In the figure, $ABCD$ is a square, $P$, $Q$, $R$, and $S$ are midpoints of the sides $AB$, $BC$, $CD$ and $DA$ respectively. Find the ratio of the shaded area to the area of the square $ABCD$.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 6


On a given circle, six points $A$, $B$, $C$, $D$, $E$, and $F$ are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles $ABC$ and $DEF$ are disjoint, i.e., have no common points.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 13


Today, Ivan the Confessor prefers continuous functions $f:[0,1] \rightarrow \mathbb{R}$ satisfying $f(x)+f(y) \geq|x-y|$ for all pairs $x, y \in[0,1]$. Find the minimum of $\int_{0}^{1} f$ over all preferred functions.
The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 5


Given real numbers $b_0,b_1,\ldots,b_{2019}$ with $b_{2019}\neq0$, let $z_1,z_2,\ldots,z_{2019}$ be the roots in the complex plane of the polynomial $$P(z)=\sum_{k=0}^{2019}b_kz^k.$$ Let $\mu=(|z_{1}|+\cdots+|z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\ldots,z_{2019}$ to the origin. Determine the largest constant $M$ such that $\mu\geq M$ for all choices of $b_0,b_1,\ldots,b_{2019}$ that satisfy $$1\leq b_0<b_1<b_2<\cdots<b_{2019}\leq2019.$$ The original answer is in the form $M=a^{b/ c}$, please provide the value of a+ b+c .

Answer: 4037


For each real number $x$, let
$$f(x)=\sum_{n\in S_x}\frac{1}{2^n},$$
where $S_x$ is the set of positive integers $n$ for which $\lfloor nx\rfloor$ is even. What is the largest real number $L$ such that $f(x) ≥ L$ for all $x ∈ [0,1)?$ The original answer is in the format \frac{m}{n}, please provide the value of m + n.

Answer: 11


Suppose that $X_1,X_2,\ldots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S= \sum _{i= 1}^{k}X_{i}/ 2^{i}$, where $k$ is the least positive integer such that $X_k<X_{k+1}$, or $k=\infty$ if there is no such integer. Find the expected value of S.  The original answer is in $ae^{b / c}-d$ format, please provide the value of a+b+c+d.

Answer: 8


Let $p$ be an odd prime number, and let ${\Bbb {F}}_p$ denote the field of integers modulo $p$. Let ${\Bbb {F}}_p[x]$ be the ring of polynomials over ${\Bbb {F}}_p$, and let $q(x)\in {\Bbb {F}}_p[x]$ be given by
$$q(x)=\sum\limits_{k=1}^{p-1}a_kx^k,$$
where
$$a_k=k^{(p-1)/2}\mod p.$$
Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in ${\Bbb {F}}_p[x]$. What is the answer when p = 3?

Answer: 1


Find the smallest constant C such that for every real polynomial $P(x)$ of degree 3 that has a root in the interval $[0,1]$, $$\int_{0}^{1} \left\vert P(x)\right\vert\, {\rm d}x\leq C\max _{x \in[0,1]} \left\vert P(x) \right\vert$$. The original answer is in \frac{m}{n} format, please give the value of m + n.

Answer: 11


The 30 edges of a regular icosahedron are distinguished by labeling them $1,2,…,30$. How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color? The standard answer format is $a^{b} c^{d}$. Please give the value of a + b + c + d.

Answer: 35


A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. The original answer is in the form \frac{a\sqrt{b}- c}{d}, please provide the value of a + b + c + d.

Answer: 14


Let $p$ be an odd prime and let $\mathbf{Z}_p$ denote (the field of) the integers modulo $p$. How many elements are in the set $$ \left\{x^2: x \in \mathbf{Z}_p\right\} \cap\left\{y^2+1: y \in \mathbf{Z}_p\right\} ? $$ Please provide the number of elements in the intersection when p = 13.

Answer: 4


Let $a$ and $b$ be positive numbers. Find the largest number $c$, in terms of $a$ and $b$, such that $$a^xb^{1-x}\leq a\frac{\sinh ux}{\sinh u}+b\frac{\sinh u(1-x)}{\sinh u}$$ for all $u$ with $0<|u| ≤c$ and for all $x,0<x<1$. (Note: $\sinh u= (e^u- e^{-u})/2.$) Please provide the sum of the bounds of $u$ when $|u| = 1 and a=b=1$.

Answer: 2


10. Given the function $f(x)=x^{3}-a x^{2}+\left(a^{2}-2\right) x+1$, if there exists $m>0$, such that $f(m) \leqslant 0$, then the maximum value of the real number $a$ is $\qquad$ .

Answer: 1


## Task B-1.5.

In a right-angled triangle $A B C$, with a right angle at vertex $C$, the length of the hypotenuse is 12. Squares $A B D E$ and $A C G F$ are constructed outward on sides $\overline{A B}$ and $\overline{A C}$. If points $D, E$, $F$ and $G$ lie on the same circle, calculate the perimeter of triangle $A B C$.

Answer: 12 + 12\sqrt{2}


Toner Drum and Celery Hilton are both running for president. A total of $2015$ people cast their vote, giving $60\%$ to Toner Drum. Let $N$ be the number of "representative'' sets of the $2015$ voters that could have been polled to correctly predict the winner of the election (i.e. more people in the set voted for Drum than Hilton). Compute the remainder when $N$ is divided by $2017$.

[i] Proposed by Ashwin Sah [/i]

Answer: 605


Given an infinite sequence $\left\{a_{n}\right\}$ where all terms are positive integers, and the sum of any consecutive terms is not equal to 100. Find the minimum value of $\max \left\{a_{n}, n \in \mathbf{N}\right\}$.

Answer: 3


16 Given that the eccentricity of an ellipse is $e$, and the two foci are $F_{1}$ and $F_{2}$, the parabola $C$ has $F_{1}$ as its vertex and $F_{2}$ as its focus. Point $P$ is an intersection point of the parabola and the ellipse. If $e\left|P F_{2}\right|=\left|P F_{1}\right|$, then $e=$ $\qquad$.

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


1. There are fewer than 500 children on the playground. The percentage of boys among all children equals the number of all girls. How many boys and how many girls are on the playground? Find all possible solutions.

Answer: (5, 20)


A graph has 8 points and no squares (a square is 4 distinct points A, B, C, D with A and C both joined to B and D). What is the largest number of edges it can have?

Answer: 11


28. Suppose $a \neq 0, b \neq 0, c \neq 0$ and $\frac{0}{b}=\frac{b}{c}=\frac{1}{a}$. Find the value of $\frac{a+b-c}{a-b+c}$.

Answer: 1


Answer: \left(\sum_{i=1}^{n} c_{i}\right)^{2} \leqslant\left(\sum_{i=1}^{n} a_{i}^{2}\right)\left(\sum_{i=1}^{n} b_{i}^{2}\right)


A positive integer $N$ is given. Panda builds a tree on $N$ vertices, and writes a real number on each vertex, so that $1$ plus the number written on each vertex is greater or equal to the average of the numbers written on the neighboring vertices. Let the maximum number written be $M$ and the minimal number written $m$. Mink then gives Panda $M-m$ kilograms of bamboo. What is the maximum amount of bamboo Panda can get?

Answer: (N-1)^2


The sum of positive numbers \( x_{1}, x_{2}, \ldots, x_{n} \) is equal to 1. Let \( s \) be the largest of the numbers

\[
\frac{x_{1}}{1+x_{1}}, \frac{x_{2}}{1+x_{1}+x_{2}}, \ldots, \frac{x_{n}}{1+x_{1}+x_{2}+\ldots+x_{n}}
\]

Find the smallest possible value of \( s \). For which values of \( x_{1}, x_{2}, \ldots, x_{n} \) is this value attained?

Answer: 1 - 2^{-1/n}


6. As shown in the figure, in $\triangle A B C$, points $D, E, F$ are on $B C, C A, A B$ respectively, and $C D=\frac{3}{5} B C, E C=\frac{1}{2} A C$, $A F=\frac{1}{3} A B$. Let $P$ be a point inside quadrilateral $A E D F$ (point $P$ is not on the boundary). If $\overrightarrow{D P}=-\frac{1}{3} \overrightarrow{D C}+x \overrightarrow{D E}$, then the range of the real number $x$ is . $\qquad$

Answer: \left( \dfrac{1}{2}, \dfrac{4}{3} \right)


Two sisters were picking currants: the older one was using a 4-liter bucket, and the younger one was using a 3.5-liter bucket. The older sister was always working faster than the younger one. When the older sister had collected three-quarters of her bucket, and the younger sister had collected more than half of hers, the girls exchanged buckets. Continuing to work at the same speeds as before exchanging buckets, they finished at the same time. How many times faster was the older sister picking berries compared to the younger sister?

Answer: \frac{3}{2}


A line $g$ is given in a plane. $n$ distinct points are chosen arbitrarily from $g$ and are named as $A_1, A_2, \ldots, A_n$. For each pair of points $A_i,A_j$, a semicircle is drawn with $A_i$ and $A_j$ as its endpoints. All semicircles lie on the same side of $g$. Determine the maximum number of points (which are not lying in $g$) of intersection of semicircles as a function of $n$.

Answer: \dfrac{n(n-1)(n-2)(n-3)}{24}


Exercise 15. For a function $f$ and $n$ a strictly positive integer, we denote $f^{n}$ the $n$-th composition of $f$ defined by $f^{n}=f \circ f \circ \ldots \circ f$ where $f$ appears $n$ times on the right-hand side.

Find all functions $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}$ such that

$$
f^{f(n)}(m)+m n=f(m) f(n)
$$

for all $m, n \in \mathbb{N}^{*}$.

Answer: f(n) = n + 1


8.1. Determine all natural numbers of the form $\overline{a b c d}$ that are divisible by 3 and simultaneously satisfy the conditions: $a+b+d=11, a+c+d=12, b+c+d=10$.

Answer: 2019


A nine-digit natural number \(A\), written in the decimal system, is obtained from a number \(B\) by moving its last digit to the first position. It is known that the number \(B\) is coprime with the number 24 and \(B > 666666666\). Find the largest and smallest numbers \(A\) that satisfy these conditions. (Two natural numbers are coprime if they have no common divisors other than one).

Answer: 166666667


XLVI OM - III - Problem 2

The diagonals of a convex pentagon divide this pentagon into a pentagon and ten triangles. What is the maximum possible number of triangles with equal areas?

Answer: 6


Shvecov D.V.

On the hypotenuse $AC$ of the right triangle $ABC$, a point $C_{1}$ is marked such that $BC = CC_{1}$. Then, on the leg $AB$, a point $C_{2}$ is marked such that

$AC_{2} = AC_{1}$; the point $A_{2}$ is defined similarly. Find the angle $AMC$, where $M$ is the midpoint of the segment $A_{2}C_{2}$.

Answer: 135^\circ


8,9 $[\quad$ Inscribed angle, based on the diameter

In triangle $A B C$, it is known that $\angle B A C=\alpha, A C=b$. The inscribed circle touches the sides $A B$ and $B C$ at points $M$ and $N$, the bisector of angle $B A C$ intersects the line $M N$ at point $K$. Find the distance from point $K$ to the line $A C$.

Answer: \dfrac{b \sin \alpha}{2}


Find and prove all such positive integers expressed in base \( n \) where all digits are different, and, excluding the left-most digit, each digit differs by exactly 1 from some digit to its left. Express the answer as a function of \( n \) in its simplest form.

Answer: 2^{n+1} - 2n - 2


## Task A-3.3.

On a board of dimensions $20 \times 19$, tiles of dimensions $3 \times 1$ are placed such that they cover exactly three squares of the board, and they do not overlap or touch each other, not even at the corners.

Determine the maximum possible number of $3 \times 1$ tiles on this board.

Answer: 50


5. There are 6 natural numbers that have the same remainder when divided by 12, and their product is 971425. Then the minimum value of the sum of these 6 natural numbers is $\qquad$ .

Answer: 150


Twelve participants are taking part in a chess tournament. After the tournament, each participant makes 12 lists as follows:
- In the 1st list, each participant writes down only their own name.
- In the 2nd list, each participant writes down their own name and the names of all the opponents they defeated.
- In the 3rd list, each participant writes down all the names from their 2nd list as well as the names of all the opponents defeated by anyone in the 2nd list.
- This process continues until the 12th list, where each participant writes down all the names from their 11th list and the names of all the opponents defeated by anyone in the 11th list.

It is known that for each participant, their 12th list contains names that did not appear in their 11th list. How many draws were there in the tournament?

Answer: 54


B3. On each of the twelve edges of a cube, we write the number 1 or -1. We then multiply the four numbers on the edges of each face of the cube and write the result on that face. Finally, we add up the eighteen written numbers.

What is the smallest (most negative) result we can get this way? In the figure, you see an example of such a cube. The numbers on the back of the cube are not visible here.

![](https://cdn.mathpix.com/cropped/2024_04_17_d68bed7f77e1173b56f6g-1.jpg?height=360&width=354&top_left_y=2213&top_left_x=1522)

Answer: -12


63rd Putnam 2002 Problem A6 b > 1 is an integer. For any positive integer n, let d(n) be the number of digits in n when it is written in base b. Define the sequence f(n) by f(1) = 1, f(2) = 2, f(n) = n f( d(n) ). For which values of b does 1/f(1) + 1/f(2) + 1/f(3) + ... converge? Solution

Answer: 2


3. Let $X=\{0,1,2,3,4,5,6,7,8,9\}$. Let $S \subseteq X$ be such that any nonnegative integer $n$ can be written as $p+q$ where the nonnegative integers $p, q$ have all their digits in $S$. Find the smallest possible number of elements in $S$.

Answer: 5


$1 \cdot 15$ Given 5 distinct positive numbers can be divided into two groups such that the sums of the numbers in each group are equal, how many different ways are there to divide these numbers into such groups?

The above text translated into English, keeping the original text's line breaks and format, is as follows:
$1 \cdot 15$ Given 5 distinct positive numbers can be divided into two groups such that the sums of the numbers in each group are equal, how many different ways are there to divide these numbers into such groups?

Answer: 1


Example 3. Find the translation trajectory of $(1,2)$ with respect to $\left(t, t^{2}\right)$.

Answer: y = x^2 - 2x + 3


## Task 1 - 170711

Matthias was in an international pioneer tent camp during the summer holidays. He reports to his classmates:

"A quarter of all participants and four pioneers came from the Soviet Union, a fifth of all participants and five pioneers from the GDR, a sixth of all participants and six pioneers from Czechoslovakia, an eighth of all participants and eight pioneers from the People's Republic of Poland, a ninth of all participants and nine pioneers from the People's Republic of Bulgaria. The remaining 21 pioneers came from the Hungarian People's Republic. In each tent of the camp, there were exactly eight pioneers."

Determine the number of tents in the camp from these details!

Answer: 45


Example 5 Suppose we have a $4 \times 4$ grid where each cell is colored differently. Each cell is filled with either 0 or 1, such that the product of the numbers in any two adjacent cells is 0. How many different ways are there to fill the grid with numbers?

(2007, Korean Mathematical Olympiad)

Answer: 1234


Under the new AMC $10, 12$ scoring method, $6$ points are given for each correct answer, $2.5$ points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between $0$ and $150$ can be obtained in only one way, for example, the only way to obtain a score of $146.5$ is to have $24$ correct answers and one unanswered question. Some scores can be obtained in exactly two ways, for example, a score of $104.5$ can be obtained with $17$ correct answers, $1$ unanswered question, and $7$ incorrect, and also with $12$ correct answers and $13$ unanswered questions. There are (three) scores that can be obtained in exactly three ways. If the sum of these scores is $x$, what is the value of $10x$?

Answer: 1885


Six cubes, each an inch on an edge, are fastened together, as shown. Find the total surface area in square inches, including the top, bottom, and sides.

Answer: 26


Let \(a\) and \(b\) be real numbers such that \(\sin{a} + \sin{b} = \frac{\sqrt{2}}{2}\) and \(\cos {a} + \cos {b} = \frac{\sqrt{6}}{2}.\) The original answer is in the form \(\frac{\sqrt{k}}{m}\). Please find the value of \(k + m\).

Answer: 5


Points \(K, L, M,\) and \(N\) lie in the plane of the square \(ABCD\) such that \(AKB\), \(BLC\), \(CMD\), and \(DNA\) are equilateral triangles. If \(ABCD\) has an area of \(16\), find the area of \(KLMN\). The original answer is in the form \(k + m\sqrt{3}\). Please give the value of \(k + m\).

Answer: 48


A point \(P\) is chosen at random in the interior of equilateral triangle \(ABC\). What is the probability that \(\triangle ABP\) has a greater area than each of \(\triangle ACP\) and \(\triangle BCP\)? Express your answer in the form \(\frac{k}{m}\), where \(k\) and \(m\) are coprime positive integers. Find the value of \(k + m\).

Answer: 4


Three semicircles of radius \(1\) are constructed on diameter \(\overline{AB}\) of a semicircle of radius \(2\). The centers of the small semicircles divide \(\overline{AB}\) into four line segments of equal length, as shown. Find the area of the shaded region that lies within the large semicircle but outside the smaller semicircles. The original answer is in the form \(\frac{k}{m}\pi - \frac{\sqrt{n}}{p}\). Please calculate the value of \(k + m + n + p\).

Answer: 18


At the "Lukomorye" station, they sell cards for one, five, and twenty rides. All cards cost a whole number of gold coins. Five cards for one ride are more expensive than one card for five rides, and four cards for five rides are more expensive than one card for twenty rides. It turned out that the cheapest way for 33 bogatyrs to travel is to buy cards for 35 rides, spending 33 gold coins on this. How much does a card for five rides cost?

Answer: 5


The altitudes of the acute-angled triangle \(ABC\) intersect at point \(H\). It is known that \(AH^{2} = BH^{2} + CH^{2}\). On the circumcircle of triangle \(ABC\), there are points \(D\) and \(E\) such that \(CE \parallel AB\) and \(BD \parallel AC\). Prove that point \(H\) lies on the line \(DE\).

Answer: H \text{ lies on the line } DE


In the 1448th exercise, let the circles centered at $A$ and $B$ be denoted by $k_{1}$ and $k_{2}$, their radii by $r_{1}$ and $r_{2}$, respectively, and the radius of the circle $k$ that touches $k$ from the inside and $k_{1}$ and $k_{2}$ from the outside by $r_{3}$. What does the ratio $r_{3} / r_{1}$ tend to, if - with fixed $A, B$ - $C$ tends to $A$, that is, $r_{1}$ tends to 0?

Answer: \dfrac{1}{2}


Problem 6. For which integers $m$ and $n$ does the equality

$$
(5+3 \sqrt{2})^{m}=(3+5 \sqrt{2})^{n} ?
$$

hold?

Answer: m = 0


Problem 6. Inside the magician's hat, there live 100 rabbits: white, blue, and green. It is known that if 81 rabbits are randomly pulled out of the hat, there will definitely be three of different colors among them. What is the minimum number of rabbits that need to be taken out of the hat to ensure that there are definitely two of different colors?

#

Answer: 61


9. In 1978, someone introduced their family by saying: "I have a son and a daughter, they are not twins. The cube of my son's age plus the square of my daughter's age is exactly my birth year. I was born after 1900. My children are both under 21 years old. I am 8 years older than my wife." Calculate the age of each family member in 1978. (15 points)

Answer: 12


The turbo multiple of a natural number is the smallest multiple of that number composed only of the digits 0 and 1. For example, the turbo multiple of 2, as well as for 5, is 10; the turbo multiple of 3 is 111, and for 110 it is itself.

a) What is the turbo multiple of 20?

b) What is the turbo multiple of 9?

c) What is the turbo multiple of 45?

d) What is the smallest natural number whose turbo multiple is 1110?

Answer: 100


Task 2 - 331032
For each positive integer \(n\), imagine another number \(n^{\prime}\) formed according to the following rule: From the digit representation of \(n\) in the decimal system, the first digit is removed and appended instead after the last digit. Then let \(n^{\prime}\) be the number with the resulting digit representation. (In this digit representation of \(n^{\prime}\), the possibility of a leading zero is also allowed if the second digit of \(n\) was a zero.)

Investigate whether there are numbers \(n\) divisible by 7 for which \(n^{\prime}=n: 7\) holds!

Determine, if such numbers exist, the smallest among them!


8.3. Find the angle $D A C$, given that $A B=B C$ and $A C=C D$, and the lines on which points $A, B, C, D$ lie are parallel, with the distances between adjacent lines being equal. Point $A$ is to the left of $B$, $C$ is to the left of $B$, and $D$ is to the right of $C$ (see figure).

![](https://cdn.mathpix.com/cropped/2024_05_06_ee373ffab09e7a30d5bdg-2.jpg?height=325&width=488&top_left_y=397&top_left_x=687)

Answer: 30^\circ


8. Let the arithmetic sequence $\left\{a_{n}\right\}$ have all terms as integers, with the first term $a_{1}=2019$, and for any positive integer $n$, there always exists a positive integer $m$ such that $a_{1}+a_{2}+\cdots+a_{n}=a_{m}$. The number of such sequences $\left\{a_{n}\right\}$ is $\qquad$

Answer: 5


35. Cubes. As is known, the entire space can be filled with equal cubes. At each vertex, eight cubes will meet. Therefore, it is possible - by appropriately cutting off the vertices of the cubes and gluing the adjacent cut parts into one body - to fill the space with regular octahedra (eight-faced bodies) and the bodies remaining from the cubes. What will these bodies be? If we maximally increase the octahedra, what part of the space will they occupy?

Answer: \dfrac{1}{6}


10. (10 points) Two people, A and B, take turns selecting numbers from the integers 1 to 17, with the rule: they cannot select numbers that have already been chosen by either party, they cannot select a number that is twice an already chosen number, and they cannot select a number that is half of an already chosen number. The person who cannot select a number loses. Now, A has already chosen 8, and B wants to ensure a certain win. The number B should choose next is $\qquad$

---

Please note that the blank space at the end (indicated by $\qquad$) is part of the original format and has been retained in the translation.

Answer: 6


87. A thief on a bus gets off when the bus arrives at a station and walks in the opposite direction of the bus's movement. After the bus continues, a passenger realizes they have been stolen from and gets off at the next station to chase the thief. If the passenger's speed is twice that of the thief, the bus's speed is ten times that of the thief, and the bus takes 40 seconds to travel between stations, then the passenger will catch up to the thief after $\qquad$ seconds of chasing.

Answer: 440


3. Vovochka approached a slot machine, on the screen of which the number 0 was glowing. The game rules stated: «The screen shows the number of points. If you throw a 1-ruble coin, the number of points will increase by 1. If you throw a 2-ruble coin, the number of points will double. If you score 50 points, the machine will give a prize. If the number exceeds 50, all the points will be lost.» What is the minimum amount of rubles Vovochka can spend to get the prize? Answer: 11 rubles.

Answer: 11


15. (5 points) It is known that Xiaoming's father and mother are of different ages, and the difference in their ages does not exceed 10 years. If last year, this year, and next year, both father and mother's ages are integer multiples of Xiaoming's age, then Xiaoming is $\qquad$ years old this year.

Answer: 2


Let $n$ be a positive integer. Assume that $n$ numbers are to be chosen from the table

$$
\begin{array}{cccc}
0 & 1 & \cdots & n-1 \\
n & n+1 & \cdots & 2 n-1 \\
\vdots & \vdots & \ddots & \vdots \\
(n-1) n & (n-1) n+1 & \cdots & n^{2}-1
\end{array}
$$

with no two of them from the same row or the same column. Find the maximal value of the product of these $n$ numbers.

Answer: (n-1)^n \cdot n!


Planet Yvette orbits its sun in an elliptical path with the sun situated at one of the foci. The closest point of Yvette to its sun (perigee) is 3 astronomical units (AU), and its furthest point (apogee) is 15 AU away. Calculate the distance of Yvette from its sun when it is one-quarter way along its orbit from perigee towards apogee.

Answer: 5 \text{ AU}


The letters of the alphabet are assigned values based on the pattern $2, 1, 2, 0, -1, -2, -1, 0$, repeated throughout the alphabet. Calculate the sum of the numeric values of the letters in the word "dynamics".

Answer: 7


Answer: 7


A circle with radius 6 cm is tangent to two longer sides and one shorter side of a rectangle. The area of the rectangle is triple the area of the circle. What is the length of the shorter side of the rectangle, in centimeters?

Answer: 12 \text{ cm}


In a diagram, there are three smaller circles positioned inside a larger circle. Each smaller circle touches the other two smaller circles exactly at the center of the larger circle. The radius of the larger circle is 9. Calculate the area of the shaded region outside the smaller circles but inside the larger circle.

[asy]
size(100);
import graph;
filldraw(Circle((0,0),3),mediumgray);
filldraw(Circle((-1.5,0),1.5),white);
filldraw(Circle((1.5,0),1.5),white);
filldraw(Circle((0,1.5*sqrt(3)),1.5),white);
[/asy]

Answer: 20.25\pi


Three identical spherical balls, each with a diameter of 8 cm, are stacked in a pyramid-shaped crate, such that each ball touches the other two. Determine the total height, \( h \), from the bottom of the bottom ball to the top of the top ball.

Answer: h = 8 + 4\sqrt{3} \text{ cm}


Wendy is trying to locate the Fermat point \( P \) of \(\triangle ABC\), where \( A \) is at the origin, \( B \) is at \( (12,0) \), and \( C \) is at \( (5,8) \). She estimates that the point is at \( P = (5,3) \), and calculates the sum of the distances from \( P \) to the vertices of \(\triangle ABC\). If she obtains \( m\sqrt{a} + n\sqrt{b} \), where \( m \), \( n \), \( a \), and \( b \) are integers, what is \( m + n \)?

Answer: 2


In a shop, $36$ lemons are required to make $48$ gallons of lemonade. For lemon tea, $20$ lemons are needed to make $10$ gallons. How many lemons are necessary to make $6$ gallons of lemonade and $5$ gallons of lemon tea?

Answer: 14.5


Consider a shaded region that consists of 13 unit squares along the $x$-axis and $y$-axis. It is divided into two regions. The bottom region has height 4 units, and the top region is directly above it with a narrow strip extending to the right. The shaded region is rotated about the $x$-axis to form a solid. Determine the volume of the resulting solid in cubic units, expressed in terms of $\pi$.

[asy]
defaultpen(linewidth(0.8));
fill((0,0)--(0,4)--(1,4)--(1,1)--(6,1)--(6,0)--cycle, gray(0.7));
draw((0,0)--(0,4)--(1,4)--(1,1)--(6,1)--(6,0)--cycle);
draw((0,1)--(6,1));
draw((0,2)--(1,2));
draw((0,3)--(1,3));
draw((1,0)--(1,1));
draw((2,0)--(2,1));
draw((3,0)--(3,1));
draw((4,0)--(4,1));
draw((5,0)--(5,1));

draw((6,0)--(7,0),EndArrow(4));
draw((0,0)--(-1,0),EndArrow(4));
label("$x$", (7,0), E);

draw((0,4)--(0,5),EndArrow(4));
draw((0,0)--(0,-1),EndArrow(4));
label("$y$", (0,5), N);
[/asy]

Answer: 21\pi


A damaged portion of a mathematical worksheet hides the coefficient of $x$ in a quadratic equation. The equation is known to have two distinct positive, integer solutions. Determine the sum of all possible integers that could be obscured by the damage.

[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
fill((0,0)--(3,0)--(3,3)--(0,3)--cycle,black);
label("$x^2+$",(1.5,1.5),W);
label("$=24$",(3,1.5),E);
[/asy]

Answer: 60


Circle $\Gamma$ is the circumcircle of $\triangle ABC$ and also serves as the incircle of $\triangle XYZ$. $X$, $Y$, and $Z$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively. If $\angle A = 50^\circ$, $\angle B = 70^\circ$, and $\angle C = 60^\circ$, find the measure of $\angle YZX$.

Answer: 60^\circ


In triangle $PQR$, the angle bisectors are $PS$, $QT$, and $RU$, which converge at the incenter $I$. Given that $\angle PRQ = 42^\circ$, find the measure of $\angle QIS$, in degrees.

Answer: 69^\circ


Lucy has a penchant for strawberry milk and decides to visit a local café for a fortnight to indulge in her favorite beverage. This café, similar to a milk bottling plant, occasionally serves caramel milk instead, with a 3/4 chance on any given day of serving strawberry milk. What is the probability that Lucy gets to drink strawberry milk on exactly 5 out of the 7 days she visits?

Answer: \frac{5103}{16384}


At the mall's food court, Crystal and her friend Amy each want to buy a meal consisting of one entree, one drink, and one dessert. The table below lists both Crystal’s and Amy’s favorite foods from the food court. Assuming they could choose from a shared assortment and wanted to try different meals from each other, how many possible distinct pairs of meals can they buy if they follow these conditions?

\begin{tabular}{ |c | c | c | }
\hline \textbf{Entrees} & \textbf{Drinks} & \textbf{Desserts} \\
\hline
Pizza & Lemonade & Frozen Yogurt \\
\hline
Chicken Teriyaki & Root Beer & Chocolate Chip Cookie \\
\hline
Corn Dog & Iced Tea & Apple Pie \\
\hline
Fish and Chips & & \\
\hline
\end{tabular}

Answer: 1260


$\left[\begin{array}{l}\text { [Inscribed, circumscribed, and exscribed circles; their radii] } \\ {[\text { Auxiliary similar triangles }}\end{array}\right]$

In parallelogram $A B C D$, side $A D$ is equal to 6. The bisector of angle $A D C$ intersects line $A B$ at point $E$. A circle is inscribed in triangle $A D E$, touching side $A E$ at point $K$ and side $A D$ at point $T, K T=3$. Find angle $B A D$.

Answer: 60^\circ


5. A5 (NET 2) ${ }^{\mathrm{IMO}}$ Let $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6}$ be a regular hexagon. Each of its diagonals $A_{i-1} A_{i+1}$ is divided into the same ratio $\frac{\lambda}{1-\lambda}$, where $0<\lambda<1$, by a point $B_{i}$ in such a way that $A_{i}, B_{i}$, and $B_{i+2}$ are collinear ( $i \equiv$ $1, \ldots, 6(\bmod 6))$. Compute $\lambda$.

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


5. In $\triangle A B C$, $A B=A C=a$, a regular triangle $B C D$ is constructed outward on side $B C, A=$
$\qquad$ when, the area of quadrilateral $A B C D$ is maximized.

Answer: 150^\circ


The distance between locations A and B is 160 kilometers. A car and a tractor leave from these two locations simultaneously and drive towards each other. After 1 hour and 20 minutes, they meet. After meeting, the tractor continues ahead while the car stops at the meeting point for 1 hour, then turns around and drives back at its original speed. After driving back for half an hour, the car catches up to the tractor. From the start to this point, the car has traveled _____ km, and the tractor has traveled _____ km.

Answer: 165


7. If the parabola $y=x^{2}-a x+3(a \in \mathbf{R})$ is always above the line $y=\frac{9}{4}$ over the interval $[a, a+1]$, then the range of values for $a$ is $\qquad$.

Answer: (-\sqrt{3}, +\infty)


5】 Given $x_{i}$ are non-negative real numbers, $i=1,2,3,4 . x_{1}+x_{2}+x_{3}+x_{4}=1$. Let $S=1-$ $\sum_{i=1}^{4} x_{i}^{3}-6 \sum_{1 \leqslant i<j<k \leqslant 4} x_{i} x_{j} x_{k}$, find the range of $S$.

Answer: [0, \dfrac{3}{4}]


3.

On the sides $AB, BC, CD$, and $AD$ of the convex quadrilateral $ABCD$, points $M, N, K$, and $L$ are located respectively, such that $AM: MB=3: 2, CN: NB=2: 3, CK=KD$, and $AL: LD=1: 2$. Find the ratio of the area of the hexagon $MBN KDL$ to the area of the quadrilateral $ABCD$.

Answer: \dfrac{4}{5}


3. (3 points) As shown in the right figure, the length of rectangle $A B C D$ is 6 cm, and the width is 2 cm. A line segment $A E$ is drawn through point $A$ to divide the rectangle into two parts, one part being a right triangle, and the other part being a trapezoid. If the area of the trapezoid is 3 times the area of the right triangle, then, the difference between the perimeter of the trapezoid and the perimeter of the right triangle is $\qquad$ cm.

Answer: 6


In this article, “ $\sum$ ” denotes the cyclic sum.
Example 1 Let $x, y, z > 0$, and $x+y+z=1$. Find
$$
f(x, y, z)=\sum \frac{x(2 y-z)}{1+x+3 y}
$$

the maximum value. ${ }^{[1]}$

Answer: \dfrac{1}{7}


Let $p$ be a prime number. Find all polynomials $P$ with integer coefficients with the following properties:
$(a)$ $P(x)>x$ for all positive integers $x$.
$(b)$ The sequence defined by $p_0:=p$, $p_{n+1}:=P(p_n)$ for all positive integers $n$, satisfies the property that for all positive integers $m$ there exists some $l\geq 0$ such that $m\mid p_l$.

Answer: P(x) = x + 1


The angle of a triangle equals the sum of its two other angles. Prove that the triangle is a right triangle.

Answer: \text{The triangle is a right triangle.}


4. The total length of all curve segments formed by a moving point on the surface of a regular quadrilateral pyramid $P-ABCD$ with lateral edge length and base edge length both being 4, and at a distance of 3 from the vertex $P$, is $\qquad$ .

Answer: 6\pi


# Problem 5. (based on materials from the Ural Tournaments)

In triangle $ABC$, angle $B$ is $80^{\circ}$. On side $BC$, point $D$ is marked such that $AB = AD = CD$. On side $AB$, point $F$ is marked such that $AF = BD$. On segment $AC$, point $E$ is marked such that $AB = AE$. Find angle $AEF$.

![](https://cdn.mathpix.com/cropped/2024_05_06_19300bfa006950daee48g-3.jpg?height=1431&width=1697&top_left_y=344&top_left_x=192)

Answer: 20^\circ


Let $p$ be an integer greater than or equal to 2. Show that $p$ is prime if and only if $(p-1)!\equiv-1 \bmod p$.

Answer: p \text{ is prime if and only if } (p-1)! \equiv -1 \pmod{p}


10.1. Having found some polynomial of the sixth degree $x^{6}+a_{1} x^{5}+\ldots+a_{5} x+a_{6}$ with integer coefficients, one of the roots of which is the number $\sqrt{2}+\sqrt[3]{5}$, write in the answer the sum of its coefficients $a_{1}+a_{2}+\ldots+a_{6}$.

Answer: -47


20. Let $a_{1}, a_{2}, \ldots$ be a sequence satisfying the condition that $a_{1}=1$ and $a_{n}=10 a_{n-1}-1$ for all $n \geq 2$. Find the minimum $n$ such that $a_{n}>10^{100}$.

Answer: 102


Maria is a bookworm who recently moved to the United States from Spain. She loves recommending books to her friends and has a goal to read and recommend a set number of books each month. Last month, Maria read 12 books, and this month, she plans to read 50% more books than last month. If Maria recommends 3 out of every 4 books she reads, how many books will she recommend this month?

Answer: 14


Alex is a proud alumnus of the University of New South Wales (UNSW) and now works as a commercial mediator. One day, Alex is helping two companies resolve a dispute over a contract. During the mediation, Alex discovers that Company A claims they are owed $15,000, while Company B contends they only owe $10,000. To find a fair resolution, Alex suggests splitting the difference between the two claims equally between the two companies. After resolving this issue, Alex decides to donate 10% of the resolved amount to the UNSW alumni fund. How much money does Alex donate to the UNSW alumni fund?

Answer: 1250


Emily, a young woman with perfect teeth, is skeptical about the need for extensive dental cosmetic procedures. Her dentist suggests a cosmetic package costing $1,200, claiming it would make her teeth even more perfect. Emily decides to compare costs before making any decisions. She finds that, on average, people spend $80 per year on regular dental check-ups and cleanings without the cosmetic procedures. Emily plans to visit her dentist for check-ups and cleanings for the next 10 years. How much more money would she spend in total if she chooses the cosmetic package now compared to just regular dental check-ups and cleanings over the 10-year period?

Answer: 1200


An immigrant artist named Sofia is preparing for her first art exhibit in her new country. She plans to create a series of paintings that express her journey of adopting a new culture. Sofia decides to make 5 paintings, each representing a different stage of her journey. She estimates that each painting will take her 3 days to complete if she works 4 hours each day.

Sofia also wants to incorporate elements of her native culture into her work, so she spends an additional 2 hours per painting researching and collecting materials that remind her of home. If Sofia works on her paintings 5 days a week, how many weeks will it take her to complete the entire series of 5 paintings?

Answer: 3.5


Jamie is a passionate dancer who grew up idolizing Janet Jackson. Every week, Jamie dedicates 3 days to practicing dance routines inspired by Janet's performances. On each of those days, Jamie practices for 2 hours in the morning, takes a 1-hour break, and then practices for another 2 hours in the afternoon. In addition to these practice sessions, Jamie also spends 1 hour each day watching Janet Jackson's performances for inspiration. How many hours in total does Jamie spend on dancing activities related to Janet Jackson's influence in a week?

Answer: 19


Maria has been living in Bakersfield, California, for over 20 years. She recently decided to visit some of the adjoining communities to explore more of the area. On a Saturday, she plans a trip to three nearby cities: Delano, Wasco, and Shafter. The distance from Bakersfield to Delano is 30 miles, from Delano to Wasco is 15 miles, and from Wasco to Shafter is 20 miles. If Maria travels from Bakersfield to Delano, then to Wasco, and finally to Shafter, before returning directly to Bakersfield, how many miles will she travel in total on her trip?

Answer: 90


An actor known for bringing a fresh and contemporary interpretation to classic roles is preparing for a new rendition of Shakespeare's "Romeo and Juliet." In this modern adaptation, each act is shortened to 20 minutes to keep the audience engaged. The play consists of 5 acts. During rehearsals, the actor spends 15 minutes discussing each act with the director before and after each rehearsal session. If the actor rehearses the play 4 times a week, how many total minutes does the actor spend rehearsing and discussing the play in a week?

Answer: 1000


During the lockdown, Alex, a small business owner, decided to help fellow entrepreneurs by organizing online workshops. Each workshop had 15 participants, and Alex held 4 workshops each month. To cover the costs of the online platform, each participant paid $20 per workshop. After covering the platform costs, Alex donated 25% of the remaining income to a local charity supporting small businesses. How much did Alex donate to the charity in a month?

Answer: 300


Shyron Ely has been a professional basketball player for many years, and you are a long-time fan of his. During a recent season, Shyron played 25 games. In each game, he scored an average of 18 points. In addition to the points he scored, he also made an average of 5 assists per game. How many total points and assists did Shyron Ely make during the entire season?

Answer: 575


Alex, a history major, is reading about the causes and consequences of a historic war. In one chapter, he learns that the war lasted for 7 years and during that time, there were 3 major battles each year. Each battle had approximately 1,200 soldiers involved from one side. If each soldier required 3 meals a day and each meal cost 2 units of currency, how much currency was spent on meals for all the soldiers involved in all the battles throughout the war?

Answer: 151200


Mr. and Mrs. Johnson are a retired couple who rarely use their car and often have it parked in their garage. They offer their driveway for street parking to help their neighbors. Their driveway can fit 2 cars. On Monday, 3 neighbors asked if they could park in the Johnsons' driveway over the week, and the Johnsons agreed. Each day, they allow only 2 cars to park from 9 AM to 5 PM. By the end of the week, each neighbor parked their car for an equal number of days. How many days did each neighbor park in the driveway?

Answer: 4


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

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

Answer: root beer


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`
 - They all have a unique favorite flower: `carnations`, `tulips`, `lilies`, `roses`, `daffodils`, `iris`
 - People have unique favorite book genres: `science fiction`, `fantasy`, `romance`, `mystery`, `biography`, `historical fiction`
 - People use unique phone models: `huawei p50`, `oneplus 9`, `iphone 13`, `xiaomi mi 11`, `samsung galaxy s21`, `google pixel 6`
 - People have unique hair colors: `black`, `gray`, `red`, `auburn`, `blonde`, `brown`
 - Each person has a favorite color: `yellow`, `green`, `red`, `white`, `blue`, `purple`

## Clues:
1. There is one house between the person who has gray hair and the person who uses an iPhone 13.
2. The person who loves the rose bouquet is the person who has blonde hair.
3. There is one house between the person who has gray hair and the person who loves yellow.
4. Carol is somewhere to the right of the person who loves a bouquet of daffodils.
5. The person who loves white is somewhere to the left of Peter.
6. The person who loves white is somewhere to the left of the person who loves science fiction books.
7. Arnold is directly left of the person who loves historical fiction books.
8. The person who uses a Huawei P50 is the person who loves a bouquet of daffodils.
9. The person who loves purple is somewhere to the left of the person who loves fantasy books.
10. Eric is the person who loves science fiction books.
11. The person who loves a bouquet of daffodils is the person who has auburn hair.
12. Alice is the person whose favorite color is green.
13. The person who loves fantasy books is in the second house.
14. The person who uses a Samsung Galaxy S21 is Carol.
15. The person who uses a Xiaomi Mi 11 is not in the fourth house.
16. Eric is the person who has auburn hair.
17. The person who loves the rose bouquet is somewhere to the left of the person who has red hair.
18. The person who loves historical fiction books is the person who loves the rose bouquet.
19. The person who uses a Xiaomi Mi 11 is somewhere to the right of the person who loves white.
20. The person who has black hair is in the fourth house.
21. The person whose favorite color is red is the person who loves mystery books.
22. The person who loves historical fiction books and the person who loves the vase of tulips are next to each other.
23. The person who has brown hair is the person who uses a OnePlus 9.
24. The person who loves romance books is somewhere to the left of the person who loves fantasy books.
25. The person who loves the boquet of lilies is in the second house.
26. The person who loves the boquet of iris is somewhere to the right of the person whose favorite color is green.
 What is the value of attribute BookGenre for the person whose attribute HairColor is auburn? Please reason step by step, and put your final answer within \boxed{}

Answer: science fiction


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`, `Carol`, `Peter`, `Bob`, `Eric`
 - The people keep unique animals: `fish`, `bird`, `rabbit`, `cat`, `horse`, `dog`
 - Each person has a unique favorite drink: `tea`, `root beer`, `water`, `milk`, `boba tea`, `coffee`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `huawei p50`, `xiaomi mi 11`, `oneplus 9`
 - Each mother is accompanied by their child: `Alice`, `Fred`, `Meredith`, `Samantha`, `Timothy`, `Bella`

## Clues:
1. The tea drinker is the person who uses an iPhone 13.
2. There is one house between the dog owner and the tea drinker.
3. The fish enthusiast is somewhere to the left of the tea drinker.
4. Eric and the person's child is named Alice are next to each other.
5. The one who only drinks water is not in the third house.
6. The root beer lover is directly left of the person's child is named Samantha.
7. The person who is the mother of Timothy is the coffee drinker.
8. The fish enthusiast is directly left of the person who likes milk.
9. Peter is the person who likes milk.
10. The person who uses a OnePlus 9 is directly left of the person's child is named Alice.
11. The person who keeps horses is not in the third house.
12. The person's child is named Meredith is directly left of Bob.
13. The person's child is named Bella is the person who uses a Google Pixel 6.
14. The bird keeper is directly left of the person's child is named Meredith.
15. The person who uses a Samsung Galaxy S21 is directly left of the person's child is named Bella.
16. The one who only drinks water is somewhere to the left of the person's child is named Fred.
17. Arnold is the tea drinker.
18. The person who keeps horses is somewhere to the left of the cat lover.
19. The person who uses a Huawei P50 and Alice are next to each other.
20. Bob is directly left of the root beer lover.
21. The person who is the mother of Timothy is Eric.
 What is the value of attribute Children for the person whose attribute Drink is tea? Please reason step by step, and put your final answer within \boxed{}

Answer: Samantha


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`, `Peter`, `Alice`, `Bob`, `Arnold`
 - The people are of nationalities: `norwegian`, `brit`, `swede`, `dane`, `german`
 - Each person prefers a unique type of vacation: `cruise`, `mountain`, `camping`, `beach`, `city`
 - Each person has a unique level of education: `bachelor`, `master`, `associate`, `doctorate`, `high school`
 - Each person has an occupation: `artist`, `doctor`, `engineer`, `teacher`, `lawyer`

## Clues:
1. The person who likes going on cruises is the person who is a lawyer.
2. The person who loves beach vacations is directly left of Arnold.
3. The person with a doctorate is somewhere to the left of Bob.
4. The person with an associate's degree is the person who likes going on cruises.
5. Peter is not in the first house.
6. The person who is an artist is Peter.
7. The person who enjoys camping trips is the person with a master's degree.
8. The Dane is somewhere to the right of the person who is a doctor.
9. The person with an associate's degree is directly left of the person who is an engineer.
10. The person who enjoys camping trips is the British person.
11. The Norwegian and the person with a bachelor's degree are next to each other.
12. The person who is an artist is the Swedish person.
13. Bob is not in the fourth house.
14. The person who enjoys camping trips is Eric.
15. Alice is the German.
16. The person who loves beach vacations is somewhere to the left of the person who prefers city breaks.
17. The person who enjoys mountain retreats is in the fifth house.
18. The person who likes going on cruises is somewhere to the right of the person who loves beach vacations.
19. The person with a bachelor's degree is in the third house.
 What is the value of attribute House for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Bob`, `Eric`, `Arnold`
 - The people keep unique animals: `fish`, `dog`, `horse`, `bird`, `cat`
 - People own unique car models: `honda civic`, `bmw 3 series`, `ford f150`, `toyota camry`, `tesla model 3`
 - Each person has a unique favorite drink: `coffee`, `tea`, `root beer`, `milk`, `water`

## Clues:
1. Arnold is not in the fifth house.
2. Alice and Bob are next to each other.
3. The person who owns a Tesla Model 3 is the person who keeps horses.
4. There are two houses between Arnold and the root beer lover.
5. Eric is in the second house.
6. The person who likes milk is the dog owner.
7. Arnold is the person who owns a Tesla Model 3.
8. The bird keeper is the person who owns a Honda Civic.
9. The one who only drinks water and the person who owns a Toyota Camry are next to each other.
10. The one who only drinks water is directly left of the cat lover.
11. There is one house between the person who owns a Ford F-150 and the fish enthusiast.
12. The dog owner is not in the fifth house.
13. The tea drinker is Bob.
 What is the value of attribute House for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - Each person has a unique type of pet: `dog`, `fish`, `cat`
 - People have unique favorite book genres: `mystery`, `romance`, `science fiction`
 - People own unique car models: `ford f150`, `tesla model 3`, `toyota camry`
 - Each person has a unique birthday month: `april`, `jan`, `sept`
 - People have unique favorite music genres: `rock`, `classical`, `pop`

## Clues:
1. The person who loves romance books is directly left of the person who owns a Ford F-150.
2. The person whose birthday is in September is the person who owns a Toyota Camry.
3. The person whose birthday is in September is Peter.
4. The person who owns a Tesla Model 3 is the person whose birthday is in January.
5. The person with an aquarium of fish is not in the second house.
6. Arnold is the person who has a cat.
7. The person who owns a Tesla Model 3 is the person who loves science fiction books.
8. The person who loves science fiction books is not in the third house.
9. The person who owns a Ford F-150 is the person who loves pop music.
10. The person who loves rock music is directly left of Eric.
 What is the value of attribute CarModel for the person whose attribute Pet is dog? Please reason step by step, and put your final answer within \boxed{}

Answer: toyota camry


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Bob`, `Arnold`, `Carol`, `Eric`, `Alice`
 - Everyone has something unique for lunch: `spaghetti`, `stir fry`, `stew`, `soup`, `grilled cheese`, `pizza`
 - Everyone has a unique favorite cigar: `blends`, `pall mall`, `dunhill`, `yellow monster`, `prince`, `blue master`
 - Each person has a unique favorite drink: `boba tea`, `water`, `coffee`, `root beer`, `milk`, `tea`
 - Each person has a unique level of education: `associate`, `bachelor`, `trade school`, `doctorate`, `high school`, `master`

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

Answer: 5


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Carol`, `Bob`, `Eric`
 - Each person has an occupation: `artist`, `nurse`, `engineer`, `lawyer`, `teacher`, `doctor`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Aniya`, `Janelle`, `Sarah`, `Kailyn`
 - People have unique favorite music genres: `pop`, `rock`, `classical`, `jazz`, `hip hop`, `country`

## Clues:
1. The person who is an engineer is directly left of the person who is an artist.
2. The person whose mother's name is Janelle is the person who is an artist.
3. The person who loves classical music is directly left of Peter.
4. The person whose mother's name is Janelle and the person who loves pop music are next to each other.
5. The person whose mother's name is Sarah is somewhere to the left of Arnold.
6. The person who loves country music is The person whose mother's name is Aniya.
7. The person whose mother's name is Penny is not in the second house.
8. Bob is the person who loves jazz music.
9. Carol is not in the sixth house.
10. The person who is an engineer is the person who loves classical music.
11. The person who is a nurse is somewhere to the right of the person who loves rock music.
12. There are two houses between Bob and The person whose mother's name is Penny.
13. The person who is a lawyer is directly left of the person who is a doctor.
14. The person whose mother's name is Penny is not in the third house.
15. The person who is an engineer is not in the fourth house.
16. The person who is a doctor is The person whose mother's name is Kailyn.
17. Carol is somewhere to the right of the person who loves country music.
18. The person who loves classical music is Alice.
19. The person who is a nurse is The person whose mother's name is Holly.
 What is the value of attribute Mother for the person whose attribute MusicGenre is classical? Please reason step by step, and put your final answer within \boxed{}

Answer: Sarah


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Alice`, `Bob`, `Arnold`, `Carol`
 - Everyone has a unique favorite cigar: `yellow monster`, `blends`, `prince`, `dunhill`, `pall mall`, `blue master`
 - People use unique phone models: `xiaomi mi 11`, `iphone 13`, `google pixel 6`, `huawei p50`, `samsung galaxy s21`, `oneplus 9`
 - People own unique car models: `bmw 3 series`, `honda civic`, `tesla model 3`, `ford f150`, `chevrolet silverado`, `toyota camry`
 - The people are of nationalities: `dane`, `brit`, `chinese`, `swede`, `norwegian`, `german`

## Clues:
1. The person who uses a Xiaomi Mi 11 is somewhere to the left of the person who uses a Google Pixel 6.
2. Carol is somewhere to the left of the person who owns a Ford F-150.
3. Peter is not in the second house.
4. There is one house between the Dane and the person who uses a Google Pixel 6.
5. The Chinese is the person who uses a Google Pixel 6.
6. The person who smokes many unique blends is somewhere to the right of the person who owns a Honda Civic.
7. The person who owns a Tesla Model 3 is in the fourth house.
8. The person who uses a Samsung Galaxy S21 is Alice.
9. There is one house between the person who uses a Xiaomi Mi 11 and the person who smokes Blue Master.
10. The person partial to Pall Mall is directly left of the person who uses a Samsung Galaxy S21.
11. The person who smokes many unique blends is the German.
12. The person who owns a Chevrolet Silverado is the Prince smoker.
13. The person who owns a BMW 3 Series is Eric.
14. The person who owns a Ford F-150 is directly left of Bob.
15. The person who smokes Yellow Monster is not in the second house.
16. The Swedish person is somewhere to the right of the person who smokes Yellow Monster.
17. The person who owns a Chevrolet Silverado is directly left of the person who uses a Huawei P50.
18. The person who smokes Blue Master is in the fifth house.
19. The person who smokes Blue Master is directly left of the person who uses a OnePlus 9.
20. The Norwegian is the person who owns a Toyota Camry.
 What is the value of attribute CarModel for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: tesla model 3


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

## Clues:
1. The person who loves jazz music is the person whose birthday is in April.
2. Alice is the person whose birthday is in April.
3. Peter is directly left of the person who loves cooking.
4. The person whose birthday is in February is Arnold.
5. The Dunhill smoker is not in the third house.
6. The Prince smoker is Arnold.
7. The person who loves jazz music is the person partial to Pall Mall.
8. The person whose birthday is in January is not in the third house.
9. The person who loves rock music is Peter.
10. The person who enjoys gardening is directly left of the photography enthusiast.
11. The person who loves classical music is in the first house.
12. The person who loves pop music is somewhere to the left of the person partial to Pall Mall.
13. The person who smokes Blue Master is the person who loves pop music.
 What is the value of attribute MusicGenre for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: jazz


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`
 - Each person has a unique type of pet: `hamster`, `bird`, `fish`, `dog`, `cat`
 - Each person has a unique favorite drink: `coffee`, `milk`, `root beer`, `water`, `tea`
 - People have unique favorite sports: `soccer`, `swimming`, `tennis`, `baseball`, `basketball`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `iphone 13`, `huawei p50`, `samsung galaxy s21`

## Clues:
1. The root beer lover is the person who owns a dog.
2. The person who uses an iPhone 13 is the root beer lover.
3. There is one house between the person who owns a dog and the person who uses a Google Pixel 6.
4. Bob is somewhere to the right of the person who uses a OnePlus 9.
5. Alice is not in the fourth house.
6. Eric is in the fifth house.
7. The person who likes milk is the person who uses a Samsung Galaxy S21.
8. The person who loves baseball is the person who uses a Google Pixel 6.
9. The coffee drinker is in the third house.
10. Peter is the tea drinker.
11. The person who loves basketball is the person who owns a dog.
12. The person with a pet hamster is the person who loves tennis.
13. There are two houses between Alice and the tea drinker.
14. The person who loves soccer is the tea drinker.
15. The person who keeps a pet bird is directly left of the person with an aquarium of fish.
16. The person with an aquarium of fish is the person who likes milk.
 What is the value of attribute PhoneModel for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: google pixel 6


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`
 - Each person has a unique type of pet: `hamster`, `bird`, `fish`, `dog`, `cat`
 - Each person has a unique favorite drink: `coffee`, `milk`, `root beer`, `water`, `tea`
 - People have unique favorite sports: `soccer`, `swimming`, `tennis`, `baseball`, `basketball`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `iphone 13`, `huawei p50`, `samsung galaxy s21`

## Clues:
1. The root beer lover is the person who owns a dog.
2. The person who uses an iPhone 13 is the root beer lover.
3. There is one house between the person who owns a dog and the person who uses a Google Pixel 6.
4. Bob is somewhere to the right of the person who uses a OnePlus 9.
5. Alice is not in the fourth house.
6. Eric is in the fifth house.
7. The person who likes milk is the person who uses a Samsung Galaxy S21.
8. The person who loves baseball is the person who uses a Google Pixel 6.
9. The coffee drinker is in the third house.
10. Peter is the tea drinker.
11. The person who loves basketball is the person who owns a dog.
12. The person with a pet hamster is the person who loves tennis.
13. There are two houses between Alice and the tea drinker.
14. The person who loves soccer is the tea drinker.
15. The person who keeps a pet bird is directly left of the person with an aquarium of fish.
16. The person with an aquarium of fish is the person who likes milk.
 What is the value of attribute Drink for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: water


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Eric`
 - Each person has a unique type of pet: `cat`, `dog`, `bird`, `fish`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`, `Kailyn`
 - Each person lives in a unique style of house: `colonial`, `ranch`, `victorian`, `craftsman`
 - People have unique favorite sports: `soccer`, `basketball`, `swimming`, `tennis`

## Clues:
1. The person who loves soccer is Alice.
2. The person whose mother's name is Kailyn is somewhere to the left of the person in a Craftsman-style house.
3. The person with an aquarium of fish and The person whose mother's name is Janelle are next to each other.
4. Alice is the person who keeps a pet bird.
5. The person who loves swimming is the person in a ranch-style home.
6. Eric is in the first house.
7. Eric is the person residing in a Victorian house.
8. Arnold is the person in a ranch-style home.
9. Arnold is not in the second house.
10. The person whose mother's name is Holly is somewhere to the left of the person who owns a dog.
11. The person whose mother's name is Janelle is in the fourth house.
12. The person whose mother's name is Kailyn is the person who loves tennis.
13. Alice is The person whose mother's name is Janelle.
 What is the value of attribute Name for the person whose attribute Mother is Kailyn? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Arnold`, `Peter`
 - Each person has a unique hobby: `painting`, `gardening`, `photography`, `cooking`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`, `fantasy`
 - Each person prefers a unique type of vacation: `city`, `beach`, `mountain`, `cruise`
 - People have unique favorite music genres: `rock`, `pop`, `classical`, `jazz`
 - Each person has a unique level of education: `high school`, `associate`, `bachelor`, `master`

## Clues:
1. The person who loves science fiction books is somewhere to the right of the person who loves fantasy books.
2. The person who paints as a hobby is the person who loves fantasy books.
3. Arnold is the person who loves fantasy books.
4. The person with a high school diploma is the person who loves pop music.
5. The person who loves cooking is somewhere to the right of Eric.
6. There are two houses between the photography enthusiast and the person who loves beach vacations.
7. The person who loves romance books is Peter.
8. The person who enjoys mountain retreats is not in the second house.
9. The person with a bachelor's degree is somewhere to the right of the person with a high school diploma.
10. The person who loves classical music is not in the third house.
11. The person who prefers city breaks is the person who loves science fiction books.
12. The person who loves rock music is Peter.
13. The person with a bachelor's degree is not in the fourth house.
14. The person with a master's degree is the person who loves jazz music.
15. The person who enjoys gardening is the person who loves romance books.
 What is the value of attribute MusicGenre for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: pop


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`, `Peter`, `Alice`, `Arnold`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`, `stew`, `stir fry`
 - The people keep unique animals: `bird`, `horse`, `cat`, `fish`, `dog`
 - Each person has a unique level of education: `master`, `doctorate`, `high school`, `associate`, `bachelor`

## Clues:
1. Alice is not in the fifth house.
2. The person who is a pizza lover is somewhere to the right of Peter.
3. Peter is in the first house.
4. The person with a bachelor's degree is the person who is a pizza lover.
5. The cat lover is the person with a master's degree.
6. The person who loves stir fry is the person with a doctorate.
7. The person with a high school diploma is directly left of the person with a master's degree.
8. The dog owner is the person who loves stir fry.
9. There is one house between Arnold and the person with an associate's degree.
10. The person who loves eating grilled cheese is the fish enthusiast.
11. Arnold is in the second house.
12. The person who keeps horses is Arnold.
13. Eric is directly left of the person with an associate's degree.
14. The person who loves the spaghetti eater is the cat lover.
 What is the value of attribute Food for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: spaghetti


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: `Eric`, `Arnold`
 - Each person has a unique level of education: `high school`, `associate`
 - Each mother is accompanied by their child: `Fred`, `Bella`
 - Each person prefers a unique type of vacation: `mountain`, `beach`
 - Each person has an occupation: `engineer`, `doctor`
 - Each person lives in a unique style of house: `victorian`, `colonial`

## Clues:
1. The person's child is named Fred is somewhere to the right of the person with an associate's degree.
2. The person residing in a Victorian house and Arnold are next to each other.
3. The person who loves beach vacations is the person who is a doctor.
4. The person with a high school diploma is Arnold.
5. The person who is a doctor is the person with an associate's degree.
 What is the value of attribute House for the person whose attribute Vacation is beach? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Arnold`, `Carol`, `Eric`, `Peter`
 - People use unique phone models: `xiaomi mi 11`, `google pixel 6`, `samsung galaxy s21`, `oneplus 9`, `huawei p50`, `iphone 13`
 - People have unique favorite book genres: `romance`, `historical fiction`, `fantasy`, `mystery`, `science fiction`, `biography`
 - They all have a unique favorite flower: `tulips`, `roses`, `carnations`, `daffodils`, `iris`, `lilies`
 - The people keep unique animals: `dog`, `cat`, `bird`, `fish`, `rabbit`, `horse`
 - Each person has a unique level of education: `trade school`, `high school`, `master`, `associate`, `bachelor`, `doctorate`

## Clues:
1. The bird keeper is the person who uses a Google Pixel 6.
2. The person who loves a bouquet of daffodils is the person with an associate's degree.
3. The person who uses a OnePlus 9 is not in the fourth house.
4. There are two houses between the person with a master's degree and the rabbit owner.
5. The person who loves a carnations arrangement is Arnold.
6. The person who loves the boquet of lilies is not in the sixth house.
7. The person who loves the vase of tulips is Carol.
8. The person with a high school diploma is Bob.
9. The person who attended trade school is somewhere to the left of the person who keeps horses.
10. The person who loves mystery books is Eric.
11. The fish enthusiast is directly left of Carol.
12. Bob is the fish enthusiast.
13. Eric is the dog owner.
14. The person who loves a carnations arrangement is the person who uses a Google Pixel 6.
15. The person who loves fantasy books is in the second house.
16. The person who uses a Samsung Galaxy S21 is the person who loves the boquet of iris.
17. Alice is the person with an associate's degree.
18. The person who uses a Google Pixel 6 is not in the second house.
19. The dog owner is the person with a doctorate.
20. The fish enthusiast is the person who uses a Samsung Galaxy S21.
21. The person who uses an iPhone 13 is somewhere to the right of the person who loves fantasy books.
22. The person who loves mystery books is somewhere to the right of the person who loves the boquet of iris.
23. Alice is the person who uses an iPhone 13.
24. The person who attended trade school is not in the fourth house.
25. The person who loves biography books is directly left of Eric.
26. There is one house between the rabbit owner and the person who loves historical fiction books.
27. The person who loves the boquet of iris is somewhere to the right of the person who uses a Xiaomi Mi 11.
28. The person who loves science fiction books is the person who uses a Samsung Galaxy S21.
29. The person who loves the boquet of lilies is not in the second house.
 What is the value of attribute Animal for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}

Answer: rabbit


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`
 - People have unique favorite book genres: `romance`, `mystery`, `fantasy`, `science fiction`
 - People have unique favorite sports: `soccer`, `basketball`, `swimming`, `tennis`

## Clues:
1. The person who loves romance books is the person who loves tennis.
2. Alice is the person who loves swimming.
3. There are two houses between the person who loves soccer and the person who loves fantasy books.
4. The person who loves swimming is in the second house.
5. Eric is the person who loves basketball.
6. The person who loves mystery books is somewhere to the left of Alice.
7. The person who loves tennis is somewhere to the right of Peter.
 What is the value of attribute House for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Eric`, `Peter`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `lilies`, `roses`
 - Everyone has a unique favorite cigar: `dunhill`, `blue master`, `pall mall`, `prince`
 - Each mother is accompanied by their child: `Meredith`, `Samantha`, `Bella`, `Fred`
 - The mothers' names in different houses are unique: `Janelle`, `Aniya`, `Holly`, `Kailyn`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `oneplus 9`, `samsung galaxy s21`

## Clues:
1. The person whose mother's name is Kailyn is the Prince smoker.
2. The person whose mother's name is Holly is the person who loves a bouquet of daffodils.
3. Arnold is in the fourth house.
4. The person's child is named Bella is somewhere to the left of the person who uses a OnePlus 9.
5. The person's child is named Fred is not in the second house.
6. The person's child is named Meredith is directly left of The person whose mother's name is Janelle.
7. The person who loves a bouquet of daffodils is Eric.
8. The person who loves the boquet of lilies is the person partial to Pall Mall.
9. The person whose mother's name is Janelle is the person who loves a carnations arrangement.
10. Alice is somewhere to the right of the person's child is named Fred.
11. The person who uses an iPhone 13 is directly left of the Dunhill smoker.
12. The person who loves the rose bouquet is somewhere to the right of the Dunhill smoker.
13. The person who loves the rose bouquet is somewhere to the left of the person who uses a Google Pixel 6.
 What is the value of attribute Flower for the person whose attribute Mother is Aniya? Please reason step by step, and put your final answer within \boxed{}

Answer: lilies


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Eric`, `Arnold`
 - People own unique car models: `ford f150`, `tesla model 3`, `toyota camry`, `honda civic`
 - People have unique favorite music genres: `pop`, `jazz`, `rock`, `classical`
 - People have unique favorite sports: `basketball`, `tennis`, `soccer`, `swimming`
 - Each person prefers a unique type of vacation: `mountain`, `beach`, `cruise`, `city`

## Clues:
1. The person who loves beach vacations is the person who loves soccer.
2. The person who likes going on cruises and Arnold are next to each other.
3. Eric is the person who owns a Toyota Camry.
4. The person who owns a Toyota Camry is the person who loves swimming.
5. The person who loves beach vacations is in the fourth house.
6. The person who loves jazz music is somewhere to the right of the person who loves rock music.
7. The person who owns a Toyota Camry is the person who loves pop music.
8. The person who prefers city breaks is the person who loves jazz music.
9. The person who loves basketball is directly left of the person who owns a Tesla Model 3.
10. The person who likes going on cruises is somewhere to the left of the person who owns a Ford F-150.
11. The person who loves jazz music is directly left of the person who enjoys mountain retreats.
12. Peter is somewhere to the right of the person who loves rock music.
 What is the value of attribute FavoriteSport for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: soccer


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

## Clues:
1. The person who loves hip-hop music is in the second house.
2. The person who loves classical music is not in the fifth house.
3. The person who loves classical music is the fish enthusiast.
4. The person who uses an iPhone 13 is the person with a doctorate.
5. The person with a master's degree is not in the second house.
6. The person who enjoys knitting is not in the first house.
7. The person with a high school diploma is somewhere to the left of the photography enthusiast.
8. The person who paints as a hobby is somewhere to the right of the person who loves rock music.
9. Arnold is the bird keeper.
10. The person with a bachelor's degree is the person who uses a Google Pixel 6.
11. The person with a bachelor's degree is somewhere to the right of the person who paints as a hobby.
12. Bob is not in the fourth house.
13. The person who loves cooking is the person who loves classical music.
14. The person who loves cooking is the person who uses a Huawei P50.
15. The person who keeps horses is somewhere to the right of Arnold.
16. The person who paints as a hobby is the person who uses a OnePlus 9.
17. The person who uses a OnePlus 9 is somewhere to the left of the person with an associate's degree.
18. The person who uses an iPhone 13 is the dog owner.
19. The person with a high school diploma is Alice.
20. Peter is the person who enjoys gardening.
21. The person who paints as a hobby is the person who loves jazz music.
 What is the value of attribute House for the person whose attribute Education is high school? 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`, `Arnold`, `Carol`, `Bob`
 - People have unique heights: `very short`, `very tall`, `short`, `super tall`, `tall`, `average`
 - Everyone has something unique for lunch: `grilled cheese`, `stir fry`, `pizza`, `spaghetti`, `soup`, `stew`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Penny`, `Aniya`, `Kailyn`, `Sarah`
 - Each person has a unique favorite drink: `tea`, `water`, `milk`, `coffee`, `boba tea`, `root beer`
 - Each person has a unique type of pet: `rabbit`, `fish`, `bird`, `hamster`, `cat`, `dog`

## Clues:
1. Arnold is the person who loves the stew.
2. The person who owns a dog is directly left of the person who has a cat.
3. The person whose mother's name is Holly is in the sixth house.
4. The tea drinker is somewhere to the left of the boba tea drinker.
5. The tea drinker is in the second house.
6. The person who is very tall is Bob.
7. The person who has a cat is The person whose mother's name is Penny.
8. The person who is very short is directly left of Alice.
9. Carol is the person who is a pizza lover.
10. The person who has an average height is not in the sixth house.
11. The person who loves stir fry is Bob.
12. The person with a pet hamster is not in the third house.
13. The person whose mother's name is Janelle is somewhere to the left of the person with a pet hamster.
14. Eric is the person who loves the soup.
15. The person whose mother's name is Kailyn is directly left of the person who keeps a pet bird.
16. The one who only drinks water is somewhere to the right of the person who likes milk.
17. The person who owns a rabbit is the person who loves the stew.
18. The person who is tall is The person whose mother's name is Aniya.
19. The person who is very short is somewhere to the left of the person who likes milk.
20. The person who is tall is not in the fifth house.
21. The person who is very short is in the second house.
22. The person who loves the stew is the root beer lover.
23. The person whose mother's name is Kailyn is the boba tea drinker.
24. The person who is tall is not in the fourth house.
25. The person who is super tall is directly left of the person who loves eating grilled cheese.
26. Carol is directly left of the root beer lover.
 What is the value of attribute Pet for the person whose attribute Drink is boba tea? Please reason step by step, and put your final answer within \boxed{}

Answer: hamster


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 Mother for the person whose attribute Children is Meredith? Please reason step by step, and put your final answer within \boxed{}

Answer: Janelle


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`, `Peter`, `Arnold`
 - Each person has a favorite color: `yellow`, `green`, `red`, `white`, `blue`
 - Each mother is accompanied by their child: `Samantha`, `Fred`, `Meredith`, `Bella`, `Timothy`
 - People have unique favorite book genres: `biography`, `mystery`, `romance`, `science fiction`, `fantasy`
 - Each person has a unique birthday month: `april`, `feb`, `sept`, `jan`, `mar`
 - People have unique favorite sports: `soccer`, `basketball`, `swimming`, `baseball`, `tennis`

## Clues:
1. The person who loves swimming is the person who loves mystery books.
2. Eric is somewhere to the left of the person who is the mother of Timothy.
3. The person who loves basketball is somewhere to the left of the person who is the mother of Timothy.
4. The person who loves soccer is the person whose birthday is in September.
5. Eric is the person who loves biography books.
6. The person who loves baseball is the person who loves white.
7. The person who loves science fiction books is the person whose birthday is in March.
8. The person who loves swimming and the person whose favorite color is red are next to each other.
9. The person who loves biography books is not in the first house.
10. The person who loves baseball is somewhere to the left of the person whose favorite color is red.
11. The person who loves fantasy books is Bob.
12. The person's child is named Bella is the person whose birthday is in February.
13. The person who loves tennis and the person's child is named Samantha are next to each other.
14. The person's child is named Bella is not in the first house.
15. The person's child is named Meredith is the person whose birthday is in March.
16. The person who loves blue is directly left of the person who loves yellow.
17. There is one house between Alice and the person whose birthday is in March.
18. The person whose favorite color is red is directly left of the person whose birthday is in January.
19. The person's child is named Meredith is not in the first house.
20. Arnold is somewhere to the right of the person who loves swimming.
21. The person who loves fantasy books is the person who is the mother of Timothy.
 What is the value of attribute Children for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{}

Answer: Timothy


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Peter`, `Carol`, `Alice`, `Eric`
 - Everyone has something unique for lunch: `stir fry`, `stew`, `soup`, `pizza`, `grilled cheese`, `spaghetti`
 - Each person has an occupation: `doctor`, `artist`, `engineer`, `teacher`, `lawyer`, `nurse`
 - Each person has a unique type of pet: `dog`, `bird`, `hamster`, `cat`, `fish`, `rabbit`
 - People have unique favorite music genres: `pop`, `country`, `jazz`, `classical`, `rock`, `hip hop`
 - Everyone has a favorite smoothie: `blueberry`, `desert`, `watermelon`, `cherry`, `lime`, `dragonfruit`

## Clues:
1. The person who loves the spaghetti eater is the Dragonfruit smoothie lover.
2. The person who drinks Lime smoothies is in the second house.
3. Carol is somewhere to the left of the Desert smoothie lover.
4. The person who is an engineer is not in the second house.
5. The person who loves the soup is in the second house.
6. The person who owns a dog is not in the second house.
7. The person who is a pizza lover is Alice.
8. Arnold is somewhere to the left of Eric.
9. The person who loves classical music is the Dragonfruit smoothie lover.
10. The person who owns a rabbit is somewhere to the left of the Desert smoothie lover.
11. The person who is a nurse is not in the sixth house.
12. There are two houses between the person who is a nurse and the person who loves the spaghetti eater.
13. The person who likes Cherry smoothies is the person with an aquarium of fish.
14. The person who has a cat is Alice.
15. The person who likes Cherry smoothies is somewhere to the right of the Desert smoothie lover.
16. There is one house between the person who drinks Blueberry smoothies and the person who loves eating grilled cheese.
17. The person who loves classical music is somewhere to the left of the person who loves the stew.
18. The person who is a lawyer is the person who loves classical music.
19. The person who loves jazz music is Bob.
20. Bob is the person who drinks Lime smoothies.
21. The person who loves pop music is the person who is a teacher.
22. The person who is an engineer is directly left of the person who loves rock music.
23. The person who loves hip-hop music is not in the first house.
24. The person with an aquarium of fish is Peter.
25. The person who drinks Blueberry smoothies is the person who keeps a pet bird.
26. The person who has a cat is in the first house.
27. The person who is a doctor is not in the sixth house.
28. The person who owns a rabbit is not in the second house.
29. Carol is not in the fourth house.
 What is the value of attribute Occupation for the person whose attribute MusicGenre is country? Please reason step by step, and put your final answer within \boxed{}

Answer: nurse


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

## Clues:
1. The person who smokes many unique blends is the person who loves fantasy books.
2. The person who loves biography books is the Desert smoothie lover.
3. There are two houses between the person who loves pop music and the person who loves country music.
4. The rabbit owner is in the third house.
5. Arnold is the bird keeper.
6. The person who loves country music is in the fifth house.
7. The person who loves mystery books is not in the first house.
8. The person who drinks Lime smoothies is not in the third house.
9. The fish enthusiast is somewhere to the right of the person who loves rock music.
10. The Prince smoker is somewhere to the left of the person who likes Cherry smoothies.
11. Bob is the cat lover.
12. Eric is the person who likes Cherry smoothies.
13. The person who smokes Yellow Monster is in the sixth house.
14. The Prince smoker is the Dragonfruit smoothie lover.
15. Carol is directly left of Arnold.
16. The person who smokes Blue Master is directly left of the person who loves classical music.
17. The person who loves classical music is the person who loves historical fiction books.
18. The Dunhill smoker and the person who loves science fiction books are next to each other.
19. The person who keeps horses is directly left of Peter.
20. The Watermelon smoothie lover is the cat lover.
21. The person who smokes many unique blends is directly left of the person who loves jazz music.
22. The cat lover is in the first house.
23. The Desert smoothie lover is the person who loves jazz music.
24. The person who drinks Lime smoothies is directly left of the fish enthusiast.
25. The person who smokes Blue Master is not in the fifth house.
 What is the value of attribute Smoothie for the person whose attribute Name is Arnold? 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: `Eric`, `Peter`, `Arnold`, `Bob`, `Alice`
 - People have unique hair colors: `brown`, `blonde`, `black`, `gray`, `red`
 - Each person prefers a unique type of vacation: `beach`, `camping`, `city`, `cruise`, `mountain`
 - Each person has an occupation: `doctor`, `lawyer`, `engineer`, `teacher`, `artist`
 - People have unique favorite music genres: `jazz`, `pop`, `hip hop`, `classical`, `rock`

## Clues:
1. Alice is the person who has brown hair.
2. The person who is a teacher is somewhere to the right of the person who is an artist.
3. The person who has blonde hair is in the first house.
4. The person who prefers city breaks is not in the fourth house.
5. The person who enjoys mountain retreats is in the third house.
6. The person who loves beach vacations is somewhere to the right of the person who is a lawyer.
7. Eric is the person who has black hair.
8. The person who is an engineer is the person who has red hair.
9. The person who loves classical music is Bob.
10. The person who loves hip-hop music is in the fifth house.
11. The person who has blonde hair is the person who enjoys camping trips.
12. The person who has red hair is in the second house.
13. The person who is a teacher is the person who has gray hair.
14. The person who loves jazz music is the person who has black hair.
15. There is one house between the person who likes going on cruises and the person who has gray hair.
16. The person who loves rock music is the person who has red hair.
17. The person who is a lawyer is not in the third house.
18. The person who has blonde hair is Arnold.
 What is the value of attribute MusicGenre for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: rock


Let $z_1,$ $z_2,$ $\dots,$ $z_{20}$ be the twenty (complex) roots of the equation
\[z^{20} - 4z^{19} + 9z^{18} - 16z^{17} + \dots + 441 = 0.\]Calculate $\cot \left( \sum_{k = 1}^{20} \operatorname{arccot} z_k \right).$  Note that the addition formula for cotangent is still valid when working with complex numbers.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 461


If two distinct members of the set $\{ 3, 7, 21, 27, 35, 42, 51 \}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 63? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 10


Two boards, one four inches wide and the other six inches wide, are nailed together to form an X. The angle at which they cross is 60 degrees. If this structure is painted and the boards are separated what is the area of the unpainted region on the four-inch board? (The holes caused by the nails are negligible.) Express your answer in simplest radical form.

[asy]
draw(6dir(150)--15dir(-30),linewidth(1));
draw((6dir(150)+12/sqrt(3)*dir(30))--(15dir(-30)+12/sqrt(3)*dir(30)),linewidth(1));

draw(6dir(210)--(0,0),linewidth(1));
draw((9dir(210)+8/sqrt(3)*dir(-30))--8/sqrt(3)*dir(-30),linewidth(1));

draw(12/sqrt(3)*dir(30)--(12/sqrt(3)+6)*dir(30),linewidth(1));
draw(12/sqrt(3)*dir(30)+8/sqrt(3)*dir(-30)--(12/sqrt(3)+9)*dir(30)+8/sqrt(3)*dir(-30),linewidth(1));

draw(2dir(150)--2dir(150)+6dir(60),dashed);
draw(2dir(210)--2dir(210)+4dir(-60),dashed);

dot((2,0));
dot((4,-1));
dot((8,1));
dot((6,2));

label("$60^{\circ}$", (11,1), E);
label(rotate(30)*"$4^{\prime\prime}$", .5*(2dir(210)+2dir(210)+4dir(-60))+(0,-.5),W);
label(rotate(-30)*"$6^{\prime\prime}$", .5*(2dir(150)+2dir(150)+6dir(60))+(1,1),W);
[/asy]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 19


\(a_{1}, a_{2}, a_{3}, \ldots\) is an increasing sequence of natural numbers. It is known that \(a_{a_{k}} = 3k\) for any \(k\). Find:
a) \(a_{100}\);
b) \(a_{1983}\).

Answer: 3762


A sequence \(a_{1}, a_{2}, a_{3}, \ldots\) of positive integers satisfies \(a_{1} > 5\) and \(a_{n+1} = 5 + 6 + \cdots + a_{n}\) for all positive integers \(n\). Determine all prime numbers \(p\) such that, regardless of the value of \(a_{1}\), this sequence must contain a multiple of \(p\).

Answer: 2