21. Alice saw a strange caterpillar while wandering in Wonderland, which was very long, with its head visible but not its tail. Alice observed carefully and found that it was like the recurring decimal 3.141592, with the head being the number 3, and each segment of its body resembling the numbers after the decimal point $14159215921592 \ldots \ldots$.
The caterpillar asked Alice: “Starting from my head, what is the sum of the first 2020 digits?” Alice should answer: “The sum is $\qquad$.”

Answer: 8577


Points $A, B, C$, and $D$ lie on a line in this order. For a point $E$ outside the line,

$$
A E B \angle=B E C \angle=C E D \angle=45^{\circ} .
$$

Let the midpoint of segment $A C$ be $F$, and the midpoint of segment $B D$ be $G$. What is the measure of angle $F E G$?

Answer: 90^\circ


## Task A-2.2.

Inside the circle $k$ with radius 20, there are a circle $k_{1}$ with radius 5 and a square $A B C D$. The circles $k$ and $k_{1}$ touch at point $P$, points $A$ and $B$ lie on the circle $k$, and the line $C D$ touches the circle $k_{1}$ at point $Q$ such that $\overline{P Q}$ is the diameter of that circle.

Determine the length of the side of the square $A B C D$.

Answer: 8 + 4\sqrt{19}


[ Average values ]

Three pirates divided the diamonds they had obtained during the day in the evening: twelve each for Bill and Sam, and the rest for John, who could not count. At night, Bill stole one diamond from Sam, Sam stole one from John, and John stole one from Bill. As a result, the average weight of Bill's diamonds decreased by one carat, Sam's decreased by two carats, but John's increased by four carats. How many diamonds did John get?

Answer: 9


Example 7. Find the Wronskian determinant for the functions $y_{1}(x)=e^{k_{1} x}$, $y_{2}(x)=e^{k_{2} x}, y_{3}(x)=e^{k_{3} x}$.

1) The linear dependence of the functions $\sin x, \sin \left(x+\frac{\pi}{8}\right), \sin \left(x-\frac{\pi}{8}\right)$ can be established by noting that $\sin \left(x+\frac{\pi}{8}\right)+\sin \left(x-\frac{\pi}{8}\right)=2 \cos \frac{\pi}{8} \sin x$, or $\sin \left(x+\frac{\pi}{8}\right)+$ $\sin \left(x-\frac{\pi}{8}\right)-2 \cos \frac{\pi}{8} \sin x \equiv 0$.

Answer: (k_2 - k_1)(k_3 - k_1)(k_3 - k_2)e^{(k_1 + k_2 + k_3)x}


3. A square table consists of 2014 rows and 2014 columns. In each cell at the intersection of the row with number $i$ and the column with number $j$, the number $a_{i, j}=(-1)^{i}(2015-i-j)^{2}$ is written. Find the sum of all the numbers in the table

Answer: 0


19.14*. Given a triangle $A B C$. Four circles of equal radius $\rho$ are constructed such that one of them touches the other three, and each of these three touches two sides of the triangle. Find $\rho$, if the radii of the inscribed and circumscribed circles of the triangle are $r$ and $R$ respectively.

Answer: \dfrac{R r}{2 r + R}


Study the sequence defined by $\mathfrak{u}_{0}=0$ and $\mathfrak{u}_{n+1}=\sqrt{12+\mathfrak{u}_{n}}$ for all $\mathfrak{n} \in \mathbb{N}$.

Regarding the second type of equations:

Definition 2.11. Let $\left(u_{n}\right)$ be a real sequence. We say it satisfies a recurrence equation of order $\mathbf{k}$ (linear) if there exists $k \in \mathbb{N}^{*}$ and real numbers $a_{0}, \cdots, a_{k-1}$ such that for all natural numbers $n$,

$$
u_{n+k}+a_{k-1} u_{n+k-1}+\cdots+a_{0} u_{n}=0
$$

Definition 2.12. The equation $X^{k}+a_{k-1} X^{k-1}+\cdots+a_{0}=0$ is called the characteristic equation.

Example 2.13. For an arithmetic sequence with common difference $a, u_{n+1}-u_{n}=u_{n+2}-u_{n+1}=a$, so $u_{n+2}-2 u_{n+1}+u_{n}=0$ is a recurrence equation satisfied by $u_{n}$.

For a geometric sequence with common ratio $a, u_{n+1}-a u_{n}=0$.

An arithmetico-geometric sequence satisfies a relation of the form: $\mathfrak{u}_{n+1}=a u_{n}+b$, and we obtain the recurrence equation $u_{n+2}-(a+1) u_{n+1}+a u_{n}=0$.

The general solution of such a recurrence equation (whose proof is unnecessary and tedious in our context) uses the concept of polynomials, discussed in the following section. Let's focus on orders 1 and 2 here.

Proposition 2.14. If for all $n \in \mathbb{N}, u_{n+1}-a u_{n}=0$, then $u_{n}=u_{0} a^{n}$.

Proposition 2.15. If for all $n \in \mathbb{N}, u_{n+2}+a_{1} u_{n+1}+a_{0} u_{n}=0$, we solve the quadratic characteristic equation $X^{2}+a_{1} X+a_{0}=0$.

- If it has two distinct solutions $r_{1}$ and $r_{2}$, then: $u_{n}=c_{1} r_{1}^{n}+c_{2} r_{2}^{n}$, where $c_{1}$ and $c_{2}$ are determined using, for example, the values of $\mathfrak{u}_{0}$ and $\mathfrak{u}_{1}$.
- If it has a double root $r$, then: $\mathfrak{u}_{n}=(a n+b) r^{n}$, and again $\mathfrak{u}_{0}$ and $\mathfrak{u}_{1}$ allow us to determine $a$ and $b$.

Would you be able to find in the previous property where the arithmetic, geometric, and arithmetico-geometric sequences are hidden?

Example 2.16. A geometric sequence corresponds to a recurrence equation of order 1. An arithmetic sequence corresponds to a recurrence equation of order 2 with 1 as a double root.

An arithmetico-geometric sequence with common ratio $a$ corresponds to a recurrence equation of order 2 whose solutions are 1 and $a$.

So much promising theory only calls out more strongly for exercises allowing us to apply these new knowledge victoriously.

Answer: 4


In an isosceles triangle, we drew one of the angle bisectors. At least one of the resulting two smaller ones  triangles is similar to the original. What can be the leg of the original triangle if the length of its base is $1$ unit?

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


We construct the following figure: Draw an isosceles triangle \( ABC \) with \( A \) as the apex, then draw line \( (d) \) perpendicular to \( (BC) \) passing through \( C \). Choose a point \( D \) on \( (d) \). Place \( E \) such that \( AEDB \) is a parallelogram. Finally, let \( M \) be the intersection point of \( (AE) \) and \( (d) \). Prove that \( M \) is the midpoint of \( [AE] \).

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


## Task A-1.1.

Determine the smallest natural number $a$ such that the value of the expression

$$
\frac{n^{8}+n^{6}+n^{4}+n^{2}+a}{n^{2}-1}
$$

for $n=2014$ is an integer divisible by 3.

Answer: 8112386


[ Text problems (other).]

Kolya and Katya study in the same class. There are twice as many boys as girls in this class. Kolya has 7 more classmates than classmates of the opposite gender. How many female classmates does Katya have?

Answer: 7


An equilateral triangle $T$ with side length $2022$ is divided into equilateral unit triangles with lines parallel to its sides to obtain a triangular grid. The grid is covered with figures shown in the image below, which consist of $4$ equilateral unit triangles and can be rotated by any angle $k \cdot 60^{\circ}$ for $k \in \{1,2,3,4,5\}$.

The covering satisfies the following conditions:

1) It is possible not to use figures of some type and it is possible to use several figures of the same type. The unit triangles in the figures correspond to the unit triangles in the grid.
2) Every unit triangle in the grid is covered, no two figures overlap, and every figure is fully contained in $T$.

Determine the smallest possible number of figures of type $1$ that can be used in such a covering.

Answer: 1011


Consider all ordered pairs \((a, b)\) of positive integers such that \(\frac{a^2 + b^2 + 2}{ab}\) is an integer and \(a \leq b\). We label all such pairs in increasing order by their distance from the origin. (It is guaranteed that no ties exist.) Thus \(P_1 = (1, 1)\), \(P_2 = (1, 3)\), and so on. If \(P_{2020} = (x, y)\), then compute the remainder when \(x + y\) is divided by \(2017\).

Answer: 52


Points $A_1, A_2, \ldots, A_{2022}$ are chosen on a plane such that no three of them are collinear. Consider all angles $A_iA_jA_k$ for distinct points $A_i, A_j, A_k$. What is the largest possible number of these angles that can be equal to $90^\circ$?

Answer: 2042220


For a positive integer $n$, let $S(n)$ denote the sum of its digits. Find the largest possible value of the expression $\frac{S(n)}{S(16n)}$.

Answer: 13


For a rational number $r$, its *period* is the length of the smallest repeating block in its decimal expansion. For example, the number $r = 0.123123123\ldots$ has period $3$. If $S$ denotes the set of all rational numbers of the form $r = \overline{abcdefgh}$ having period $8$, find the sum of all elements in $S$.

Answer: 49995000


Let $ABCD$ be an isosceles trapezoid with $AD \parallel BC$. Points $P$ and $Q$ are placed on segments $\overline{CD}$ and $\overline{DA}$ respectively such that $AP \perp CD$ and $BQ \perp DA$, and point $X$ is the intersection of these two altitudes. Suppose that $BX = 3$ and $XQ = 1$. Compute the largest possible area of $ABCD$.

Answer: 32


Let $N$ be the number of distinct rearrangements of the 34 letters in "SUPERCALIFRAGILISTICEXPIALIDOCIOUS". How many positive factors does $N$ have?

Answer: 3225600


The integers $1, 2, \dots, 20$ are written on the blackboard. Consider the following operation as one step: choose two integers $a$ and $b$ such that $a - b \ge 2$ and replace them with $a - 1$ and $b + 1$. Determine the maximum number of steps that can be done.

Answer: 330


The diagram below shows a large square with each of its sides divided into four equal segments. The shaded square, whose sides are diagonals drawn to these division points, has an area of $13$. Find the area of the large square.

Answer: 104


Consider a sequence $x_1, x_2, \cdots, x_{12}$ of real numbers such that $x_1 = 1$ and for $n = 1, 2, \dots, 10$, let \[ x_{n+2} = \frac{(x_{n+1} + 1)(x_{n+1} - 1)}{x_n}. \] Suppose $x_n > 0$ for $n = 1, 2, \dots, 11$ and $x_{12} = 0$. Then the value of $x_2$ can be written as $\frac{\sqrt{a} + \sqrt{b}}{c}$ for positive integers $a, b, c$ with $a > b$ and no square dividing $a$ or $b$. Find $100a + 10b + c$.

Answer: 622


Let $\triangle ABC$ be a triangle with circumcenter $O$ and orthocenter $H$. Let $D$ be a point on the circumcircle of $ABC$ such that $AD \perp BC$. Suppose that $AB = 6$, $DB = 2$, and the ratio $\frac{\text{area}(\triangle ABC)}{\text{area}(\triangle HBC)}=5$. Then, if $OA$ is the length of the circumradius, then $OA^2$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Answer: 29


64 numbers (not necessarily distinct) are placed on the squares of a chessboard such that the sum of the numbers in every $2\times2$ square is 7. What is the sum of the four numbers in the corners of the board?

Answer: 7


A subset $S$ of the set of integers from 0 to 99 is said to have property $A$ if it is impossible to fill a crossword puzzle with 2 rows and 2 columns using numbers from $S$ (where 0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in the set $S$ that has property $A$.

Answer: 25


Let $ABCD$ be an isosceles trapezoid with $\overline{AD} \parallel \overline{BC}$. The incircle of $\triangle ABC$ has center $I$ and is tangent to $\overline{BC}$ at $P$. The incircle of $\triangle ABD$ has center $J$ and is tangent to $\overline{AD}$ at $Q$. If $PI = 8$, $IJ = 25$, and $JQ = 15$, compute the greatest integer less than or equal to the area of $ABCD$.

Answer: 1728


Let there be $A = 1^{a_1} \cdot 2^{a_2} \cdot \ldots \cdot 100^{a_{100}}$ and $B = 1^{b_1} \cdot 2^{b_2} \cdot \ldots \cdot 100^{b_{100}}$, where $a_i, b_i \in \mathbb{N}$ and $a_i + b_i = 101 - i$ for $i = 1, 2, \ldots, 100$. Find the last 1124 digits of $P = A \times B$.

Answer: 0


Initially, a natural number $n$ is written on the blackboard. Then, at each minute, Esmeralda chooses a divisor $d > 1$ of $n$, erases $n$, and writes $n + d$. If the initial number on the board is $2022$, what is the largest composite number that Esmeralda will never be able to write on the blackboard?

Answer: 2023


Triangle $ABC$ has $\angle A = 90^\circ$, $\angle C = 30^\circ$, and $AC = 12$. Let the circumcircle of this triangle be $W$. Define $D$ to be the point on arc $BC$ not containing $A$ so that $\angle CAD = 60^\circ$. Define points $E$ and $F$ to be the feet of the perpendiculars from $D$ to lines $AB$ and $AC$, respectively. Let $J$ be the intersection of line $EF$ with $W$, where $J$ is on the minor arc $AC$. The line $DF$ intersects $W$ at $H$ other than $D$. The area of the triangle $FHJ$ is in the form $\frac{a}{b}(\sqrt{c}-\sqrt{d})$ for positive integers $a,b,c,d$, where $a,b$ are relatively prime, and the sum of $a,b,c,d$ is minimal. Find $a+b+c+d$.

Answer: 23


The list of all numbers formed using the digits \(1, 2, 3, 4,\) and \(5\), without repetition, is created. These numbers are then ordered from least to greatest. Find the number that occupies the \(100^{th}\) position in this ordered list.

Answer: 51342


There is a graph with 30 vertices. If any 26 of its vertices, along with their outgoing edges, are deleted, the remaining graph is a connected graph with 4 vertices. What is the smallest number of edges in the initial graph with 30 vertices?

Answer: 405


Five individuals are assigned to five specific seats, but they do not know their assigned numbers. When these five individuals sit randomly in these five seats, find:
(1) The probability that exactly 3 out of the 5 individuals sit in their assigned seats.
(2) If the probability that all 5 individuals sit in their assigned seats is not less than $\frac{1}{6}$, what is the maximum number of people who can sit in their assigned seats?

Answer: 2


For each positive integer \( n \), define the function
\[
f(n) = \begin{cases}
0, & \text{if } n \text{ is a perfect square}, \\
\left\lfloor \frac{1}{\{\sqrt{n}\}} \right\rfloor, & \text{if } n \text{ is not a perfect square}.
\end{cases}
\]
(where \( \left\lfloor x \right\rfloor \) denotes the greatest integer less than or equal to \( x \), and \( \{x\} = x - \left\lfloor x \right\rfloor \) ). Find the value of \( \sum_{k=1}^{240} f(k) \).

Answer: 768


Point \( I \) is the intersection point of the angle bisectors of triangle \( ABC \). Point \( X \) is marked on side \( BC \). It turns out that \( AI = BX \), \( AC = CX \), and \(\angle ABC = 42^\circ\).

(a) (2 points) How many degrees is angle \( BIX \)?

(b) (2 points) How many degrees is angle \( BCA \)?

Answer: 54


Let
$$
P=\prod_{i=0}^{2016}\left(i^{3}-i-1\right)^{2} .
$$

The remainder when \( P \) is divided by the prime 2017 is not zero. Compute this remainder.

Answer: 1994


In the following figure, the square \(ABCD\) is divided into three rectangles of equal area.

If the length of the segment \(BM\) is equal to 4, calculate the area of the square \(ABCD\).

Answer: 144


In a single round-robin table tennis tournament, a) 10 players, b) 11 players competed. What is the minimum number of matches the champion must have won, given that there was no tie for first place?

Answer: 6


Figure 1 is a rhombus-shaped piece of paper composed of 2 small equilateral triangles. Figure 2 is a fixed regular hexagonal chessboard $ABCDEF$, consisting of 24 small equilateral triangles of the same size. Now, 12 rhombus-shaped paper pieces are used to completely cover the hexagonal chessboard. There are $\qquad$ different ways to cover it.

Answer: 20


Immediately after harvest, the water content in 1 ton of tomatoes was $99 \%$. By the time of sale, the water content decreased by $4 \%$ (after drying). What is the total weight of the tomatoes now?

Answer: 200


Consider the following two-player game. Player 1 starts with a number, \( N \). He then subtracts a proper divisor of \( N \) from \( N \) and gives the result to Player 2 (a proper divisor of \( N \) is a positive divisor of \( N \) that is not equal to 1 or \( N \)). Player 2 does the same thing with the number she gets from Player 1, and gives the result back to Player 1. The two players continue until a player is given a prime number, at which point that player loses. For how many values of \( N \) between 2 and 100 inclusive does Player 1 have a winning strategy?

Answer: 47


Each square on an $8 \times 8$ checkers board contains either one or zero checkers. The number of checkers in each row is a multiple of $3$, and the number of checkers in each column is a multiple of $5$. Assuming the top left corner of the board is shown below, how many checkers are used in total?

Answer: 30


Michael's task is to determine how many 7-digit telephone numbers can exist such that all seven digits are distinct, the first four digits are in descending order, and the last four digits are in ascending order.

Given a 7-digit number format, consider the number $987-1234$ where the first four digits $(9-8-7-1)$ descend and the last four digits $(1-2-3-4)$ ascend.

Calculate the total number of 7-digit telephone numbers possible under these conditions.

Answer: 2400


A *strip* is the region between two parallel lines. Let $A$ and $B$ be two strips in a plane. The intersection of strips $A$ and $B$ is a parallelogram $P$. Let $A'$ be a rotation of $A$ in the plane by $60^\circ$. The intersection of strips $A'$ and $B$ is a parallelogram with the same area as $P$. Let $x^\circ$ be the measure (in degrees) of one interior angle of $P$. What is the greatest possible value of the number $x$?

Answer: 150


Three uncoordinated aliens launch a three-day attack on four galaxies. Each day, each of the three aliens chooses a galaxy uniformly at random from the remaining galaxies and destroys it. They make their choice simultaneously and independently, so two aliens could destroy the same galaxy. If the probability that every galaxy is destroyed by the end of the attack can be expressed as \( \frac{m}{n} \) for relatively prime positive integers \( m \) and \( n \), what is \( m+n \)?

Answer: 1151


Each number in the list $1, 2, 3, \ldots, 10$ is either colored red or blue. Numbers are colored independently, and both colors are equally probable. The expected value of the number of positive integers expressible as a sum of a red integer and a blue integer can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. What is $m+n$?

Answer: 455


Let $s_n$ be the number of solutions to the equation \(a_1 + a_2 + a_3 + a_4 + b_1 + b_2 = n\), where \(a_1, a_2, a_3,\) and \(a_4\) are elements of the set \(\{2, 3, 5, 7\}\) and \(b_1\) and \(b_2\) are elements of the set \(\{1, 2, 3, 4\}\). Find the number of \(n\) for which \(s_n\) is odd.

Answer: 12


The $12$-sided polygon below was created by placing three $3 \times 3$ squares with their sides parallel so that vertices of two of the squares are at the center of the third square. Find the perimeter of this $12$-sided polygon.

Answer: 24


Let $\pi$ be a permutation of the numbers from $1$ through $2012$. What is the maximum possible number of integers $n$ with $1 \le n \le 2011$ such that $\pi(n)$ divides $\pi(n + 1)$?

Answer: 1006


Consider the graph in 3-space of
\[0 = xyz(x + y)(y + z)(z + x)(x - y)(y - z)(z - x).\]
This graph divides 3-space into \(N\) connected regions. What is \(N\)?

Answer: 48


Vasya has $n$ candies of several types, where $n > 145$. It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of $n$.

Answer: 160


In the diagram below, triangle $ABC$ has been reflected over its median $\overline{AM}$ to produce triangle $AB'C'$.  If $AE = 6$, $EC =12$, and $BD = 10$, then find $AB$.

[asy]
size(250);
pair A,B,C,D,M,BB,CC,EE;
B = (0,0);
D = (10,0);
M = (15,0);
C=2*M;
A = D + (scale(1.2)*rotate(aCos((225-144-25)/120))*(M-D));
CC = D + D + D - A - A;
BB = reflect(A,M)*B;
EE = reflect(A,M)*D;
draw(M--A--BB--CC--A--B--C--A);
label("$M$",M,SE);
label("$A$",A,N);
label("$B$",B,SW);
label("$C$",C,SE);
label("$C'$",CC,S);
label("$B'$",BB,E);
label("$D$",D,NW);
label("$E$",EE,N);
label("$12$",(EE+C)/2,N);
label("$6$",(A+EE)/2,S);
label("$10$",D/2,S);
[/asy]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 11


Compute the value of the infinite series \[
\sum_{n=2}^{\infty} \frac{n^4+3n^2+10n+10}{2^n \cdot \left(n^4+4\right)}
\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 21


The diameter $AB$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and line $ED$ is perpendicular to line $AD$. Segment $AE$ intersects the circle at a point $C$ between $A$ and $E$. What is the area of $\triangle  ABC$?
$\textbf{(A)}\ \frac{120}{37}\qquad\textbf{(B)}\ \frac{140}{39}\qquad\textbf{(C)}\ \frac{145}{39}\qquad\textbf{(D)}\ \frac{140}{37}\qquad\textbf{(E)}\ \frac{120}{31}$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 177


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`
 - Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `desert`, `watermelon`
 - Everyone has a unique favorite cigar: `blue master`, `pall mall`, `dunhill`, `prince`
 - People have unique heights: `tall`, `average`, `short`, `very short`
 - People use unique phone models: `google pixel 6`, `samsung galaxy s21`, `iphone 13`, `oneplus 9`

## Clues:
1. The Dragonfruit smoothie lover is Eric.
2. The Dunhill smoker is the person who likes Cherry smoothies.
3. The person who uses a Samsung Galaxy S21 is directly left of the person who uses an iPhone 13.
4. The Dunhill smoker is somewhere to the right of the person who is very short.
5. The Watermelon smoothie lover is somewhere to the right of the Desert smoothie lover.
6. The Prince smoker is the person who uses a OnePlus 9.
7. The person who is tall is in the third house.
8. The person who is very short is the person who uses an iPhone 13.
9. The person who smokes Blue Master is not in the first house.
10. The Dunhill smoker is the person who is short.
11. Peter is not in the third house.
12. Arnold is the person who uses a Google Pixel 6.
13. The Dragonfruit smoothie lover is the person partial to Pall Mall.
 What is the value of attribute Smoothie for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: watermelon


There are 2 houses, numbered 1 to 2 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`
 - Each person has an occupation: `engineer`, `doctor`
 - Each person has a unique birthday month: `april`, `sept`
 - Each person lives in a unique style of house: `victorian`, `colonial`
 - People have unique heights: `very short`, `short`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`

## Clues:
1. The person who is an engineer is in the first house.
2. The person whose birthday is in April and the person who is a doctor are next to each other.
3. The person living in a colonial-style house is the person who is an engineer.
4. The person who is very short is the person who is an engineer.
5. The person who is short is the person partial to Pall Mall.
6. The person who is an engineer is Eric.
 What is the value of attribute Cigar for the person whose attribute HouseStyle is victorian? Please reason step by step, and put your final answer within \boxed{}

Answer: pall mall


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`
 - Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `desert`, `watermelon`
 - People have unique favorite sports: `soccer`, `tennis`, `basketball`, `swimming`
 - People own unique car models: `tesla model 3`, `toyota camry`, `honda civic`, `ford f150`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`

## Clues:
1. The person who owns a Tesla Model 3 is the person who loves the rose bouquet.
2. Peter is the Dragonfruit smoothie lover.
3. The Desert smoothie lover is the person who owns a Toyota Camry.
4. The person who loves tennis is in the first house.
5. The person who owns a Toyota Camry and the person who loves basketball are next to each other.
6. Arnold is the person who loves basketball.
7. The person who owns a Honda Civic is the person who loves a bouquet of daffodils.
8. Eric is the person who loves the rose bouquet.
9. The Watermelon smoothie lover is not in the first house.
10. The person who owns a Honda Civic is somewhere to the right of the Desert smoothie lover.
11. The person who loves basketball is the person who loves the boquet of lilies.
12. The person who loves tennis and the person who loves soccer are next to each other.
 What is the value of attribute House for the person whose attribute CarModel is tesla model 3? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`
 - The mothers' names in different houses are unique: `Janelle`, `Aniya`, `Holly`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`
 - Each person has a unique type of pet: `cat`, `dog`, `fish`

## Clues:
1. The person who owns a dog is in the first house.
2. The person whose mother's name is Janelle is the person with an aquarium of fish.
3. There is one house between Eric and The person whose mother's name is Holly.
4. The person who loves mystery books is in the first house.
5. The person who loves romance books is directly left of Peter.
 What is the value of attribute BookGenre for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: mystery


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

## Clues:
1. The one who only drinks water is in the first house.
2. The person who loves tennis is Peter.
3. The person who loves a bouquet of daffodils is The person whose mother's name is Aniya.
4. The person who loves soccer is Arnold.
5. Arnold is directly left of The person whose mother's name is Janelle.
6. The person who loves basketball is not in the third house.
7. The person whose mother's name is Janelle is the person who likes milk.
8. The person whose mother's name is Aniya is in the second house.
9. There is one house between the person who loves a carnations arrangement and the person who loves tennis.
 What is the value of attribute Drink for the person whose attribute Flower is daffodils? Please reason step by step, and put your final answer within \boxed{}

Answer: tea


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

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

Answer: stew


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Bob`, `Eric`, `Arnold`
 - The people are of nationalities: `dane`, `swede`, `norwegian`, `brit`, `german`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`, `dragonfruit`, `lime`
 - They all have a unique favorite flower: `tulips`, `roses`, `daffodils`, `carnations`, `lilies`
 - Each person lives in a unique style of house: `ranch`, `modern`, `victorian`, `colonial`, `craftsman`
 - The people keep unique animals: `dog`, `horse`, `bird`, `fish`, `cat`

## Clues:
1. The person living in a colonial-style house is Peter.
2. The person who keeps horses is the person who drinks Lime smoothies.
3. Alice is the British person.
4. The Swedish person is in the first house.
5. The person who keeps horses and the dog owner are next to each other.
6. There is one house between the person in a Craftsman-style house and the Dane.
7. The person who loves the boquet of lilies is directly left of the fish enthusiast.
8. The person who loves the boquet of lilies is the person in a modern-style house.
9. The Desert smoothie lover is in the first house.
10. The person residing in a Victorian house is somewhere to the right of the Watermelon smoothie lover.
11. The person who loves a bouquet of daffodils is Alice.
12. The Dragonfruit smoothie lover is not in the third house.
13. The person who loves a carnations arrangement is the German.
14. The Dragonfruit smoothie lover is Eric.
15. Bob is somewhere to the right of the person who loves the rose bouquet.
16. The Watermelon smoothie lover is the Norwegian.
17. Bob is the person in a modern-style house.
18. The person residing in a Victorian house is not in the fifth house.
19. The person who loves a carnations arrangement is somewhere to the left of the person in a modern-style house.
20. The Desert smoothie lover is the bird keeper.
 What is the value of attribute House for the person whose attribute Smoothie is watermelon? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`
 - Each person has a unique type of pet: `cat`, `fish`, `dog`
 - People have unique favorite book genres: `science fiction`, `mystery`, `romance`
 - Everyone has something unique for lunch: `spaghetti`, `pizza`, `grilled cheese`

## Clues:
1. The person who owns a Ford F-150 is not in the first house.
2. The person who owns a Ford F-150 is not in the third house.
3. The person who loves science fiction books is somewhere to the right of the person who is a pizza lover.
4. The person with an aquarium of fish is the person who is a pizza lover.
5. The person who is a pizza lover is somewhere to the left of Peter.
6. Peter is the person who owns a Tesla Model 3.
7. The person who loves science fiction books is directly left of the person who loves mystery books.
8. Arnold is the person who loves the spaghetti eater.
9. The person who has a cat is the person who loves the spaghetti eater.
 What is the value of attribute BookGenre for the person whose attribute Pet is fish? Please reason step by step, and put your final answer within \boxed{}

Answer: romance


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`
 - Everyone has a unique favorite cigar: `blue master`, `prince`, `pall mall`
 - Each person has a unique hobby: `photography`, `gardening`, `cooking`
 - Each person has a unique level of education: `high school`, `associate`, `bachelor`
 - Each person has a unique favorite drink: `tea`, `milk`, `water`

## Clues:
1. The person partial to Pall Mall is Peter.
2. The person who likes milk is directly left of the person with a high school diploma.
3. Eric is the tea drinker.
4. Arnold and the Prince smoker are next to each other.
5. The person who enjoys gardening is somewhere to the left of the Prince smoker.
6. The person who likes milk is the person with an associate's degree.
7. The person with a bachelor's degree is directly left of the photography enthusiast.
 What is the value of attribute Cigar for the person whose attribute Drink is tea? Please reason step by step, and put your final answer within \boxed{}

Answer: prince


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`, `honda civic`, `bmw 3 series`
 - The people are of nationalities: `brit`, `norwegian`, `german`, `swede`, `dane`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `lilies`, `tulips`, `roses`

## Clues:
1. The person who owns a Tesla Model 3 is the person who loves a carnations arrangement.
2. The person who owns a Toyota Camry is Alice.
3. The person who owns a BMW 3 Series is somewhere to the right of Arnold.
4. Arnold is not in the third house.
5. The Swedish person is in the first house.
6. The German is the person who loves a bouquet of daffodils.
7. Arnold is not in the second house.
8. The Norwegian is the person who owns a Tesla Model 3.
9. The Dane is somewhere to the left of the person who owns a Ford F-150.
10. The person who owns a BMW 3 Series is somewhere to the right of the person who loves a carnations arrangement.
11. Alice is directly left of the person who loves the rose bouquet.
12. Peter and the person who loves the vase of tulips are next to each other.
13. The British person is somewhere to the right of the Norwegian.
14. The person who loves the vase of tulips is Eric.
15. The person who owns a Ford F-150 is Eric.
 What is the value of attribute Flower for the person whose attribute House is 5? Please reason step by step, and put your final answer within \boxed{}

Answer: tulips


Lisa writes a positive whole number in the decimal system on the blackboard and now makes in each turn the following:
The last digit is deleted from the number on the board and then the remaining shorter number (or 0 if the number was one digit) becomes four times the number deleted number added. The number on the board is now replaced by the result of this calculation.
Lisa repeats this until she gets a number for the first time was on the board.
(a) Show that the sequence of moves always ends.
(b) If Lisa begins with the number $53^{2022} - 1$, what is the last number on the board?

Example: If Lisa starts with the number $2022$, she gets $202 + 4\cdot 2 = 210$ in the first move and overall the result $$2022 \to 210 \to 21 \to 6 \to 24 \to 18 \to 33 \to 15 \to 21$$.
Since Lisa gets $21$ for the second time, the turn order ends.

[i](Stephan Pfannerer)[/i]

Answer: 39


1. As shown in the figure, $\angle A C B=90^{\circ}, A C=B C, A D \perp C E$, $B E \perp C E$, with the feet of the perpendiculars being $D, E$ respectively. Given $A D=8$, $B E=3$, then $D E=$ $\qquad$

Answer: 5


3. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{n+1}=a_{n}^{2}-2\left(n \in \mathbf{N}_{+}\right) \text {, }
$$

and $a_{1}=a, a_{2012}=b(a 、 b>2)$.
Then $a_{1} a_{2} \cdots a_{2011}=$
(express in terms of $a, b$).

Answer: \sqrt{\dfrac{b^2 - 4}{a^2 - 4}}


Let \( x \) and \( y \) be strictly positive real numbers. Show that \( x+\frac{y^{2}}{x} \geqslant 2y \) and find the cases of equality.

Answer: x + \frac{y^2}{x} \geq 2y


$3-$

The city plan has a layout represented by a $5 \times 10$ rectangular grid. One-way traffic is in effect on the streets: travel is only allowed to the right and upwards. How many different routes are there from the bottom-left corner to the top-right corner?

Answer: 3003


3.52 A passenger on a train knows that the speed of this train on this section of the track is 40 km/h. As soon as a passing train started to go by the window, the passenger started a stopwatch and noticed that the passing train went by the window for 3 seconds. Determine the speed of the passing train, given that its length is 75 m.

## Work

## Paбoma

## Work

Answer: 50


9.1. There are seven cards on the table. In one move, it is allowed to flip any five cards. What is the minimum number of moves required to flip all the cards?

Answer: 3


# Problem 2.

In a football tournament, each team is supposed to play one match against each of the others. But during the tournament, half of all the teams were disqualified and withdrew from further participation. As a result, 77 matches were played, and the teams that withdrew had managed to play all their matches among themselves, with each of the withdrawn teams playing the same number of matches. How many teams were there at the beginning of the tournament?

Answer: 14


Task 5. (20 points) At the first deposit, equipment of the highest class was used, and at the second deposit, equipment of the first class was used, with the highest class being less than the first. Initially, $40 \%$ of the equipment from the first deposit was transferred to the second. Then, $20 \%$ of the equipment that ended up on the second deposit was transferred back to the first, with half of the transferred equipment being of the first class. After this, the equipment of the highest class on the first deposit was 26 units more than on the second, and the total amount of equipment on the second deposit increased by more than $5 \%$ compared to the initial amount. Find the total amount of equipment of the first class.

Answer: 60


Let $f:\mathbb{N}\mapsto\mathbb{R}$ be the function \[f(n)=\sum_{k=1}^\infty\dfrac{1}{\operatorname{lcm}(k,n)^2}.\] It is well-known that $f(1)=\tfrac{\pi^2}6$.  What is the smallest positive integer $m$ such that $m\cdot f(10)$ is the square of a rational multiple of $\pi$?

Answer: 42


Jackson begins at $1$ on the number line. At each step, he remains in place with probability $85\%$ and increases his position on the number line by $1$ with probability $15\%$. Let $d_n$ be his position on the number line after $n$ steps, and let $E_n$ be the expected value of $\tfrac{1}{d_n}$. Find the least $n$ such that $\tfrac{1}{E_n}
> 2017$.

Answer: 13446


8. If integers $x, y$ satisfy $\frac{1}{x}-\frac{1}{y}=\frac{1}{2023}$, then the number of such integer pairs $(x, y)$ is $\qquad$ pairs.

Answer: 29


5. Vova has 19 math grades in his journal, all twos and threes, and the first four grades are twos. It turned out that among the quartets of consecutive grades, all 16 possible combinations of four twos and threes are present. What are Vova's last four grades?

Answer: 3222


5. Given a numerical riddle: TETA + BETA = GAMMA. (Different letters correspond to different digits, the same letters correspond to the same digits.) Find all its solutions and prove that there are no others.

Answer: 4940 + 5940 = 10880


11.3. For what smallest $n$ is the following condition satisfied: if $n$ crosses are placed in some cells of a $6 \times 6$ table in any order (no more than one per cell), then there will definitely be three cells forming a strip of length 3, either vertical or horizontal, each containing a cross?

Answer: 13


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`, `Bob`, `Alice`
 - People use unique phone models: `google pixel 6`, `samsung galaxy s21`, `huawei p50`, `oneplus 9`, `iphone 13`
 - People have unique favorite book genres: `science fiction`, `romance`, `biography`, `mystery`, `fantasy`

## Clues:
1. Peter is the person who uses a Huawei P50.
2. The person who loves fantasy books is directly left of Alice.
3. Bob is in the second house.
4. There are two houses between the person who uses a OnePlus 9 and the person who loves biography books.
5. Arnold is the person who loves romance books.
6. The person who loves fantasy books is the person who uses a OnePlus 9.
7. The person who uses a Samsung Galaxy S21 is not in the fifth house.
8. The person who uses an iPhone 13 is the person who loves science fiction books.
9. Peter is directly left of the person who uses a OnePlus 9.
 What is the value of attribute House for the person whose attribute PhoneModel is huawei p50? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`
 - Each person has a unique hobby: `gardening`, `photography`, `cooking`
 - Each person prefers a unique type of vacation: `mountain`, `city`, `beach`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`, `blue master`
 - The mothers' names in different houses are unique: `Janelle`, `Holly`, `Aniya`
 - People have unique heights: `short`, `average`, `very short`

## Clues:
1. The person who loves beach vacations is somewhere to the right of the person who is very short.
2. The person whose mother's name is Aniya is somewhere to the right of the person who loves cooking.
3. Eric is not in the second house.
4. The person who is short is in the third house.
5. The person who prefers city breaks is the photography enthusiast.
6. The person who enjoys gardening is the person who enjoys mountain retreats.
7. The person partial to Pall Mall is the photography enthusiast.
8. The person who enjoys mountain retreats is The person whose mother's name is Holly.
9. The person who enjoys gardening is the person who smokes Blue Master.
10. Peter is The person whose mother's name is Aniya.
 What is the value of attribute Height for the person whose attribute Hobby is gardening? Please reason step by step, and put your final answer within \boxed{}

Answer: very short


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`
 - People have unique favorite sports: `soccer`, `basketball`, `tennis`
 - Each person lives in a unique style of house: `victorian`, `colonial`, `ranch`
 - Each person has a unique type of pet: `fish`, `dog`, `cat`
 - Each person has a unique birthday month: `sept`, `april`, `jan`
 - The people are of nationalities: `swede`, `dane`, `brit`

## Clues:
1. The person who loves soccer is the person residing in a Victorian house.
2. The person who owns a dog is the person whose birthday is in April.
3. The person who owns a dog is the person residing in a Victorian house.
4. The person living in a colonial-style house is Arnold.
5. The person living in a colonial-style house is the person whose birthday is in January.
6. Arnold is directly left of the person whose birthday is in April.
7. The British person is the person who loves basketball.
8. The Swedish person is the person whose birthday is in April.
9. The Dane is Eric.
10. The person who has a cat is the person whose birthday is in January.
11. The person who loves soccer is in the third house.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 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 prefers a unique type of vacation: `mountain`, `city`, `beach`
 - People have unique heights: `very short`, `average`, `short`
 - They all have a unique favorite flower: `carnations`, `daffodils`, `lilies`
 - People have unique hair colors: `brown`, `black`, `blonde`
 - Each person has a unique level of education: `associate`, `bachelor`, `high school`

## Clues:
1. Peter is the person who has an average height.
2. The person who loves a bouquet of daffodils is Arnold.
3. The person who is very short is not in the second house.
4. The person who loves beach vacations is in the first house.
5. The person with a high school diploma is in the third house.
6. The person who is short is somewhere to the right of the person who is very short.
7. The person who loves the boquet of lilies is Eric.
8. The person who loves the boquet of lilies is the person with a bachelor's degree.
9. The person who prefers city breaks is somewhere to the right of Peter.
10. The person who has blonde hair is in the third house.
11. The person who loves beach vacations is the person who has brown hair.
 What is the value of attribute Height for the person whose attribute Vacation is city? Please reason step by step, and put your final answer within \boxed{}

Answer: short


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`
 - Each person prefers a unique type of vacation: `mountain`, `cruise`, `beach`, `city`
 - Each person has a unique birthday month: `jan`, `feb`, `sept`, `april`
 - The people are of nationalities: `swede`, `dane`, `brit`, `norwegian`

## Clues:
1. The person who enjoys mountain retreats is Alice.
2. The Norwegian is somewhere to the left of the person whose birthday is in September.
3. The person who prefers city breaks is the British person.
4. Alice and Eric are next to each other.
5. The person who likes going on cruises is not in the fourth house.
6. Alice is somewhere to the left of the person whose birthday is in January.
7. The person who enjoys mountain retreats is the Norwegian.
8. Alice is directly left of the Swedish person.
9. Eric is in the third house.
10. The person whose birthday is in April is Peter.
11. The Dane is the person whose birthday is in September.
 What is the value of attribute Vacation for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: mountain


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

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

Answer: Peter


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Arnold`, `Peter`, `Alice`
 - Each person has a unique hobby: `painting`, `knitting`, `cooking`, `photography`, `gardening`

## Clues:
1. Alice is in the fifth house.
2. Bob is directly left of the person who enjoys gardening.
3. The person who loves cooking and the person who paints as a hobby are next to each other.
4. The person who paints as a hobby is not in the third house.
5. The photography enthusiast is not in the third house.
6. Arnold is in the fourth house.
7. Bob is in the first house.
8. The person who enjoys knitting is directly left of Eric.
 What is the value of attribute Name for the person whose attribute House is 3? 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`, `Arnold`, `Eric`, `Peter`
 - Each mother is accompanied by their child: `Samantha`, `Meredith`, `Fred`, `Bella`
 - People have unique hair colors: `red`, `black`, `brown`, `blonde`

## Clues:
1. The person who has red hair is directly left of Eric.
2. The person's child is named Meredith is directly left of the person who has brown hair.
3. The person who has blonde hair is the person's child is named Fred.
4. Peter is in the fourth house.
5. The person who has black hair is directly left of Arnold.
6. The person's child is named Bella is Arnold.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Peter`, `Alice`, `Carol`, `Eric`
 - Everyone has something unique for lunch: `stew`, `grilled cheese`, `stir fry`, `soup`, `pizza`, `spaghetti`
 - People have unique heights: `tall`, `average`, `super tall`, `very short`, `very tall`, `short`
 - Each person has a unique favorite drink: `root beer`, `boba tea`, `coffee`, `water`, `tea`, `milk`
 - Each person has a unique type of pet: `hamster`, `fish`, `cat`, `dog`, `bird`, `rabbit`
 - People use unique phone models: `samsung galaxy s21`, `xiaomi mi 11`, `google pixel 6`, `iphone 13`, `huawei p50`, `oneplus 9`

## Clues:
1. The person who uses an iPhone 13 is in the third house.
2. Bob is the person who is tall.
3. The person who loves the soup is in the second house.
4. The root beer lover is directly left of the person who uses a Xiaomi Mi 11.
5. The person who uses a Huawei P50 is directly left of the person who loves eating grilled cheese.
6. The person who loves stir fry is the person who likes milk.
7. The person who loves eating grilled cheese is the person who is tall.
8. The person who uses a Xiaomi Mi 11 is the coffee drinker.
9. The person who uses a OnePlus 9 is Arnold.
10. The person who owns a rabbit is not in the fifth house.
11. The person with a pet hamster is somewhere to the right of the person who uses a Google Pixel 6.
12. The person who is super tall is the person with an aquarium of fish.
13. The person with an aquarium of fish is Alice.
14. The tea drinker is directly left of the person who is a pizza lover.
15. The person who uses a Samsung Galaxy S21 is Carol.
16. The person who is a pizza lover is the person who is short.
17. Arnold is the person who is very tall.
18. The person who loves the spaghetti eater is the person who uses a Google Pixel 6.
19. The boba tea drinker is somewhere to the right of the person who loves the soup.
20. The person with a pet hamster is not in the fifth house.
21. The person who is very tall is not in the second house.
22. The person who is super tall is somewhere to the left of Peter.
23. The person who is very short is the person who loves the spaghetti eater.
24. The person who keeps a pet bird is somewhere to the left of the person who loves the spaghetti eater.
25. The person with an aquarium of fish is directly left of Eric.
26. The person who owns a dog is the person who likes milk.
 What is the value of attribute Pet for the person whose attribute PhoneModel is iphone 13? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Alice`, `Arnold`
 - People have unique hair colors: `brown`, `black`, `blonde`, `red`
 - People have unique favorite sports: `tennis`, `basketball`, `soccer`, `swimming`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `dragonfruit`, `desert`
 - Each person has a unique birthday month: `april`, `sept`, `feb`, `jan`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `iphone 13`, `google pixel 6`

## Clues:
1. The person whose birthday is in January is somewhere to the left of the person who loves basketball.
2. The person whose birthday is in February is directly left of the person who has red hair.
3. Eric is the person who uses a OnePlus 9.
4. The person whose birthday is in April is Arnold.
5. The person who uses a Samsung Galaxy S21 and the person who likes Cherry smoothies are next to each other.
6. The person who has blonde hair and Peter are next to each other.
7. The person who has black hair is somewhere to the right of the person who has brown hair.
8. The person who has black hair is the person whose birthday is in February.
9. The Dragonfruit smoothie lover is in the second house.
10. The person who loves soccer is somewhere to the right of the person who uses a OnePlus 9.
11. There are two houses between the person who loves tennis and the person who loves basketball.
12. The person who has black hair is the person who uses a Samsung Galaxy S21.
13. Alice is the person who likes Cherry smoothies.
14. The Desert smoothie lover is the person who uses a Google Pixel 6.
 What is the value of attribute PhoneModel for the person whose attribute FavoriteSport is tennis? Please reason step by step, and put your final answer within \boxed{}

Answer: google pixel 6


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Eric`, `Peter`, `Alice`, `Bob`, `Arnold`
 - Everyone has something unique for lunch: `spaghetti`, `soup`, `grilled cheese`, `stir fry`, `stew`, `pizza`
 - Each person has a favorite color: `green`, `purple`, `blue`, `red`, `white`, `yellow`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Aniya`, `Kailyn`, `Sarah`, `Janelle`
 - Each person lives in a unique style of house: `colonial`, `craftsman`, `modern`, `ranch`, `victorian`, `mediterranean`

## Clues:
1. The person whose mother's name is Janelle is the person living in a colonial-style house.
2. Peter is the person whose favorite color is green.
3. The person who loves blue is directly left of the person who loves the spaghetti eater.
4. The person whose mother's name is Penny is directly left of the person whose favorite color is red.
5. The person who loves blue is The person whose mother's name is Penny.
6. The person whose mother's name is Aniya is directly left of the person whose favorite color is green.
7. The person who loves white is somewhere to the left of The person whose mother's name is Janelle.
8. Eric is the person who loves eating grilled cheese.
9. The person whose mother's name is Holly is directly left of Alice.
10. The person whose mother's name is Holly and the person who loves the stew are next to each other.
11. Alice is the person in a ranch-style home.
12. Carol is not in the sixth house.
13. The person who loves stir fry is the person who loves purple.
14. The person whose favorite color is red and the person living in a colonial-style house are next to each other.
15. The person who loves the soup is Peter.
16. The person who is a pizza lover is somewhere to the left of The person whose mother's name is Sarah.
17. The person residing in a Victorian house is the person who loves eating grilled cheese.
18. The person who loves the spaghetti eater is not in the fifth house.
19. There are two houses between the person who loves eating grilled cheese and The person whose mother's name is Aniya.
20. Arnold is the person whose favorite color is red.
21. The person in a modern-style house is Arnold.
22. The person who loves yellow is the person in a Mediterranean-style villa.
 What is the value of attribute House for the person whose attribute HouseStyle is colonial? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


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

## Clues:
1. The person who loves beach vacations is the person who loves yellow.
2. Alice is the person who loves white.
3. The person whose favorite color is green is somewhere to the right of the person who likes going on cruises.
4. The person whose favorite color is red is somewhere to the right of the person whose favorite color is green.
5. Arnold is the person whose favorite color is red.
6. There are two houses between Eric and the person who prefers city breaks.
 What is the value of attribute Color for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: yellow


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

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

Answer: swede


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

Answer: very short


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`, `Bob`
 - Everyone has something unique for lunch: `stir fry`, `spaghetti`, `stew`, `grilled cheese`, `pizza`
 - People own unique car models: `ford f150`, `tesla model 3`, `bmw 3 series`, `toyota camry`, `honda civic`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `oneplus 9`, `huawei p50`
 - Each person has an occupation: `teacher`, `lawyer`, `doctor`, `artist`, `engineer`
 - Each person has a unique favorite drink: `tea`, `milk`, `water`, `root beer`, `coffee`

## Clues:
1. The root beer lover is the person who owns a Honda Civic.
2. The person who likes milk is directly left of the person who loves eating grilled cheese.
3. Alice is the person who uses a Samsung Galaxy S21.
4. Alice is the person who loves stir fry.
5. The tea drinker is not in the fifth house.
6. The person who owns a BMW 3 Series is somewhere to the left of the tea drinker.
7. The person who is a doctor is Arnold.
8. The person who uses an iPhone 13 is the coffee drinker.
9. The person who is an engineer is the person who owns a BMW 3 Series.
10. The person who loves the stew is the person who uses an iPhone 13.
11. The person who is a doctor is directly left of the person who uses a OnePlus 9.
12. The person who owns a Honda Civic is directly left of the person who loves the spaghetti eater.
13. The person who uses a Google Pixel 6 is the tea drinker.
14. Alice is the person who is an artist.
15. There is one house between Alice and the person who owns a Ford F-150.
16. Arnold is the person who owns a Toyota Camry.
17. Eric is in the fourth house.
18. The person who uses a OnePlus 9 is the person who is a lawyer.
19. The person who loves eating grilled cheese is Peter.
 What is the value of attribute CarModel for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: toyota camry


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`
 - The mothers' names in different houses are unique: `Holly`, `Kailyn`, `Janelle`, `Aniya`
 - They all have a unique favorite flower: `carnations`, `roses`, `lilies`, `daffodils`

## Clues:
1. Alice is The person whose mother's name is Kailyn.
2. The person whose mother's name is Janelle is somewhere to the right of Arnold.
3. Peter is somewhere to the right of the person who loves a carnations arrangement.
4. Eric is the person who loves a bouquet of daffodils.
5. Arnold is The person whose mother's name is Holly.
6. The person who loves a carnations arrangement is somewhere to the right of The person whose mother's name is Holly.
7. The person who loves the boquet of lilies is directly left of Alice.
8. Alice is in the third house.
 What is the value of attribute House for the person whose attribute Mother is Janelle? Please reason step by step, and put your final answer within \boxed{}

Answer: 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`, `Peter`, `Eric`
 - Each mother is accompanied by their child: `Bella`, `Meredith`, `Fred`, `Samantha`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`, `stew`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`

## Clues:
1. The person who is a pizza lover is somewhere to the left of the person's child is named Samantha.
2. The person whose favorite color is red is directly left of the person who loves white.
3. Eric is somewhere to the right of the person who loves the stew.
4. The person who loves white is in the fourth house.
5. The person whose favorite color is green is not in the second house.
6. The person whose favorite color is red is the person's child is named Bella.
7. The person who loves eating grilled cheese is in the second house.
8. There is one house between the person's child is named Meredith and Alice.
9. There is one house between Peter and Alice.
10. Alice is not in the third house.
 What is the value of attribute Children for the person whose attribute Color is yellow? Please reason step by step, and put your final answer within \boxed{}

Answer: Meredith


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `pizza`
 - Each person has a unique birthday month: `april`, `jan`, `sept`

## Clues:
1. The person whose birthday is in April is Arnold.
2. The person whose birthday is in January is somewhere to the left of the person who is a pizza lover.
3. Peter is the person who loves eating grilled cheese.
4. The person who loves the spaghetti eater is Arnold.
5. Peter is directly left of Arnold.
 What is the value of attribute Birthday for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Peter`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `cherry`, `dragonfruit`
 - Each person lives in a unique style of house: `craftsman`, `colonial`, `ranch`, `victorian`
 - Each person has a unique hobby: `cooking`, `photography`, `gardening`, `painting`
 - Each person has a unique favorite drink: `coffee`, `tea`, `milk`, `water`

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

Answer: watermelon


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Alice`, `Arnold`, `Carol`, `Peter`
 - The people are of nationalities: `swede`, `norwegian`, `german`, `chinese`, `dane`, `brit`
 - They all have a unique favorite flower: `daffodils`, `roses`, `carnations`, `lilies`, `tulips`, `iris`
 - Each person has a unique favorite drink: `water`, `tea`, `root beer`, `coffee`, `boba tea`, `milk`
 - Each person has a favorite color: `green`, `blue`, `purple`, `red`, `yellow`, `white`
 - Each person has a unique level of education: `high school`, `associate`, `bachelor`, `doctorate`, `trade school`, `master`

## Clues:
1. Arnold is somewhere to the left of the person who loves a carnations arrangement.
2. The coffee drinker is the person who loves the vase of tulips.
3. The German is directly left of the one who only drinks water.
4. The person whose favorite color is green is Alice.
5. The person who loves white is the coffee drinker.
6. The coffee drinker is the person with a bachelor's degree.
7. The one who only drinks water is the person who loves purple.
8. The person who loves a bouquet of daffodils is the person who loves blue.
9. The tea drinker is directly left of the person who loves a carnations arrangement.
10. The person with a high school diploma is Carol.
11. There is one house between the person who loves the boquet of iris and Peter.
12. The person with an associate's degree is the person whose favorite color is green.
13. Arnold is the Chinese.
14. The person whose favorite color is red is in the third house.
15. The boba tea drinker and the person who loves the vase of tulips are next to each other.
16. Alice is directly left of the Swedish person.
17. There is one house between the British person and the Chinese.
18. The person who loves the rose bouquet is somewhere to the left of the person with a bachelor's degree.
19. The person who loves yellow is not in the first house.
20. There are two houses between the person who likes milk and Peter.
21. Bob is the person with a master's degree.
22. The Swedish person is the person with a master's degree.
23. The person who attended trade school is somewhere to the left of the Chinese.
24. The German is in the fifth house.
25. The Norwegian is the one who only drinks water.
 What is the value of attribute Nationality for the person whose attribute Drink is tea? Please reason step by step, and put your final answer within \boxed{}

Answer: german


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Bob`, `Eric`, `Carol`, `Alice`, `Arnold`
 - The people keep unique animals: `rabbit`, `dog`, `horse`, `fish`, `bird`, `cat`
 - People have unique favorite sports: `basketball`, `swimming`, `tennis`, `soccer`, `baseball`, `volleyball`
 - People have unique hair colors: `blonde`, `gray`, `auburn`, `red`, `black`, `brown`
 - People have unique heights: `average`, `very short`, `very tall`, `tall`, `super tall`, `short`
 - Everyone has a unique favorite cigar: `pall mall`, `yellow monster`, `blue master`, `prince`, `dunhill`, `blends`

## Clues:
1. The person who is very tall and the dog owner are next to each other.
2. There are two houses between the person who loves tennis and the person who has an average height.
3. The cat lover is the person who loves baseball.
4. The person who is very short is not in the fifth house.
5. There is one house between the fish enthusiast and the person who has red hair.
6. The cat lover is somewhere to the left of the Dunhill smoker.
7. The rabbit owner is somewhere to the left of the person who is super tall.
8. The Dunhill smoker is Alice.
9. The person who has red hair is the Dunhill smoker.
10. The person who is super tall is not in the fifth house.
11. The Prince smoker is somewhere to the left of the person who smokes Yellow Monster.
12. The person who loves volleyball is somewhere to the right of the person who smokes Yellow Monster.
13. The person who smokes Blue Master is directly left of the person who has black hair.
14. There is one house between the person who loves basketball and the person who smokes Blue Master.
15. The person who smokes Yellow Monster is the person who is tall.
16. Bob and the fish enthusiast are next to each other.
17. The Dunhill smoker is the person who loves soccer.
18. The person who loves tennis is not in the third house.
19. Arnold is the person who smokes many unique blends.
20. The Prince smoker is Peter.
21. There is one house between the person who has blonde hair and the person who is tall.
22. The person who loves swimming is directly left of Alice.
23. The person who is super tall is Eric.
24. The person who is short is the rabbit owner.
25. The person who keeps horses is not in the fifth house.
26. The cat lover and the person who has brown hair are next to each other.
27. The person who is short and the person who has auburn hair are next to each other.
 What is the value of attribute House for the person whose attribute HairColor is red? Please reason step by step, and put your final answer within \boxed{}

Answer: 6


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Alice`
 - Each person has a unique birthday month: `april`, `sept`, `jan`, `feb`
 - Everyone has a unique favorite cigar: `dunhill`, `prince`, `pall mall`, `blue master`
 - People use unique phone models: `google pixel 6`, `samsung galaxy s21`, `oneplus 9`, `iphone 13`
 - They all have a unique favorite flower: `lilies`, `roses`, `carnations`, `daffodils`
 - People have unique heights: `average`, `short`, `tall`, `very short`

## Clues:
1. The person who loves a bouquet of daffodils is Arnold.
2. The person whose birthday is in April is directly left of the person who smokes Blue Master.
3. The person who is short is the person who uses an iPhone 13.
4. The person who uses a Samsung Galaxy S21 is the person who is tall.
5. Eric is the person who uses a OnePlus 9.
6. The person who uses a Samsung Galaxy S21 is directly left of the person who uses a Google Pixel 6.
7. The person whose birthday is in February is Alice.
8. The Prince smoker is somewhere to the right of the person who is short.
9. The Prince smoker is the person whose birthday is in February.
10. The person who uses a OnePlus 9 is directly left of the person who loves the rose bouquet.
11. The person partial to Pall Mall is somewhere to the left of the person who loves the boquet of lilies.
12. The person who has an average height and the person whose birthday is in September are next to each other.
13. The person who loves a bouquet of daffodils is in the fourth house.
 What is the value of attribute Birthday for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: jan


A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point \(A\) in the figure on the right. The box has base length \(w\) and height \(h\). Find the area of the sheet of wrapping paper in terms of \(w\) and \(h\). The answer should be in the form \(a(w+h)^b\). Please find the value of \(a + b\).

Answer: 4


Let \(ABCDEF\) be a regular hexagon with side length \(1\). Denote by \(X\), \(Y\), and \(Z\) the midpoints of sides \(\overline{AB}\), \(\overline{CD}\), and \(\overline{EF}\), respectively. Find the area of the convex hexagon whose interior is the intersection of the interiors of \(\triangle ACE\) and \(\triangle XYZ\). The original answer is in the form \(\frac{k}{m}\sqrt{n}\), where \(\frac{k}{m}\) is in its simplest form. Please give the value of \(k + m + n\).

Answer: 50


Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. Find the number of real numbers $x$ that satisfy the equation $x^2 + 10,000\lfloor x \rfloor = 10,000x$.

Answer: 199


A space agency discovers 13 habitable planets: 7 Earth-like and 6 Mars-like. Each Earth-like planet requires 3 units of colonization, and each Mars-like requires 1 unit. If 18 total units of colonies are available, in how many different ways can the agency choose a combination of planets to colonize, assuming all planets are unique and different from each other, with the condition that at least one Earth-like planet must be colonized?

Answer: 462


Define a regular 12-pointed double-layered star where each vertex connects to two others, skipping some vertices between connections, and no three points of connections are collinear. Vertices are labeled $1, 2, \ldots, 12$. For each vertex like $P_k$, connect it to $P_{k+m \mod 12}$ and $P_{k+2m \mod 12}$ where $m$ is chosen such that $\gcd(m, 12) = 1$. Determine how many non-similar regular 12-pointed double-layered stars can be made.

Answer: 2


Determine all possible values of \( r \) so that for every \( s > 0 \), the inequality \( \frac{4(r s^2 + r^2 s + 4 s^2 + 4 r s)}{r+s} \leq 3 r^2 s \) holds true. Provide your answer in interval notation.

Answer: \left[\frac{2+2\sqrt{13}}{3}, \infty\right)


A right pyramid has a square base where each side measures 8 cm. Its peak is 15 cm vertically above the center of its base. Calculate the sum of the lengths of the pyramid's edges and also find the volume of the pyramid.

Answer: 320 \text{ cm}^3


In the diagram, $O$ is the center of the larger circle with an area of $81\pi$. A smaller circle is inside the larger one and its topmost point touches the center of the larger circle. Each circle is divided into two equal areas. Calculate the total area of the shaded regions.

Answer: 50.625\pi


Let $AB$ be a diameter of a circle centered at $O$. Let $E$ be a point on the circle, and let the tangent at $B$ intersect the tangent at $A$ and $AE$ at $C$ and $D$, respectively. If $\angle BAE = 53^\circ$, find $\angle CED$, in degrees.

Answer: 37^\circ


John has $25$ dollars and $45$ cents. He wants to buy as many notebooks as possible, where each notebook costs $2$ dollars and $35$ cents. However, for every $5$ notebooks he buys, he gets a $15$ cents discount on each notebook. How many notebooks can John buy?

Answer: 11


A soccer team has 23 players in total. 12 players start the game, while the remaining 11 are available as substitutes. The coach is allowed to make up to 5 substitutions during the game. Any one of the players in the game can be replaced by one of the substitutes, but substituting players cannot return to the game after being removed. The order of substitutions and players involved is considered in calculating the number of possible substitutions. Let $m$ be the number of ways the coach can make substitutions during the game (including the possibility of making no substitutions). Calculate the remainder when $m$ is divided by $100$.

Answer: 93


Consider the parabolas defined by $y = 2(x - 2)^2$ and $x - 1 = 2(y + 1)^2$. Find the sum of all $x$ and $y$ coordinates of their intersection points.

Answer: 4


A rectangle with dimensions 15 cm by 12 cm has four quarter circles drawn with centers at the four corners. Additionally, a circle of radius 3 cm is drawn at the center of the rectangle. Calculate the area of the shaded region outside the central circle and inside the quarter circles. Express your answer in terms of $\pi$.

Answer: 27\pi \text{ cm}^2


Answer: 22


On a similar one-way single-lane highway, cars travel at the same speed and obey a modified safety rule: the distance from the back of the car ahead to the front of the car behind is exactly two car lengths for each 20 kilometers per hour of speed or fraction thereof. Assuming each car is 5 meters long and that cars can travel at any speed, let $N$ be the maximum whole number of cars that can pass a photoelectric eye in one hour. Find the quotient when $N$ is divided by 10.

Answer: 200


A club has 30 members and needs to elect 3 officers: president, secretary, and treasurer. Each person can hold at most one office. Three of the members, Alice, Bob, and Charlie, will only serve as officers if at least two of them are elected as officers. In how many ways can the club elect its officers?

Answer: 18042


A box contains 3 red balls, 3 white balls, and 3 black balls. They are drawn one at a time. What is the probability that all of the draws alternate in colors without repeating any color consecutively?

Answer: \frac{1}{280}


In an isosceles right triangle, one of the angles opposite the equal sides measures $45^{\circ}$. What is the measure of the third angle?

Answer: 90^\circ


35. In triangle $ABC$, points $M, N$, and $P$ are taken: $M$ and $N$ are on sides $AC$ and $BC$, and $P$ is on segment $MN$, such that

$$
\frac{|AM|}{|MC|}=\frac{|CN|}{|NB|}=\frac{|MP|}{|PN|}
$$

Find the area of triangle $ABC$, if the areas of triangles $AMP$ and $BNP$ are $T$ and $Q$.

Answer: (\sqrt[3]{T} + \sqrt[3]{Q})^3


Task 1. Let $n$ be a positive integer. Given are circular discs with radii $1, 2, \ldots, n$. For each size, we have two discs: one transparent and one opaque. Each disc has a hole in the middle, allowing us to stack the discs on a vertical rod. We want to create stacks that meet the following conditions:

- For each size, there is exactly one disc on the stack.
- When viewed from directly above, the outer edges of all $n$ discs on the stack can be seen. (This means that if there is an opaque disc on the stack, then no smaller discs may lie below it.)

Determine the number of different stacks that can be made under these conditions. (Two stacks are different if they do not use exactly the same set of discs, but also if they use exactly the same set of discs but not in the same order.)

Answer: (n+1)!


A bag contains exactly 15 marbles of which 3 are red, 5 are blue, and 7 are green. The marbles are chosen at random and removed one at a time from the bag until all of the marbles are removed. One colour of marble is the first to have 0 remaining in the bag. What is the probability that this colour is red?

## PART B

For each question in Part B, your solution must be well-organized and contain words of explanation or justification. Marks are awarded for completeness, clarity, and style of presentation. A correct solution, poorly presented, will not earn full marks.

## Useful Fact for Part B:

The sum of the first $k$ perfect squares is equal to $\frac{k(k+1)(2 k+1)}{6}$.

That is, $1^{2}+2^{2}+3^{2}+\cdots+k^{2}=\frac{k(k+1)(2 k+1)}{6}$.

Answer: \dfrac{21}{40}


6. [5] The elliptic curve $y^{2}=x^{3}+1$ is tangent to a circle centered at $(4,0)$ at the point $\left(x_{0}, y_{0}\right)$. Determine the sum of all possible values of $x_{0}$.

Answer: \dfrac{1}{3}


14. In the figure below, $B A D, B C E, A C F$ and $D E F$ are straight lines. It is given that $B A=B C, A D=A F, E B=E D$. If $\angle B E D=x^{\circ}$, find the value of $x$.

Answer: 108


## Task 1 - 080621

In a 6th grade class, no student received a final grade of 5 in mathematics, every ninth student received a grade of 1, every third student received a grade of 2, and every sixth student received a grade of 4.

The number of students $n$ in this class is known to be: $20<n<40$.

Calculate the number of students who received a final grade of 3!

Answer: 14


2. Let $a$, $b$, $c$ be integers, and for all real numbers $x$,
$$
(x-a)(x-8)+1=(x-b)(x-c)
$$

always holds. Then the value of $a+b+c$ is $\qquad$.

Answer: 20


E csudaszép emlék födi szent porait Diophantosnak $S$ éveinek számát hirdeti a fölirat.

This beautiful monument covers the sacred ashes of Diophantus, the inscription on the monument proclaims the number of his years $S$.

Éltének hatodát boldog gyermekkorda kapta, tizenkettede szőtt álmokat ifjúkorán.

He spent one-sixth of his life in a joyful childhood, and one-twelfth of his life dreaming in his youth.

Majd hetedét tölté el, mennyekzője mikor lőn; $\mathrm{S}$ az ötödik tavaszon kis fia is született.

Then he spent one-seventh of his life, when he became a groom; $S$ on the fifth spring, his little son was born.

Hajh, de szegény éppen csak még félannyi időt élt, Mint a bús apa, kit ... bánata sírba vive ...

Alas, but the poor child lived only half as long as his grieving father, who ... was carried to the grave by his sorrow ...

Négy évig hordván gyötrelmét a szerető szív. Élete hosszát ím: - látod e bölcs sorokon.

For four years he bore his grief in a loving heart. The length of his life is thus: - see it in these wise lines.

Answer: 84


3-4. In how many different ways can 1000000 be represented as a product of three natural ${ }^{1}$ numbers? Products that differ only in the order of the factors are considered identical.

(This problem was not solved by any of the olympiad participants.)

Answer: 139


9. For what values of the parameter $\boldsymbol{a}$ does the equation $f(x)=p(x)$ have one solution, if $f(x)=\left|\frac{2 x^{3}-x^{2}-18 x+9}{(1.5 x+1)^{2}-(0.5 x-2)^{2}}\right|, p(x)=|-2 x+2|+a$. If there are more than one value of the parameter, indicate their sum in the answer.

Answer: \dfrac{3}{2}


6. 6.1. The figure "lame rook" can move to an adjacent cell in one move. On a $20 \times 20$ board, crosses were placed in all cells that the "lame rook" can reach from the top-left corner in exactly 10 moves. How many cells were marked with a cross?

Answer: 36


In triangle $ABC$, point $D$ lies on side $AC$ and point $E$ lies on side $BC$. The measures of angles $ABD, BAE, CAE$, and $CBD$ are $30^{\circ}, 60^{\circ}, 20^{\circ}$, and $30^{\circ}$, respectively.

Determine the measure of angle $AED$.

![](https://cdn.mathpix.com/cropped/2024_04_17_6d7ca75cbeb5035abbabg-4.jpg?height=525&width=528&top_left_y=443&top_left_x=770)

Note: The image is for illustration only.

(A. Bohiniková)

Hint. Look for congruent angles.

Answer: 20


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

[i]Problem proposed by B. Dejean[/i]

Answer: 33\%


Example 6. A football invitational tournament has 16 cities participating, each city sending Team A and Team B. According to the competition rules, each pair of teams plays at most one match, and the two teams from the same city do not play against each other. If, after two days of the tournament, it is found that except for Team A of City $A$, all other teams have played a different number of matches. How many matches has Team B of City $A$ played? Please prove your conclusion. (1985, National High School League)

Answer: 15


9. Given a positive integer $n \geqslant 3$, the function $f:\{1,2, \cdots, n\} \rightarrow\{1,2, \cdots, n\}$ satisfies $f(f(k))=(f(k))^{3}-$ $6(f(k))^{2}+12 f(k)-6, \forall k \in\{1,2, \cdots, n\}$, find the number of all such functions.

Answer: 3^{n-3} + 3 \cdot 2^{n-2} + 3


1. Given a natural number n. A square with side length n is divided into $n^{2}$ unit squares. The distance between two squares is considered to be the distance between their centers. Determine the number of pairs of squares whose distance is 5.

Answer: 6n^2 - 38n + 48


1. In the isosceles triangle \(ABC\), given are \(\overline{AC} = \overline{BC} = 2b\) and \(\measuredangle CAB = 2\alpha\). Let \(P_1\) be the area of the inscribed circle \(k_1\) in triangle \(ABC\), \(P_2\) be the area of the circle that touches the circle \(k_1\) and the legs of triangle \(ABC\), \(\ldots\), \(P_i\) be the area of the circle that touches the circle \(k_{i-1}\) and the legs of triangle \(ABC\), and so on. Calculate the sum \(P_1 + P_2 + \ldots + P_n + \ldots\)

Answer: \pi b^2 \sin^2 2\alpha \cos 2\alpha


Sergeev I.N.

At the base $A_{1} A_{2} \ldots A_{n}$ of the pyramid $S A_{1} A_{2} \ldots A_{n}$ lies a point $O$, such that $S A_{1}=S A_{2}=\ldots=S A_{n}$ and $\angle S A_{1} O=\angle S A_{2} O=$ $\ldots=\angle S A_{n} O$.

For what least value of $n$ does it follow from this that $S O$ is the height of the pyramid?

Answer: 5


Let \( \triangle ABC \) be an acute-angled triangle with orthocenter \( H \), and let \( W \) be a point on side \( BC \). Denote by \( M \) and \( N \) the feet of the altitudes from \( B \) and \( C \), respectively. Denote by \( \omega_1 \) the circumcircle of \( BWN \), and let \( X \) be the point on \( \omega_1 \) which is diametrically opposite to \( W \). Analogously, denote by \( \omega_2 \) the circumcircle of \( CWM \), and let \( Y \) be the point on \( \omega_2 \) which is diametrically opposite to \( W \). Prove that \( X \), \( Y \), and \( H \) are collinear.

Answer: X, Y, \text{ and } H \text{ are collinear}


8. Given 5 points $A, B, C, D, E$ on a plane, no three of which are collinear. By connecting these points with 4 line segments, each point being an endpoint of at least one line segment, how many different connection methods are there?

Answer: 135


## 6. Cube from sticks

![](https://cdn.mathpix.com/cropped/2024_05_30_12924d306592e3a90c60g-05.jpg?height=392&width=371&top_left_y=1186&top_left_x=180)

Result:

Cubes are made from sticks. For a cube $1 \times 1 \times 1$, 12 sticks are needed. To complete this cube to a cube $2 \times 2 \times 2$, 42 additional sticks are required (see the image). How many additional sticks are needed to complete a cube $8 \times 8 \times 8$ to a cube $9 \times 9 \times 9$?

Answer: 756


3. Annemiek and Bart each wrote three different positive integers on a piece of paper. It turns out that there is exactly one number that appears on both of their papers. Furthermore, if you take two different numbers from Annemiek's paper and add them together, the result is always a number on Bart's paper. One of the three numbers on Annemiek's paper is her favorite number, and if you multiply that by 3, you also get a number on Bart's paper. On Bart's paper, his favorite number is 25.

What is Annemiek's favorite number?

Answer: 5


Given a positive integer \( n \geq 2 \), consider the matrices
\[
A = \begin{pmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn}
\end{pmatrix}, \quad
B = \begin{pmatrix}
b_{11} & b_{12} & \cdots & b_{1n} \\
b_{21} & b_{22} & \cdots & b_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
b_{n1} & b_{n2} & \cdots & b_{nn}
\end{pmatrix}
\]

and they satisfy
\[
\{a_{ij} \mid 1 \leq i, j \leq n\} = \{b_{ij} \mid 1 \leq i, j \leq n\} = \{1, 2, \ldots, n^2\}.
\]

An operation can be performed on \( A \) as follows: select two numbers located in the same row or column and swap their positions, while the remaining \( n^2 - 2 \) numbers remain unchanged. This operation is called a single swap. Determine the smallest positive integer \( m \) such that for any \( A \) and \( B \), \( A \) can be transformed into \( B \) by performing no more than \( m \) swaps.

Answer: n^2


17.1.17 $\underset{\star}{\star \star}$ Let $a_{i}=\min \left\{k+\frac{i}{k}, k \right.$ is a positive integer $\}$. Try to find $S_{n^{2}}=\left[a_{1}\right]+\left[a_{2}\right]+\cdots$ $+\left[a_{n^{2}}\right]$, where $n \geqslant 2$.

Answer: \dfrac{8n^3 - 3n^2 + 13n - 6}{6}


6. Each point on a circle is colored one of three colors: red, yellow, or blue, and all three colors are used. Now, take any $n$ points from the circle. If among these points, there always exist three points that form a triangle with all vertices of the same color and the triangle is obtuse, then the minimum possible value of $n$ is $\qquad$

Answer: 13


A restaurant can offer 9 types of appetizers, 9 types of main courses, 9 types of desserts, and 9 types of wines. A company is having a dinner party at this restaurant, and each guest can choose one appetizer, one main course, one dessert, and one type of wine. It is known that any two people's choices of the four dishes are not completely identical, and it is also impossible to find four people on the spot who have three identical choices but differ pairwise in the fourth choice (for example, there are no 9 people who have the same appetizer, main course, and dessert, but differ pairwise in the wine). Then, at most how many guests can there be?

Answer: 5832


Gardener Mr. Malina was selling strawberries. In the last nine crates, he had 28, 51, 135, 67, 123, 29, 56, 38, and 79 strawberry plants, respectively. He sold the crates whole, never removing any plants from the crates. The gardener wanted to sell the crates to three customers so that nothing was left and each of these customers had the same number of plants.

How could he do it? Provide two possibilities.

(L. Hozová)

Answer: 135 + 67


3. Let point $Q$ be in the plane $\alpha$ of $\triangle A B C$, and point $P$ be outside the plane $\alpha$. If for any real numbers $x$ and $y, |\overrightarrow{A P}-x \overrightarrow{A B}-y \overrightarrow{A C}| \geq|\overrightarrow{P Q}|$, then the angle $\theta$ between vector $\overrightarrow{P Q}$ and $\overrightarrow{B C}$ is $\qquad$.

Answer: \dfrac{\pi}{2}