At a large family reunion, there are 8 married couples. Every participant shakes hands exactly once with every other person except for their spouse and any of their family members (defined here as the other three couples they are related to through marriage but not direct siblings). How many handshakes occur among the people at the reunion?

Answer: 64


13. $A B C D$ is a convex quadrilateral in which $A C$ and $B D$ meet at $P$. Given $P A=1$, $P B=2, P C=6$ and $P D=3$. Let $O$ be the circumcentre of $\triangle P B C$. If $O A$ is perpendicular to $A D$, find the circumradius of $\triangle P B C$.
(2 marks)
$A B C D$ is a convex quadrilateral, in which $A C$ and $B D$ intersect at $P$. Given $P A=1$, $P B=2$, $P C=6$, and $P D=3$. Let $O$ be the circumcenter of $\triangle P B C$. If $O A$ is perpendicular to $A D$, find the circumradius of $\triangle P B C$.

Answer: 3


3. Represent the number 1947 as the sum or difference of the squares of three integers. Prove that any integer can be represented as the sum or difference of the squares of four integers.

Answer: 1947 = 43^2 + 7^2 + 7^2


$1 \cdot 111$ An increasing integer sequence, if its 1st term is odd, the 2nd term is even, the 3rd term is odd, the 4th term is even, and so on, is called an alternating sequence. The empty set is also considered an alternating sequence. The number of all alternating sequences with each term taken from the set $\{1,2, \cdots, n\}$ is denoted as $A(n)$. Clearly, $A(1)=2, A(2)=3$, find $A(20)$ and explain the reason.

Answer: 17711


In a set of 100 numbers. If one number is removed, the median of the remaining numbers will be 78. If another number is removed, the median of the remaining numbers will be 66. Find the median of the entire set.

#

Answer: 72


2. Find the greatest three-digit number $n$, for which the sum

$$
1^{2}+2^{3}+3^{4}+4^{5}+\ldots+n^{n+1}
$$

is divisible by three.

Answer: 997


2. Given points $A(1,3), B(5,4), C(3,7), D(7,1), E(10,2), F(8,6)$, if line $l \parallel DF$, and the shortest distance from line $l$ to any point within (including the boundary) $\triangle ABC$ equals the shortest distance from line $l$ to any point within (including the boundary) $\triangle DEF$, then the equation of line $l$ is $\qquad$ .

Answer: 10x - 2y - 55 = 0


Let  $f$  be a function from R to R. Suppose we have:
(1)  $f(0)=0$

(2) For all  $x, y \in (-\infty, -1) \cup (1, \infty)$ , we have  $f(\frac{1}{x})+f(\frac{1}{y})=f(\frac{x+y}{1+xy})$ .

(3) If  $x \in (-1,0)$ , then  $f(x) > 0$ .

Prove:  $\sum_{n=1}^{+\infty} f(\frac{1}{n^2+7n+11}) > f(\frac12)$  with  $n \in N^+$ .

Answer: \sum_{n=1}^{+\infty} f\left(\frac{1}{n^2 + 7n + 11}\right) > f\left(\frac{1}{2}\right)


184. The King traversed an $8 \times 8$ chessboard, visiting each square exactly once and returning to the starting square on the last move (the King moves according to the usual rules). When his path was drawn, connecting the centers of the squares he sequentially passed through with line segments, a closed broken line without self-intersections was obtained.

a) Provide an example showing that the King could have made exactly 28 moves horizontally and vertically.

b) Prove that he could not have made fewer than 28 such moves.

c) What is the greatest and the least length that the King's path can have if the side of a cell is 1?

Answer: 28


4. Given a triangle $A B C$, where side $B C$ is twice the length of side $A B$, and a bisector $B M$ is drawn. a) Find the ratio $R_{1} / R_{2}$, where $R_{1}, R_{2}$ are the radii of the circumcircles of triangles $A B M$ and $C B M$ respectively. b) Prove that $3 / 4 < r_{1} / r_{2} < 1$, where $r_{1}, r_{2}$ are the radii of the incircles of triangles $A B M$ and $C B M$ respectively.

Answer: \dfrac{1}{2}


8 Let $X=\{1,2, \cdots, 1995\}, A$ be a subset of $X$, such that when $x \in A$, $19 x \notin A$. Find the maximum value of $|A|$.

When $x \in A$, $19 x \notin A$.

Answer: 1895


Let $ABC$ be a triangle with $\angle C=90^\circ$, and $A_0$, $B_0$, $C_0$ be the mid-points of sides $BC$, $CA$, $AB$ respectively. Two regular triangles $AB_0C_1$ and $BA_0C_2$ are constructed outside $ABC$. Find the angle $C_0C_1C_2$.

Answer: 30^\circ


11. In the Cartesian coordinate system $x O y$, on the parabola $\Gamma: y^{2}=2 p x(p>0)$, there are three points $A, B, C$ whose orthocenter is the focus $F$. Let the centroid of $\triangle A B C$ be $G$. Find the range of the slope $k$ of the line $F G$.

Answer: \left[ -\dfrac{\sqrt{7}}{7}, \dfrac{\sqrt{7}}{7} \right]


Let $N$ be a positive integer. Suppose given any real $x\in (0,1)$ with decimal representation $0.a_1a_2a_3a_4\cdots$, one can color the digits $a_1,a_2,\cdots$ with $N$ colors so that the following hold:
    1. each color is used at least once;
    2. for any color, if we delete all the digits in $x$ except those of this color, the resulting decimal number is rational.
Find the least possible value of $N$.

[i]~Sutanay Bhattacharya[/i]

Answer: 10


Answer: 361


6. The base of the pyramid $TABC$ is the triangle $ABC$, all sides of which are equal to 3, and the height of the pyramid, equal to $\sqrt{3}$, coincides with the lateral edge $TA$. Find the area of the section of the pyramid by a plane that passes through the center of the sphere circumscribed around the pyramid, is parallel to the median $AD$ of the base, and forms an angle of $60^{\circ}$ with the base plane.

Answer: \dfrac{11\sqrt{3}}{10}


On one of the cells of an $8 \times 8$ grid, a treasure is buried. You are located with a metal detector at the center of one of the corner cells of this grid and move by transitioning to the centers of adjacent cells by side. The metal detector triggers if you have arrived at the cell where the treasure is buried, or one of the cells adjacent by side to it. Can you guarantee to specify the cell where the treasure is buried, covering no more than a distance of 26?

Answer: Yes


Problem 9.2. A triangle $A B C$ with centroid $G$ and incenter $I$ is given. If $A B=42, G I=2$ and $A B \| G I$, find $A C$ and $B C$.

Ivailo Kortezov

Answer: 36


A1. The numbers 1, 2, 3, 4, 5, 6, 7, and 8 must be distributed over the circles in the figure such that the sum of the three numbers on each of the four sides of the square is the same.

Provide a solution where this sum is minimized.

![](https://cdn.mathpix.com/cropped/2024_04_17_0b81ff2ef64511bfe8eeg-1.jpg?height=243&width=243&top_left_y=1078&top_left_x=1569)

Answer: 12


3. Equilateral $\triangle A B C$ and square $A B D E$ share a common side $A B$, the cosine of the dihedral angle $C-A B-D$ is $\frac{\sqrt{3}}{3}, M, N$ are the midpoints of $A C, B C$ respectively, then the cosine of the angle formed by $E M, A N$ is $\qquad$ .

Answer: \dfrac{1}{6}


On a $16 \times 16$ torus as shown all 512 edges are colored red or blue. A coloring is good if every vertex is an endpoint of an even number of red edges. A move consists of switching the color of each of the 4 edges of an arbitrary cell. What is the largest number of good colorings such that none of them can be converted to another by a sequence of moves?

#

Answer: 4


9. (16 points) In the $x O y$ plane, draw two lines $A_{1} B_{1}$ and $A_{2} B_{2}$ through a point $P$ on the $x$-axis. Points $A_{1}$, $B_{1}$, $A_{2}$, and $B_{2}$ are all on the parabola $\Gamma: y^{2}=x$. Let $A_{1} B_{2}$ and $A_{2} B_{1}$ intersect the $x$-axis at points $S$ and $T$, respectively. Compare $\overrightarrow{O S} \cdot \overrightarrow{O T}$ with $|O P|^{2}$.

Answer: \overrightarrow{O S} \cdot \overrightarrow{O T} = |O P|^{2}


7. Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, point $P$ is on edge $B C$, and point $Q$ is the midpoint of edge $C C_{1}$. If the plane passing through points $A, P, Q$ intersects the cube to form a pentagon, then the range of $B P$ is $\qquad$ .

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


1. Two circles of equal radii touch each other externally. Tangents are drawn from the center of one circle to the other circle. Determine the ratio of the area of the quadrilateral whose vertices are the centers of the circles and the points of tangency to the area of the region between the tangents and the circles determined by the given circles (shaded in the figure).

![](https://cdn.mathpix.com/cropped/2024_06_03_5d6b13b2bd1879787abdg-4.jpg?height=532&width=736&top_left_y=1136&top_left_x=657)

Answer: \dfrac{2\sqrt{3}}{2\sqrt{3} - \pi}


A rook moves across an $n \times n$ board, visiting each cell exactly once, with each move being exactly to an adjacent cell (one cell away). The cells are numbered from 1 to $n^2$ in the order the rook visits them. Let $M$ be the maximum difference between the numbers of neighboring (side-adjacent) cells. What is the smallest possible value of $M$?

Answer: 2n - 1


4. Let $S=\left\{A=\left(a_{1}, \cdots, a_{8}\right) \mid a_{i}=0\right.$ or $1, i=1, \cdots$, 8\}. For two elements $A=\left(a_{1}, \cdots, a_{8}\right)$ and $B=\left(b_{1}\right.$, $\cdots, b_{8}$ ) in $S$, denote
$$
d(A, B)=\sum_{i=1}^{\delta}\left|a_{i}-b_{i}\right|,
$$

and call it the distance between $A$ and $B$. What is the minimum number of elements that can be selected from $S$ such that the distance between any two of them is $\geqslant 5$?

Answer: 4


8. Find the minimum value of the discriminant of a quadratic trinomial, the graph of which has no common points with the regions located below the x-axis and above the graph of the function $y=\frac{1}{\sqrt{1-x^{2}}}$.

Answer: -4


The distance between the centers \( O_{1} \) and \( O_{2} \) of the circles \( \omega_{1} \) and \( \omega_{2} \) is \( 10r \), and their radii are \( 5r \) and \( 6r \) respectively. A line intersects circle \( \omega_{1} \) at points \( M \) and \( N \) and is tangent to circle \( \omega_{2} \) at point \( K \), with \( MN = 2NK \). Find the length of the chord \( MN \).

Answer: 2\sqrt{21}r


19. Person A and Person B start from points $A$ and $B$ respectively at the same time, heading towards each other. The speed ratio of A to B is 5:3. After they meet, they continue to their respective destinations, with A reaching point $B$ and B reaching point $A$, and then both immediately turn back along the same route. If the second meeting point is 50 kilometers away from the first meeting point, then the distance between $A$ and $B$ is $\qquad$ kilometers.

Answer: 100


\section*{Exercise 4 - 221014}

Let \(r\) be the radius of the circumcircle of a regular decagon \(P_{1} P_{2} \ldots P_{10}\) and \(s\) the length of one side of this decagon.

Calculate \(s\) in terms of \(r\)!

Answer: \dfrac{r(\sqrt{5} - 1)}{2}


$\left.\frac{\text { Auxiliary similar triangles }}{[\quad \text { Law of Cosines }}\right]$

In triangle $ABC$, a point $D$ is taken on side $AC$, such that $AD=3$, $\cos \angle BDC=13/20$, and $\angle B+\angle ADB=180^{\circ}$. Find the perimeter of triangle $ABC$ if $BC=2$.

Answer: 11


2. In an isosceles right $\triangle ABC$, $\angle C=90^{\circ}$, points $D$, $E$, and $F$ are on sides $AB$, $BC$, and $CA$ respectively, $FD=FE=\frac{AC}{2}$, $\angle DFE=90^{\circ}$. Then $FC: CE: EF=$ $\qquad$ .

Answer: 3 : 4 : 5


11. If the real and imaginary parts of $\frac{z}{20}$ and $\frac{20}{z}$ are all between 0 and 1, then the area of the figure formed by the points corresponding to such complex numbers $z$ is $\qquad$ .

Answer: 50(6-\pi)


. Let $A=\{1,2,3,4,5\}$. How many functions $f: \mathscr{P}(A) \backslash\{\varnothing\} \rightarrow A$ satisfy $f(B) \in B$ and $f(B \cup C) \in\{f(B), f(C)\}$ for all non-empty subsets $B$ and $C$ of $A$?

Answer: 120


Let's inscribe a rectangle with the largest area in the ellipse $b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}(a>b)$ (the sides of the rectangle are parallel to the major axes of the ellipse). In this rectangle, inscribe an ellipse whose major axes are equal to the sides of the rectangle. In the thus obtained ellipse, inscribe a rectangle with the largest area, and so on. What is the sum of the areas of the infinitely many rectangles and ellipses thus obtained?

Answer: 2ab(\pi + 2)


II. (25 points) Several boxes are unloaded from a cargo ship, with a total weight of 10 tons, and the weight of each box does not exceed 1 ton. To ensure that these boxes can be transported away in one go, the question is: what is the minimum number of trucks with a carrying capacity of 3 tons needed?

Answer: 5


7. Let the pair of positive integers $(x, y)$ satisfy $\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}=\frac{1}{\sqrt{20}}$. Then $xy$ has $\qquad$ different possible values.

Answer: 2


Given a polynomial \( P(x)=x^{k}+c_{k-1} x^{k-1}+c_{k-2} x^{k-2}+\cdots+c_{1} x+c_{0} \) that divides the polynomial \( (x + 1)^n - 1 \), where \( k \) is an even number and all coefficients \( c_{0}, c_{1}, \cdots, c_{k-1} \) are odd. Prove that \( k + 1 \) divides \( n \).

Answer: k + 1 \text{ divides } n


18. (USA 5) Inside triangle $A B C$ there are three circles $k_{1}, k_{2}, k_{3}$ each of which is tangent to two sides of the triangle and to its incircle $k$. The radii of $k_{1}, k_{2}, k_{3}$ are 1,4 , and 9 . Determine the radius of $k$.

Answer: 11


4th Centromerican 2002 Problem B1 ABC is a triangle. D is the midpoint of BC. E is a point on the side AC such that BE = 2AD. BE and AD meet at F and ∠FAE = 60 o . Find ∠FEA.

Answer: 60^\circ


A bowl-shaped pot with a hemispherical shape has a volume of 8 liters. How much soup fills the pot to half its height?

Answer: \frac{5}{2}


2. Integers, the decimal representation of which reads the same from left to right and from right to left, we will call symmetric. For example, the number 513315 is symmetric, while 513325 is not. How many six-digit symmetric numbers exist such that adding 110 to them leaves them symmetric?

Answer: 81


4. Let $a, b$ be unequal real numbers, and $a^{2}+2 a=$ $5 \cdot b^{2}+2 b=5$. Then $a^{2} b+a b^{2}=$

Answer: 10


3. Given the equalities $\log _{a} b+\log _{b} c+\log _{c} a=\log _{b} a+\log _{c} b+\log _{a} c=-1$. Find $\log _{a} b$.

If there are multiple answers, list them in ascending order separated by a semicolon.

Answer: -1


$\left.\begin{array}{l}\text { [ Law of Cosines } \\ \text { [ Angles subtending equal arcs and equal chords }]\end{array}\right]$

An equilateral triangle $ABC$ with a side length of 3 is inscribed in a circle. Point $D$ lies on the circle, and the chord $AD$ is equal to $\sqrt{3}$. Find the chords $BD$ and $CD$.

Answer: \sqrt{3}


Shveiov D.V.

The circle inscribed in the right triangle $ABC\left(\angle ABC=90^{\circ}\right)$ touches the sides $AB, BC, AC$ at points $C_{1}, A_{1}, B_{1}$ respectively. The excircle opposite to $A$ touches the side $BC$ at point $A_{2}$. $A_{0}$ is the center of the circumcircle of triangle $A_{1} A_{2} B_{1}$; the point $C_{0}$ is defined similarly. Find the angle $A_{0} B C_{0}$.

Answer: 45^{\circ}


5. The sequence of real numbers $\left\{a_{n}\right\}$ satisfies the conditions: $a_{1}=\frac{\sqrt{2}}{2}+1, a_{2}=\sqrt{2}+1, a_{n+1}+a_{n-1}=\frac{n}{a_{n}-a_{n-1}}+2$ $(n \geqslant 2)$, then the general term formula $a_{n}=(n \geqslant 1)$.

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


Question 191: A die is rolled three times, the first time it shows $a$, the second time it shows $b$, and the third time it shows $c$. Assuming the die shows numbers from 1 to 6 with equal probability,
(1) Find the probability that $a b + 2 c \geq a b c$;
(2) Find the probability that $a b + 2 c$ and $2 a b c$ are coprime.

Answer: \dfrac{29}{108}


25. For a real number $x$, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$. Consider the function
$$
f(x, y)=\sqrt{M(M+1)}(|x-m|+|y-m|),
$$
where $M=\max (\lfloor x\rfloor,\lfloor y\rfloor)$ and $m=\min (\lfloor x\rfloor,\lfloor y\rfloor)$. The set of all real numbers $(x, y)$ such that $2 \leq x, y \leq 2022$ and $f(x, y) \leq 2$ can be expressed as a finite union of disjoint regions in the plane. The sum of the areas of these regions can be expressed as a fraction $a / b$ in lowest terms. What is the value of $a+b$ ?

Answer: 2021


In a two-story house, which is inhabited not only on both floors but also in the basement, 35 people live above someone and 45 people live below someone. At the same time, one third of all the people living in the house live on the 1st floor.

How many people live in the house in total?

(L. Hozová)

Answer: 60


91. A wild rabbit escapes 100 steps before the hunting dog starts to chase it. The rabbit runs 8 steps in the distance the hunting dog runs 3 steps, and in the time the hunting dog runs 4 steps, the rabbit can run 9 steps. The hunting dog has to run at least $\qquad$ steps to catch up with the rabbit.

Answer: 240


一、(50 分) In $\triangle A B C$, the angle bisectors $B D$ and $C E$ intersect at $I$, with $D$ and $E$ on $A C$ and $A B$ respectively. The perpendicular from $I$ to $D E$ meets $D E$ at $P$. The extension of $P I$ meets $B C$ at $Q$. If $I Q=2 I P^{\prime}$, find $\angle A$.

Answer: 60^\circ


397. Measuring Water. A maid was sent to a spring with two vessels of 7 and 11 pints capacity. She needs to bring back exactly 2 pints of water.

What is the minimum number of operations in this case? By "operation" we mean either filling a vessel, emptying it, or pouring water from one vessel to another.

Answer: 14


1. Each cell of a $68 \times 68$ table is to be colored with one of three colors (red, blue, white). In how many ways can this be done so that every triplet of adjacent cells in each row and each column contains cells of all three colors?

(Josef Tkadlec)

Answer: 12


Vertex \(E\) of equilateral \(\triangle{ABE}\) is in the interior of unit square \(ABCD\). Let \(R\) be the region consisting of all points inside \(ABCD\) and outside \(\triangle{ABE}\) whose distance from \(AD\) is between \(\frac{1}{3}\) and \(\frac{2}{3}\). The area of \(R\) is given in the form \(\frac{k-m\sqrt{n}}{p}\), where \(n\) is a positive integer. Find the value of \(k + m + n + p\).

Answer: 56


Two circles that share the same center have radii $10$ meters and $20$ meters. An aardvark runs along the path shown, starting at $A$ and ending at $K$. How many meters does the aardvark run? Express your answer in the form of $a\pi + b$, where $a$ and $b$ are integers. Please provide the value of a + b.

Answer: 60


A function \(f\) is defined by \(f(z) = (4 + i) z^2 + \alpha z + \gamma\) for all complex numbers \(z\), where \(\alpha\) and \(\gamma\) are complex numbers and \(i^2 = -1\). Suppose that \(f(1)\) and \(f(i)\) are both real. The original answer is in the form \(\sqrt{m}\). Find the smallest possible value of \(m\).

Answer: 2


A permutation $(a_1,a_2,a_3,a_4,a_5)$ of $(1,2,3,4,5)$ is heavy-tailed if $a_1 + a_2 < a_4 + a_5$. Find the number of heavy-tailed permutations.

Answer: 48


Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color? Express your answer in the form \(\frac{k}{m}\) where the fraction is in its simplest form, and find the value of \(k + m\).

Answer: 7


Segment $BD$ and $AE$ intersect at $C$, as shown, $AB=BC=CD=CE$, and $\angle A = \frac{5}{2} \angle B$. What is the value of $\angle D$ in degrees? If the answer is $x^\circ$, what is the value of $10x$?

Answer: 525


Let $P(x,\ y)\ (x>0,\ y>0)$ be a point on the curve $C: x^2-y^2=1$. If $x=\frac{e^u+e^{-u}}{2}\ (u\geq 0)$, then find the area bounded by the line $OP$, the $x$ axis and the curve $C$ in terms of $u$.

Answer: \dfrac{u}{2}


[ Chessboards and chess pieces $]$ [ Examples and counterexamples. Constructions ] [ Evenness and oddness

Authors: Pechkovsky A.N., Itenberg I.

Given an infinite grid paper with a cell side equal to one. The distance between two cells is defined as the length of the shortest path of a rook from one cell to another (the path of the center of the rook is considered). What is the minimum number of colors needed to color the board (each cell is colored with one color) so that two cells, located at a distance of 6, are always colored differently?

#

Answer: 4


Example 6. The chord $\mathrm{AB}$ of the ellipse $\frac{x^{2}}{\mathrm{a}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1$ passes through a fixed point $\mathrm{M}\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)$, find the locus of the intersection point $\mathrm{P}$ of the two tangents through $\mathrm{A}$ and $\mathrm{B}$.

Answer: \dfrac{x x_0}{a^2} + \dfrac{y y_0}{b^2} = 1


3. What is the smallest radius of a sphere from which a regular quadrilateral pyramid with a base edge of 14 and an apothem of 12 can be cut?

Answer: 7\sqrt{2}


13.266. Two athletes start running simultaneously - the first from $A$ to $B$, the second from $B$ to $A$. They run at different but constant speeds and meet at a distance of 300 m from $A$. After running the entire path $A B$, each immediately turns around and meets the other at a distance of 400 m from $B$. Find the length of $A B$.

Answer: 500


Consider a chessboard that is infinite in all directions. Alex the T-rex wishes to place a positive integer in each square in such a way that:
[list]
[*] No two numbers are equal.
[*] If a number $m$ is placed on square $C$, then at least $k$ of the squares orthogonally adjacent to $C$ have a multiple of $m$ written on them.
[/list]
What is the greatest value of $k$ for which this is possible?

Answer: 2


3. As shown in the figure,
$A B C D$ is a square,
$E$ is the midpoint of $A B$. If
$\triangle D A E$ and
$\triangle C B E$ are folded along the dotted lines $D E$ and $C E$ respectively, so that $A E$ and $B E$ coincide, and the point where $A$ and $B$ coincide is denoted as $P$, then the dihedral angle between plane $P C D$ and plane $E C D$ is $\qquad$ degrees.

Answer: 30


## Task $6 / 71$

The cubic equation with real coefficients $p, q$, and $r$

$$
x^{3}+p x^{2}+q x+r=0
$$

has three real solutions. What condition must the coefficients $p, q$, and $r$ satisfy if the solutions are to be the lengths of the sides of a plane triangle?

Answer: p^3 - 4pq + 8r > 0


What is the smallest number of circles of radius $\sqrt{2}$ that are needed to cover a rectangle

(a) of size $6 \times 3$ ?

(b) of size $5 \times 3$ ?

Answer: (a) Six circles, (b) five circles.

Answer: 6


## Task 12/78

When asked how old he is, someone answers that in the year $x^{2}$ he was exactly $x$ years old. In which year was he born?

Answer: 1892


5. In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, $E$ is the midpoint of $B C$, and $F$ is on $A A_{1}$ such that $A_{1} F: F A=1: 2$. Find the dihedral angle formed by the plane $B_{1} E F$ and the base $A_{1} B_{1} C_{1} D_{1}$.
(1985 National High School League Problem)

Answer: \arctan \frac{\sqrt{37}}{3}


9. [7] Let $A B C D$ be a trapezoid such that $A B \| C D, \angle B A C=25^{\circ}, \angle A B C=125^{\circ}$, and $A B+A D=C D$. Compute $\angle A D C$.

Answer: 70^\circ


The circle $k_{1}$ touches the circle $k_{2}$ internally at point $P$. The chord $AB$ of the larger circle touches the smaller circle at point $C$, and $AP$ and $BP$ intersect the smaller circle at points $D$ and $E$, respectively. We know that $AB=84$, $PD=11$, and $PE=10$. What is the length of $AC$?

Answer: 44


Four congruent semicircles are drawn on the surface of a sphere with radius $2$ , as
shown, creating a close curve that divides the surface into two congruent regions.
The length of the curve is $\pi\sqrt{n}$ . What is $n$

Answer: 32


Task 3. Sasha chose five numbers from the numbers 1, 2, 3, 4, 5, 6, and 7 and told Anna their product. Based on this information, Anna realized that she could not uniquely determine the parity of the sum of the numbers chosen by Sasha. What number did Sasha tell Anna? (20 points)

Answer: 420


40. The difference equals the quotient. Find two numbers whose difference and quotient are both equal to 5.

Answer: \frac{25}{4}


Each cell of an $100\times 100$ board is divided into two triangles by drawing some diagonal. What is the smallest number of colors in which it is always possible to paint these triangles so that any two triangles having a common side or vertex have different colors?

Answer: 8


Folklore

In a cube with an edge length of 1, two sections in the form of regular hexagons were made. Find the length of the segment where these sections intersect.

Answer: \sqrt{2}


Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function.

[i]Proposed by James Rickards, Canada[/i]

Answer: 2


1. Let $A B$ and $C D$ be the bases of trapezoid $A B C D$, and $E$ the midpoint of leg $B C$. If $A E=10, D E=8$ and $\measuredangle A E D=30^{\circ}$, determine the area of this trapezoid.

Answer: 40


2. In the Martian parliament, deputies from three parties are in session: "Vowels," "Consonants," and "Sibilants" - 50 deputies from each party. A bill "On the Reconstruction of Martian Canals" was put to a vote. After the vote, 30 deputies from each party said they voted "for," 10 said they voted against, and the rest said they abstained. It is known that among the "consonant" deputies, those who supported the bill were the only ones who told the truth, among the "vowel" deputies - those who voted against, and among the "sibilant" deputies - those who abstained. The bill is considered adopted if at least $50 \%$ of the votes are in favor. Was the bill adopted?

Answer: \text{No}


10. A city adopts a lottery system for car license plates, with 300,000 people participating in the lottery, and 30,000 slots available each month.
(1) If those who win the lottery each month exit the lottery, and those who do not win continue to the next month's lottery with an additional 30,000 people joining each month, how long on average will it take for each person to win a slot?
(2) Under the conditions of (1), if the traffic management department can control the proportion of people who win, such that the probability of winning in the first month of each quarter is $\frac{1}{11}$, the second month is $\frac{1}{10}$, and the third month is $\frac{1}{9}$, how long on average will it take for each person to win a slot?

Answer: 10


7. A token is placed on the B1 square of a chessboard. The token can be moved diagonally one square, either up-left or up-right, but it must not leave the board. In how many ways can the token be moved to the C8 square?

![](https://cdn.mathpix.com/cropped/2024_06_03_47603db0a075197c1d6ag-9.jpg?height=716&width=717&top_left_y=1561&top_left_x=681)

The use of a pocket calculator or any reference materials is not allowed.

Answer: 28


9. Given a square piece of paper $A B C D$ with an area of $2007 \mathrm{~cm}^{2}$. Now fold the paper along a line segment (as shown in Figure 1), so that point $D$ lands on point $D^{\prime}$ on side $B C$, and point $A$ lands on point $A^{\prime}$, with $A^{\prime} D^{\prime}$ intersecting $A B$ at point $E$. Then the perimeter of $\triangle B D^{\prime} E$ is $\mathrm{cm}$.

Answer: 2\sqrt{2007}


12 、At 8:08 AM, Xiaoming set off from home on a bicycle. 8 minutes later, his father set off on a motorcycle to catch up with him. He caught up with Xiaoming 4 kilometers away from home, then immediately returned home. After arriving home, he immediately turned around to catch Xiaoming again, and caught up with him again 8 kilometers away from home. What time was it then?

Answer: 8:32


We are looking for a six-digit numerical code, about which we know that:

- no digit in it is repeated,
- it contains 0, which is not in the second-to-last position,
- in its notation, there are never two odd or two even digits next to each other,
- adjacent single-digit numbers differ by at least 3,
- the numbers obtained by reading the first and second pairs are both multiples of the number formed by reading the third, i.e., the last pair.

Determine the sought code.

(M. Volfová)

Hint. Focus on what the individual pairs look like, especially the last one.

Answer: 903618


9. Use the digits $1 \sim 9$ to form three three-digit numbers, each digit used only once, and all three numbers are divisible by 4. Then the largest of these three numbers is at least $\qquad$ .

Answer: 548


Example 2 Find the largest real number $m$, such that as long as the quadratic function $f(x)=a x^{2}+b x+c$ satisfies the conditions
(1) $a>0$;
(2) $|f(x)| \leqslant 1$, when $x \in[-1,1]$;
(3) $x_{0}=-\frac{b}{2 a} \in[-2,-1) \cup(1,2]$,

it must be true that $f(x) \geqslant m(x \in[-2,2])$.
Analysis: $m_{\text {max }}=f(x)_{\text {min }}=f\left(x_{0}\right)$, so, the quadratic function can be written in vertex form. Without loss of generality, we can assume $x_{0}=-\frac{b}{2 a} \in(1,2]$. From the graph of $f(x)=a x^{2}+b x+c$, we know that $f(x)$ is monotonically decreasing on $[-1,1]$.
Therefore, condition $(2) \Leftrightarrow\left\{\begin{array}{l}f(1) \geqslant-1, \\ f(-1) \leqslant 1 .\end{array}\right.$

Answer: -\dfrac{5}{4}


14.Workdays. Three workers are digging a pit. They work in turns, and each works as long as the other two need to dig half the pit. Working this way, they dug the pit. How many times faster would they have finished the work if they had worked simultaneously

## 43 Languages of Mathematics

Answer: \dfrac{5}{2}


18 Let the sum of $m$ distinct positive even numbers and $n$ distinct positive odd numbers be 1987. For all such $m$ and $n$, what is the maximum value of $3m + 4n$?

Answer: 221


13. (12 points) There are 16 students, who form a $4 \times 4$ square array. In a certain exam, their scores are all different. After the scores are announced, each student compares their score with their adjacent classmates (adjacent means front, back, left, and right, for example, a student sitting at a corner has only 2 classmates adjacent to him). If at most only 1 classmate's score is higher than his, he will consider himself “happy”. Then, the maximum number of students who will consider themselves “happy” is $\qquad$.

Answer: 12


3.19. In the rhombus $A B C D$ and the triangle $A B C$, which contains its larger diagonal, circles are inscribed. Find the ratio of the radii of these circles if the acute angle of the rhombus is $\alpha$.

Answer: 1 + \cos \frac{\alpha}{2}


59. An electronic watch uses 5 two-digit numbers (including two-digit numbers with a leading 0) to indicate time and date, such as 15:23:45/06/18 representing June 18, 15:23:45. There are some special moments when all ten digits on the electronic watch are different. Among these special moments, the maximum sum of the 5 two-digit numbers on the electronic watch is $\qquad$ .

Answer: 153


3. Given an integer $n \geqslant 2$. Find the smallest positive real number $c$, such that for any complex numbers $z_{1}, z_{2}, \cdots, z_{n}$, we have
$$
\left|\sum_{i=1}^{n} z_{i}\right|+c \sum_{1 \leqslant i<j \leqslant n}\left|z_{i}-z_{j}\right| \geqslant \sum_{i=1}^{n}\left|z_{i}\right| .
$$
(Supplied by Zhang Duanyang)

Answer: \dfrac{2}{n}


# Problem 5. (3 points)

In the cells of a $7 \times 7$ table, pairwise distinct non-negative integers are written. It turns out that for any two numbers in the same row or column, the integer parts of their quotients when divided by 8 are different. What is the smallest value that the largest number in the table can take?

Answer: 54


8. (Mathematics for Middle School, Issue 1, 2004 Olympiad Training Problem) Let $n$ be a given positive integer, and $n \geqslant 3$. For $n$ real numbers $x_{1}$, $x_{2}, \cdots, x_{n}$, denote the minimum value of $\left|x_{i}-x_{j}\right|(1 \leqslant i<j \leqslant n)$ as $m$. If $x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1$, try to find the maximum value of $m$.

Answer: \sqrt{\dfrac{12}{n(n^2-1)}}


## Task A-2.3.

Determine all values of the parameter $a$ for which the system

$$
\begin{aligned}
2^{|x|}+|x| & =x^{2}+y+a \\
x^{2}+y^{2} & =1
\end{aligned}
$$

has exactly one solution $(x, y) \in \mathbb{R}^{2}$.

Answer: 0


3. The $n$th term of a sequence is the first non-zero digit of the decimal expansion of $\frac{1}{\sqrt{n}}$. How many of the first one million terms of the sequence are equal to 1 ?

Answer: 757576


a) Three rays emerge from one point in space. Prove that at least one of the pairwise angles between these rays does not exceed $120^{\circ}$; however, the rays can be positioned such that none of the pairwise angles between them is less than $120^{\circ}$.

b) Four rays emerge from one point in space. Prove that at least one of the pairwise angles between these rays does not exceed $\alpha$, where $\cos \alpha = -\frac{1}{3}$ (and hence, $\alpha \approx 109^{\circ} 28' 16''$); however, the rays can be positioned such that none of the pairwise angles between them is less than $\alpha$.

c) Five rays emerge from one point in space. Prove that at least one of the pairwise angles between these rays does not exceed $90^{\circ}$; however, the rays can be positioned such that none of the angles between them is acute.

d) Prove that the statement in problem (c) remains valid for six rays in space.

Answer: 120^\circ


11. If positive real numbers $x, y, z$ satisfy $x^{3}+y^{3}+z^{3}-3 x y z=1$, find the minimum value of $x^{2}+y^{2}+z^{2}$.

Answer: 1


Let $n$ be an integer bigger than $0$. Let $\mathbb{A}= ( a_1,a_2,...,a_n )$ be a set of real numbers. Find the number of functions $f:A \rightarrow A$ such that $f(f(x))-f(f(y)) \ge x-y$ for any $x,y \in \mathbb{A}$, with $x>y$.

Answer: \sum_{k=0}^{\lfloor n/2 \rfloor} \frac{n!}{k! \cdot 2^k \cdot (n - 2k)!}


[Central Angle. Arc Length and Circumference]

Two concentric circles are given. A tangent to the smaller circle divides the arc length of the larger circle in the ratio $1: 5$. Find the ratio of the areas of the circles bounded by these circles.

#

Answer: \dfrac{3}{4}


7-6. In an ornithological park, there are several species of birds, a total of 2021 individuals. The birds sat in a row, and it turned out that between any two birds of the same species, there was an even number of birds. What is the smallest number of bird species that could have been there?

Answer: 1011


Find the maximum and minimum values of the expression \(2x + y\), given that \((x, y)\) are related by the equation \(\sqrt{x-1} + \sqrt{y-4} = 2\). For which \((x, y)\) are these values achieved?

Answer: 14


I1.1 If $|x+\sqrt{5}|+|y-\sqrt{5}|+|z|=0$, determine $\alpha=x^{2}+y^{2}+z^{2}$.
I1.2 If $\beta$ is the sum of all digits of the product $\underbrace{11111 \cdots 11}_{\alpha 1_{1}^{\prime} s} \times \underbrace{99999 \ldots 99}_{\alpha 9 \text { s's }}$, determine the value of $\beta$.
I1.3 Suppose that the real function $f(x)$ satisfies $f(x y)=f(x) f(y)$ for all real numbers $x$ and $y$, and $f(1)<1$. Determine the value of $\gamma=f(\beta)+100-\beta$.

I1.4 If $n$ is a positive integer and $f(n)=2^{n}+2^{n-1}+2^{n-2}+\ldots+2^{2}+2^{1}+1$, determine the value of $\delta=f(\gamma)$.

Answer: 2047


6.42 12 players participated in a chess tournament. After the tournament, each of them made 12 lists. On the 1st list, they only wrote their own name; on the 2nd list, they wrote their own name and the names of all the opponents they defeated; on the 3rd list, they wrote all the names on the 2nd list and the names of all the opponents these people defeated; and so on, until on the 12th list, they wrote all the names on the 11th list and the names of all the opponents these people defeated. It is known that for each player, on their 12th list, there are names that did not appear on their 11th list. How many draws were there in the tournament?

Answer: 54


11 Given a positive integer $n$, among its positive divisors, there is at least one positive integer ending in each of the digits $0,1,2, \cdots, 9$. Find the smallest $n$ that satisfies this condition.

Answer: 270


5. As shown in Figure 3, on each side of the square $A B C D$ with side length 1, segments $A E=B F$ $=C G=D H=x$ are cut off, and lines $A F 、 B G 、 C H 、 D E$ are drawn to form quadrilateral $P Q R S$. Express the area $S$ of quadrilateral $P Q R S$ in terms of $x$, then $S=$

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


Given that $x_{1}, x_{2}, \cdots, x_{n}$ are all real numbers, where $n \geq 2$, and \(\sum_{k=1}^{n}\left|x_{k}\right|=1, \sum_{i=1}^{n} x_{i}=0\). Prove: \(\left|\sum_{k=1}^{n} \frac{x_{k}}{k}\right| \leq \frac{1}{2}-\frac{1}{2n}\).

Answer: \left|\sum_{k=1}^{n} \frac{x_{k}}{k}\right| \leq \frac{1}{2} - \frac{1}{2n}


It is given an acute triangle  $ABC$ , $AB \neq AC$  where the feet of altitude from $A$ its $H$.  In the extensions of the sides $AB$ and $AC$  (in the direction of $B$ and $C$) we take the points $P$ and $Q$ respectively such that $HP=HQ$ and the points $B,C,P,Q$ are concyclic.
Find the ratio $\tfrac{HP}{HA}$.

Answer: 1


5. Let positive numbers $p, q, r$ satisfy $\frac{(p+q+r)^{2}}{p^{2}}=$ $\frac{2(p+q+r)^{2}}{q^{2}}=\frac{3(p+q+r)^{2}}{r^{2}}=k$, then the value of $k$ is $\qquad$.

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


14. (15 points) The main staircase of a mall hall is shown in the figure. There are 15 steps from the 1st floor to the 2nd floor, each step is 16 cm high and 26 cm deep. It is known that the staircase is 3 meters wide. To lay a carpet on the stairs from the 1st floor to the 2nd floor, which costs 80 yuan per square meter, how much money is needed at least to buy the carpet?

Answer: 1512


a) Two circles with the same radius of 13 intersect at points \(A\) and \(B\). Point \(C\) is chosen on the first circle, and point \(D\) is chosen on the second circle. It turns out that point \(B\) lies on segment \(CD\), and \(\angle CAD = 90^\circ\). On the perpendicular to \(CD\) passing through point \(B\), point \(F\) is chosen such that \(BF = BD\) (points \(A\) and \(F\) are on the same side of line \(CD\)). Find the length of segment \(CF\).

b) Additionally, suppose that \(BC = 10\). Find the area of triangle \(ACF\).

Answer: 26


In triangle \(ABC\), \(AD\) is a median, and \(E\) is a point on \(AD\). The circle \(\odot O_1\) passing through \(B\) and \(E\) touches \(BC\) at \(B\), and the circle \(\odot O_2\) passing through \(C\) and \(E\) touches \(BC\) at \(C\). Moreover, \(\odot O_1\) and \(\odot O_2\) intersect \(AB\) and \(AC\) at \(M\) and \(N\), respectively. Prove that the circumcircle of \(\triangle AMN\) is tangent to both \(\odot O_1\) and \(\odot O_2\).

Answer: \text{The circumcircle of } \triangle AMN \text{ is tangent to both } \odot O_1 \text{ and } \odot O_2


Let $R$ be a unit square region and $n \geq 4$ an integer. A point $X$ in the interior of $R$ is called n-ray partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are $100$ -ray partitional but not $60$ -ray partitional?

Answer: 2320


6. When four dice are rolled randomly, the probability that the product of the four numbers obtained is a perfect square is $\qquad$ .

Answer: \dfrac{25}{162}


Example 11 (25th IMO Shortlist) Find the last two digits of $\left[(\sqrt{29}+\sqrt{21})^{1984}\right]$.

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

Answer: 71


3. Given the radii of $\odot O_{1}$ and $\odot O_{2}$ are 3 and 5, respectively, and $O_{1} O_{2}=10$. Then the area of the triangle formed by the two inner common tangents and one outer common tangent of the two circles is

Answer: \dfrac{45}{4}


8.3. Krosch, Losyash, and Sovunya participated in races. Krosch started first, but during the race, he was overtaken, or he overtook others exactly 12 times. Sovunya started last, but during the race, she was overtaken, or she overtook others exactly 10 times. In what order did the participants finish, given that Losyash finished the race before Krosch?

Answer: Sovunya, Losyash, Krosch


Answer: \begin{pmatrix} 0 & 7 & 9 & 14 \\ 11 & 12 & 2 & 5 \\ 6 & 1 & 15 & 8 \\ 13 & 10 & 4 & 3 \end{pmatrix}


The center of the circle inscribed in a quadrilateral lies on its diagonal, which is equal to 5. It is known that the perimeter of the quadrilateral is 14, and the area is 12. Find the second diagonal and the sides of the quadrilateral.

Answer: \dfrac{24}{5}


## Exercise 3

## Statement.

Find all integers $x, y, z$ that satisfy: $1<x<y<z$ and $x+y+z+xy+yz+zx+xyz=2009$.

Answer: (2, 4, 133)


The squadron of $10$ powerful destroyers and $20$ small boats is about to attack the island. All ships are positioned on the straight line, and are equally spaced.

Two torpedo boats with $10$ torpedoes each want to protect the island. However, the first torpedo boat can shoot only $10$ successive boats, whereas the second one can shoot $10$ targets which are next by one. Note that they have to shoot at the same moment, so that some targets may be hit by both torpedoes.

What is the biggest number of destroyers that can avoid the torpedoes no matter which targets the torpedo boats choose?

[i] Proposed by Bohdan Rublyov [/i]

Answer: 3


1. On the table lie 54 piles of stones with $1,2,3, \ldots, 54$ stones. In each step, we select any pile, say with $k$ stones, and remove it from the table along with $k$ stones from each pile that has at least $k$ stones. For example, after the first step, if we select the pile with 52 stones, the piles remaining on the table will have $1,2,3, \ldots, 51,1$ and 2 stones. Suppose that after a certain number of steps, only one pile remains on the table. Justify how many stones can be in it.

Answer: 1


13. Given three different prime numbers $p$, $q$, and $r$ that satisfy $p q r = 18 \underbrace{99 \ldots 9}_{n \uparrow 9} 62$, define $f(n)$ as the sum of the digits of the natural number $n$. Then $f(p) + f(q) + f(r) - f(p q r) =$ $\qquad$

Answer: 8


The base of pyramid \( SABC \) is a right triangle \( ABC \). The edge \( SA \) is perpendicular to the base plane, with \( |SA| = |AB| = |BC| = a \). A cutting plane is passed through the midpoint of edge \( AB \), perpendicular to edge \( SC \), and we need to find the area of the resulting cross-section.

Given: \( SABC \) is a pyramid with vertex \( S \), \( [SA] \perp (ABC) \), \( [AB] \perp [BC] \), \( |SA| = |AB| = |BC| = a \), \( |AM| = |BM| \), and the cross-section \( (MKLN) \) is perpendicular to \( (SC) \).

Find: \( S_{MKLN} \).

Answer: \dfrac{\sqrt{3}}{8} a^2


Raskin M.A.

A square frame was placed on the ground, and a vertical pole was set up in the center of the square. When a fabric was stretched over this structure, a small tent was formed. If two such frames are placed side by side, a vertical pole of the same length is placed in the center of each, and a fabric is stretched over the top, a larger tent is formed. It took 4 square meters of fabric for the small tent. How much fabric will be needed for the large tent?
![](https://cdn.mathpix.com/cropped/2024_05_06_ccb5665c9ecd710ea159g-34.jpg?height=576&width=1316&top_left_y=892&top_left_x=382)

Answer: 8


Pedro decided to take all his children, boys and girls, for ice cream at the Ice Cream Math parlor. At the parlor, there are 12 different ice cream flavors, and each child ordered a combo with 3 scoops of ice cream. After leaving the parlor, Pedro realized that exactly two scoops of each available flavor were ordered in total.

a) Knowing that Pedro did not have ice cream, what is the total number of his children (girls and boys)?

b) Pedro looked more closely at the flavors each one ordered and noticed that no flavor was chosen by both a boy and a girl, that is, if a boy chose a flavor, no girl chose that same flavor. Knowing that at least one of his children is a boy and that he has more daughters than sons, determine the number of his daughters.

Answer: 8


49*. On the board, there are 128 ones. In one move, you can replace a pair of numbers $a$ and $b$ with the number $a b + 1$. Let $A$ be the maximum number that can appear on the board after 127 such operations. What is its last digit?

## 11th grade

Answer: 2


2. Bear Big and Bear Small agreed to climb Baiyun Mountain along the same route on the weekend. They started from the foot of the mountain at the same time and immediately went down after reaching the top. Both of their downhill speeds were twice their respective uphill speeds, and Bear Big was faster than Bear Small. After 2 hours of departure, Bear Big and Bear Small met 1600 meters away from the top; when Bear Small reached the top, Bear Big had just descended to the halfway point. How many hours did Bear Big spend from departure to returning to the foot of the mountain?

Answer: \dfrac{14}{5}


9. $[\mathbf{7}]$ Let $x(t)$ be a solution to the differential equation
$$
\left(x+x^{\prime}\right)^{2}+x \cdot x^{\prime \prime}=\cos t
$$
with $x(0)=x^{\prime}(0)=\sqrt{\frac{2}{5}}$. Compute $x\left(\frac{\pi}{4}\right)$.

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


3. Which positive integers have an odd number of positive divisors?

Answer: n^2


Four, on a circular road, there are 4 middle schools arranged clockwise: $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$. They have 15, 8, 5, and 12 color TVs, respectively. To make the number of color TVs in each school the same, some schools are allowed to transfer color TVs to adjacent schools. How should the TVs be transferred to minimize the total number of TVs transferred? And what is the minimum total number of TVs transferred?

Answer: 10


Given the complex number $z$ has a modulus of 1. Find
$$
u=\frac{(z+4)^{2}-(\bar{z}+4)^{2}}{4 i} \text {. }
$$

the maximum value.

Answer: \dfrac{\sqrt{9 + 24\sqrt{6}}}{2}


Given real numbers \( x \) and \( y \) satisfy \( n \leqslant x < y \leqslant n+1 \). If there exist \( x \), \( y \), and \( m \in(x, y) \) such that \( |y| = |m| + |x| \), where \( m \) and \( n \) are real numbers, determine the possible range of \( n \).

Answer: (-1, 1)


The teacher wrote the following numbers on the board:

$$
1,4,7,10,13,16,19,22,25,28,31,34,37,40,43
$$

Any two adjacent numbers differ by the same value, in this case, 3. Then she erased all the numbers from the board except for 1, 19, and 43. She further added several integers between these three numbers such that any two adjacent numbers again differed by the same value, and no number was written more than once.

In how many ways could the teacher have completed the numbers?

(K. Pazourek)

Hint. Could 5 be among the added numbers?

Answer: 4


The diameter of a circle of radius $R$ is divided into $2n$ equal parts. The point $M$ is taken on the circle. Prove that the sum of the squares of the distances from the point $M$ to the points of division (together with the ends of the diameter) does not depend on the choice of the point $M$. Calculate this sum.

Answer: \dfrac{(2n + 1)(4n + 1)}{3n} R^2


Problem 8.3. Masha has three identical dice, on the faces of each of which six different prime numbers are written, with a total sum of 87.

Masha rolled all three dice twice. The first time, the sum of the numbers that came up was 10, and the second time, the sum was 62.

Exactly one of the six numbers did not come up at all. Which one?

Answer: 17


Each cell of a beehive is constructed from a right regular 6-angled prism, open at the bottom and closed on the top by a regular 3-sided pyramidical mantle. The edges of this pyramid are connected to three of the rising edges of the prism and its apex $T$ is on the perpendicular line through the center $O$ of the base of the prism (see figure). Let $s$ denote the side of the base, $h$ the height of the cell and $\theta$ the angle between the line $TO$ and $TV$.

(a) Prove that the surface of the cell consists of 6 congruent trapezoids and 3 congruent rhombi.
(b) the total surface area of the cell is given by the formula $6sh - \dfrac{9s^2}{2\tan\theta} + \dfrac{s^2 3\sqrt{3}}{2\sin\theta}$

[img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=286[/img]

Answer: 6 \text{ congruent trapezoids and } 3 \text{ congruent rhombi}


4. Given that $M$ is the least common multiple of 15 consecutive natural numbers $1,2, \cdots, 15$. If a divisor of $M$ is divisible by exactly 14 of these 15 natural numbers, it is called a "good number" of $M$. Then the number of good numbers of $M$ is $\qquad$.

Answer: 4


12. If $\sec \alpha \sqrt{1+\tan ^{2} \alpha}+\tan \alpha \sqrt{\csc ^{2} \alpha-1}=\tan ^{2} \alpha$, determine the sign of the product $\sin (\cos \alpha) \cdot \cos (\sin \alpha)$.

Answer: positive


9. Bags in the Shed. A truck delivered 4 bags of cement. They are stacked in the truck bed. The worker can carry one bag at a time from the truck to the gate, or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it where needed, and placing it on top of the stack (if there are already bags). If there is a choice to carry a bag from the truck or from the gate, the worker chooses each option with a probability of 0.5. In the end, all the bags end up in the shed.

a) (for 7th grade, 1 point). What is the probability that the bags will end up in the shed in the reverse order compared to how they were stacked in the truck?

b) (for 7th grade, 1 point). What is the probability that the bag that was second from the bottom in the truck will end up at the bottom in the shed?

Answer: \dfrac{1}{8}


# Problem 6.

In-1 In an irregular pyramid $A B C D$, the sum of the planar angles at vertex $A$ is $180^{\circ}$. Find the surface area of this pyramid if the area of face $B C D$ is $s$ and $A B=$ $C D, A D=B C$.

Answer: 4s


Let \(ABC\) be an acute-angled triangle with \(AB \neq AC\) and orthocenter \(H\). Let the midpoint of side \(BC\) be \(M\). Let points \(D\) on \(AB\) and \(E\) on \(AC\) be such that \(AE = AD\) and \(D, H, E\) lie on a straight line. Show that \(HM\) and the common chord of the circumcircles of the two triangles \(ABC\) and \(ADE\) are perpendicular to each other.

Answer: HM \text{ and the common chord are perpendicular}


5. (20 points) Students' written work is evaluated on a two-point grading system, i.e., a work will either be credited if it is done well, or not credited if it is done poorly. The works are first checked by a neural network, which gives an incorrect answer in $10 \%$ of cases, and then all works deemed uncredited are rechecked manually by experts, who do not make mistakes. The neural network can both incorrectly classify a good work as uncredited and a bad work as credited. It is known that among all the works submitted by students, $20 \%$ are actually bad. What is the smallest percentage of bad works that can be among those that experts recheck after the neural network's selection? Provide the integer part of the number.

Answer: 66


2. From the 4040 integers ranging from -2020 to 2019, choose 3 numbers to multiply. Let the smallest result be $m$, and the largest result be $n$. Then $\frac{m}{n}=$ $\qquad$ . (The result should be expressed as a simplified fraction)

Answer: -\dfrac{2018}{2019}


9.6 There is a red card box and $k$ blue card boxes $(k>1)$, and a deck of cards, totaling $2n$ cards, which are numbered from 1 to $2n$. Initially, this deck of cards is stacked in the red card box in any order. From any card box, the top card can be taken out, either placed in an empty box, or placed on top of a card with a number exactly one less. For what maximum $n$ can all the cards be moved to one of the blue card boxes?

Answer: k-1


$\therefore 、 m$ distinct positive even numbers and $n$ distinct positive odd numbers have a total sum of 1987, for all such $m$ and $n$, what is the maximum value of $3 m+4 n$? Please prove your conclusion.

The sum of $m$ distinct positive even numbers and $n$ distinct positive odd numbers is 1987, for all such $m$ and $n$, what is the maximum value of $3 m+4 n$? Please prove your conclusion.

Answer: 221


39. Aladdin used the magic lamp to summon the genie, who said to Aladdin: “You can choose some numbers from the natural numbers 1, 2, 3, 4, 5, $\qquad$ 2019, 2020 such that the sum of any two numbers is not an integer multiple of their difference. The number of numbers you choose, I will give you that many gold coins.” Therefore, the maximum number of gold coins Aladdin can get is $\qquad$.

Answer: 674


4. A test paper has 4 multiple-choice questions, each with three options (A), (B), (C). Several students take the exam, and after grading, it is found that: any 3 students have 1 question where their answers are all different. How many students can take the exam at most?

Answer: 9


For example. Given that all edges of a parallelepiped are equal, and its four diagonals are $a$, $b$, $c$, and $d$, find the edge length of this parallelepiped.

Answer: \sqrt{\dfrac{a^2 + b^2 + c^2 + d^2}{12}}


\section*{Problem 6 - 290936}

Let \(k_{1}\) and \(k_{2}\) be two circles that touch each other externally. For their radii \(r_{1}\) and \(r_{2}\), it holds that \(r_{1}>r_{2}\).

A line that touches \(k_{1}\) and \(k_{2}\) at two different points is denoted by \(t\). The tangent to \(k_{1}\) that is parallel to \(t\) and different from \(t\) is denoted by \(u\).

Determine the radius \(r_{3}\) of the circle \(k_{3}\) that touches \(u\) and also touches \(k_{1}\) and \(k_{2}\) externally, in terms of \(r_{1}\) and \(r_{2}\)!

Answer: \dfrac{r_{1}^2}{4r_{2}}


As illustrated, a pattern is formed using six squares, six triangles, and a regular hexagon. The side length of each square is $2 \mathrm{~cm}$. The perimeter of this pattern is $\qquad$ $\mathrm{cm}$.

Answer: 24


6. As shown in Figure 2, in rectangle $A B C D$, $A B=5, B C=$ 8, points $E, F, G, H$ are on sides $A B, B C, C D, D A$ respectively, such that $A E=2, B F=5, D G=3, A H=3$. Point $O$ is on line segment $H F$ such that the area of quadrilateral $A E O H$ is 9. Then the area of quadrilateral $O F C G$ is

Answer: \dfrac{13}{2}


Problem 9.5. Given a right triangle $A B C\left(\Varangle A C B=90^{\circ}\right)$, let $C H, H \in A B$, be the altitude to $A B$ and $P$ and $Q$ be the tangent points of the incircle of $\triangle A B C$ to $A C$ and $B C$, respectively. If $A Q \perp H P$ find the ratio $\frac{A H}{B H}$.

Stoyan Atanasov

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


97. We call a four-digit number with this property a "central four-digit number": by rearranging the four digits of this number in any order, and arranging all the four-digit numbers (at least 2) formed in ascending order, the original four-digit number is exactly in the middle position. For example, 2021 is a "central four-digit number". Therefore, including 2021, there are $\qquad$ such "central four-digit numbers" in total.

Answer: 90


15. Let $\boldsymbol{i}, \boldsymbol{j}$ be the unit vectors in the positive directions of the $x, y$ axes in a Cartesian coordinate system. If the vectors $\boldsymbol{a}=(x+2) \boldsymbol{i}+y \boldsymbol{j}$, $\boldsymbol{b}=(x-2) \boldsymbol{i}+y \boldsymbol{j}$, and $|\boldsymbol{a}|-|\boldsymbol{b}|=2$.
(1) Find the equation of the locus of point $P(x, y)$ that satisfies the above conditions;
(2) Let $A(-1,0), F(2,0)$. Does there exist a constant $\lambda(\lambda>0)$ such that $\angle P F A=\lambda \angle P A F$ always holds? Prove your conclusion.

Answer: x^2 - \dfrac{y^2}{3} = 1


In the quadrilateral $A B C D$, $A B=A C=D B$, and the diagonals are perpendicular to each other. Determine the sum of the angles $A C B$ and $A D B$.

Answer: 135^\circ


There are many $a\times b$ rectangular cardboard pieces ($a,b\in\mathbb{N}$ such that $a<b$). It is given that by putting such pieces together without overlapping one can make $49\times 51$ rectangle, and $99\times 101$ rectangle. Can one uniquely determine $a,b$ from this?

Answer: (1,3)


76. Fill a 5x5 table with 5 ones, 5 twos, 5 threes, 5 fours, and 5 fives (one number per cell), such that the absolute difference between any two numbers in the same column does not exceed 2. Consider the sum of the numbers in each column, and let the minimum of these 5 sums be \( M \). Then the maximum value of \( M \) is \(\qquad\)

Answer: 10


Given a graph \( G \) with \( e \) edges and \( n \) vertices, the degrees of the vertices are \( d_{1}, d_{2}, \cdots, d_{n} \). Let \( k \) be an integer such that
\[
k < \min \left\{d_{1}, d_{2}, \cdots, d_{n}\right\}.
\]
Prove that graph \( G \) contains an induced subgraph \( H \) (i.e., if two vertices in \( H \) are connected by an edge in \( G \), then they are also connected by an edge in \( H \)) such that \( H \) does not contain \( K_{k+1} \) (a complete graph with \( k+1 \) vertices, where each pair of vertices is connected by an edge) and \( H \) has at least \(\frac{k n^{2}}{2 e + n}\) vertices.

Answer: \frac{k n^2}{2 e + n}


The function \( f(x) \) is differentiable on the interval \([a, b]\), and \( f^{\prime}(x) = 0 \) for all points \( x \) in the interval \([a, b]\). Prove that the function \( f(x) \) is constant on the interval \([a, b]\).

Answer: f(x) \text{ is constant on } [a, b]


Answer: 3 < x_{n+1} < x_n \text{ for all } n \geq 1


B1. Determine the exact upper bound of the sequence with the general term $a_{n}=n\left(\frac{20}{23}\right)^{n}$ for all $n \in \mathbb{N}$.

Answer: \dfrac{7 \times 20^{7}}{23^{7}}


4. As shown in the figure, $AB$ is the diameter of the semicircle $\odot O$, $CD \perp AB$. Let $\angle COD = \theta$, then $\frac{AD}{BD}$ - $\tan^2 \frac{\theta}{2}=$ $\qquad$

Answer: 1


In a $6 \times 4$ grid ( $6$ rows, $4$ columns), $12$ of the $24$ squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column.  Let $N$ be the number of shadings with this property.  Find the remainder when $N$ is divided by $1000$ . AIME I 2007-10.png
Please give the answer directly without any intermediate steps.

Answer: 860


【Question 19】
There are two isosceles triangles with a leg length of 1, one with a vertex angle of $30^{\circ}$ (an acute isosceles triangle), and the other with a vertex angle of $90^{\circ}$ (a right isosceles triangle). As shown in the figure below, these two types of triangles are placed around a point in a clockwise direction, following these rules: the 1st and 2nd are acute isosceles triangles, the 3rd is a right isosceles triangle; the 4th and 5th are acute isosceles triangles, the 6th is a right isosceles triangle, and so on. If the $n$-th triangle exactly overlaps with the 1st triangle, find the smallest value of $n$.

Answer: 23


4. How many sets of three consecutive three-digit natural numbers have a product that is divisible by 120?

Answer: 336


We place a coin in one of three identical boxes and put these three boxes into two accounts so that each account gets a box. What is the probability that if someone randomly selects an account and takes a box from it, they will find the coin,

a) if the box containing the coin is alone in an account?

b) if the box containing the coin is with another box in an account?

c) if the 3 boxes are placed randomly into the two accounts, without knowing which box contains the coin?

Answer: \dfrac{1}{2}


a) What is the last digit of the number

$$
\left(\ldots\left(\left(\left(7^{7}\right)^{7}\right)^{7}\right)^{\ldots 7}\right)
$$

(where raising to the power of 7 is repeated 1000 times)? What are the last two digits of this number?

b) What is the last digit of the number

$$
7\left(7^{\left(.7^{\left(7^{7}\right)}\right)} \ldots\right)
$$

written using 1001 sevens, similar to the number in part a), but with a different order of exponentiation? What are the last two digits of this number?

Answer: 7


$23$ frat brothers are sitting in a circle. One, call him Alex, starts with a gallon of water. On the first turn, Alex gives each person in the circle some rational fraction of his water. On each subsequent turn, every person with water uses the same scheme as Alex did to distribute his water, but in relation to themselves. For instance, suppose Alex gave $\frac{1}{2}$ and $\frac{1}{6}$ of his water to his left and right neighbors respectively on the first turn and kept $\frac{1}{3}$ for himself. On each subsequent turn everyone gives $\frac{1}{2}$ and $\frac{1}{6}$ of the water they started the turn with to their left and right neighbors, respectively, and keep the final third for themselves. After $23$ turns, Alex again has a gallon of water. What possibilities are there for the scheme he used in the first turn?
(Note: you may find it useful to know that $1+x+x^2+\cdot +x^{23}$ has no polynomial factors with rational coefficients)

Answer: 1


1. Given $a>b>2$, and $a+b$, $a-b$, $a b$, $\frac{b}{a}$ form a geometric sequence in a certain order. Then $a=$ $\qquad$ .

Answer: 7 + 5\sqrt{2}


11.5. A square plot of 14 by 14 cells needs to be paved with rectangular tiles of size $1 \times 4$. The tiles can only be laid along the grid (not diagonally), and the tiles cannot be broken. What is the maximum number of tiles required? Will there be any uncovered area left?

Answer: 48


We are given five watches which can be winded forward.  What is the smallest sum of winding intervals which allows us to set them to the same time, no matter how they were set initially?

Answer: 24


B2. In the figure, you see an equilateral triangle $A B C$ and points $D$ and $E$ on sides $B C$ and $A C$. If you fold the triangle along the line $D E$, then vertex $C$ lands exactly on line $A B$, at point $C^{\prime}$. It is given that $\angle D C^{\prime} B=90^{\circ}$.

How large is $\angle D E C^{\prime}$?

Note: the picture is not drawn to scale.

![](https://cdn.mathpix.com/cropped/2024_04_17_8b0e5b2f470f23b328a2g-1.jpg?height=377&width=483&top_left_y=1345&top_left_x=1346)

Answer: 45^\circ


3. Between the digits of the number 987654321, signs + should be placed so that the sum is 99. In how many ways can this be achieved?

Answer: 2


Let $S^{}_{}$ be a subset of $\{1,2,3^{}_{},\ldots,1989\}$ such that no two members of $S^{}_{}$ differ by $4^{}_{}$ or $7^{}_{}$ . What is the largest number of elements $S^{}_{}$ can have?
Please give the answer directly without any intermediate steps.

Answer: 905


4. Let $a_{1}, a_{2}, \cdots, a_{20} \in\{1,2, \cdots, 5\}, b_{1}, b_{2}, \cdots, b_{20} \in\{1,2, \cdots, 10\}$, and the set $X=$ $\left\{(i, j) \mid 1 \leqslant i<j \leqslant 20,\left(a_{i}-a_{j}\right)\left(b_{i}-b_{j}\right)<0\right\}$. Find the maximum number of elements in $X$.

Answer: 160


1. Write seven different digits instead of the seven asterisks so that the correct equation is obtained: ****+**+* = 2015.

Answer: 1987 + 23 + 5 = 2015


B4 In a cube with edge 6, there is a sphere with diameter 6. The sphere thus touches the six faces of the cube internally. We imagine the cube divided into 216 unit cubes of $1 \times 1 \times 1$. How many of these unit cubes lie entirely within the sphere? A unit cube with a vertex on the sphere's surface and otherwise inside the sphere is counted as lying entirely within the sphere.

The work of the Stichting Nederlandse Wiskunde Olympiade is made possible by financial contributions and support from:

The Ministry of Education, Culture and Science

The Dutch Association of Mathematics Teachers

The Mathematical Society

Eindhoven University of Technology

Citogroep

Utrecht University of Applied Sciences

Educatieve Partners Nederland

Woltersgroep Groningen

AKZO/NOBEL

Answer: 56


3.82. The ratio of the area of the diagonal section of a regular quadrilateral pyramid to the area of its base is $k$. Find the cosine of the plane angle at the vertex of the pyramid.

Answer: \dfrac{4k^2}{1 + 4k^2}


33 friends are collecting stickers for a 2011-sticker album. A distribution of stickers among the 33 friends is incomplete when there is a sticker that no friend has. Determine the least  $m$  with the following property: every distribution of stickers among the 33 friends such that, for any two friends, there are at least  $m$  stickers both don't have, is incomplete.

Answer: 1890


A chessboard  $1000 \times 1000$  is covered by dominoes  $1 \times 10$  that can be rotated. We don't know which is the cover, but we are looking for it. For this reason, we choose a few  $N$  cells of the chessboard, for which we know the position of the dominoes that cover them.
Which is the minimum  $N$  such that after the choice of  $N$  and knowing the dominoed that cover them, we can be sure and for the rest of the cover?

(Bulgaria)

Answer: 100,000


There are  $2022$  equally spaced points on a circular track  $\gamma$  of circumference  $2022$ . The points are labeled  $A_1, A_2, \ldots, A_{2022}$  in some order, each label used once. Initially, Bunbun the Bunny begins at  $A_1$ . She hops along  $\gamma$  from  $A_1$  to  $A_2$ , then from  $A_2$  to  $A_3$ , until she reaches  $A_{2022}$ , after which she hops back to  $A_1$ . When hopping from  $P$  to  $Q$ , she always hops along the shorter of the two arcs  $\widehat{PQ}$  of  $\gamma$ ; if  $\overline{PQ}$  is a diameter of  $\gamma$ , she moves along either semicircle.

Determine the maximal possible sum of the lengths of the  $2022$  arcs which Bunbun traveled, over all possible labellings of the  $2022$  points.

*Kevin Cong*

Answer: 2042222


There are 15 cities, and there is a train line between each pair operated by either the Carnegie Rail Corporation or the Mellon Transportation Company. A tourist wants to visit exactly three cities by travelling in a loop, all by travelling on one line. What is the minimum number of such 3-city loops?

Answer: 88


Suppose that  $a$  is real number. Sequence  $a_1,a_2,a_3,....$  satisfies  $$ a_1=a, a_{n+1} =
\begin{cases}
a_n - \frac{1}{a_n}, & a_n\ne 0
0, & a_n=0
\end{cases}
(n=1,2,3,..) $$  Find all possible values of  $a$  such that  $|a_n|<1$  for all positive integer  $n$ .

Answer: 0


Positive integers 1 to 9 are written in each square of a  $ 3 \times 3 $  table. Let us define an operation as follows: Take an arbitrary row or column and replace these numbers  $ a, b, c$  with either non-negative numbers  $ a-x, b-x, c+x $  or  $ a+x, b-x, c-x$ , where  $ x $  is a positive number and can vary in each operation.
(1) Does there exist a series of operations such that all 9 numbers turn out to be equal from the following initial arrangement a)? b)?
\[ a) \begin{array}{ccc} 1 & 2 & 3  4 & 5 & 6  7 & 8 & 9 \end{array} )\]
\[ b) \begin{array}{ccc} 2 & 8 & 5  9 & 3 & 4  6 & 7 & 1 \end{array} )\]
(2) Determine the maximum value which all 9 numbers turn out to be equal to after some steps.

Answer: 5


Let  $ABCD$  be an isosceles trapezoid with  $AB=5$ ,  $CD = 8$ , and  $BC = DA = 6$ . There exists an angle  $\theta$  such that there is only one point  $X$  satisfying  $\angle AXD = 180^{\circ} - \angle BXC = \theta$ . Find  $\sin(\theta)^2$ .

Answer: \frac{27}{32}


A group of 6 students decided to make *study groups* and *service activity groups* 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  $n$  cards such that for each  $i=1,2, \cdots n$ , there are exactly one card labeled  $i$ . Initially the cards are piled with increasing order from top to bottom. There are two operations:

- $A$  : One can take the top card of the pile and move it to the bottom;
- $B$  : One can remove the top card from the pile.

The operation  $ABBABBABBABB \cdots $  is repeated until only one card gets left. Let  $L(n)$  be the labeled number on the final pile. Find all integers  $k$  such that  $L(3k)=k$ .

Answer: 1


Let  $H$  be a rectangle with angle between two diagonal  $\leq 45^{0}$ . Rotation  $H$  around the its center with angle  $0^{0}\leq x\leq 360^{0}$  we have rectangle  $H_{x}$ . Find  $x$  such that  $[H\cap H_{x}]$  minimum, where  $[S]$  is area of  $S$ .

Answer: 45


 $k$  marbles are placed onto the cells of a  $2024 \times 2024$  grid such that each cell has at most one marble and there are no two marbles are placed onto two neighboring cells (neighboring cells are defined as cells having an edge in common).

a) Assume that  $k=2024$ . Find a way to place the marbles satisfying the conditions above, such that moving any placed marble to any of its neighboring cells will give an arrangement that does not satisfy both the conditions.
b) Determine the largest value of  $k$  such that for all arrangements of  $k$  marbles satisfying the conditions above, we can move one of the placed marble onto one of its neighboring cells and the new arrangement satisfies the conditions above.

Answer: 2023


Some squares of a  $1999\times 1999$  board are occupied with pawns. Find the smallest number of pawns for which it is possible that for each empty square, the total number of pawns in the row or column of that square is at least  $1999$ .

Answer: 1998001


Square  $ABCD$  has side length  $30$ . Point  $P$  lies inside the square so that  $AP = 12$  and  $BP = 26$ . The centroids of  $\triangle{ABP}$ ,  $\triangle{BCP}$ ,  $\triangle{CDP}$ , and  $\triangle{DAP}$  are the vertices of a convex quadrilateral. What is the area of that quadrilateral?

[asy]
unitsize(120);
pair B = (0, 0), A = (0, 1), D = (1, 1), C = (1, 0), P = (1/4, 2/3);
draw(A--B--C--D--cycle);
dot(P);
defaultpen(fontsize(10pt));
draw(A--P--B);
draw(C--P--D);
label(" $A$ ", A, W);
label(" $B$ ", B, W);
label(" $C$ ", C, E);
label(" $D$ ", D, E);
label(" $P$ ", P, N*1.5+E*0.5);
dot(A);
dot(B);
dot(C);
dot(D);
[/asy] $\textbf{(A) }100\sqrt{2}\qquad\textbf{(B) }100\sqrt{3}\qquad\textbf{(C) }200\qquad\textbf{(D) }200\sqrt{2}\qquad\textbf{(E) }200\sqrt{3}$

Answer: 200


A chess knight has injured his leg and is limping. He alternates between a normal move and a short move where he moves to any diagonally neighbouring cell.
The limping knight moves on a  $5 \times 6$  cell chessboard starting with a normal move. What is the largest number of moves he can make if he is starting from a cell of his own choice and is not allowed to visit any cell (including the initial cell) more than once?

Answer: 24


Cyclic quadrilateral  $ABCD$  satisfies  $\angle ADC = 2 \cdot \angle BAD = 80^\circ$  and  $\overline{BC} = \overline{CD}$ . Let the angle bisector of  $\angle BCD$  meet  $AD$  at  $P$ . What is the measure, in degrees, of  $\angle BP D$ ?

Answer: 60


Regular octagon \(ABCDEFGH\) has area \(n\). Let \(m\) be the area of quadrilateral \(ACEG\). The answer to the question is in the form \(\frac{\sqrt{k}}{m}\). What is the value of \(k + m\)?

Answer: 4


Steve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ digits, beginning $123451234512\ldots.$ He then erased every third digit from his list (that is, the $3$rd, $6$th, $9$th, $\ldots$ digits from the left), then erased every fourth digit from the resulting list (that is, the $4$th, $8$th, $12$th, $\ldots$ digits from the left in what remained), and then erased every fifth digit from what remained at that point. Find the sum of the three digits that were then in the positions $2019, 2020, 2021$.

Answer: 11


Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. Find the number of the $125$ sequences of three of these transformations (not necessarily distinct) that will return $T$ to its original position.

Answer: 12


Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. Find the probability that after they are painted, the cubes can be rotated to be identical in appearance. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 1171


In square \(ABCD\), points \(E\) and \(H\) lie on \(\overline{AB}\) and \(\overline{DA}\), respectively, so that \(AE=AH.\) Points \(F\) and \(G\) lie on \(\overline{BC}\) and \(\overline{CD}\), respectively, and points \(I\) and \(J\) lie on \(\overline{EH}\) so that \(\overline{FI} \perp \overline{EH}\) and \(\overline{GJ} \perp \overline{EH}\). See the figure below. Triangle \(AEH\), quadrilateral \(BFIE\), quadrilateral \(DHJG\), and pentagon \(FCGJI\) each has area \(1.\) The original answer is in the form \(k - m\sqrt{n}\), please give the value of \(k + m + n\).

Answer: 14


Square \(ABCD\) in the coordinate plane has vertices at the points \(A(1,1), B(-1,1), C(-1,-1),\) and \(D(1,-1).\) Consider the following four transformations:

\(\quad\bullet\qquad\) \(L,\) a rotation of \(90^{\circ}\) counterclockwise around the origin;

\(\quad\bullet\qquad\) \(R,\) a rotation of \(90^{\circ}\) clockwise around the origin;

\(\quad\bullet\qquad\) \(H,\) a reflection across the \(x\)-axis; and

\(\quad\bullet\qquad\) \(V,\) a reflection across the \(y\)-axis.

Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying \(R\) and then \(V\) would send the vertex \(A\) at \((1,1)\) to \((-1,-1)\) and would send the vertex \(B\) at \((-1,1)\) to itself. Find how many sequences of \(20\) transformations chosen from \(\{L, R, H, V\}\) will send all of the labeled vertices back to their original positions. The original answer is in the form \(x * 2^y\), please give the value of \(x + y\).

Answer: 39


For each real number \(a\) with \(0 \leq a \leq 1\), let numbers \(x\) and \(y\) be chosen independently at random from the intervals \([0, a]\) and \([0, 1]\), respectively, and let \(P(a)\) be the probability that
\(\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1\).
Find the maximum value of \(P(a)\). The original answer is in the form \(k - \sqrt{m}\), please give the value of \(k + m\).

Answer: 4


Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. Find the number of distinct integers that can be written as the sum of two, not necessarily different, special fractions.

Answer: 11


Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. Find how many distinct integers can be written as the sum of two, not necessarily different, special fractions.

Answer: 11


A regular hexagon of side length $1$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. Find the area of the region bounded by these $6$ reflected arcs. The original answer is in the form of k\sqrt{m} - n\pi, please provide the value of k + m + n.

Answer: 7


Shapovalov A.V.

A banker learned that among identical-looking coins, one is counterfeit (lighter). He asked an expert to identify this coin using a balance scale without weights, and required that each coin participate in weighings no more than twice. What is the maximum number of coins the banker can have so that the expert can definitely identify the counterfeit one in $n$ weighings?

Answer: 2n^2+1


30. Solve the equation $p^{2}-6 p q+q^{2}+3 q-1=0$ in prime numbers.

Answer: p=17,q=3


8. A deck of playing cards has a total of 54 cards, including 2 jokers, and 13 cards of each of the four suits: spades, hearts, clubs, and diamonds. If you randomly draw some cards from the deck, to ensure that the drawn cards include at least two suits, and that each of these suits has at least 6 cards, what is the minimum number of cards you need to draw? $\qquad$

Answer: 31


## Task 6

Determine the number that is 7 less than 14! Write an equation!

Answer: 7


7. We call a four-digit number with this property a "central four-digit number": by rearranging the four digits of this number in any order, and arranging all the four-digit numbers (at least 2) formed in ascending order, the original four-digit number is exactly in the middle position. For example, 2021 is a "central four-digit number". Therefore, including 2021, there are $\qquad$ such "central four-digit numbers" in total.

Answer: 90


Four, on a plane there are $n(n \geqslant 4)$ lines. For lines $a$ and $b$, among the remaining $n-2$ lines, if at least two lines intersect with both lines $a$ and $b$, then lines $a$ and $b$ are called a "congruent line pair"; otherwise, they are called a "separated line pair". If the number of congruent line pairs among the $n$ lines is 2012 more than the number of separated line pairs, find the minimum possible value of $n$ (the order of the lines in a pair does not matter).

Answer: 72


15. A 33-story building has an elevator that starts on the first floor. It can accommodate a maximum of 32 people and can only stop once at one of the floors from the 2nd to the 33rd. For each person, walking down one floor of stairs results in 1 point of dissatisfaction, and walking up one floor of stairs results in 3 points of dissatisfaction. Now, there are 32 people on the first floor, and they each live on one of the floors from the 2nd to the 33rd. Question: On which floor should the elevator stop to minimize the total dissatisfaction score of these 32 people? What is the minimum value? (Some people can choose to walk up the stairs instead of taking the elevator).

Answer: 316


19. Monochromatic pairs. In a box, there are $n$ white and $n$ black balls. Balls are drawn from the box in pairs - random pairs.

a) (from 9th grade. 2 points). Find the expected number of mixed-color pairs drawn from the box by the time it is empty.

b) (from 9th grade. 5 points). Suppose that if a pair is mixed-color, it is set aside, and if it is monochromatic, it is returned to the box. Such an operation is called an attempt. Find the expected number of attempts needed to set all the balls aside, leaving nothing in the box.

Answer: 2n-H_n


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Carol`, `Eric`, `Peter`, `Arnold`, `Bob`
 - Everyone has a unique favorite cigar: `pall mall`, `yellow monster`, `dunhill`, `blends`, `blue master`, `prince`
 - Everyone has a favorite smoothie: `watermelon`, `dragonfruit`, `lime`, `desert`, `blueberry`, `cherry`

## Clues:
1. The person who drinks Lime smoothies is in the third house.
2. The person partial to Pall Mall is the Watermelon smoothie lover.
3. The Desert smoothie lover is Arnold.
4. The Watermelon smoothie lover is somewhere to the left of Eric.
5. There is one house between Peter and Carol.
6. The person who drinks Blueberry smoothies is somewhere to the left of the person who drinks Lime smoothies.
7. The Watermelon smoothie lover is somewhere to the left of the person who smokes many unique blends.
8. The Dragonfruit smoothie lover and Carol are next to each other.
9. Bob is directly left of Carol.
10. The person who likes Cherry smoothies is the Dunhill smoker.
11. The Dragonfruit smoothie lover is the Prince smoker.
12. The person who smokes Blue Master is Peter.
13. The Watermelon smoothie lover and the Desert smoothie lover are next to each other.
 What is the value of attribute Cigar for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}

Answer: yellow monster


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Peter`, `Arnold`, `Bob`
 - Each person lives in a unique style of house: `craftsman`, `modern`, `victorian`, `ranch`, `colonial`
 - Each person has a unique type of pet: `cat`, `bird`, `dog`, `hamster`, `fish`
 - Each person has a unique favorite drink: `coffee`, `milk`, `tea`, `water`, `root beer`

## Clues:
1. The person who has a cat is in the third house.
2. The person with an aquarium of fish is Arnold.
3. The root beer lover is in the fourth house.
4. There is one house between the person in a Craftsman-style house and the person residing in a Victorian house.
5. The person who has a cat is Bob.
6. Peter is the person with a pet hamster.
7. The person in a modern-style house is the person who has a cat.
8. The coffee drinker is not in the fifth house.
9. The one who only drinks water is the person who owns a dog.
10. The person living in a colonial-style house and the person with an aquarium of fish are next to each other.
11. The person with an aquarium of fish is the person in a Craftsman-style house.
12. The tea drinker is in the third house.
13. The person residing in a Victorian house is somewhere to the right of Alice.
14. The person in a ranch-style home is somewhere to the right of Eric.
 What is the value of attribute Drink for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: water


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`, `Bob`, `Arnold`, `Alice`, `Eric`
 - Each person has an occupation: `lawyer`, `teacher`, `artist`, `engineer`, `doctor`
 - Each person lives in a unique style of house: `victorian`, `craftsman`, `modern`, `ranch`, `colonial`
 - People have unique hair colors: `blonde`, `gray`, `red`, `brown`, `black`
 - People have unique favorite book genres: `romance`, `biography`, `science fiction`, `mystery`, `fantasy`

## Clues:
1. Alice is not in the fifth house.
2. Peter is not in the second house.
3. Peter is directly left of Eric.
4. The person living in a colonial-style house is somewhere to the left of the person who loves fantasy books.
5. Arnold is the person who is a doctor.
6. The person who has gray hair is the person who loves fantasy books.
7. The person who loves science fiction books is not in the fifth house.
8. Peter is the person who has red hair.
9. The person who loves romance books is the person who has black hair.
10. Eric is the person who has black hair.
11. The person who is a doctor is somewhere to the right of Peter.
12. The person who has blonde hair is the person who is a teacher.
13. Peter is the person who loves mystery books.
14. The person who is a lawyer is Eric.
15. The person living in a colonial-style house is the person who is an engineer.
16. The person residing in a Victorian house is the person who loves mystery books.
17. The person who is a teacher is not in the fourth house.
18. The person who is an artist is somewhere to the left of the person in a modern-style house.
19. The person who is a teacher is not in the third house.
20. The person in a ranch-style home is directly left of the person in a Craftsman-style house.
 What is the value of attribute HairColor for the person whose attribute HouseStyle is ranch? Please reason step by step, and put your final answer within \boxed{}

Answer: gray


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`, `Eric`, `Arnold`, `Bob`, `Alice`
 - Each person has a unique favorite drink: `milk`, `tea`, `root beer`, `water`, `coffee`
 - Each person has a favorite color: `blue`, `white`, `red`, `green`, `yellow`
 - Each mother is accompanied by their child: `Samantha`, `Meredith`, `Timothy`, `Bella`, `Fred`
 - Each person lives in a unique style of house: `modern`, `ranch`, `victorian`, `colonial`, `craftsman`
 - People use unique phone models: `samsung galaxy s21`, `iphone 13`, `huawei p50`, `oneplus 9`, `google pixel 6`

## Clues:
1. The person in a Craftsman-style house is directly left of the person living in a colonial-style house.
2. The person living in a colonial-style house is somewhere to the left of the person who uses a Samsung Galaxy S21.
3. The person who uses a Huawei P50 is the person in a Craftsman-style house.
4. The root beer lover is somewhere to the left of the person residing in a Victorian house.
5. There is one house between the person in a ranch-style home and the root beer lover.
6. The person who uses a OnePlus 9 is the person whose favorite color is red.
7. The tea drinker is not in the fourth house.
8. The person living in a colonial-style house and Peter are next to each other.
9. The person in a ranch-style home is directly left of the person whose favorite color is green.
10. The person who is the mother of Timothy and the person's child is named Meredith are next to each other.
11. The person who loves blue is the person residing in a Victorian house.
12. Bob is the person residing in a Victorian house.
13. The person's child is named Bella is the person who uses a Google Pixel 6.
14. The person who loves blue is the one who only drinks water.
15. The person who likes milk is the person who is the mother of Timothy.
16. The person who loves white is Arnold.
17. The root beer lover is directly left of the person who uses a Samsung Galaxy S21.
18. The person who likes milk is Alice.
19. The person who is the mother of Timothy is the person whose favorite color is green.
20. The person who loves white is not in the first house.
21. The person's child is named Samantha is the person whose favorite color is red.
 What is the value of attribute PhoneModel for the person whose attribute Color is yellow? Please reason step by step, and put your final answer within \boxed{}

Answer: samsung galaxy s21


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`, `Eric`, `Alice`, `Arnold`, `Bob`
 - The people keep unique animals: `fish`, `dog`, `bird`, `cat`, `horse`
 - Each person prefers a unique type of vacation: `beach`, `city`, `mountain`, `camping`, `cruise`
 - People use unique phone models: `google pixel 6`, `huawei p50`, `samsung galaxy s21`, `iphone 13`, `oneplus 9`

## Clues:
1. Alice is not in the fifth house.
2. Arnold is not in the fifth house.
3. The person who uses a OnePlus 9 is the person who enjoys mountain retreats.
4. Bob is the person who loves beach vacations.
5. The cat lover is in the second house.
6. There are two houses between the person who uses a Google Pixel 6 and the fish enthusiast.
7. The person who enjoys camping trips is the person who keeps horses.
8. The fish enthusiast and the person who enjoys mountain retreats are next to each other.
9. The dog owner is not in the fifth house.
10. The fish enthusiast is Alice.
11. The person who prefers city breaks is the person who uses a Huawei P50.
12. There are two houses between Arnold and the person who uses a Samsung Galaxy S21.
13. The person who loves beach vacations is in the third house.
14. Eric is somewhere to the left of the person who enjoys mountain retreats.
 What is the value of attribute Animal for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`
 - Each person has a unique birthday month: `sept`, `jan`, `april`
 - Each person has a unique hobby: `photography`, `cooking`, `gardening`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `pizza`
 - People have unique favorite book genres: `science fiction`, `romance`, `mystery`
 - The mothers' names in different houses are unique: `Janelle`, `Holly`, `Aniya`

## Clues:
1. The person whose birthday is in April is the person who enjoys gardening.
2. The person whose mother's name is Janelle is not in the third house.
3. The photography enthusiast is in the first house.
4. The person whose mother's name is Janelle is the person who loves mystery books.
5. The person who loves science fiction books and Eric are next to each other.
6. Arnold is not in the third house.
7. The person whose mother's name is Holly is the person whose birthday is in September.
8. The person whose birthday is in January is The person whose mother's name is Aniya.
9. The person whose mother's name is Holly is the person who loves science fiction books.
10. The person who loves eating grilled cheese is The person whose mother's name is Aniya.
11. The person who loves the spaghetti eater is Arnold.
 What is the value of attribute Mother for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{}

Answer: Holly


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`, `Alice`
 - Each person has a unique hobby: `painting`, `gardening`, `photography`, `cooking`
 - The people keep unique animals: `horse`, `bird`, `fish`, `cat`

## Clues:
1. Peter is the cat lover.
2. There is one house between the person who paints as a hobby and Peter.
3. Alice is the fish enthusiast.
4. The person who enjoys gardening is somewhere to the right of the person who paints as a hobby.
5. Arnold is not in the second house.
6. The photography enthusiast is not in the third house.
7. The bird keeper is the person who enjoys gardening.
8. Eric is the bird keeper.
9. Alice is in the first house.
 What is the value of attribute House for the person whose attribute Hobby is gardening? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Carol`, `Peter`, `Bob`, `Arnold`, `Eric`
 - They all have a unique favorite flower: `carnations`, `iris`, `tulips`, `daffodils`, `lilies`, `roses`
 - Each person prefers a unique type of vacation: `camping`, `mountain`, `city`, `beach`, `cultural`, `cruise`
 - Each person lives in a unique style of house: `colonial`, `modern`, `mediterranean`, `craftsman`, `ranch`, `victorian`
 - People have unique heights: `short`, `very short`, `average`, `tall`, `super tall`, `very tall`
 - People have unique hair colors: `red`, `black`, `gray`, `brown`, `auburn`, `blonde`

## Clues:
1. The person who loves the boquet of lilies is not in the fifth house.
2. The person who is very short is the person who has auburn hair.
3. Arnold is the person who enjoys camping trips.
4. The person who has auburn hair is Eric.
5. The person who goes on cultural tours is somewhere to the left of the person who is super tall.
6. There are two houses between the person who has gray hair and the person who is short.
7. The person who is tall is somewhere to the left of the person who has brown hair.
8. The person in a Craftsman-style house is Bob.
9. The person who is very short is somewhere to the left of the person who has blonde hair.
10. The person who is very tall is Bob.
11. There are two houses between the person who loves the rose bouquet and Peter.
12. The person in a modern-style house is not in the fourth house.
13. The person who prefers city breaks is the person who is short.
14. The person living in a colonial-style house is somewhere to the left of the person who loves the vase of tulips.
15. The person who loves a carnations arrangement is the person who has brown hair.
16. The person who has red hair is the person who loves the boquet of lilies.
17. The person who enjoys mountain retreats is the person residing in a Victorian house.
18. The person who loves a bouquet of daffodils is the person in a modern-style house.
19. Arnold is the person who has an average height.
20. The person in a Mediterranean-style villa is not in the first house.
21. The person who is tall is not in the fifth house.
22. The person who loves the vase of tulips is not in the fourth house.
23. Carol is directly left of Bob.
24. The person who is very tall is the person who goes on cultural tours.
25. The person in a Mediterranean-style villa is the person who has red hair.
26. The person who is super tall is the person who has black hair.
27. Eric is the person who loves the rose bouquet.
28. The person who enjoys mountain retreats is directly left of the person who likes going on cruises.
 What is the value of attribute House for the person whose attribute HouseStyle is ranch? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Alice`
 - People have unique favorite music genres: `pop`, `classical`, `jazz`, `rock`
 - Each person has a unique hobby: `cooking`, `gardening`, `painting`, `photography`
 - Each person has a unique level of education: `master`, `associate`, `high school`, `bachelor`
 - Each person has a unique type of pet: `fish`, `bird`, `dog`, `cat`

## Clues:
1. The person who loves classical music is not in the fourth house.
2. Arnold is the person with a master's degree.
3. Alice is the person who owns a dog.
4. The person who enjoys gardening is not in the third house.
5. The person with a bachelor's degree is Alice.
6. Alice is the photography enthusiast.
7. Arnold is directly left of the person who loves pop music.
8. The person with an associate's degree is directly left of the photography enthusiast.
9. The person who loves rock music is the person who loves cooking.
10. The person with an associate's degree is Eric.
11. The person who has a cat is the person who loves jazz music.
12. Arnold is somewhere to the right of the person with an aquarium of fish.
 What is the value of attribute Name for the person whose attribute Pet is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Peter`, `Bob`, `Carol`
 - Everyone has a favorite smoothie: `cherry`, `blueberry`, `desert`, `lime`, `dragonfruit`, `watermelon`
 - People have unique favorite music genres: `rock`, `classical`, `jazz`, `pop`, `country`, `hip hop`
 - People have unique favorite book genres: `science fiction`, `romance`, `historical fiction`, `fantasy`, `mystery`, `biography`
 - Each person has a favorite color: `yellow`, `white`, `green`, `blue`, `red`, `purple`

## Clues:
1. The person who loves purple and the Dragonfruit smoothie lover are next to each other.
2. The person who loves classical music is in the fifth house.
3. Eric is somewhere to the left of the person who loves classical music.
4. The person who loves hip-hop music is directly left of the Desert smoothie lover.
5. The person who loves yellow is the person who loves science fiction books.
6. Arnold is the person who loves rock music.
7. Carol is the person who likes Cherry smoothies.
8. There are two houses between the person who loves jazz music and the Watermelon smoothie lover.
9. The person whose favorite color is green is somewhere to the right of the person who loves jazz music.
10. The person who drinks Blueberry smoothies is directly left of the Watermelon smoothie lover.
11. Arnold is somewhere to the left of the Watermelon smoothie lover.
12. Bob is somewhere to the left of the person who loves yellow.
13. Bob is the person who loves historical fiction books.
14. The person who loves biography books is the person who loves pop music.
15. The person who loves romance books is in the fifth house.
16. There are two houses between Bob and the person who loves mystery books.
17. The person who loves white is the person who loves historical fiction books.
18. Peter is somewhere to the left of the person who loves blue.
19. There is one house between the person who loves biography books and the person who loves yellow.
 What is the value of attribute House for the person whose attribute MusicGenre is jazz? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Bob`, `Peter`, `Carol`
 - Each person has a unique hobby: `woodworking`, `knitting`, `cooking`, `gardening`, `painting`, `photography`
 - Everyone has a unique favorite cigar: `pall mall`, `blue master`, `blends`, `prince`, `dunhill`, `yellow monster`

## Clues:
1. Alice is the person who loves cooking.
2. Bob is not in the second house.
3. The person who enjoys gardening is the person who smokes Blue Master.
4. The person partial to Pall Mall is the person who enjoys knitting.
5. The person who enjoys gardening is directly left of the Dunhill smoker.
6. The person who smokes many unique blends is directly left of Carol.
7. The person partial to Pall Mall is directly left of Bob.
8. The Dunhill smoker is the person who paints as a hobby.
9. The person who smokes Yellow Monster is in the second house.
10. The woodworking hobbyist is somewhere to the right of the Dunhill smoker.
11. Eric is somewhere to the left of the person who enjoys gardening.
12. There are two houses between the photography enthusiast and Peter.
 What is the value of attribute Cigar for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}

Answer: yellow monster