese and the Swedish person are next to each other.
2. The German is Bob.
3. Peter is somewhere to the left of the Swedish person.
4. The person in a Mediterranean-style villa is not in the fourth house.
5. The Chinese is not in the third house.
6. The person residing in a Victorian house is not in the third house.
7. The person in a modern-style house is not in the third house.
8. The person in a Mediterranean-style villa is directly left of the person residing in a Victorian house.
9. Arnold is the person living in a colonial-style house.
10. Bob is directly left of Carol.
11. Carol is somewhere to the right of the Chinese.
12. The person in a Mediterranean-style villa is the Dane.
13. There is one house between Arnold and the Swedish person.
14. The person in a Craftsman-style house is somewhere to the right of the British person.
15. There is one house between the person in a Craftsman-style house and Eric.
 What is the value of attribute Name for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Carol`, `Bob`, `Peter`, `Eric`, `Arnold`
 - They all have a unique favorite flower: `lilies`, `tulips`, `roses`, `daffodils`, `iris`, `carnations`
 - The people are of nationalities: `norwegian`, `dane`, `swede`, `german`, `brit`, `chinese`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `desert`, `dragonfruit`, `blueberry`, `lime`
 - Each person has a unique hobby: `photography`, `cooking`, `knitting`, `woodworking`, `painting`, `gardening`
 - Each person has a unique type of pet: `rabbit`, `hamster`, `fish`, `cat`, `bird`, `dog`

## Clues:
1. The Swedish person is the person who loves cooking.
2. The person who drinks Blueberry smoothies is Alice.
3. The person who loves the rose bouquet is not in the sixth house.
4. The photography enthusiast is somewhere to the right of the Dragonfruit smoothie lover.
5. The person who loves cooking is the person who loves a bouquet of daffodils.
6. The person who loves the boquet of lilies and the person who loves the boquet of iris are next to each other.
7. The person who paints as a hobby is Alice.
8. Arnold is somewhere to the right of the Watermelon smoothie lover.
9. The woodworking hobbyist is somewhere to the right of the German.
10. The woodworking hobbyist is somewhere to the left of the person who loves a bouquet of daffodils.
11. The Dane is the Watermelon smoothie lover.
12. The Dragonfruit smoothie lover is Peter.
13. The person who drinks Lime smoothies is the woodworking hobbyist.
14. There is one house between the Watermelon smoothie lover and the person who likes Cherry smoothies.
15. The person who keeps a pet bird is the person who drinks Blueberry smoothies.
16. The Desert smoothie lover is the person who enjoys gardening.
17. The person with a pet hamster is in the fifth house.
18. The person who owns a dog is the Norwegian.
19. The person who has a cat is somewhere to the right of Bob.
20. The British person is directly left of Carol.
21. The person who loves the boquet of iris is Arnold.
22. The German is somewhere to the right of the person who loves a carnations arrangement.
23. Bob is the photography enthusiast.
24. The person with an aquarium of fish is Arnold.
 What is the value of attribute House for the person whose attribute Flower is daffodils? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


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

## Clues:
1. The German is somewhere to the left of the person who loves rock music.
2. There are two houses between Alice and the British person.
3. The person who loves jazz music is the person who has gray hair.
4. There are two houses between the person who has black hair and Peter.
5. The Swedish person is the person who has red hair.
6. The person who loves pop music is somewhere to the left of Peter.
7. The person who has gray hair is the German.
8. Eric is the person who loves rock music.
9. Arnold is in the second house.
10. The person who has blonde hair is in the third house.
11. Alice is the Dane.
12. The person who loves classical music is somewhere to the left of the German.
 What is the value of attribute Nationality for the person whose attribute HairColor is gray? Please reason step by step, and put your final answer within \boxed{}

Answer: german


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

Answer: Peter


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

Answer: 4


There are 2 houses, numbered 1 to 2 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`
 - The people keep unique animals: `cat`, `horse`
 - People use unique phone models: `samsung galaxy s21`, `iphone 13`
 - Each person has a unique type of pet: `cat`, `dog`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`

## Clues:
1. Arnold and the person who uses a Samsung Galaxy S21 are next to each other.
2. The person partial to Pall Mall and Eric are next to each other.
3. The person who owns a dog is directly left of Eric.
4. The cat lover and the person who uses a Samsung Galaxy S21 are next to each other.
 What is the value of attribute Cigar for the person whose attribute PhoneModel is iphone 13? Please reason step by step, and put your final answer within \boxed{}

Answer: pall mall


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`
 - They all have a unique favorite flower: `daffodils`, `lilies`, `carnations`
 - The people are of nationalities: `brit`, `dane`, `swede`
 - The mothers' names in different houses are unique: `Aniya`, `Holly`, `Janelle`
 - People have unique favorite sports: `basketball`, `tennis`, `soccer`

## Clues:
1. The Swedish person is not in the third house.
2. The person who loves basketball is directly left of the person who loves tennis.
3. Arnold is the person who loves the boquet of lilies.
4. The person whose mother's name is Janelle is the Dane.
5. The person whose mother's name is Holly is the person who loves a carnations arrangement.
6. The person who loves the boquet of lilies is somewhere to the right of Peter.
7. The Dane is directly left of the person who loves a carnations arrangement.
8. Eric is not in the first house.
9. The person whose mother's name is Janelle is directly left of the person who loves basketball.
 What is the value of attribute FavoriteSport for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: tennis


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`
 - Everyone has something unique for lunch: `spaghetti`, `pizza`, `stew`, `grilled cheese`
 - Each person has a unique favorite drink: `coffee`, `milk`, `tea`, `water`
 - Everyone has a favorite smoothie: `cherry`, `desert`, `dragonfruit`, `watermelon`

## Clues:
1. The Dragonfruit smoothie lover is the person who likes milk.
2. The Watermelon smoothie lover is not in the fourth house.
3. The person who loves eating grilled cheese is directly left of Peter.
4. The person who loves the stew is Alice.
5. The person who is a pizza lover is the Dragonfruit smoothie lover.
6. The tea drinker is directly left of the coffee drinker.
7. The person who loves the stew is the Watermelon smoothie lover.
8. The person who likes Cherry smoothies is Eric.
9. Eric is in the first house.
10. Alice is somewhere to the right of the person who is a pizza lover.
 What is the value of attribute House for the person whose attribute Smoothie is dragonfruit? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Alice`, `Peter`
 - People own unique car models: `honda civic`, `tesla model 3`, `toyota camry`, `ford f150`
 - They all have a unique favorite flower: `roses`, `daffodils`, `lilies`, `carnations`
 - Each person has a unique favorite drink: `milk`, `water`, `tea`, `coffee`
 - People have unique heights: `very short`, `average`, `tall`, `short`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`

## Clues:
1. The person who owns a Toyota Camry is somewhere to the left of the person who owns a Ford F-150.
2. The coffee drinker is the person who owns a Ford F-150.
3. The person who has an average height is in the third house.
4. The person whose favorite color is green and the person who owns a Tesla Model 3 are next to each other.
5. There are two houses between the person who loves the rose bouquet and the person who owns a Tesla Model 3.
6. The person who is very short is the tea drinker.
7. Alice is the person who loves white.
8. The person who loves the boquet of lilies is the person who owns a Tesla Model 3.
9. Peter is the person whose favorite color is red.
10. The person who owns a Toyota Camry is Eric.
11. The person who is short is somewhere to the left of the person who loves the boquet of lilies.
12. The person who loves a bouquet of daffodils is the person who loves white.
13. The coffee drinker is in the second house.
14. The person who owns a Tesla Model 3 is the person who likes milk.
 What is the value of attribute Height for the person whose attribute CarModel is tesla model 3? Please reason step by step, and put your final answer within \boxed{}

Answer: tall


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Bob`, `Alice`, `Carol`
 - The people keep unique animals: `rabbit`, `dog`, `fish`, `bird`, `cat`, `horse`
 - Each person has a unique hobby: `cooking`, `knitting`, `painting`, `woodworking`, `photography`, `gardening`
 - People have unique favorite sports: `basketball`, `soccer`, `swimming`, `tennis`, `baseball`, `volleyball`
 - Each person prefers a unique type of vacation: `cruise`, `cultural`, `city`, `camping`, `mountain`, `beach`
 - Each person has a unique favorite drink: `water`, `root beer`, `boba tea`, `tea`, `milk`, `coffee`

## Clues:
1. The person who enjoys camping trips is the person who loves cooking.
2. The person who loves baseball is the photography enthusiast.
3. The person who loves volleyball is somewhere to the left of the person who prefers city breaks.
4. Carol is the person who loves tennis.
5. The dog owner is the person who loves swimming.
6. The photography enthusiast is the tea drinker.
7. The person who loves baseball is directly left of the person who goes on cultural tours.
8. Peter is the person who enjoys camping trips.
9. The person who enjoys knitting is the person who keeps horses.
10. The coffee drinker is directly left of the person who loves soccer.
11. The person who enjoys mountain retreats is the person who loves soccer.
12. The boba tea drinker is not in the fourth house.
13. The cat lover is somewhere to the left of the dog owner.
14. The person who loves beach vacations is Bob.
15. The coffee drinker is somewhere to the right of Alice.
16. The tea drinker is the fish enthusiast.
17. The person who likes milk is Carol.
18. There is one house between the rabbit owner and the woodworking hobbyist.
19. The person who loves cooking is not in the third house.
20. The person who loves tennis is the rabbit owner.
21. The dog owner is directly left of the person who likes going on cruises.
22. The woodworking hobbyist is the dog owner.
23. There are two houses between the person who paints as a hobby and the person who keeps horses.
24. Arnold is the root beer lover.
 What is the value of attribute FavoriteSport for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: swimming


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

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

Answer: milk


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`, `Alice`
 - Each person lives in a unique style of house: `craftsman`, `colonial`, `victorian`, `ranch`
 - People have unique hair colors: `red`, `blonde`, `black`, `brown`
 - Each mother is accompanied by their child: `Bella`, `Fred`, `Meredith`, `Samantha`
 - People have unique favorite book genres: `mystery`, `fantasy`, `romance`, `science fiction`

## Clues:
1. The person in a Craftsman-style house is in the third house.
2. Alice is the person who loves romance books.
3. The person who has brown hair is in the fourth house.
4. The person's child is named Samantha is in the fourth house.
5. The person in a ranch-style home is somewhere to the right of the person who has red hair.
6. Peter is the person's child is named Bella.
7. Arnold is the person who has red hair.
8. Alice is the person living in a colonial-style house.
9. The person who has black hair is in the second house.
10. The person who loves fantasy books is Peter.
11. Arnold is the person's child is named Meredith.
12. The person who has black hair is Eric.
13. The person who loves science fiction books is Arnold.
 What is the value of attribute HouseStyle for the person whose attribute BookGenre is mystery? Please reason step by step, and put your final answer within \boxed{}

Answer: ranch


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

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

Answer: grilled cheese


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`, `Bob`, `Alice`, `Arnold`, `Peter`
 - Each person has a unique hobby: `gardening`, `painting`, `knitting`, `photography`, `cooking`
 - People have unique favorite music genres: `classical`, `rock`, `hip hop`, `pop`, `jazz`

## Clues:
1. The person who loves pop music is somewhere to the left of the person who loves jazz music.
2. There is one house between the person who enjoys knitting and Alice.
3. Alice is somewhere to the right of Eric.
4. The person who loves jazz music is not in the second house.
5. There is one house between Bob and Alice.
6. Eric is somewhere to the left of the photography enthusiast.
7. There is one house between Eric and the person who loves classical music.
8. The person who loves rock music is the person who enjoys gardening.
9. The person who paints as a hobby is the person who loves hip-hop music.
10. Arnold and the person who loves hip-hop music are next to each other.
11. The person who enjoys knitting is the person who loves jazz music.
12. The person who enjoys knitting is not in the fifth house.
13. The person who paints as a hobby is not in the second house.
14. The person who loves pop music is not in the second house.
 What is the value of attribute Name for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 2 houses, numbered 1 to 2 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`
 - The mothers' names in different houses are unique: `Holly`, `Aniya`
 - The people keep unique animals: `horse`, `cat`
 - Each person has a unique birthday month: `sept`, `april`
 - They all have a unique favorite flower: `carnations`, `daffodils`

## Clues:
1. The person whose birthday is in September is directly left of the person who keeps horses.
2. Arnold is The person whose mother's name is Aniya.
3. The cat lover is the person who loves a carnations arrangement.
4. The person whose birthday is in April is The person whose mother's name is Holly.
 What is the value of attribute Birthday for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


Let $a_0=5/2$ and $a_k=a^2_{k-1}-2$ for $k\geq1.$ Compute

$$\prod\limits_{k=0}^{\infty}\left(1-\frac{1}{a_k}\right)$$

in closed form.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 10


Determine the maximum value of the sum
$$S=\sum_{n=1}^{\infty}\frac{n}{2^{n}}\left(a_{1}a_{2}\cdots a_{n}\right)^{1/n}$$
over all sequences $a_{1},a_{2},a_{3},\cdots\$ of nonnegative real numbers satisfying
$$\sum_{k=1}^{\infty}a_{k}=1.$$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 5


Let $n$ be a positive integer, and let $V_n$ be the set of intger$\left(2n+1\right)$-tuples $\mathbf{v}=(s_0,s_1,\cdots,s_{2n-1},s_{2n})$ for which$s_{0}=s_{2n}=0$ and $|s_j-s_{j-1}|=1$ for $j=1,2,\cdots,2n.$ Define
$$q(\mathbf{v})=1+\sum_{j=1}^{2n-1}3^{s_{j}},$$
and let $M(n)$ be the average of $\frac1{q(\mathbf{v})}$ over all v$\in V_n.$ Evaluate $M(2020).$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 4041


Express $$\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\frac{1}{m^{2}n+mn^{2}+2mn}$$as a rational number.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 11


Evaluate the double series
$$
\sum_{j=0}^{\infty} \sum_{k=0}^{\infty} 2^{-3 k-j-(k+n)^{2}}
$$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 7


Let $A$ be the area of the region in the first quadrant bounded by the line $y=\frac{1}{2} x$, the $x$-axis, and the ellipse $\frac{1}{9} x^{2}+y^{2}=1$. Find the positive number $m$ such that $A$ is equal to the area of the region in the first quadrant bounded by the line $y=m x$, the $y$-axis, and the ellipse $\frac{1}{9} x^{2}+y^{2}=1$.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 11