The $n$ contestants of an EGMO are named $C_{1}, \ldots, C_{n}$. After the competition they queue in front of the restaurant according to the following rules.
- The Jury chooses the initial order of the contestants in the queue.
- Every minute, the Jury chooses an integer $i$ with $1 \leq i \leq n$.
- If contestant $C_{i}$ has at least $i$ other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly $i$ positions.
- If contestant $C_{i}$ has fewer than $i$ other contestants in front of her, the restaurant opens and the process ends.
Determine for every $n$ the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves.
Please provide the value when n = 9.

Answer: 502


Let $n$ be a positive integer and fix $2 n$ distinct points on a circumference. Split these points into $n$ pairs and join the points in each pair by an arrow (i.e., an oriented line segment). The resulting configuration is good if no two arrows cross, and there are no arrows $\overrightarrow{A B}$ and $\overrightarrow{C D}$ such that $A B C D$ is a convex quadrangle oriented clockwise. Determine when n=6， the number of good configurations.

Answer: 924


Let $a_{1}, a_{2} \ldots, a_{51}$ be non-zero elements of a field. We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence $b_{1} \ldots, b_{51}$. If this new sequence is a permutation of the original one, what can be the characteristic of the field? Please provide the sum of the possible characteristics.

Answer: 9


How many nonzero coefficients can a polynomial $P(z)$ have if its coefficients are integers and $|P(z)| \leq 2$ for any complex number $z$ of unit length? Please provide the sum of all possible numbers of nonzero coefficients.

Answer: 3


For each positive integer $k$, find the smallest number $n_{k}$ for which there exist real $n_{k} \times n_{k}$ matrices $A_{1}, A_{2}, \ldots, A_{k}$ such that all of the following conditions hold:
(1) $A_{1}^{2}=A_{2}^{2}=\ldots=A_{k}^{2}=0$,
(2) $A_{i} A_{j}=A_{j} A_{i}$ for all $1 \leq i, j \leq k$, and
(3) $A_{1} A_{2} \ldots A_{k} \neq 0$.
Please provide the sum of the smallest $n_k$ values for $k = 2$ and $k = 3$.

Answer: 12


(a) A sequence $x_{1}, x_{2}, \ldots$ of real numbers satisfies $$ x_{n+1}=x_{n} \cos x_{n} \quad \text { for all } \quad n \geq 1 $$ Determine if this sequence converges for all initial values $x_{1}$. (b) A sequence $y_{1}, y_{2}, \ldots$ of real numbers satisfies $$ y_{n+1}=y_{n} \sin y_{n} \quad \text { for all } \quad n \geq 1 $$ Determine if this sequence converges for all initial values $y_{1}$. If the answer to (a) is NO and (b) is YES, provide the sum of their binary representations as a single integer.

Answer: 1


Given an integer $n>1$, let $S_{n}$ be the group of permutations of the numbers $1,2, \\ldots, n$. Two players, A and B, play the following game. Taking turns, they select elements (one element at a time) from the group $S_{n}$. It is forbidden to select an element that has already been selected. The game ends when the selected elements generate the whole group $S_{n}$. The player who made the last move loses the game. The first move is made by A. For which values of n does player A have a winning strategy? Please provide the sum of these values of n.

Answer: 5


We say that a subset of $\mathbb{R}^{n}$ is $k$-almost contained by a hyperplane if there are less than $k$ points in that set which do not belong to the hyperplane. We call a finite set of points $k$-generic if there is no hyperplane that $k$-almost contains the set. For each pair of positive integers $k$ and $n$, find the minimal number $d(k, n)$ such that every finite $k$-generic set in $\mathbb{R}^{n}$ contains a $k$-generic subset with at most $d(k, n)$ elements. Calculate the value of $d(k, n)$ when $k = 2$ and $n = 2$, and provide the result of k + n + d(k, n).

Answer: 8


Let $M$ be a subset of the set of $2021$ integers $\{1, 2, 3, \ldots, 2021\}$ such that for any three elements (not necessarily distinct) $a, b, c$ of $M$, we have $|a + b - c| > 10$. Determine the largest possible number of elements of $M$.

Answer: 1006


Each number from the set $\{1, 2, 3, 4, 5, 6, 7\}$ must be written in each circle of the diagram, so that the sum of any three aligned numbers is the same (e.g., $A+D+E = D+C+B$). What number cannot be placed on the circle $E$?

Answer: 4


When counting from $3$ to $201$, $53$ is the $51^{\text{st}}$ number counted. When counting backwards from $201$ to $3$, $53$ is the $n^{\text{th}}$ number counted. What is $n$?

\[ \textbf{(A) } 146 \quad \textbf{(B) } 147 \quad \textbf{(C) } 148 \quad \textbf{(D) } 149 \quad \textbf{(E) } 150 \]

Answer: 149


Define the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$ for all positive real numbers $x$. How many integers from $1$ to $2023$ inclusive are in the range of $f(x)$? Note that $\lfloor x\rfloor$ is known as the \textit{floor} function, which returns the greatest integer less than or equal to $x$.

Answer: 1973


There are $2024$ people, consisting of knights and liars, and some of them are friends. Each person was asked how many friends they have, and the possible answers were $0, 1, \ldots, 2023$. Every knight answered truthfully, while every liar altered the real answer by exactly $1$. What is the minimal number of liars?

Answer: 1012


It is given that $2^{333}$ is a 101-digit number whose first digit is 1. How many of the numbers $2^k$, where $1 \leq k \leq 332$, have the first digit as 4?

Answer: 32


In triangle $ABC$, angle $A$ is twice angle $B$, angle $C$ is obtuse, and the three side lengths $a, b, c$ are integers. Determine, with proof, the minimum possible perimeter.

Answer: 77


The diagram shows a $20 \times 20$ square $ABCD$. The points $E$, $F$, and $G$ are equally spaced on side $BC$. The points $H$, $I$, $J$, and $K$ on side $DA$ are placed so that the triangles $BKE$, $EJF$, $FIG$, and $GHC$ are isosceles. Points $L$ and $M$ are midpoints of the sides $AB$ and $CD$, respectively. Find the total area of the shaded regions.

Answer: 200


It is well-known that the $n^{\text{th}}$ triangular number can be given by the formula $\frac{n(n+1)}{2}$. A Pythagorean triple of \textit{square numbers} is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of \textit{triangular numbers} (a PTTN) be an ordered triple of positive integers $(a,b,c)$ such that $a\leq b<c$ and
\[\frac{a(a+1)}{2}+\frac{b(b+1)}{2}=\frac{c(c+1)}{2}.\]
For instance, $(3,5,6)$ is a PTTN ($6+15=21$). Here we call both $a$ and $b$ \textit{legs} of the PTTN. Find the smallest natural number $n$ such that $n$ is a leg of \textit{at least} six distinct PTTNs.

Answer: 14


Let $x$ be a positive real number. Define
\[
  A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad
  B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad \text{and} \quad
  C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2)!}.
\]
Given that $A^3+B^3+C^3 + 8ABC = 2014$, compute $ABC$.

Answer: 183


How many ways can you fill a table of size $n \times n$ with integers such that each cell contains the total number of even numbers in its row and column, excluding itself? Two tables are considered different if they differ in at least one cell.

Answer: 1


There are $100$ people standing in a line from left to right. Half of them are randomly chosen to face right (with all $\binom{100}{50}$ possible choices being equally likely), and the others face left. Then, while there is a pair of people who are facing each other and have no one between them, the leftmost such pair leaves the line. Compute the expected number of people remaining once this process terminates.

Answer: 50


Consider polynomials $P$ of degree $2015$, all of whose coefficients are in the set \{0,1,\dots,2010\}. Call such a polynomial "good" if for every integer $m$, one of the numbers $P(m)-20$, $P(m)-15$, $P(m)-1234$ is divisible by $2011$, and there exist integers $m_{20}, m_{15}, m_{1234}$ such that $P(m_{20})-20$, $P(m_{15})-15$, $P(m_{1234})-1234$ are all multiples of $2011$. Let $N$ be the number of good polynomials. Find the remainder when $N$ is divided by $1000$.

Answer: 460


Let $p$ be an odd prime number less than $10^5$. Granite and Pomegranate play a game. First, Granite picks an integer $c \in \{2,3,\dots,p-1\}$. Pomegranate then picks two integers $d$ and $x$, defines $f(t) = ct + d$, and writes $x$ on a sheet of paper. Next, Granite writes $f(x)$ on the paper, Pomegranate writes $f(f(x))$, Granite writes $f(f(f(x)))$, and so on, with the players taking turns writing. The game ends when two numbers appear on the paper whose difference is a multiple of $p$, and the player who wrote the most recent number wins. Find the sum of all $p$ for which Pomegranate has a winning strategy.

Answer: 65819


Let $T$ denote the 15-element set $\{10a+b : a,b \in \mathbb{Z}, 1 \le a < b \le 6\}$. Let $S$ be a subset of $T$ in which all six digits $1,2,\ldots ,6$ appear and in which no three elements together use all these six digits. Determine the largest possible size of $S$.

Answer: 9


Let $f(x) = x^4 + 14x^3 + 52x^2 + 56x + 16$. Let $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible value of $|z_a z_b + z_c z_d|$ where $\{a, b, c, d\} = \{1, 2, 3, 4\}$.

Answer: 8


The incircle $\Gamma$ of a scalene triangle $ABC$ touches $BC$ at $D$, $CA$ at $E$, and $AB$ at $F$. Let $r_A$ be the radius of the circle inside $\triangle ABC$ which is tangent to $\Gamma$ and the sides $AB$ and $AC$. Define $r_B$ and $r_C$ similarly. If $r_A = 16$, $r_B = 25$, and $r_C = 36$, determine the radius of $\Gamma$.

Answer: 74


We call $A_1, A_2, \ldots, A_n$ an $n$-division of $A$ if:

1. $A_1 \cap A_2 \cap \cdots \cap A_n = A$,
2. $A_i \cap A_j \neq \emptyset$ for $i \neq j$.

Find the smallest positive integer $m$ such that for any $14$-division $A_1, A_2, \ldots, A_{14}$ of $A = \{1, 2, \ldots, m\}$, there exists a set $A_i$ ($1 \leq i \leq 14$) such that there are two elements $a, b$ of $A_i$ where $b < a \leq \frac{4}{3}b$.

Answer: 56


There are infinitely many boxes - initially, one of them contains $n$ balls, and all others are empty. In a single move, we take some balls from a non-empty box and put them into an empty box. On a sheet of paper, we write down the product of the resulting number of balls in the two boxes. After some moves, the sum of all numbers on the sheet of paper became $2023$. What is the smallest possible value of $n$?

Answer: 65


Suppose $P(x)$ is a monic polynomial of degree $2023$ such that $P(k) = k^{2023}P(1-\frac{1}{k})$ for every positive integer $1 \leq k \leq 2023$. Then $P(-1) = \frac{a}{b}$ where $a$ and $b$ are relatively prime integers. Compute the unique integer $0 \leq n < 2027$ such that $bn-a$ is divisible by the prime $2027$.

Answer: 0


10. (12 points) In the $3 \times 3$ table of Figure 1, numbers have already been filled in. Choose a black cell as the starting point. If the number in this black cell and the number in an adjacent white cell undergo one of the four arithmetic operations (addition, subtraction, multiplication, or division, with the larger number always first), and the result is an integer multiple of the number in another adjacent black cell, you can move from this white cell to the next black cell. Each cell must be visited exactly once. (For example, in Figure 2, from 7 through 8 to 5, and the direction of the arrows in Figure 2 is a correct path). Find the correct path in Figure 1. If the first 3 cells in the correct path are filled with $A$, $B$, and $C$ respectively, then the three-digit number $\overline{\mathrm{ABC}}=$ $\qquad$

Answer: 834


B3. We have a sheet of graph paper of size $7 \times 7$ squares and an equally large plastic graph stencil, on which some squares are green, and the rest are transparent. If we place the stencil on the sheet of paper so that the sides of the stencil align with the sides of the paper, the grid pattern on the stencil matches the grid pattern on the paper. If we choose any square on the paper and color it red, then we can place the stencil on the paper so that its sides align with the sides of the paper and the red square on the paper is covered by one of the green squares on the stencil (the stencil can be rotated and flipped over as needed). What is the minimum number of green squares on the stencil?

67th Mathematical Competition for High School Students in Slovenia National Competition, April 22, 2023

# Problems for 3rd Year

Time for solving: 180 minutes. Each problem in section A has exactly one correct answer. In section A, we will award two points for the correct answer, and deduct one point for an incorrect answer. Write the answers for section A in the left column of the answer sheet, and leave the right column blank. The committee will consider only what is written on the answer sheets, marked with the contestant's code.

Answer: 10


16. Fill in each square in $\square \square+\square \square=\square \square \square$ with a digit from $0, 1, 2, \cdots \cdots, 9$ (digits in the squares can be the same, and the leading digit of any number cannot be 0), so that the equation holds.

There are

Answer: 4860


If we walk along the fence from north to south, the distances between its posts are initially the same. From a certain post, the distance decreases to 2.9 meters and remains so until the southern end of the fence. Between the 1st and 16th posts (counted from the north), the distances do not change, and the distance between these two posts is 48 meters. The distance between the 16th and 28th posts is 36 meters.

Which post has different distances from its neighboring posts?

(L. Šimünek)

Answer: 20


For an integer $n \ge 3$ we consider a circle with $n$ points on it.
We place a positive integer at each point, where the numbers are not necessary need to be different. Such placement of numbers is called [i]stable [/i] as three numbers next to always have product $n$ each other.
For how many values of $n$ with $3 \le n \le 2020$ is it possible to place numbers in a stable way?

Answer: 680


The side lengths $a,b,c$ of a triangle $ABC$ are positive integers. Let:\\

\[T_{n}=(a+b+c)^{2n}-(a-b+c)^{2n}-(a+b-c)^{2n}+(a-b-c)^{2n}\]

for any positive integer $n$. If $\frac{T_{2}}{2T_{1}}=2023$ and $a>b>c$ , determine all possible perimeters of the triangle $ABC$.

Answer: 49


3. Given $a, b, c \in \mathbf{R}_{+}$, and $abc=4$. Then the minimum value of the algebraic expression $a^{a+b} b^{3b} c^{c+b}$ is $\qquad$

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

Answer: 64


15. Let $M$ be a set composed of a finite number of positive integers
$$
\begin{array}{l}
\text { such that, } M=\bigcup_{i=1}^{20} A_{i}=\bigcup_{i=1}^{20} B_{i}, \\
A_{i} \neq \varnothing, B_{i} \neq \varnothing(i=1,2, \cdots, 20),
\end{array}
$$

and satisfies:
(1) For any $1 \leqslant i<j \leqslant 20$,
$$
A_{i} \cap A_{j}=\varnothing, B_{i} \cap B_{j}=\varnothing \text {; }
$$
(2) For any $1 \leqslant i \leqslant 20,1 \leqslant j \leqslant 20$, if $A_{i} \cap B_{j}=\varnothing$, then $\left|A_{i} \cup B_{j}\right| \geqslant 18$.

Find the minimum number of elements in the set $M$ ( $|X|$ denotes the number of elements in the set $X$).

Answer: 180


Example 9 For any positive integer $n$, let $x_{n}=\mathrm{C}_{2 n}^{n}$. Find all positive integers $h(h>1)$, such that the sequence of remainders of $\left\{x_{n}\right\}$ modulo $h$ is eventually periodic.

Answer: 2


In the universe of Pi Zone, points are labeled with $2 \times 2$ arrays of positive reals. One can teleport from point $M$ to point $M'$ if $M$ can be obtained from $M'$ by multiplying either a row or column by some positive real. For example, one can teleport from $\left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right)$ to $\left( \begin{array}{cc} 1 & 20 \\ 3 & 40 \end{array} \right)$ and then to $\left( \begin{array}{cc} 1 & 20 \\ 6 & 80 \end{array} \right)$.

A [i]tourist attraction[/i] is a point where each of the entries of the associated array is either $1$, $2$, $4$, $8$ or $16$.  A company wishes to build a hotel on each of several points so that at least one hotel is accessible from every tourist attraction by teleporting, possibly multiple times. What is the minimum number of hotels necessary?

[i]Proposed by Michael Kural[/i]

Answer: 17


In triangle \(ABC\), the altitude \(BM\) is drawn, and a circle is constructed with diameter \(BM\). This circle intersects side \(AB\) at point \(K\) and side \(BC\) at point \(Z\). Find the ratio of the area of triangle \(KZM\) to the area of triangle \(ABC\), given that \(\angle A = \alpha\) and \(\angle C = \beta\).

Answer: \dfrac{\sin 2\alpha \sin 2\beta}{4}


12. A wristwatch is 5 minutes slow per hour; 5.5 hours ago they were set to the correct time. Now, on the clock showing the correct time, it is 1 PM. In how many minutes will the wristwatch show 1 PM?

Answer: 30


8. Let the integer sequence $a_{1}, a_{2}, \cdots, a_{10}$ satisfy:
$$
a_{10}=3 a_{1}, a_{2}+a_{8}=2 a_{5} \text {, }
$$

and $a_{i+1} \in\left\{1+a_{i}, 2+a_{i}\right\}(i=1,2, \cdots, 9)$.
Then the number of such sequences is $\qquad$

Answer: 80


Arrange $\frac{n(n+1)}{2}$ different numbers randomly into a triangular array. Let $M_k$ be the maximum number in the $k$-th row (from top to bottom). Find the probability that $M_1 < M_2 < \cdots < M_n$.

Answer: \dfrac{2^n}{(n + 1)!}


5. (1996 Beijing Middle School Mathematics Competition) The function $f(x)$ is defined on $\mathbf{N}$, taking values in $\mathbf{N}$, and is a strictly increasing function (if for any $x_{1}, x_{2} \in A$, when $x_{1}<x_{2}$, we have $f\left(x_{1}\right)<f\left(x_{2}\right)$, then $f(x)$ is called a strictly increasing function on $A$), and satisfies the condition $f(f(k))=3 k$. Try to find: the value of $f(1)+f(9)+f(96)$.

Answer: 197


12. Let $[x]$ denote the greatest integer not exceeding $x$, and let the natural number $n$ satisfy $\left[\frac{1}{15}\right]+\left[\frac{2}{15}\right]+\left[\frac{3}{15}\right]+\cdots+\left[\frac{\mathrm{n}-1}{15}\right]+\left[\frac{\mathrm{n}}{15}\right]>$ 2011, then what is the minimum value of $n$?

Answer: 253


5. Given is a regular pentagon $A B C D E$. Determine the least value of the expression

$$
\frac{P A+P B}{P C+P D+P E},
$$

where $P$ is an arbitrary point lying in the plane of the pentagon $A B C D E$.

Answer: \sqrt{5} - 2


4. A circle touches the sides of an angle at points $A$ and $B$. A point $M$ is chosen on the circle. The distances from $M$ to the sides of the angle are 24 and 6. Find the distance from $M$ to the line $A B$.

![](https://cdn.mathpix.com/cropped/2024_05_06_bb70a8d18e0320708328g-3.jpg?height=691&width=1102&top_left_y=1057&top_left_x=451)

Answer: 12


[ Sphere inscribed in a pyramid ]
[Volume helps solve the problem.]

In a triangular pyramid, two opposite edges are equal to 12 and 4, and the other edges are equal to 7. A sphere is inscribed in the pyramid. Find the distance from the center of the sphere to the edge equal to 12.

Answer: \dfrac{39 - 3\sqrt{65}}{8}


5. Given point $A$ is a point inside the unit circle with center $O$, satisfying $|\overrightarrow{O A}|=\frac{1}{2}, B, C$ are any two points on the unit circle $O$, then the range of $\overrightarrow{A C} \cdot \overrightarrow{B C}$ is $\qquad$ .

Answer: \left[ -\dfrac{1}{8}, 3 \right]


Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$.  The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$.  Compute the smallest possible perimeter of $ABCD$.

[i]Proposed by Evan Chen[/i]

Answer: 118


Determine the number of positive integers that, in their base \( n \) representation, have all different digits and every digit except the leftmost one differs from some digit to its left by \( \pm 1 \). Express the answer as a simple function of \( n \) and provide a proof of your result.

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


Thirty numbers are arranged on a circle in such a way that each number equals the absolute difference of its two neighbors. Given that the sum of the numbers is $2000$, determine the numbers.

Answer: 0


3. Let $\mathbb{R}_{+}$ be the set of positive real numbers. Find all functions $f: \mathbb{R}_{+}^{3} \rightarrow \mathbb{R}_{+}$ such that for all positive real numbers $x, y, z$ and $k$ the following three conditions hold:

(i) $x f(x, y, z)=z f(z, y, x)$;

(ii) $f\left(x, k y, k^{2} z\right)=k f(x, y, z)$;

(iii) $f(1, k, k+1)=k+1$.

(Great Britain)

Answer: \dfrac{y + \sqrt{y^2 + 4xz}}{2x}


Problem 10.7. Nikita schematically drew the graph of the quadratic polynomial $y=a x^{2}+b x+c$. It turned out that $A B=C D=1$. Consider four numbers $-a, b, c$ and the discriminant of the quadratic polynomial. It is known that three of them are equal in some order to $1 / 4, -1, -3 / 2$. Find what the fourth number is.

![](https://cdn.mathpix.com/cropped/2024_05_06_62ad5020cf8c9b2474f9g-4.jpg?height=258&width=418&top_left_y=1500&top_left_x=518)

Answer: -\dfrac{1}{2}


$\mathrm{Az} r_{1}$ and $r_{2}$ radius circles touch each other externally. The segment of their common external tangent that lies between the points of tangency is rotated around the line connecting the centers of the circles. Express the area of the frustum of the cone generated by this rotation in terms of $r_{1}$ and $r_{2}$.

Answer: 4\pi r_1 r_2


1. As shown in Figure 14, quadrilateral $A B C D$ is a square, and an equilateral triangle $\triangle A B E$ is constructed outward from side $A B$. $C E$ intersects $B D$ at point $F$. Then $\angle A F D=$ $\qquad$ degrees.

Answer: 60


6. All natural numbers, the sum of the digits of each of which is equal to 5, were arranged in ascending order. What number is in the 125th place

Answer: 41000


7.85 Given the following $5 \times 5$ number table
$$
\left(\begin{array}{ccccc}
11 & 17 & 25 & 19 & 16 \\
24 & 10 & 13 & 15 & 3 \\
12 & 5 & 14 & 2 & 18 \\
23 & 4 & 1 & 8 & 22 \\
6 & 20 & 7 & 21 & 9
\end{array}\right)
$$

Try to select 5 elements from this number table such that no two of them are in the same row or column, and make the smallest one as large as possible. Prove the correctness of your selected answer.

Answer: 15


9. Given $z \in \mathbf{C}$, if the equation $4 x^{2}-8 z x+4 \mathrm{i}+3=0$ (where $\mathrm{i}$ is the imaginary unit) has real roots, then the minimum value of the modulus of the complex number $z$ $|z|$ is $\qquad$ .

Answer: 1


6. Let $k$ be a real number, in the Cartesian coordinate system $x O y$ there are two point sets $A=\left\{(x, y) \mid x^{2}+y^{2}=\right.$ $2(x+y)\}$ and $B=\{(x, y) \mid k x-y+k+3 \geqslant 0\}$. If $A \cap B$ is a singleton set, then the value of $k$ is $\qquad$

Answer: -2 - \sqrt{3}


Joseane's calculator is malfunctioning: each digit pressed appears doubled on the screen. The addition, subtraction, multiplication, and division operation keys work normally and cannot be pressed consecutively twice. For example, a permissible sequence of operations is pressing $2 \rightarrow \times \rightarrow 3$, which produces the number $4 \cdot 6=24$.

a) How can she make 80 appear by pressing 3 keys?

b) How can she make 50 appear by pressing 3 number keys and two operation keys alternately?

c) What is the minimum number of keys she must press to obtain the number 23?

Answer: 4 \times 5


3. Circles $\omega_{1}$ and $\omega_{2}$ with centers $O_{1}$ and $O_{2}$ respectively intersect at point $B$. The extension of segment $O_{2} B$ beyond point $B$ intersects circle $\omega_{1}$ at point $K$, and the extension of segment $O_{1} B$ beyond point $B$ intersects circle $\omega_{2}$ at point $L$. The line passing through point $B$ parallel to $K L$ intersects circles $\omega_{1}$ and $\omega_{2}$ again at points $A$ and $C$ respectively. The rays $A K$ and $C L$ intersect at point $N$. Find the angle between the lines $O_{1} N$ and $O_{2} B$.

Answer: 90^\circ


Example 7 Let $S=\{1,2,3,4\}$. An $n$-term sequence: $q_{1}, q_{2}, \cdots, q_{n}$ has the following property: For any non-empty subset $B$ of $S$ (the number of elements in $B$ is denoted by $|B|$), there are adjacent $|B|$ terms in the sequence that exactly form the set $B$. Find the minimum value of $n$.
(1997, Shanghai High School Mathematics Competition)

Answer: 8


Let $n, m$ be positive integers such that
\[n(4n+1)=m(5m+1)\]
(a) Show that the difference $n-m$ is a perfect square of a positive integer.
(b) Find a pair of positive integers $(n, m)$ which satisfies the above relation.

Additional part (not asked in the TST): Find all such pairs $(n,m)$.

Answer: (38, 34)


Assume $A,B,C$ are three collinear points that $B \in [AC]$. Suppose $AA'$ and $BB'$
are to parrallel lines that $A'$, $B'$ and $C$ are not collinear. Suppose $O_1$ is circumcenter of circle passing through $A$, $A'$ and $C$. Also $O_2$ is circumcenter of circle passing through $B$, $B'$ and $C$. If area of $A'CB'$ is equal to area of $O_1CO_2$, then find all possible values for $\angle CAA'$

Answer: 30^\circ


6. In square $A B C D$, there is a point $M$, point $P$ is on $A B$, point $Q$ is on $C D$, and $\angle A P M=150^{\circ}$, $\angle M Q C=35^{\circ}$, then $\angle P M Q=$ $\qquad$ $\circ$.

Answer: 65


Let $E F G H$ be a convex quadrilateral, and $K, L, M, N$ be the midpoints of segments $E F, F G, G H, H E$ respectively; $O$ be the intersection point of segments $K M$ and $L N$. It is known that $\angle L O M = 90^{\circ}$, $K M = 3 L N$, and the area of quadrilateral $K L M N$ is $S$. Find the diagonals of quadrilateral $E F G H$.

Answer: 2\sqrt{\dfrac{5S}{3}}


Let \( G \) be a graph with \( n \) vertices such that \( G \) contains no triangle (cycle of length 3). Prove that \( G \) has at most \( \left\lfloor \frac{n^2}{4} \right\rfloor \) edges. Show that this result is optimal.

Bonus: Show that in a graph having \( k \) edges, there are at most \( \left\lfloor \frac{\sqrt{2}}{3} k^{3/2} \right\rfloor \) triangles.

Answer: \left\lfloor \frac{n^2}{4} \right\rfloor


## Problem 4.

An encyclopedia consists of 2000 numbered volumes. The volumes are stacked in order with number 1 on top and 2000 in the bottom. One may perform two operations with the stack:

(i) For $n$ even, one may take the top $n$ volumes and put them in the bottom of the stack without changing the order.

(ii) For $n$ odd, one may take the top $n$ volumes, turn the order around and put them on top of the stack again.

How many different permutations of the volumes can be obtained by using these two operations repeatedly?

Answer: (1000!)^2


10.373 In a triangle with sides 6, 10, and 12 cm, a circle is inscribed. A tangent is drawn to the circle such that it intersects the two larger sides. Find the perimeter of the cut-off triangle.

Answer: 16


1. Using the digits $4,5,6,7,8$ and 9, a six-digit number was written. Zoran, Darko, and Nikola guessed the number. Zoran: 574698, Darko: 786945, Nikola: 456789. It turned out that Zoran correctly guessed the positions of three digits. Darko also guessed the positions of three digits, and Nikola guessed the position of only one digit. Determine the six-digit number.

Answer: 576948


6. Let $A$ and $B$ be non-empty subsets of the set $\{1,2, \cdots, 10\}$, and the smallest element in set $A$ is not less than the largest element in set $B$. Then the number of such pairs $(A, B)$ is.

Answer: 9217


Let $n$ be a positive integer. Now, a frog starts jumping from the origin of the number line and makes $2^{n}-1$ jumps. The process satisfies the following conditions:

(1) The frog will jump to each point in the set $\left\{1,2,3, \cdots, 2^{n}-1\right\}$ exactly once, without missing any.

(2) Each time the frog jumps, it can choose a step length from the set $\left\{2^{0}, 2^{1}, 2^{2}, \cdots\right\}$, and it can jump either left or right.

Let $T$ be the reciprocal sum of the step lengths of the frog. When $n=2024$, the minimum value of $T$ is \_\_\_\_\_.

Answer: 2024


4. A $10 \times 10$ table is filled with numbers 1 and -1 such that the sum of the numbers in each row, except one, is equal to zero, and the sum of the numbers in each column, except one, is also equal to zero. Determine the maximum possible sum of all the numbers in the table.

(Patrik Bak)

Answer: 10


Example 3. As shown in Figure 3, the equilateral triangles $\triangle A B C$ and $\triangle A_{1} B_{1} C_{1}$ have their sides $A C$ and $A_{1} C_{1}$ bisected at $O$. Then $A A_{1}: B B_{1}=$ $\qquad$

Answer: 1 : \sqrt{3}


[ The ratio in which the bisector divides the side.] Pythagorean Theorem (direct and inverse). $\quad]$

In a right triangle $ABC$, the bisector $AP$ of the acute angle $A$ is divided by the center $O$ of the inscribed circle in the ratio

$AO: OP=(\sqrt{3}+1):(\sqrt{3}-1)$. Find the acute angles of the triangle.

Answer: 30^\circ


8. Given that $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ is a permutation of $1,2,3,4,5$, and satisfies $\left|a_{i}-a_{i+1}\right| \neq 1(i=1,2,3,4)$. Then the number of permutations $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ that meet the condition is $\qquad$.

Answer: 14


# 3. Clone 1

A rope was divided into 19 equal parts and arranged in a snake-like pattern. After that, a cut was made along the dotted line. The rope split into 20 pieces: the longest of them is 8 meters, and the shortest is 2 meters. What was the length of the rope before it was cut? Express your answer in meters.

![](https://cdn.mathpix.com/cropped/2024_05_06_d0dc74dfba1714359e24g-04.jpg?height=483&width=714&top_left_y=2100&top_left_x=725)

Answer: 152


Assume that all angles of a triangle \(ABC\) are acute. Let \(D\) and \(E\) be points on the sides \(AC\) and \(BC\) of the triangle such that \(A, B, D,\) and \(E\) lie on the same circle. Further suppose the circle through \(D, E,\) and \(C\) intersects the side \(AB\) in two points \(X\) and \(Y\). Show that the midpoint of \(XY\) is the foot of the altitude from \(C\) to \(AB\).

Answer: M = H


Let $(x, y, z, t) \in\left(\mathbb{N}^{*}\right)^{4}$ be a quadruple satisfying $x+y=z+t$ and $2xy=zt$, with $x \geq y$. Find the largest value of $m$ such that $m \leq x / y$ for sure.

Answer: 3 + 2\sqrt{2}


Numbers 1, 2, 3, ..., 10 are written in a circle in some order. Petya calculated the sums of all triplets of neighboring numbers and wrote the smallest of these sums on the board. What is the largest number that could have been written on the board?

Answer: 15


Five integers were written on the board. By summing them in pairs, the following ten numbers were obtained:

$$
0, 2, 4, 4, 6, 8, 9, 11, 13, 15
$$

What are the five numbers that were written on the board? Is it possible to obtain the following ten numbers by summing five integers in this way:

$$
12, 13, 14, 15, 16, 16, 17, 17, 18, 20?
$$

Answer: \text{No}


Out of 24 matchsticks of the same length, use some of them to form 6 triangles in a plane where each side of the equilateral triangle is one matchstick. Then, use the remaining matchsticks to form squares in the plane where each side of the square is one matchstick. What is the maximum number of such squares that can be formed?

Answer: 4


A regular triangular prism $ABC A_1B_1C_1$ is inscribed in a sphere, with base $ABC$ and lateral edges $AA_1, BB_1, CC_1$. Segment $CD$ is a diameter of this sphere, and point $K$ is the midpoint of edge $AA_1$. Find the volume of the prism if $CK = 2 \sqrt{6}$ and $DK = 4$.

Answer: 36


Initially, there are three cards in a box, each with numbers $1, 2, 3$. Each time, two cards are drawn from the box, and their sum is written on another blank card, then one card is put back into the box. After performing this operation 5 times, all cards except the last one have been drawn at least once but not more than twice. What is the maximum number on the card in the box at this point?

Answer: 28


A $7 \times 7$ grid is colored black and white. If the number of columns with fewer black squares than white squares is $m$, and the number of rows with more black squares than white squares is $n$, find the maximum value of $m + n$.

Answer: 12


Let $z$ be a complex number such that $z^{23} = 1$ and $z \neq 1.$  Find
\[\sum_{n = 0}^{22} \frac{1}{1 + z^n + z^{2n}}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 49


Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\frac{n}{d}$ where $n$ and $d$ are integers with $1 \le d \le 5$. What is the probability that \[(\text{cos}(a\pi)+i\text{sin}(b\pi))^4\]is a real number?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 31


The equation \[2000x^6+100x^5+10x^3+x-2=0\]has two real roots. Compute the square of the difference between them.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 561


Consider a cubic polynomial \( x^3 - sx^2 + px - q \) whose roots \( r_1, r_2, \) and \( r_3 \) satisfy \( r_1 + r_2 + r_3 = r_1^2 + r_2^2 + r_3^2 \). Calculate the maximum possible value of \( \frac{1}{r_1^3} + \frac{1}{r_2^3} + \frac{1}{r_3^3} \).

Answer: 3


A line contains the points $(7, -3)$ and $(3, 1)$. Where does this line intersect the $x$-axis?

Answer: (4, 0)


When $600_{10}$ is expressed in a certain base, it has four digits in the form ABBA, where A and B are different digits. What is this base?

Answer: 7


A framed artwork consists of a painting surrounded by a uniform frame. The outer dimensions of the framed artwork are 90 cm by 120 cm, and the width of the frame is 15 cm. What is the area of the painting inside the frame?

Answer: 5400 \text{ cm}^2


The isosceles triangle and the rectangle shown here have the same area in square units. What is the height of the triangle, $h$, in terms of the length $l$ and width $w$ of the rectangle?

Answer: 2w


Two triangles, $\triangle ABC$ and $\triangle DEF$, are drawn such that $\triangle ABC \sim \triangle DEF$. The lengths of segments $BC = 8$ cm and $AB = 10$ cm in $\triangle ABC$, and $DE = 24$ cm in $\triangle DEF$. If angle $BAC = 90^\circ$ and angle $EDF = 90^\circ$, what is the length of segment $EF$?

Answer: 19.2 \text{ cm}


In a public park, a circular arrangement is designed as follows: at the center, there is a circular water fountain surrounded by a ring-shaped flower garden. Outside the garden, there is a walking path and then another inner walking path adjacent to the garden but inside it. The water fountain has a diameter of 12 feet. The flower garden ring is 9 feet wide, the outer walking path is 7 feet wide, and the inner walking path is 3 feet wide. Calculate the diameter of the outermost boundary of the outer walking path.

Answer: 50


A triangle has sides of length 10 cm, 13 cm, and 7 cm. Find the perimeter of the triangle and also calculate its area using Heron's formula.

Answer: 20\sqrt{3} \text{ cm}^2


A right triangle has one leg of 15 inches and the hypotenuse of 34 inches. A second triangle, which is similar to the first, has a hypotenuse that is twice the length of the first triangle’s hypotenuse. What is the length of the second triangle's shortest leg, if the longer leg of the second triangle is 33% larger than the corresponding leg of the first triangle?

Answer: 30 \text{ inches}


The sum of the lengths of all the edges of a rectangular prism is 88 cm. The prism's length is twice its width, and its height is half of its width. Find the volume of the rectangular prism.

Answer: V = \frac{85184}{343} \text{ cm}^3


The planet Zorion travels in an elliptical orbit around its sun, which is positioned at one focus of the ellipse. Zorion's closest approach to its sun is 3 astronomical units (AU), and its furthest point is 15 AU away. If the entire orbit of Zorion is tilted by 30 degrees from its original plane, calculate the distance from Zorion to its sun when Zorion is exactly halfway through its orbit from perigee to apogee.

Answer: 9 \text{ AU}


Elmo makes $N$ sandwiches for a school event. For each sandwich, he uses $B$ dollops of peanut butter at $3$ cents each and $J$ spoonfuls of jelly at $7$ cents each. The total cost for the peanut butter and jelly to make all the sandwiches is $\$3.78$. Assume that $B$, $J$, and $N$ are positive integers with $N>1$. What is the cost, in dollars, of the jelly Elmo uses to make the sandwiches?

Answer: \$2.94


7.37. What is the maximum number of planes of symmetry that a spatial figure consisting of three pairwise non-parallel lines can have?

Symmetry with respect to a line $l$ is a transformation of space that maps a point $X$ to a point $X^{\prime}$ such that the line $l$ passes through the midpoint of the segment $X X^{\prime}$ and is perpendicular to it. This transformation is also called axial symmetry, and $l$ is called the axis of symmetry:

Answer: 9


Task 3.

## Maximum 10 points

In the Country of Wonders, a pre-election campaign is being held for the position of the best tea lover, in which the Mad Hatter, March Hare, and Dormouse are participating. According to a survey, $20 \%$ of the residents plan to vote for the Mad Hatter, $25 \%$ for the March Hare, and $30 \%$ for the Dormouse. The rest of the residents are undecided. Determine the smallest percentage of the undecided voters that the Mad Hatter must attract to ensure he does not lose to the March Hare and the Dormouse (under any distribution of votes), knowing that each of the undecided voters will vote for one of the candidates. The winner is determined by a simple majority of votes. Justify your answer.

Answer: 70


3. If $x \in(-1,1)$, then
$$
f(x)=x^{2}-a x+\frac{a}{2}
$$

is always positive, the range of the real number $a$ is $\qquad$ $ـ$.

Answer: (0,2]


Emília's tree grows according to the following rule: two weeks after a branch appears, that branch produces a new branch every week, and the original branch continues to grow. After five weeks, the tree has five branches, as shown in the figure. How many branches, including the main branch, will the tree have at the end of eight weeks?

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-018.jpg?height=320&width=275&top_left_y=1165&top_left_x=1553)

Answer: 21


8. A thin beam of light falls normally on a plane-parallel glass plate. Behind the plate, at some distance from it, stands an ideal mirror (its reflection coefficient is equal to one). The plane of the mirror is parallel to the plate. It is known that the intensity of the beam that has passed through this system is 256 times less than the intensity of the incident beam. The reflection coefficient at the glass-air boundary is assumed to be constant regardless of the direction of the beam. Neglect absorption and scattering of light in air and glass. Find the reflection coefficient at the glass-air boundary under these conditions. (10 points)

Answer: 0.75


13. Ring Line (from 8th grade. 3 points). On weekdays, the Absent-Minded Scientist travels to work on the ring line of the Moscow metro from the station "Taganskaya" to the station "Kievskaya", and back in the evening (see the diagram).

Upon entering the station, the Scientist boards the first train that arrives. It is known that trains run at approximately equal intervals in both directions, and that the train traveling on the northern route (through "Belorusskaya") takes 17 minutes to travel from "Kievskaya" to "Taganskaya" or vice versa, while the train on the southern route (through "Paveletskaya") takes 11 minutes.

Out of habit, the Scientist always calculates everything. Once, he calculated that over many years of observation:

- the train traveling counterclockwise arrives at "Kievskaya" on average 1 minute and 15 seconds after the train traveling clockwise arrives at the same station. The same is true for "Taganskaya";

- the average travel time from home to work is 1 minute less than the travel time from work to home.

Find the expected interval between trains traveling in the same direction.

![](https://cdn.mathpix.com/cropped/2024_05_06_0312347ad1209bb36d0eg-08.jpg?height=785&width=785&top_left_y=1606&top_left_x=1138)

Answer: 3


Problem 2. A square canvas is divided into 100 identical squares, arranged in 10 rows and 10 columns. We have 10 cards, numbered differently with digits from 0 to 9. On the canvas, we need to place two cards, whose sum is 10, in squares located on different rows and columns. Determine the number of possibilities for placing these cards.

Answer: 32400


6. Given that $a$ is a real number, and for any $k \in$ $[-1,1]$, when $x \in(0,6]$, we have
$$
6 \ln x+x^{2}-8 x+a \leqslant k x \text {. }
$$

Then the maximum value of $a$ is $\qquad$ .

Answer: 6 - 6 \ln 6


2. A stack of A4 sheets was folded in half and folded in two (resulting in an A5 booklet). After that, the pages of the resulting booklet were renumbered: $1,2,3, \ldots$ It turned out that the sum of the numbers on one of the sheets was 74. How many sheets were in the stack?

Answer: 9


5. A regular triangle with a side length of 2 is divided into four smaller triangles by its median lines. Except for the central small triangle, three other small triangles are used as bases to construct three regular tetrahedrons of height 2 on the same side. There is a sphere that is tangent to the plane of the original triangle and also tangent to one side face of each tetrahedron. What is the radius of this small sphere?

Answer: \dfrac{1}{3}


Integer $n>2$ is given. Find the biggest integer $d$, for which holds, that from any set $S$ consisting of $n$ integers, we can find three different (but not necesarilly disjoint) nonempty subsets, such that sum of elements of each of them is divisible by $d$.

Answer: n-1


A square is inscribed in a triangle such that two of its vertices lie on the base, and the other two vertices lie on the sides of the triangle. Prove that the side of the square is less than $2r$ but greater than $\sqrt{2}r$, where $r$ is the radius of the circle inscribed in the triangle.

Answer: \sqrt{2}r < s < 2r


6. Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $C: \frac{x^{2}}{4}-\frac{y^{2}}{5}=1$, respectively. Point $P$ is on the right branch of the hyperbola $C$, and the excenter of $\triangle P F_{1} F_{2}$ opposite to $\angle P F_{1} F_{2}$ is $I$. The line $P I$ intersects the $x$-axis at point $Q$. Then
$$
\frac{|P Q|}{|P I|}+\frac{\left|F_{1} Q\right|}{\left|F_{1} P\right|}=
$$
$\qquad$

Answer: 4


10,11 |

A sphere is constructed with the height of a cone as its diameter. The surface area of the part of the sphere lying inside the cone is equal to the surface area of the part of the cone lying inside the sphere. Find the angle in the axial section of the cone.

Answer: 2 \arcsin\left( \frac{\sqrt{5} - 1}{2} \right)


13.260. Two material particles, being 295 m apart from each other, started moving towards each other simultaneously. The first particle moves uniformly at a speed of $15 \mathrm{~m} / \mathrm{c}$, while the second particle moved 1 m in the first second and 3 m more in each subsequent second than in the previous one. Through what angle will the second hand of a clock move during the time that has passed from the start of the particles' movement until their meeting?

Answer: 60


5. The power in the kingdom of gnomes was seized by giants. The giants decided to get rid of the gnomes and told them the following: "Tomorrow we will line you up so that each of you will see those who stand after and not see those who stand before (i.e., the 1st sees everyone, the last sees no one). We will put either a black or a white hat on each of you (equally likely, each will have either a black or a white hat) and ask what color it is. Those who answer correctly will be released, and those who answer incorrectly will be executed." How many gnomes can be risked at a minimum with certain agreements before the execution, if there are p gnomes in the kingdom and p $<\infty$. Justify your answer. (12 points)

#

Answer: 1


Regarding natural numbers \( n \) and \( m \), it is known that \(\sqrt{3} - \frac{m}{n} > 0\).

Prove that \(\sqrt{3} - \frac{m}{n} > \frac{1}{2 m n}\).

Answer: \sqrt{3} - \frac{m}{n} > \frac{1}{2mn}


## Task B-4.5.

In triangle $A B C$, $\varangle C A B=50^{\circ}$ and $\varangle A B C=60^{\circ}$. On side $\overline{A B}$ there is a point $D$, and on side $\overline{B C}$ there is a point $E$ such that $\varangle C A E=\varangle A C D=30^{\circ}$. Calculate the measure of angle $\varangle C D E$.

Answer: 40^\circ


A circle $k$ has its center at $O$, and points $A, B, C, D$ are consecutive points on the circumference of $k$, such that $\angle AOB = \angle BOC = \angle COD = \alpha < 60^{\circ}$. The projection of $D$ onto the diameter $AO$ is $E$, and the point $F$ is the closer trisection point of segment $DE$ to $E$. Finally, the intersection of $AO$ and $BF$ is $G$. What does the size of angle $OGD$ approach as $\alpha \rightarrow 60^{\circ}$?

Answer: 60^\circ


3. Two spheres $O_{1}$ and $O_{2}$, both with a radius of 1, are tangent to each other, and they are also tangent to the two half-planes of a dihedral angle $\alpha-l-\beta$ of $60^{\circ}$. There is another larger sphere $O$ that is tangent to both half-planes of the dihedral angle and is externally tangent to spheres $O_{1}$ and $O_{2}$. What is the radius $R$ of sphere $O$?

Answer: \dfrac{5 + \sqrt{13}}{3}


Through the vertex \( A \) of the rectangle \( ABCD \), a line \( \ell \) is drawn, as shown in the figure. From points \( B \) and \( D \), perpendiculars \( BX \) and \( DY \) are dropped onto the line \( \ell \). Find the length of segment \( XY \) if it is known that \( BX = 4 \), \( DY = 10 \), and \( BC = 2 AB \).

Answer: 13


Let the set \( M = \{1, 2, \cdots, 2020\} \). The subset \( A \subseteq M \) is such that for any element \( x \) in \( A \), \( 4x \) is not in \( A \). What is the maximum number of elements in the set \( A \)?

Answer: 1616


As shown in the figure, Rourou's garden is a large rectangle composed of 4 square plots of land and 1 small rectangular pool. If the area of each plot of land is 20 square meters and the length of the garden is 9 meters, what is the perimeter of the pool (the shaded area in the figure)? ___ meters.

Answer: 18


One hundred mathematicians attending an international combinatorial conference are being accommodated in a hotel where the rooms are numbered from 1 to 100. The reception plans to assign the mathematicians to rooms according to their arrival order and their corresponding room numbers. However, the first guest forgets the instruction and randomly chooses a room. The receptionist then instructs the subsequent guests to take the room corresponding to their arrival number sequentially and, if that room is occupied, to choose any free room that they like. In how many different ways could the guests occupy the rooms?

Answer: 2^{99}


In triangle ABC, the median BM is drawn. It is known that \(\angle \mathrm{BAC} = 30^\circ\) and \(\angle \mathrm{BMC} = 45^\circ\). Find angle BAC.

Answer: 105


In the pentagon \( U V W X Y \), \(\angle U = \angle V = 90^\circ \), \( U Y = V W \), and \( Y X = X W \). Four equally spaced points are marked between \( U \) and \( V \), and perpendiculars are drawn through each point. The dark shaded region has an area of \( 13 \mathrm{~cm}^2 \) and the light shaded region has an area of \( 10 \mathrm{~cm}^2 \). What is the area, in \(\mathrm{cm}^2\), of the entire pentagon?
A) 45
B) 47
C) 49
D) 58
E) 60

Answer: 45


Dinesh has several squares and regular pentagons, all with side length $ 1$. He wants to arrange the shapes alternately to form a closed loop (see diagram). How many pentagons would Dinesh need to do so?
[img]https://cdn.artofproblemsolving.com/attachments/8/9/6345d7150298fe26cfcfba554656804ed25a6d.jpg[/img]

Answer: 10


Example: There are 3 bags with 10 balls each of red, white, and black. If 16 balls are drawn from them, and it is required that all three colors are included, how many different ways are there to do this?

Answer: 75


5. Let $P$ be a point inside an equilateral $\triangle A B C$, with $A P=3$, $B P=4$, $C P=5$. Then the area of quadrilateral $A B C P$ is $\qquad$

Answer: 4\sqrt{3} +6


## Task 5 - 261245

Determine all natural numbers $n \geq 3$ for which the following statement holds:

Every plane convex $n$-gon $A_{1} A_{2} \ldots A_{n}$ is completely covered by the areas of the $n$ circles that have the segments $A_{i} A_{i+1}$ as diameters $\left(i=1,2, \ldots, n\right.$; set $A_{n+1}=A_{1}$).

Here, each $n$-gon area and each circle area are understood to include their boundary points.

Answer: 4


# Assignment 4. 20 points

## Variant 1

In country Alpha, only two goods, X and Y, are produced from a single production factor - factor Z, of which there are exactly 100 units in Alpha. The amount of good X produced from factor Z can be described by the function $X=\frac{\sqrt{Z_{X}}}{2}$, and one unit of good Y is produced from each unit of factor Z. Country Alpha can buy and sell both goods on the world market at prices $P_{X}=8$ and $P_{Y}=1$. The residents of the country always consume goods in sets: for every 3 units of good X, 184 units of good Y are consumed. The government of the country is concerned only with ensuring that the residents can consume as many such sets of goods as possible.

(a) Find the quantities of goods X and Y that will be produced and consumed by the residents of this country. Graphically represent the found values and the production possibilities frontier of the country on one diagram.

(b) On the world market, an increase in the price of good X by a factor of $\alpha$ is expected, while it is known that the price of good Y will not change. Do you think that country Alpha can benefit from the increase in the price of good X? If it can, find all such values of $\alpha$ for which the country will benefit, or explain why Alpha cannot benefit from the increase in the price of good X.

Answer: X = 1


In the right-angled triangle $\mathrm{ABC}$, the angle at vertex $B$ is $30^{\circ}$. The center of the square constructed outward on the hypotenuse $\mathrm{ABC}$ is $D$. What is the measure of the angle $A D B$?

Answer: 60^\circ


## Task 3 - 050713

The driver of a car registered in the GDR fled the scene after a traffic accident. After questioning several witnesses, the following information was obtained about the police registration number of the car:

a) The two letters of the license plate were AB or AD.

b) The two front digits were the same and different from the last two digits.

c) The number formed by the last two digits was 69 or 96.

What is the maximum possible number of cars that can meet these conditions?

Answer: 32


$\mathbf{N 4 2}$ (37-4, Russia) Let positive integers $a, b$ be such that $15a + 16b$ and $16a - 15b$ are both squares of positive integers. Find the smallest value that the smaller of these two squares can take.

Answer: 231361


1. Let $A B C D$ be a quadrilateral inscribed in a circle. Suppose $A B=$ $\sqrt{2+\sqrt{2}}$ and $A B$ subtends $135^{\circ}$ at the centre of the circle. Find the maximum possible area of $A B C D$.
![](https://cdn.mathpix.com/cropped/2024_06_05_177a0e3dd76b8909557ag-1.jpg?height=360&width=1090&top_left_y=478&top_left_x=515)

Answer: \dfrac{5\sqrt{2} + 3\sqrt{6}}{8}


Let $S$ be a finite set of points in the plane, such that for each $2$ points $A$ and $B$ in $S$, the segment $AB$ is a side of a regular polygon all of whose vertices are contained in $S$. Find all possible values for the number of elements of $S$.

Proposed by [i]Viktor Simjanoski, Macedonia[/i]

Answer: 3


Let $p$ be a prime number. How many monic polynomials are there in $\mathbb{Z} / p \mathbb{Z}$ of degree $p-2$ that have exactly $p-2$ distinct roots, and whose coefficients are all distinct and non-zero?

## Irreducibility

##

The translation is provided as requested, maintaining the original text's line breaks and format.

Answer: \phi(p-1)


Let $y(x)$ be the unique solution of the differential equation
$$
\frac{\mathrm{d} y}{\mathrm{~d} x}=\log _{e} \frac{y}{x}, \quad \text { where } x>0 \text { and } y>0,
$$
with the initial condition $y(1)=2018$.
How many positive real numbers $x$ satisfy the equation $y(x)=2000$ ?

Answer: 1


Let \( a_n \) be the recurrence sequence defined by \( a_0 = 2 \) and \( a_{n+1} = 2a_n^2 - 1 \). Let \( N \geq 1 \) and \( p \) be a prime divisor of \( a_N \). Suppose there exists an integer \( x \) such that \( x^2 \equiv 3 \ (\bmod \ p) \). Show that \( 2^{N+2} \) divides \( p-1 \).

Answer: 2^{N+2} \text{ divides } p-1


On an $n \times k$ chessboard, a bishop moves. The bishop starts from one of the light-colored corners and moves diagonally. When it reaches the edge of the board, it "bounces back". The movement ends when it reaches a corner again. For which pairs $(n, k)$ is it true that the bishop covers all light-colored squares?

Answer: \gcd(n-1, k-1) = 1


2. Usually, Dima leaves home at $8:10$ AM, gets into Uncle Vanya's car, who delivers him to school by a certain time. But on Thursday, Dima left home at 7:20 and ran in the opposite direction. Uncle Vanya waited for him and at $8:20$ drove after him, caught up with Dima, turned around, and delivered him to school 26 minutes late. How many times faster was Uncle Vanya's car speed compared to Dima's running speed?

Answer: \dfrac{17}{2}


10. Right triangle $X Y Z$ has right angle at $Y$ and $X Y=228, Y Z=$ 2004. Angle $Y$ is trisected, and the angle trisectors intersect $X Z$ at $P$ and $Q$ so that $X, P, Q, Z$ lie on $X Z$ in that order. Find the value of $(P Y+Y Z)(Q Y+X Y)$.

Answer: 1370736


The 79 interns of the Animath internship each choose an activity for the free afternoon from 5 proposed activities. We know that:

- The swimming pool was at least as popular as football
- Students went shopping in groups of 5
- At most 4 students played cards
- At most one student stayed in their room

We write next to each activity the number of students who participated in it. How many different lists could we have written?

## Combinatorial identities

Answer: 3240


4. Find the number of all 6-digit natural numbers such that the sum of their digits is 10 and each of the digits $0,1,2,3$ occurs at least once in them.

Answer: 490


8. Find all integer values of the parameter \(a\) for which the system has at least one solution

\[
\left\{\begin{array}{l}
y-2=x(x+2) \\
x^{2}+a^{2}+2 x=y(2 a-y)
\end{array}\right.
\]

In the answer, specify the sum of the found values of the parameter \(a\).

Answer: 3


It is a custom of five good friends that they wear a different tie each day, each one the one they wore the longest ago. Each of them has at least two ties, but none of them has a dozen. None of them has two ties of the same color, and there are no two of them who have the same number of ties. We have the following data from the last month of last year:

1. On December 1, Aladár wore a blue, Bandi and Feri wore red, Pista wore green, and Géza wore a yellow tie.
2. On December 19, Pista wore green, Géza wore yellow, Feri wore blue, and the other two wore red ties.
3. On December 23, Pista wore a white tie, and on December 26, he wore a yellow tie.
4. The colors of the ties worn on December 11 were: yellow, red, blue, green, and white.
5. On New Year's Eve, all five wore ties of the same color as on December 1.

Question: What tie did Bandi wear on January 1 of this year?

Answer: green


3.55. A truncated cone is described around a sphere, with the area of one base being 4 times larger than the area of the other. Find the angle between the slant height of the cone and the plane of its base.

## Group B

Answer: \arctan(2\sqrt{2})


## Task B-3.4.

While preparing for the competition, Matko discovered a bookstore with good mathematical literature. The bookstore offers 7 different books with problems only in geometry, 4 only in number theory, and 5 only in combinatorics. Furthermore, the store also offers books with problems from exactly two areas. Thus, there are 6 different books with problems in number theory and combinatorics, and 7 with problems in geometry and combinatorics. In how many ways can Matko choose literature from two of all the mentioned areas if he can buy at most two books?

Answer: 270


7. The base $ABCD$ of the quadrilateral pyramid $P-ABCD$ is a rhombus with a top angle of $60^{\circ}$, and the angle between each side face and the base is $60^{\circ}$. There is a point $M$ inside the pyramid that is 1 unit away from the base and each side face. Then the volume of the pyramid is

Answer: 8\sqrt{3}


2. Let $a_{1}, a_{2}, \ldots$ be a sequence of integers defined by $a_{1}=3, a_{2}=3$, and
$$
a_{n+2}=a_{n+1} a_{n}-a_{n+1}-a_{n}+2
$$
for all $n \geq 1$. Find the remainder when $a_{2020}$ is divided by 22 .

Answer: 11


Let $K$ and $N > K$ be fixed positive integers. Let $n$ be a positive integer and let $a_1, a_2, ..., a_n$ be distinct integers. Suppose that whenever $m_1, m_2, ..., m_n$ are integers, not all equal to $0$, such that $\mid{m_i}\mid \le K$ for each $i$, then the sum
$$\sum_{i = 1}^{n} m_ia_i$$
is not divisible by $N$. What is the largest possible value of $n$?

[i]Proposed by Ilija Jovcevski, North Macedonia[/i]

Answer: \left\lfloor \log_{K+1} N \right\rfloor


10. Let the edge length of the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ be $1, \alpha$ be the plane passing through the line $B D_{1}$. Then the range of the area of the section cut by $\alpha$ on the cube is $\qquad$

Answer: \left[ \dfrac{\sqrt{6}}{2}, \sqrt{2} \right]


7. Given $S_{n}$ as the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$, with the rule $S_{0}=0$. If for any $n \in \mathbf{Z}_{+}$, we have
$$
\begin{array}{l}
\frac{a_{n}}{2017}=-\frac{2017+S_{n-1}}{n}, \\
\text { then } \sum_{n=1}^{2017} 2^{n} a_{n}=
\end{array}
$$

Answer: -4034


In parallelogram $ABCD,$ let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$ . Angles $CAB$ and $DBC$ are each twice as large as angle $DBA,$ and angle $ACB$ is $r$ times as large as angle $AOB$ . Find the greatest integer that does not exceed $1000r$ .
Please give the answer directly without any intermediate steps.

Answer: 777


1. As shown in the figure, divide a line segment of length 1 into segments $x$ and $y$. Then, bend the segment of length $x$ into a semicircle $A C B$, and fold the segment of length $y$ into three sides $(B D, D E, E A)$ of a rectangle $A B D E$ to form a closed "curved polygon" $A C B D E A$. The maximum value of the area of this curved polygon is . $\qquad$

Answer: \dfrac{1}{2(\pi +4)}


Bethany is told to create an expression from $2 \square 0 \square 1 \square 7$ by putting a + in one box, a in another, and $a x$ in the remaining box. There are 6 ways in which she can do this. She calculates the value of each expression and obtains a maximum value of $M$ and a minimum value of $m$. What is $M-m$ ?

Answer: 15


3. In quadrilateral $A B C D$, it is known that $B C=8, C D=12$, $A D=10, \angle A=\angle B=60^{\circ}$. Then $A B=$

Answer: 9 + \sqrt{141}


A group of 6 students decided to make [i]study groups[/i] and [i]service activity groups[/i] according to the following principle:

Each group must have exactly 3 members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members.

Supposing there are at least one group and no three students belong to the same study group and service activity group, find the minimum number of groups.

Answer: 8


There are 2011 positive numbers with both their sum and the sum of their reciprocals equal to 2012.  Let $x$ be one of these numbers.  Find the maximum value of $x + \frac{1}{x}.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 10057


15. Three rectangular pieces of paper (Jia, Bing, Ding) and one square piece (Yi) can be combined to form a large rectangle with an area of 480 square centimeters. It is known that the areas of Yi, Bing, and Ding are all three times that of Jia. The total perimeter of the four rectangles Jia, Yi, Bing, and Ding is $\qquad$ centimeters.

Answer: 184


520. A circle of radius $r$ is inscribed in a triangle with perimeter $p$ and area $S$. How are these three quantities related?

Answer: S = \frac{r \cdot p}{2}


61st Putnam 2000 Problem A3 An octagon is incribed in a circle. One set of alternate vertices forms a square area 5. The other set forms a rectangle area 4. What is the maximum possible area for the octagon? Solution

Answer: 3\sqrt{5}


4. For a truncated triangular pyramid, the sides of the smaller base are $7 \mathrm{~cm}$, $5 \mathrm{~cm}$, and $3 \mathrm{~cm}$, and the lateral edges form an angle of $45^{\circ}$ with the plane of the larger base. Calculate the volume of the truncated pyramid if its height is $\frac{2}{3}$ of the height of the corresponding pyramid.

Answer: \dfrac{455}{2}


4. Given 100 lines on a plane, let $T$ denote the set of right-angled triangles formed by some three of these lines. Find the maximum value of $|T|$. (Supplied by Zou Jin)

Answer: 62500


12. Chris planned a $210 \mathrm{~km}$ bike ride. However, he rode $5 \mathrm{~km} / \mathrm{h}$ faster than he planned and finished his ride 1 hour earlier than he planned. His average speed for the ride was $x \mathrm{~km} / \mathrm{h}$. What is the value of $x$ ?

Answer: 35


19. As shown in the figure, in the dihedral angle $\alpha-E F-\beta$, $A E \subset \alpha, B F \subset \beta$, and $A E \perp E F, B F \perp E F, E F=1, A E=2, A B=$ $\sqrt{2}$, find the maximum volume of the tetrahedron $A B E F$.

Answer: \dfrac{1}{3}


Exercise 4. We want to color the three-element subsets of $\{1,2,3,4,5,6,7\}$ such that if two of these subsets have no element in common, then they must be of different colors. What is the minimum number of colors needed to achieve this goal?

Answer: 3


4. The base of the oblique prism is a kite, which has a shorter diagonal of length e. The internal angles of the kite at the endpoints of the longer diagonal measure $90^{\circ}$ and $60^{\circ}$. The height of the prism is equal to the longer diagonal of the kite. Express the volume of the prism in terms of e. The result should be exact.

Answer: \dfrac{(2 + \sqrt{3})e^3}{4}


11. As shown in Figure 5, in quadrilateral $A B C D$, $\angle A=$ $\angle B C D=90^{\circ}, B C=$ $C D, E$ is a point on the extension of $A D$. If $D E=A B$ $=3, C E=4 \sqrt{2}$, then the length of $A D$ is

Answer: 5


2.2. Given a convex pentagon $A B C D E$, such that

$$
A B=A E=D C=B C+D E=1 \text { and } \angle A B C=D E A=90^{\circ} .
$$

What is the area of this pentagon?

Answer: 1


II. Fill-in-the-blank Questions (Full marks 54 points, each question 9 points)
1. Given a positive integer $n$ does not exceed 2000, and can be expressed as the sum of at least 60 consecutive positive integers, then the number of such $n$ is $\qquad$.

Answer: 6


Answer: 36


Say a real number $r$ is \emph{repetitive} if there exist two distinct complex numbers $z_1, z_2$ with $|z_1| = |z_2| = 1$ and $\{z_1, z_2\} \neq \{-i, i\}$ such that:
\[
z_1(z_1^3 + z_1^2 + rz_1 + 1) = z_2(z_2^3 + z_2^2 + rz_2 + 1).
\]
There exist real numbers $a, b$ such that a real number $r$ is \emph{repetitive} if and only if $a < r \le b$. If the value of $|a| + |b|$ can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$, find $100p + q$.

Answer: 2504


Suppose that a planet contains $(CCAMATHBONANZA_{71})^{100}$ people ($100$ in decimal), where in base $71$ the digits $A, B, C, \ldots, Z$ represent the decimal numbers $10, 11, 12, \ldots, 35$, respectively. Suppose that one person on this planet is snapping, and each time they snap, at least half of the current population disappears. Estimate the largest number of times that this person can snap without disappearing. An estimate of $E$ earns $2^{1-\frac{1}{200}\left|A-E\right|}$ points, where $A$ is the actual answer.

Answer: 8355


In triangle $ABC$ with centroid $G$ and circumcircle $\omega$, line $\overline{AG}$ intersects $BC$ at $D$ and $\omega$ at $P$. Given that $GD = DP = 3$, and $GC = 4$, find $AB^2$.

Answer: 168


Tanya wrote numbers in the form $n^7 - 1$ for $n = 2, 3, \ldots$ and noticed that for $n = 8$, she obtained a number divisible by $337$. For what minimal $n$ did she get a number divisible by $2022$?

Answer: 79


You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out, you must open all the doors and disarm all the mines. In the room, there is a panel with $3$ buttons, which conveniently contains an instruction manual:

- The red button arms a mine.
- The yellow button disarms two mines and closes a door.
- The green button opens two doors.

Initially, $3$ doors are closed, and $3$ mines are armed. The manual warns that attempting to disarm two mines or open two doors when only one is armed/closed will reset the system to its initial state. What is the minimum number of buttons you must push to get out?

Answer: 9


Given a $2018 \times 4$ grid, tint the cells with red and blue such that each row and each column has an equal number of red and blue cells. Determine the number of ways, $M$, to tint the grid with the mentioned requirement and find $M \pmod{2018}$.

Answer: 6


For a natural number $n$, define the function $f(n)$ as the sum of the digits of the number $n$. For example, $f(16) = 7$, $f(f(78)) = 6$, and $f(f(f(5978))) = 2$.

Find the smallest natural number $n$ such that $f(f(f(n)))$ is not a one-digit number.


A trifecta is an ordered triple of positive integers \((a, b, c)\) with \(a < b < c\) such that \(a\) divides \(b\), \(b\) divides \(c\), and \(c\) divides \(ab\). What is the largest possible sum \(a + b + c\) over all trifectas of three-digit integers?

Answer: 1736


Let $S_0 = 0$ and let $S_k$ equal $a_1 + 2a_2 + \ldots + ka_k$ for $k \geq 1$. Define $a_i$ to be $1$ if $S_{i-1} < i$ and $-1$ if $S_{i-1} \geq i$. What is the largest $k \leq 2010$ such that $S_k = 0$?

Answer: 1092


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

## Clues:
1. The person whose favorite color is green is somewhere to the left of the person with a high school diploma.
2. Eric is the person whose favorite color is red.
3. There are two houses between Alice and the person who loves white.
4. The person who smokes Blue Master and the person partial to Pall Mall are next to each other.
5. The person who loves white is the person with a master's degree.
6. Arnold is in the fourth house.
7. The person who loves yellow is the person who smokes Blue Master.
8. The Dunhill smoker is Alice.
9. The person partial to Pall Mall is the person with a master's degree.
10. The person who loves yellow is the person with a bachelor's degree.
 What is the value of attribute Name for the person whose attribute Education is bachelor? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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

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

Answer: Fred


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Eric`, `Peter`, `Alice`, `Arnold`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`, `stew`, `stir fry`
 - The people keep unique animals: `bird`, `horse`, `cat`, `fish`, `dog`
 - Each person has a unique level of education: `master`, `doctorate`, `high school`, `associate`, `bachelor`

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

Answer: stir fry


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

## Clues:
1. Bob is not in the second house.
2. The person who smokes many unique blends is somewhere to the left of the person who is a pizza lover.
3. The person partial to Pall Mall is somewhere to the right of the Dunhill smoker.
4. The person who loves the soup is not in the first house.
5. The person who likes milk is directly left of the person who loves the stew.
6. The person who loves the soup is the person with a doctorate.
7. The coffee drinker is the person with a high school diploma.
8. There are two houses between the person who loves the spaghetti eater and the person with an associate's degree.
9. The Prince smoker is not in the third house.
10. The Dunhill smoker and Bob are next to each other.
11. There are two houses between Alice and the boba tea drinker.
12. The person with a high school diploma is Eric.
13. The person who attended trade school is the Prince smoker.
14. There is one house between the Dunhill smoker and the person who loves stir fry.
15. The Dunhill smoker is directly left of the person who is a pizza lover.
16. There is one house between the person partial to Pall Mall and the person who smokes Blue Master.
17. The person who smokes Blue Master is the coffee drinker.
18. The person with a bachelor's degree is the Dunhill smoker.
19. Alice is the tea drinker.
20. Arnold is directly left of the root beer lover.
21. The person with a master's degree is Peter.
22. Arnold is somewhere to the left of the person who likes milk.
 What is the value of attribute Food for the person whose attribute Cigar is blue master? 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: `Arnold`, `Eric`, `Peter`, `Bob`, `Alice`
 - Each person has an occupation: `engineer`, `doctor`, `artist`, `lawyer`, `teacher`
 - Each person prefers a unique type of vacation: `cruise`, `beach`, `city`, `camping`, `mountain`

## Clues:
1. The person who prefers city breaks is somewhere to the left of the person who is an engineer.
2. There are two houses between the person who likes going on cruises and the person who enjoys camping trips.
3. Bob is in the fourth house.
4. The person who is a doctor is not in the fifth house.
5. The person who likes going on cruises is in the first house.
6. There are two houses between the person who is a doctor and the person who is a lawyer.
7. The person who is a lawyer and Arnold are next to each other.
8. There is one house between Peter and the person who is a teacher.
9. The person who enjoys mountain retreats is somewhere to the left of the person who loves beach vacations.
10. Alice is somewhere to the left of the person who is a teacher.
11. There is one house between the person who prefers city breaks and Arnold.
 What is the value of attribute House for the person whose attribute Occupation is doctor? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


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

## Clues:
1. There is one house between the boba tea drinker and Bob.
2. There are two houses between the person who uses a Google Pixel 6 and the person who enjoys camping trips.
3. The person who goes on cultural tours is Carol.
4. The person who enjoys camping trips is Eric.
5. The person who uses a Google Pixel 6 is not in the fifth house.
6. The person who uses a OnePlus 9 and the person who goes on cultural tours are next to each other.
7. The person who has gray hair and the person who goes on cultural tours are next to each other.
8. The person who uses a Samsung Galaxy S21 is somewhere to the right of the person who has red hair.
9. The person who enjoys mountain retreats is somewhere to the right of the person who uses a Xiaomi Mi 11.
10. The person who uses a OnePlus 9 is Peter.
11. The person who likes going on cruises is the person who has brown hair.
12. The coffee drinker is the person who goes on cultural tours.
13. The person who likes milk is directly left of the person who has black hair.
14. The person who uses a Xiaomi Mi 11 is directly left of the person who uses a Samsung Galaxy S21.
15. The person who prefers city breaks is not in the sixth house.
16. The person who has blonde hair is Arnold.
17. The person who goes on cultural tours is not in the sixth house.
18. The one who only drinks water and the person who goes on cultural tours are next to each other.
19. The tea drinker is somewhere to the left of the person who enjoys mountain retreats.
20. The person who likes milk is the person who uses a Huawei P50.
21. The person who has blonde hair is directly left of the person who has gray hair.
22. Bob is not in the third house.
23. Peter is the person who prefers city breaks.
 What is the value of attribute HairColor for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}

Answer: red


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`, `Arnold`, `Carol`, `Alice`, `Eric`, `Bob`
 - People have unique favorite music genres: `pop`, `hip hop`, `rock`, `jazz`, `classical`, `country`
 - The people keep unique animals: `fish`, `dog`, `cat`, `rabbit`, `horse`, `bird`
 - Each person lives in a unique style of house: `modern`, `colonial`, `mediterranean`, `ranch`, `victorian`, `craftsman`
 - Each person has a unique hobby: `gardening`, `woodworking`, `knitting`, `cooking`, `painting`, `photography`
 - The people are of nationalities: `brit`, `german`, `norwegian`, `swede`, `dane`, `chinese`

## Clues:
1. The cat lover is the person in a Mediterranean-style villa.
2. The person who loves country music is somewhere to the right of the rabbit owner.
3. The person who loves pop music is not in the second house.
4. Alice is somewhere to the right of Arnold.
5. The person who loves jazz music and the photography enthusiast are next to each other.
6. The person who loves rock music is the German.
7. The woodworking hobbyist is somewhere to the right of the person in a modern-style house.
8. The person who loves hip-hop music is the Norwegian.
9. The cat lover is somewhere to the left of the woodworking hobbyist.
10. Alice is the person who paints as a hobby.
11. The person in a Craftsman-style house is Carol.
12. Peter is the fish enthusiast.
13. The person who loves hip-hop music is not in the fifth house.
14. The person who loves pop music is not in the sixth house.
15. The person who loves cooking is somewhere to the left of the Swedish person.
16. The Dane is directly left of the person who keeps horses.
17. The rabbit owner is somewhere to the left of the British person.
18. There is one house between Alice and the person who loves country music.
19. The person who paints as a hobby is the person living in a colonial-style house.
20. The German is somewhere to the left of the photography enthusiast.
21. Eric is the Norwegian.
22. The person residing in a Victorian house is directly left of the person who enjoys gardening.
23. There are two houses between the person who loves cooking and the person who loves classical music.
24. The bird keeper is the person who loves cooking.
25. The cat lover is directly left of Bob.
26. Eric is not in the first house.
27. Carol is somewhere to the right of the German.
28. The rabbit owner is directly left of the person in a modern-style house.
 What is the value of attribute House for the person whose attribute HouseStyle is colonial? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `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 HouseStyle for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: modern


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`, `Arnold`, `Eric`, `Alice`, `Bob`
 - Each person has a unique level of education: `associate`, `doctorate`, `high school`, `bachelor`, `master`
 - Each mother is accompanied by their child: `Fred`, `Meredith`, `Bella`, `Timothy`, `Samantha`
 - Everyone has a unique favorite cigar: `prince`, `pall mall`, `blue master`, `dunhill`, `blends`
 - The people are of nationalities: `swede`, `brit`, `dane`, `german`, `norwegian`
 - People have unique favorite book genres: `romance`, `mystery`, `biography`, `science fiction`, `fantasy`

## Clues:
1. The person with a master's degree is the person partial to Pall Mall.
2. The person with a high school diploma is in the fourth house.
3. The person's child is named Samantha is in the first house.
4. The Norwegian and the person with a master's degree are next to each other.
5. The German is the Dunhill smoker.
6. The person who is the mother of Timothy is directly left of the person with a master's degree.
7. Arnold and the Dane are next to each other.
8. The Swedish person is the person who loves mystery books.
9. The Swedish person is Alice.
10. Arnold is the person with an associate's degree.
11. The person with a doctorate is in the second house.
12. Peter is somewhere to the right of the person who loves biography books.
13. The person who smokes many unique blends is Alice.
14. The person who loves fantasy books is in the first house.
15. Peter is the Norwegian.
16. The person with a doctorate is the person's child is named Meredith.
17. The Prince smoker is somewhere to the left of the person who smokes many unique blends.
18. The person partial to Pall Mall is the person's child is named Bella.
19. The person who loves romance books is Bob.
 What is the value of attribute Name for the person whose attribute Nationality is norwegian? 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: `Eric`, `Alice`, `Peter`, `Arnold`
 - People have unique hair colors: `black`, `blonde`, `brown`, `red`
 - People own unique car models: `tesla model 3`, `ford f150`, `honda civic`, `toyota camry`

## Clues:
1. Eric is somewhere to the right of Alice.
2. The person who has red hair is somewhere to the right of Eric.
3. Alice is not in the second house.
4. The person who owns a Honda Civic is somewhere to the left of the person who has brown hair.
5. The person who has red hair is in the fourth house.
6. The person who has blonde hair is the person who owns a Tesla Model 3.
7. Arnold is not in the second house.
8. Arnold is not in the fourth house.
9. The person who owns a Ford F-150 is in the second 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: 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: `Peter`, `Arnold`, `Eric`
 - Everyone has a unique favorite cigar: `prince`, `blue master`, `pall mall`

## Clues:
1. Arnold is not in the third house.
2. Arnold is somewhere to the right of the person partial to Pall Mall.
3. There is one house between Eric and the person who smokes Blue Master.
 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 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`
 - The people are of nationalities: `dane`, `swede`, `german`, `brit`, `norwegian`
 - Each person has a unique hobby: `gardening`, `painting`, `knitting`, `photography`, `cooking`
 - Everyone has a favorite smoothie: `dragonfruit`, `lime`, `watermelon`, `desert`, `cherry`
 - People have unique hair colors: `brown`, `gray`, `black`, `blonde`, `red`
 - People have unique favorite sports: `baseball`, `soccer`, `swimming`, `tennis`, `basketball`

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

Answer: soccer


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`, `Arnold`, `Bob`, `Alice`, `Eric`, `Carol`
 - The mothers' names in different houses are unique: `Holly`, `Aniya`, `Kailyn`, `Janelle`, `Penny`, `Sarah`
 - Everyone has something unique for lunch: `stir fry`, `stew`, `spaghetti`, `soup`, `pizza`, `grilled cheese`
 - The people keep unique animals: `dog`, `horse`, `cat`, `fish`, `rabbit`, `bird`
 - Everyone has a favorite smoothie: `watermelon`, `desert`, `lime`, `cherry`, `blueberry`, `dragonfruit`
 - Each person lives in a unique style of house: `mediterranean`, `craftsman`, `victorian`, `colonial`, `modern`, `ranch`

## Clues:
1. The person whose mother's name is Penny is the person residing in a Victorian house.
2. The person in a modern-style house is The person whose mother's name is Kailyn.
3. The person in a modern-style house is somewhere to the right of the person residing in a Victorian house.
4. The person who drinks Blueberry smoothies is not in the sixth house.
5. The person whose mother's name is Sarah is the person who drinks Blueberry smoothies.
6. The person who likes Cherry smoothies is not in the third house.
7. The person who likes Cherry smoothies is Eric.
8. The person whose mother's name is Janelle is not in the third house.
9. The person who loves the spaghetti eater is in the third house.
10. The fish enthusiast is The person whose mother's name is Holly.
11. The person who keeps horses is in the first house.
12. The person in a Mediterranean-style villa is the person who loves the stew.
13. The bird keeper is in the third house.
14. The person living in a colonial-style house is the person who drinks Blueberry smoothies.
15. Carol is the person who keeps horses.
16. There are two houses between the Desert smoothie lover and Peter.
17. The person who loves eating grilled cheese is the Dragonfruit smoothie lover.
18. The person who is a pizza lover is somewhere to the right of the Dragonfruit smoothie lover.
19. The person whose mother's name is Janelle is somewhere to the left of Arnold.
20. Carol is directly left of The person whose mother's name is Aniya.
21. The person in a ranch-style home is The person whose mother's name is Aniya.
22. Arnold is somewhere to the right of the person in a Craftsman-style house.
23. The cat lover is Bob.
24. There is one house between the Watermelon smoothie lover and the person who is a pizza lover.
25. There are two houses between the cat lover and the person who loves stir fry.
26. Arnold is the rabbit owner.
27. Peter is The person whose mother's name is Janelle.
 What is the value of attribute House for the person whose attribute Smoothie is dragonfruit? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 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`, `Peter`, `Alice`, `Eric`
 - Each person has a unique favorite drink: `milk`, `root beer`, `coffee`, `tea`, `water`
 - Each person has a favorite color: `blue`, `green`, `white`, `yellow`, `red`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `tulips`, `carnations`
 - Each person has a unique hobby: `painting`, `cooking`, `photography`, `gardening`, `knitting`

## Clues:
1. Alice is not in the fourth house.
2. The root beer lover is the person who enjoys gardening.
3. The person whose favorite color is green is the coffee drinker.
4. The person whose favorite color is green is the person who loves the boquet of lilies.
5. The person who loves blue is somewhere to the right of the person who loves a bouquet of daffodils.
6. The person who loves cooking is the person who loves blue.
7. Eric is directly left of the tea drinker.
8. The one who only drinks water is Peter.
9. Arnold is the photography enthusiast.
10. The person who loves white is the person who loves the rose bouquet.
11. There is one house between the person who loves a carnations arrangement and the person whose favorite color is red.
12. The person who loves cooking is somewhere to the left of the person who paints as a hobby.
13. The one who only drinks water is in the third house.
14. The person who loves a carnations arrangement is the root beer lover.
15. The person who loves white is in the second house.
 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: `Alice`, `Carol`, `Bob`, `Peter`, `Eric`, `Arnold`
 - They all have a unique favorite flower: `lilies`, `tulips`, `roses`, `daffodils`, `iris`, `carnations`
 - The people are of nationalities: `norwegian`, `dane`, `swede`, `german`, `brit`, `chinese`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `desert`, `dragonfruit`, `blueberry`, `lime`
 - Each person has a unique hobby: `photography`, `cooking`, `knitting`, `woodworking`, `painting`, `gardening`
 - Each person has a unique type of pet: `rabbit`, `hamster`, `fish`, `cat`, `bird`, `dog`

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

Answer: 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`, `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`
 - People use unique phone models: `xiaomi mi 11`, `iphone 13`, `huawei p50`, `google pixel 6`, `oneplus 9`, `samsung galaxy s21`
 - People have unique favorite book genres: `science fiction`, `fantasy`, `mystery`, `historical fiction`, `romance`, `biography`

## Clues:
1. Bob is somewhere to the left of the person who uses a Samsung Galaxy S21.
2. The person who uses a OnePlus 9 is Alice.
3. The person who loves romance books is Alice.
4. The person who loves biography books is Bob.
5. The person who loves science fiction books is Arnold.
6. Eric is somewhere to the left of the person who uses a Google Pixel 6.
7. There is one house between Eric and Peter.
8. The person who loves fantasy books is Carol.
9. The person who loves science fiction books and the person who uses a Huawei P50 are next to each other.
10. The person who uses a Huawei P50 and the person who loves fantasy books are next to each other.
11. The person who loves romance books is directly left of the person who loves science fiction books.
12. The person who uses a Xiaomi Mi 11 is in the sixth house.
13. The person who loves historical fiction books is not in the fourth house.
 What is the value of attribute House for the person whose attribute PhoneModel is samsung galaxy s21? Please reason step by step, and put your final answer within \boxed{}

Answer: 3