Given $n$ points in a plane, each pair of points has a distance between them, with the ratio of the longest distance to the shortest distance denoted as $\lambda_{n}$. Prove that $\lambda_{5} \geqslant 2 \sin 54^{\circ}$.

Answer: \lambda_5 \geqslant 2 \sin 54^\circ


7. A digital clock displays hours, minutes, and seconds using two digits each (e.g., 10:09:18). Between 05:00:00 and 22:59:59 on the same day, the probability that all six digits on the clock are different is

Answer: \dfrac{16}{135}


3. From the center of the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$, two perpendicular chords $A C$ and $B D$ are drawn. Connecting $A, B, C, D$ in sequence forms a quadrilateral. Then, the maximum value of the area $S$ of quadrilateral $A B C D$ is

Answer: 12


A [i]quadratic[/i] number is a real root of the equations $ax^2 + bx + c = 0$ where $|a|,|b|,|c|\in\{1,2,\ldots,10\}$. Find the smallest positive integer $n$ for which at least one of the intervals$$\left(n-\dfrac{1}{3}, n\right)\quad \text{and}\quad\left(n, n+\dfrac{1}{3}\right)$$does not contain any quadratic number.

Answer: 11


We have a portable radio that operates with two AA batteries. In the drawer, there are 8 AA batteries, out of which 4 are drained. The good and bad batteries are mixed together. There is no other way to test the batteries except by placing two of them in the device. If the radio works, both batteries are good; if it does not work, at least one of them is bad. What is the minimum number of trials needed to ensure the radio will operate?

Answer: 7


29. As shown in the figure, $P$ is the midpoint of $AC$, $BM=NC=2$ cm, $MN=4$ cm. It is known that the area of triangle $ABC$ is 16 square cm, then the area of $\triangle PMN$ is $\qquad$ square cm.

Answer: 4


4. For a set $X$, denote $\mathscr{P}(X)=\{Y: Y \subseteq X\}$. (For example: $\mathscr{P}(\{1\})=$ $\{\varnothing,\{1\}\}$, because the subsets of the set $\{1\}$ are the sets $\varnothing$ and $\{1\} ; \mathscr{P}(\varnothing)=\{\varnothing\}$, because the set $\varnothing$ has exactly one subset, which is $\varnothing$.) We denote $\mathscr{P}^{n}(X)$ as the expression $\mathscr{P}(\mathscr{P}(\ldots \mathscr{P}(X) \ldots))$, where $\mathscr{P}$ is applied $n$ times. Find all two-element subsets $A$ of the set $\bigcup_{n=1}^{2018} \mathscr{P}^{n}(\varnothing)$ such that $A \subseteq \mathscr{P}(A)$.

Answer: \{\varnothing, \{\varnothing\}\}


2・109 Let $S=\left\{A=\left(a_{1}, \cdots, a_{8}\right) \mid a_{i}=0\right.$ or $\left.1, i=1,2, \cdots, 8\right\}$. For two elements $A=\left(a_{1}, \cdots, a_{8}\right)$ and $B=\left(b_{1}, \cdots, b_{8}\right)$ in $S$, let
$$
d(A, B)=\sum_{i=1}^{8}\left|a_{i}-b_{i}\right|,
$$

and call it the distance between $A$ and $B$. How many elements can be selected from $S$ at most, such that the distance between any two of them is $\geqslant 5$?

Answer: 16


2. Let $ABCD$ be a convex quadrilateral such that

$$
\measuredangle ABC=90^{\circ}, \measuredangle DAB=60^{\circ} \text{ and } \measuredangle BCD=120^{\circ}.
$$

Let the diagonals $AC$ and $BD$ intersect at point $M$, such that $\overline{BM}=1$ and $\overline{MD}=2$. Calculate the area of quadrilateral $ABCD$.

Answer: \dfrac{9}{2}


3. In a square $ABCD$ with side length 2, $E$ is the midpoint of $AB$. Now, fold the figure along segments $EC$ and $ED$ so that segments $EA$ and $EB$ coincide (point $B$ overlaps with $A$), forming a tetrahedron $CDEA$. Then the volume of this tetrahedron is $\qquad$

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


In a $k$-player tournament for $k > 1$, every player plays every other player exactly once. Find with proof the smallest value of $k$ such that it is possible that for any two players, there was a third player who beat both of them.

Answer: 7


14. An equilateral triangle is divided into three congruent trapezoids, as shown in the figure. If the perimeter of each trapezoid is $10+5 \sqrt{3}$, what is the side length of the triangle?

Answer: 6 + 3\sqrt{3}


The base of the right prism $A B C A_{1} B_{1} C_{1}$ is an isosceles triangle $A B C$, where $A B=B C=5$, $\angle A B C=2 \arcsin 3 / 5$. A plane perpendicular to the line $A_{1} C$ intersects the edges $A C$ and $A_{1} C_{1}$ at points $D$ and $E$ respectively, such that $A D=1 / 3 A C, E C_{1}=1 / 3 A_{1} C_{1}$. Find the area of the section of the prism by this plane.

Answer: \dfrac{40}{3}


4. On the board, the numbers $2,3,4, \ldots ., n+1$ are written. Then the products of every two of them are written, followed by the products of every three of them, and the process continues until the product of all the numbers 2,3,4,...., $n+1$ is written. Calculate the sum of the reciprocals of all the numbers written on the board.

Answer: \dfrac{n}{2}


17. Let \(\xi_{1}, \ldots, \xi_{n}\) be independent and identically distributed random variables with a continuous distribution function \(F=F(x)\). Given that \(\mathrm{P}\left(\xi_{i}=\xi_{j}\right)=0\) for \(i \neq j\), show the following:

(a) Prove that the distribution function \(F_{r: n}=F_{r: n}(x)\) of \(\xi_{r: n}\) (the \(r\)-th order statistic) can be expressed via the incomplete beta function as follows:

\[
F_{r: n}(x)=\sum_{i=r}^{n} \binom{n}{i} [F(x)]^{i}[1-F(x)]^{n-i} = \frac{n!}{(r-1)!(n-r)!} \int_{0}^{F(x)} z^{r-1}(1-z)^{n-r} dz
\]

Assuming now that the distribution function \(F\) has a probability density function \(f=f(x)\):

(b) Show that the probability density function of \(\xi_{r: n}\) is given by the formula:

\[
\frac{n!}{(r-1)!(n-r)!} F(x)^{r-1}[1-F(x)]^{n-r} f(x)
\]

(c) The joint density \(f_{r, s: n}(x, y)\) of \(\xi_{r: n}\) and \(\xi_{s: n}\) (\(r<s\)) is:

\[
\frac{n!}{(r-1)!(s-r-1)!(n-s)!} F(x)^{r-1}[F(y)-F(x)]^{s-r-1} F(y)^{n-s} f(x) f(y)
\]

for \(x < y\) and is zero otherwise.

(d) The joint density \(f_{(n)}\left(x_{1}, \ldots, x_{n}\right)\) of \(\xi_{1: n}, \ldots, \xi_{n: n}\) is given by:

\[
f_{(n)}\left(x_{1}, \ldots, x_{n}\right)=
\begin{cases}
n!f\left(x_{1}\right) \ldots f\left(x_{n}\right), & \text{if } x_{1}<\ldots<x_{n} \\
0 & \text{otherwise}
\end{cases}
\]

Answer: F_{r: n}(x) = \frac{n!}{(r-1)!(n-r)!} \int_{0}^{F(x)} z^{r-1}(1-z)^{n-r} dz


We form a number from three consecutive tickets. If we write these digits in reverse order and subtract the original number from the resulting number, we always get a difference of 198.

Answer: 198


$\begin{aligned} & {\left[\begin{array}{l}\text { Linear dependence of vectors } \\ \text { [ angles between lines and planes }\end{array}\right]}\end{aligned}$

The side of the base of a regular quadrilateral pyramid $\operatorname{ABCDP}$ ( $P$ - apex) is $4 \sqrt{2}$, and the angle between adjacent lateral faces is $120^{\circ}$. Find the area of the section of the pyramid by a plane passing through the diagonal $B D$ of the base parallel to the lateral edge $C P$.

Answer: 4\sqrt{6}


6. There are 900 three-digit numbers $(100,101, \cdots, 999)$. If these three-digit numbers are printed on cards, with one number per card, some cards, when flipped, still show a three-digit number, such as 198, which when flipped shows 861. Therefore, at most, $\qquad$ fewer cards can be printed.

Answer: 34


Mice have built an underground house consisting of chambers and tunnels:

- each tunnel leads from one chamber to another (i.e., none are dead ends),
- from each chamber, exactly three tunnels lead to three different chambers,
- from each chamber, one can reach any other chamber via tunnels,
- the house has exactly one tunnel such that its collapse would divide the house into two separate parts.

What is the minimum number of chambers the mouse house could have? Sketch how the chambers could be connected.

(K. Jasenčáková)

Hint. Start with the critical tunnel.

Answer: 10


4. A cinema was visited by 50 viewers, the total age of whom is 1555 years, and among them, there are no viewers of the same age. For what largest $k$ can we guarantee to select 16 viewers whose total age is not less than $k$ years?

Answer: 776


Several identical boxes weigh a total of 10 tons, with each box weighing no more than 1 ton. What is the minimum number of 3-ton trucks needed to transport all this cargo in one trip?

Answer: 5


11. (20 points) Through the point \( P\left(1, \frac{1}{2}\right) \) inside the ellipse \(\frac{x^{2}}{6}+\frac{y^{2}}{3}=1\), draw a line that does not pass through the origin, intersecting the ellipse at points \( A \) and \( B \). Find the maximum value of \( S_{\triangle O A B} \).

Answer: \dfrac{3\sqrt{6}}{4}


A sphere tangent to the edges of a $4 \mathrm{~cm}$ cube is inscribed. One vertex of the cube is $A$. What is the volume of the region consisting of points that are inside the cube, outside the sphere, and closest to $A$ among the vertices of the cube?

Answer: 8 - \dfrac{4}{3}\pi


8. (5 points) Cut out 4 equally sized isosceles right triangles from a rectangular sheet of paper that is 12 cm long and 8 cm wide. The minimum area of the remaining part is $\qquad$ square centimeters.

Answer: 24


Exercise 3. We have 100 light bulbs, numbered from 1 to 100, each of which can be either on or off. These light bulbs are connected to three switches A, B, and C.

Pressing A changes the state of all the light bulbs: those that were on turn off, and those that were off turn on.

Pressing B changes the state only of the light bulbs with odd numbers.

Pressing C changes the state only of the light bulbs with numbers of the form 3n+1.

At the beginning of the evening, all the light bulbs were on. However, during the party and carried away by his enthusiasm, Igor pressed the switches a total of 1000 times, randomly. It turns out that the light bulbs numbered 95 and 96 are off.

How many light bulbs are still on?

Answer: 34


1. As shown in Figure $3 . \triangle P A B 、 \triangle P C D$ are both equilateral triangles. $M 、 N$ are the midpoints of $P B 、 C D$ respectively, and $P Q \perp A C$ at $Q$. Then $\angle M Q N=$

Answer: 30^\circ


Problem 6.7. Petya thinks of a four-digit number of the form $\overline{20 * *}$.

Vasya sequentially checks if the number Petya thought of is divisible by 1, 3, 5, 7, 9, 11, and if it is, Vasya pays Petya 1, 3, 5, 7, 9, or 11 rubles respectively. For example, for the number 2000, Vasya would have paid Petya $1+5=6$ rubles.

What is the maximum number of rubles Petya can receive?

Answer: 31


Portraits of famous scientists hang on a wall. The scientists lived between 1600 and 2008, and none of them lived longer than 80 years. Vasya multiplied the years of birth of these scientists, and Petya multiplied the years of their death. Petya's result is exactly $ 5\over 4$ times greater than  Vasya's result. What minimum number of portraits can be on the wall?

[i]Author: V. Frank[/i]

Answer: 5


## Task 6 - 020816

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0580.jpg?height=306&width=377&top_left_y=2234&top_left_x=314)

Given a rectangle $A B C D$, whose sides are all divided in the ratio $1: 2$ as shown in the figure. We call the division points $P, Q, R, S$ and connect them in sequence.

a) Perform this construction for the rectangle with sides $A B = 10 \mathrm{~cm}$ and $B C = 7 \mathrm{~cm}$!

b) What kind of quadrilateral is the quadrilateral $P Q R S$? (Proof!)

c) How does the area of the quadrilateral $P Q R S$ compare to that of the rectangle $A B C D$? Does this result hold for other similarly divided rectangles? (Justification!)

Answer: \text{Parallelogram}


Example 22. Using the lower base of a frustum as the base, and the center of the upper base as the vertex, construct a cone. If the lateral surface of this cone divides the volume of the frustum into two equal parts, find the ratio of the radii of the upper and lower bases of the frustum.

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


9. If the equation $\tan \alpha+\tan \beta+\tan \gamma=\tan \alpha \tan \beta \tan \gamma$ holds, what algebraic relationship should $\alpha, \beta, \gamma$ satisfy?

Answer: \alpha + \beta + \gamma = k\pi


2. For a finite set of points $M$ in the plane, it has the property: for any two points $A, B$ in $M$, there must exist a third point $C \in M$ such that $\triangle A B C$ is an equilateral triangle. Find the maximum value of $|M|$.

Answer: 3


3. In a $9 \times 9$ grid, there are 81 small squares. In each small square, write a number. If in every row and every column, there are at most three different numbers, it can be guaranteed that there is a number in the grid that appears at least $n$ times in some row and at least $n$ times in some column. What is the maximum value of $n$? Prove your conclusion.

In a $9 \times 9$ grid, there are 81 small squares. In each small square, write a number. If in every row and every column, there are at most three different numbers, it can be guaranteed that there is a number in the grid that appears at least $n$ times in some row and at least $n$ times in some column. What is the maximum value of $n$? Prove your conclusion.

Answer: 3


Given a right triangle $A B C$. From the vertex $B$ of the right angle, a median $B D$ is drawn. Let $K$ be the point of tangency of side $A D$ of triangle $A B D$ with the inscribed circle of this triangle. Find the acute angles of triangle $A B C$, if $K$ bisects $A D$.

Answer: 30^\circ


10 students are arranged in a row. Every minute, a new student is inserted in the row (which can occur in the front and in the back as well, hence $11$ possible places) with a uniform $\tfrac{1}{11}$ probability of each location. Then, either the frontmost or the backmost student is removed from the row (each with a $\tfrac{1}{2}$ probability).

Suppose you are the eighth in the line from the front. The probability that you exit the row from the front rather than the back is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m+n$.

[i]Proposed by Lewis Chen[/i]

Answer: 828


11. Given an integer $n(n \geqslant 2)$. For a $2 n$-tuple ordered array $T=\left(a_{1}, b_{1}, a_{2}, b_{2}, \cdots, a_{n}, b_{n}\right)$, if each component of $T$ is 0 or 1, and for any $p, q(1 \leqslant p<q \leqslant n)$, we have $\left(a_{p}, b_{p}, b_{q}\right) \neq (1,0,1),\left(a_{q}, b_{q}, a_{p}\right) \neq(1,0,0)$, then $T$ is called a "fun array". Find the number of fun arrays.

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


On a circle of length 2013, there are 2013 points marked, dividing it into equal arcs. At each marked point, there is a token. The distance between two points is defined as the length of the shorter arc between these points. For what largest $n$ can the tokens be rearranged such that each marked point still has a token, and the distance between any two tokens, originally separated by no more than $n$, increases?

Answer: 670


11. (20 points) Given non-negative real numbers $x, y, z$ satisfying $x+y+z=1$. Find the maximum and minimum values of $x^{3}+2 y^{2}+\frac{10}{3} z$.

Answer: \dfrac{10}{3}


Austin and Joshua play a game. Austin chooses a random number equal to 1, 2, 3, 4, or 5 . Joshua then chooses randomly from the remaining four numbers. Joshua's first round score is equal to the product of his number and Austin's number. Austin then chooses randomly from the remaining three numbers, and his first round score is the product of his second number and Joshua's first number. The process is repeated until each of the five numbers has been chosen. The sum of each player's two scores is their final score and the player with the highest final score wins. If Austin chooses 2 to start and Joshua then chooses 3 (making Joshua's first round score 6), what is the probability that Austin will win?

Answer: \dfrac{2}{3}


Problem 8. In 28 examination tickets, each includes two theoretical questions and one problem. The student has prepared 50 theoretical questions and can solve the problems in 22 tickets. What is the probability that, by randomly picking one ticket, the student will be able to answer all the questions in the ticket?

Answer: \dfrac{5}{8}


Let $ \omega$ be a circle with center O and radius 10, and let H be a point such that $OH = 6$. A point P is called snug if, for all triangles ABC with circumcircle ω and orthocenter $H$,
we have that P lies on $\triangle$ABC or in the interior of $\triangle$ABC. Find the area of the region consisting of all snug points.

Answer: 20\pi


Positive real numbers $a$ and $b$ verify $a^5+b^5=a^3+b^3$. Find the greatest possible value of the expression $E=a^2-ab+b^2$.

Answer: 1


In the parallelogram \(ABCD\), \(AC = AB \sqrt{2}\). Prove that the angles between the diagonals are equal to the angles between the sides.

Answer: \text{The angles between the diagonals are equal to the angles between the sides.}


5. In an isosceles trapezoid $M N K L$ with bases $M L, N K$, the diagonals are perpendicular to the sides $M N$, $K L$ and intersect at an angle of $15^{\circ}$. Find the height of the trapezoid if the length $N Q=5$, where $Q-$ is the midpoint of the larger base.

Answer: \dfrac{5(\sqrt{6} - \sqrt{2})}{4}


Example 16 (1999 Canadian Mathematical Olympiad) Find all real solutions to the equation $4 x^{2}-40[x]+51=0$.

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

(Note: The provided text is already in English, so no translation is needed. If you intended to have the problem solved, please let me know!)

Answer: \dfrac{\sqrt{29}}{2}


[ Rectangles and squares. Properties and characteristics ]  [ Equilateral (equiangular) triangle ]

On the side $A B$ of the square $A B C D$, an equilateral triangle $A B M$ is constructed. Find the angle $D M C$.

#

Answer: 30^\circ


333.

a) On the circle circumscribing a regular $n$-gon $A_{1} A_{2} \ldots A_{n}$, a point $M$ is chosen. Prove that the sum of the squares of the distances from this point to all vertices of the $n$-gon does not depend on the position of the point on the circle and is equal to $2nR^2$, where $R$ is the radius of the circle.

b) Prove that the sum of the squares of the distances from an arbitrary point $M$ taken in the plane of the regular $n$-gon $A_{1} A_{2} \ldots A_{n}$ to all vertices of the $n$-gon depends only on the distance $l$ of point $M$ from the center $O$ of the polygon and is equal to $n(R^2 + l^2)$, where $R$ is the radius of the circumscribed circle around the $n$-gon.

c) Prove that the statement of the previous problem remains valid even if the point $M$ does not lie in the plane of the $n$-gon $A_{1} A_{2} \ldots A_{n}$.

Answer: 2nR^2


Three. (50 points) Given a finite set of planar vectors $M$, for any three elements chosen from $M$, there always exist two elements $\boldsymbol{a}, \boldsymbol{b}$ such that $\boldsymbol{a}+\boldsymbol{b} \in M$. Try to find the maximum number of elements in $M$.

Answer: 7


Four vehicles start from $A$ at the same time, and arrive at $B$ in succession at equal time intervals. The speeds of the fastest and the slowest vehicle are $v_{1}$ and $v_{4}$, respectively. What are the speeds of the other two vehicles?

Answer: \dfrac{3v_1v_4}{v_1 + 2v_4}


In $\square A B C D$, $C E \perp A B, C F \perp A D$, with feet of the perpendiculars at $E$ and $F$. Let $E F$ intersect the diagonal $B D$ at point $P$. If $A B: A D=2: 3$, find $P F: P E$.

Answer: \dfrac{4}{9}


## Task 1 - 210921

In the division problem $a: b=c$, $a, b, c$ are to be replaced by natural numbers so that a correctly calculated division problem results, where only the digits 1, 2, 3, 4, 5, and each exactly once, are to be used.

Determine all triples $(a ; b ; c)$ of natural numbers that meet these requirements!

Answer: (52; 4; 13)


10. Among $1,2,3,4, \cdots, 1000$, the numbers that can be written in the form $a^{2}-b^{2}+1(a, b \in \mathbf{N})$ and are not divisible by 3 are $\qquad$ in number.

Answer: 501


4. In Rt $\triangle A B C$, $\angle A C B=90^{\circ}$, on the hypotenuse $A B$ respectively intercept $A D=$
$$
A C, B E=B C, D E=6 \text{, }
$$
$O$ is the circumcenter of $\triangle C D E$, as shown in Figure 2. Then the sum of the distances from $O$ to the three sides of $\triangle A B C$ is . $\qquad$

Answer: 9


As shown in Figure 1, in $\triangle A B C$, $A C-A B=\frac{\sqrt{2}}{2} B C$. Point $D$ is a point inside $\triangle A B C$ such that $A D=A B$. A perpendicular line is drawn from point $D$ to $B C$, with the foot of the perpendicular being $E$. If $3 B E=C E$, find the measure of $\angle A D E$.

Answer: 135^\circ


Problem 3.6. Petya can draw only 4 things: the sun, a ball, a tomato, and a banana. But he does it extremely realistically! Today he drew several things, among which there were exactly 15 yellow, 18 round, and 13 edible. What is the maximum number of balls he could have drawn?

Petya believes that all tomatoes are round and red, all balls are round and can be of any color, and all bananas are yellow and not round.

Answer: 18


7. The equation of the hyperbola $C$ is $x^{2}-\frac{y^{2}}{3}=1$, with the left and right foci being $F_{1}$ and $F_{2}$, respectively. A line is drawn through point $F_{2}$ intersecting the right branch of the hyperbola $C$ at points $P$ and $Q$, such that $\angle F_{1} P Q=90^{\circ}$. Then, the radius of the incircle of $\triangle F_{1} P Q$ is

Answer: \sqrt{7} -1


Three regular polygons are inscribed in a circle, the number of sides of each subsequent one being twice that of the previous one. The areas of the first two are $S_{1}$ and $S_{2}$. Find the area of the third.

Answer: \sqrt{\dfrac{2 S_2^3}{S_1 + S_2}}


## Task 30/74

All pairs of three-digit natural numbers $(x ; y)$ are sought that have the following properties:

1. The numbers of a pair differ in the last two digits exactly by the order of the digits.
2. The difference $x-y$ is 18.
3. Each of the two numbers of a pair is the product of exactly two two-digit prime numbers.

Answer: (731, 713)


Let $n$ points with integer coordinates be given in the $xy$-plane. What is the minimum value of $n$ which will ensure that three of the points are the vertices of a triangel with integer (possibly, 0) area?

Answer: 5


There are three equal-radius, tangent circles. What is the area of the lobe between the circles if the radius of each circle is $r$?

Answer: \left( \sqrt{3} - \frac{\pi}{2} \right) r^2


134. Solve the equation $y^{\prime \prime}-3 y^{\prime}+2 y=0$.

Answer: y = C_1 e^x + C_2 e^{2x}


6. Given the ellipse $C: \frac{x^{2}}{9}+\frac{y^{2}}{8}=1$ with left and right foci $F_{1}$ and $F_{2}$, and left and right vertices $A$ and $B$, the line $l: x=m y+1$ passing through the right focus $F_{2}$ intersects the ellipse $C$ at points $M\left(x_{1}, y_{1}\right)$ and $N\left(x_{2}, y_{2}\right)\left(y_{1}>0, y_{2}<0\right)$. If $M A$ $\perp N F_{1}$, then the real number $m=$ $\qquad$.

Answer: \dfrac{\sqrt{3}}{12}


10.409 A triangle with sides 13, 14, and 15 is divided into three equal areas by lines perpendicular to the longest side. Find the distances from these lines to the nearest vertices of the triangle located on the longest side.

Answer: \sqrt{42}


On a planet, there are $2^{N}$ countries $(N \geqslant 5)$. Each country has a flag composed of a line of $N$ squares of side 1, each being yellow or blue. The flags are all distinct from each other. A set of $N$ flags is diverse if they can be placed in a certain order to form an $N \times N$ square such that the $N$ squares on the main diagonal are of the same color. Determine the smallest integer $M$ such that any set of $M$ flags contains a diverse set of $N$ flags.

## Directed Graphs

Answer: 2^{N-2} + 1


Problem 9.6. Given an obtuse triangle $ABC$ with an obtuse angle $C$. On its sides $AB$ and $BC$, points $P$ and $Q$ are marked such that $\angle ACP = CPQ = 90^\circ$. Find the length of the segment $PQ$, if it is known that $AC = 25, CP = 20, \angle APC = \angle A + \angle B$.

![](https://cdn.mathpix.com/cropped/2024_05_06_a324d57f9e21fe03fd09g-3.jpg?height=397&width=679&top_left_y=730&top_left_x=371)

Answer: 16


Let $ t$ be a positive number. Draw two tangent lines from the point $ (t, \minus{} 1)$ to the parabpla $ y \equal{} x^2$. Denote $ S(t)$ the area bounded by the tangents line and the parabola. Find the minimum value of $ \frac {S(t)}{\sqrt {t}}$.

Answer: \dfrac{4\sqrt{6}}{5^{5/4}}


Six. (16 points) As shown in Figure 3, there is a fixed point $P$ inside $\angle M A N$. It is known that $\tan \angle M A N=3, P$ is at a distance $P D=12$ from the line $A N$, and $A D=30$. A line through $P$ intersects $A N$ and $A M$ at points $B$ and $C$ respectively. Find the minimum area of $\triangle A B C$.

---

The translation is provided as requested, maintaining the original formatting and structure.

Answer: 624


9.2. On a line, there are blue and red points, with no fewer than 5 red points. It is known that on any segment with endpoints at red points, containing a red point inside, there are at least 4 blue points. And on any segment with endpoints at blue points, containing 3 blue points inside, there are at least 2 red points. What is the maximum number of blue points that can be on a segment with endpoints at red points, not containing other red points?

Answer: 4


On each face of a cube, we marked the points that divide the midlines in the ratio $1: 3$. How does the volume of the convex body determined by these points compare to the volume of the cube?

Answer: \dfrac{1}{2}


3. Given is $\triangle A B C$ where $\measuredangle C$ is obtuse. Points $D \in A C, E \in A B$, and $F \in B C$ are marked on its sides such that quadrilateral $C D E F$ is a rhombus. If $A E=140, B E=84$, and $r=15 \sqrt{3}$, where $r$ is the radius of the circle inscribed in the rhombus $C D E F$, calculate the area of $\triangle A B C$.

Answer: 3840\sqrt{3}


# Task 5. (12 points)

Asha is learning to write and can write three letters A, S, and Y. Her mother suggested she write seven letters in a row. In the resulting "word," three consecutive letters formed the name "ASY". How many such different seven-letter "words" exist?

Answer: 396


10. Every natural number has its digit sum, for example:
45 has a digit sum of $4+5=9$, and 102 has a digit sum of $1+0+2=3$.
First, calculate the digit sum of each natural number from 1 to 2020, then add these 2020 digit sums together, the result is . $\qquad$

Answer: 28144


Example 3 Given an integer $n \geqslant 3$. Find the largest real number $M$ such that for any sequence of positive real numbers $x_{1}, x_{2}, \cdots, x_{n}$, there exists a permutation $y_{1}, y_{2}, \cdots, y_{n}$, satisfying
$$
\sum_{i=1}^{n} \frac{y_{i}^{2}}{y_{i+1}^{2}-y_{i+1} y_{i+2}+y_{i+2}^{2}} \geqslant M,
$$

where, $y_{n+1}=y_{1}, y_{n+2}=y_{2}$.

Answer: n-1


7.3 On the Island of Truth and Lies, there are knights who always tell the truth, and liars who always lie. One day, 17 residents of the island lined up by height (from tallest to shortest, the tallest being the first) for a game. Each had to say one of the phrases: "There is a liar below me" or "There is a knight above me." In the end, those standing from the third to the sixth position said the first phrase, while the rest said the second. How many knights were among these 17 people, if all residents have different heights?

Answer: 14


16. Fill in 1 and 6 in the three equations on the right, using each number exactly once, so that $\mathrm{A}+\mathrm{B}$ is a multiple of 2, $\mathrm{C}+\mathrm{D}$ is a multiple of 3, and $\mathrm{E}+\mathrm{F}$ is a multiple of 5. Then the smaller number in $\mathrm{C} 、 \mathrm{D}$ is $\qquad$ (fill in the specific number)

Answer: 1


Let $S$ be the set of all permutations of $\{1, 2, 3, 4, 5\}$. For $s = (a_1, a_2,a_3,a_4,a_5) \in S$, define $\text{nimo}(s)$ to be the sum of all indices $i \in \{1, 2, 3, 4\}$ for which $a_i > a_{i+1}$. For instance, if $s=(2,3,1,5,4)$, then $\text{nimo}(s)=2+4=6$. Compute \[\sum_{s\in S}2^{\text{nimo}(s)}.\]

[i]Proposed by Mehtaab Sawhney[/i]

Answer: 9765


4th Brazil 1982 Problem 3 S is a (k+1) x (k+1) array of lattice points. How many squares have their vertices in S?

Answer: \dfrac{k(k + 1)^2(k + 2)}{12}


1. Evan's analog clock displays the time $12: 13$; the number of seconds is not shown. After 10 seconds elapse, it is still $12: 13$. What is the expected number of seconds until $12: 14$ ?

Answer: 25


3. Find the minimum value of the real number $k$ such that for any real numbers $x, y, z$ not all positive, the inequality
$$
\begin{array}{l}
k\left(x^{2}-x+1\right)\left(y^{2}-y+1\right)\left(z^{2}-z+1\right) \\
\geqslant(x y z)^{2}-x y z+1
\end{array}
$$

always holds.

Answer: \dfrac{16}{9}


Three. (20 points) Given the function $f(x)$ for any real numbers $x, y$ has
$$
f(x+y)=f(x)+f(y)+2(x+y)+1,
$$

and $f(1)=1$.
If $x \in \mathbf{N}_{+}$ and $x \geqslant 2$, the inequality
$$
f(x) \geqslant(a+7) x-(a+10)
$$

always holds, try to find the range of the real number $a$.

Answer: (-\infty, 2]


Task B-3.5. An isosceles triangle with the length of the legs $b=10 \mathrm{~cm}$, and the measure of the angle between the legs $\alpha=30^{\circ}$, has a circumscribed circle. Let $t$ be the tangent to this circle that is parallel to the height to the base. Calculate the volume of the solid formed by rotating the given triangle around the tangent $t$.

Answer: 250\pi\left(\sqrt{6} - \sqrt{2}\right)


One. (20 points) Let $k, t$ be constants, and the equation about $x$
$$
k x^{2}+(3-3 k) x+2 k-6=0
$$

has only integer roots. The quadratic equation about $y$
$$
(k+3) y^{2}-15 y+t=0
$$

has two positive integer roots $y_{1}, y_{2}$.
(1) Find the value of $k$;
(2) For what value of $t$ is $y_{1}^{2}+y_{2}^{2}$ minimized?

Answer: \dfrac{3}{4}


In a singing competition, the participants were the Rooster, the Crow, and the Cuckoo. Each jury member voted for one of the three performers. Woodpecker counted that there were 59 judges, with a total of 15 votes for the Rooster and the Crow, 18 votes for the Crow and the Cuckoo, and 20 votes for the Cuckoo and the Rooster. Woodpecker counts poorly, but each of the four numbers he called differs from the correct value by no more than 13. How many judges voted for the Crow? [6 points] (I.V. Raskina)

Answer: 13


2. In triangle $ABC$, the altitudes $AA_{0}$, $BB_{0}$, and $CC_{0}$ are dropped. Let $H$ be the orthocenter. If $\overline{A H}: \overline{H A_{0}}=1: 1$ and $\overline{B H}: \overline{H B_{0}}=2: 1$, then what is the ratio $\overline{C H}: \overline{H C_{0}}$?

Answer: 5


Let $ABCDEF$ be an equiangular hexagon such that $AB=6, BC=8, CD=10$ , and $DE=12$ . Denote $d$ the diameter of the largest circle that fits inside the hexagon. Find $d^2$ .
Please give the answer directly without any intermediate steps.

Answer: 147


4. In a coffee shop, 55 Indians and Turks met, each drinking tea or coffee. All Indians tell the truth when drinking tea and lie when drinking coffee, while all Turks do the opposite. When asked "Are you drinking coffee?" 44 people answered "yes," when asked "Are you a Turk?" 33 people answered "yes," and 22 people agreed with the statement "It is raining outside." How many Indians in the coffee shop are drinking tea?

Answer: 0


2. In the USA, the date is typically written as the month number, followed by the day number, and then the year. In Europe, however, the day comes first, followed by the month and the year. How many days in a year cannot be read unambiguously without knowing which format it is written in?

Answer: 132


Let $N$ be the number of ways of distributing $52$ identical balls into $4$ distinguishable boxes such that no box is empty and the difference between the number of balls in any two of the boxes is not a multiple of $6$. If $N = 100a + b$, where $a, b$ are positive integers less than $100$, find $a + b$.

Answer: 81


Let $a_1, a_2, \ldots$ be an infinite sequence of integers such that $a_i$ divides $a_{i+1}$ for all $i \geq 1$. Let $b_i$ be the remainder when $a_i$ is divided by $210$. What is the maximal number of distinct terms in the sequence $b_1, b_2, \ldots$?

Answer: 16


Triangle $ABC$ has side lengths $AB=231$, $BC=160$, and $AC=281$. Point $D$ is constructed on the opposite side of line $AC$ as point $B$ such that $AD=178$ and $CD=153$. Compute the distance from $B$ to the midpoint of segment $AD$.

Answer: 208


3. In the coordinate plane, there are two regions $M$ and $N$. $M$ is defined by
\[
\left\{\begin{array}{l}
y \geqslant 0, \\
y \leqslant x, \\
y \leqslant 2-x,
\end{array} \quad N\right.
\]
is a region that varies with $t$, determined by the inequality $t \leqslant x \leqslant t+1$, where the range of $t$ is $0 \leqslant t \leqslant 1$. Then the area of the common part of $M$ and $N$ is a function $f(x)=$ $\qquad$

Answer: -t^2 + t + \frac{1}{2}


5. The diagonals of quadrilateral $A B C D$ intersect at point $O$. It is known that $A B=B C=$ $=C D, A O=8$ and $\angle B O C=120^{\circ}$. What is $D O ?$

Answer: 8


5. At the ceremony marking the City Day, a tribune was set up where the seats were properly arranged in rows and columns (the number of seats in all rows is the same, the number of seats in all columns is the same). Each spectator has either a cap or a hat on their head. The mayor noticed that in each row there are exactly 8 spectators with caps, and in each column there are exactly 9 spectators with hats. There are a total of 12 empty seats on the tribune.

a) What is the minimum number of seats that can be on the tribune?

b) Show an example of a seating arrangement that meets the conditions of the problem.

Answer: 315


6. From a deck of cards, 4 aces and 4 kings were removed. These cards were shuffled and laid out in a row. What is the probability that all 4 kings will be located next to each other?

Answer: \dfrac{1}{14}


118. A circle is inscribed in a square, and then a new square is inscribed in the circle. Find the ratio of the areas of these squares.

Answer: 2


9.3. Inside a semicircle of radius 12, there are a circle of radius 6 and a small semicircle, each touching the others pairwise, as shown in the figure. Find the radius of the small semicircle.

Answer: 4


8,9 |

Two circles touch each other externally at point $A$. Find the radii of the circles if the chords connecting point $A$ with the points of tangency from one of the common external tangents are 6 and 8.

Answer: \dfrac{15}{4}


\( a \) is a fixed odd positive integer. Find the largest positive integer \( n \) for which there are no positive integers \( x, y, z \) such that \( ax + (a + 1)y + (a + 2)z = n \).

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


## Task 2 - 110512

Rolf claims that an addition problem with a sum of 1000 can be formed, where all addends are natural numbers, and in their decimal representation, only the digit 8 appears, and exactly 8 times in total.

Determine whether Rolf's claim is correct!

If it is, provide all such addition problems and order the addends from largest to smallest!

Answer: 888 + 88 + 8 + 8 + 8 = 1000


47. As shown in the figure, point $D$ is on the right-angle side $BC$ of Rt $\triangle ABC$, and $BD=2, DC=3$. If $AB=m, AD=n$, then $m^{2}-n^{2}=$ $\qquad$

Answer: 16


How many real numbers $x^{}_{}$ satisfy the equation $\frac{1}{5}\log_2 x = \sin (5\pi x)$ ?
Please give the answer directly without any intermediate steps.

Answer: 159


The sum of all eight numbers \( a_{1}, a_{2}, \ldots, a_{8} \) is \(\frac{4}{3}\), and the sum of any seven of them is positive. Find all possible values of the smallest of these numbers.

Answer: (-8, \frac{1}{6}]


2. A truck traveling at $60 \mathrm{~km} / \mathrm{h}$ started from city $A$ to city $B$. After some time, a car traveling at $90 \mathrm{~km} / \mathrm{h}$ also started from city $A$ to city $B$. It was planned for the car to catch up with the truck in city $B$. However, after covering $\frac{2}{3}$ of the distance, the truck had to reduce its speed to $30 \mathrm{~km} / \mathrm{h}$ (due to a malfunction). As a result, the car caught up with the truck $50 \mathrm{~km}$ before city $B$. Determine the length of the road between cities $A$ and $B$.

Answer: 200


Blinkov A. A:

The teams held a football tournament in a round-robin format (each team played one match against every other team, with 3 points for a win, 1 point for a draw, and 0 points for a loss). It turned out that the sole winner scored less than $50 \%$ of the maximum possible points for one participant. What is the minimum number of teams that could have participated in the tournament?

Answer: 6


5. In a hat, there are 11 slips of paper with numbers $0,1,2,3,4,5,6,7,8,9,10$ (one with each number). Each of the eleven students takes one slip. Then they announce in alphabetical order the sentence "so far, at least $k$ students have told a lie," where $k$ is the number on the drawn slip. How many of these 11 statements can be true at most?

Answer: 6


In a corner stand, there are three shelves measuring $30 \mathrm{~cm} \times 40 \mathrm{~cm}$, and the distance between adjacent shelves is equal. Where the two walls and the middle shelf meet, three spiders have made their nests. Once, one of them climbed diagonally up one wall to the corner of the top shelf, another crawled diagonally down the other wall to the corner of the bottom shelf. The third spider remained in place and observed that its companions appeared at an angle of $120^{\circ}$. What is the distance between the shelves? (The adjacent shelves are the same distance apart.)

Answer: 35


4. Students $A$ and $B$ each have 3 cards. They play a game by tossing a fair coin, where if the coin lands heads up, $A$ wins a card from $B$, and if it lands tails up, $B$ wins a card from $A$. The game ends when one person has won all the cards. What is the probability that the game ends exactly after 5 coin tosses? $\qquad$

Answer: \dfrac{3}{16}


5. Given the equation of the ellipse $\Gamma$ as $\frac{x^{2}}{9}+\frac{y^{2}}{5}=1$, a line passing through the left focus $F(-2,0)$ with a slope of $k_{1}\left(k_{1} \neq 0\right)$ intersects the ellipse at points $A$ and $B$. Let $R(1,0)$, and extend $A R$ and $B R$ to intersect the ellipse at points $C$ and $D$, respectively. The slope of line $C D$ is $k_{2}$. Then $\frac{k_{1}}{k_{2}}=$ $\qquad$ .

Answer: \dfrac{4}{7}


Let $p$ and $q$ be two given positive integers. A set of $p+q$ real numbers $a_1<a_2<\cdots <a_{p+q}$ is said to be balanced iff $a_1,\ldots,a_p$ were an arithmetic progression with common difference $q$ and $a_p,\ldots,a_{p+q}$ where an arithmetic progression with common difference $p$. Find the maximum possible number of balanced sets, so that any two of them have nonempty intersection.

Comment: The intended problem also had "$p$ and $q$ are coprime" in the hypothesis. A typo when the problems where written made it appear like that in the exam (as if it were the only typo in the olympiad). Fortunately, the problem can be solved even if we didn't suppose that and it can be further generalized: we may suppose that a balanced set has $m+n$ reals $a_1<\cdots <a_{m+n-1}$ so that $a_1,\ldots,a_m$ is an arithmetic progression with common difference $p$ and $a_m,\ldots,a_{m+n-1}$ is an arithmetic progression with common difference $q$.

Answer: p + q


4. Find all pairs $(p, q)$ of real numbers such that the polynomial $x^{2}+p x+q$ is a divisor of the polynomial $x^{4}+p x^{2}+q$.

---

Note: The translation preserves the original formatting and structure of the text.

Answer: (0, -1)


Given a trapezoid \(ABCD\). Points \(Q\) and \(S\) are selected on the bases \(BC\) and \(AD\) respectively. The segments \(AQ\) and \(BS\) intersect at point \(P\), and the segments \(CS\) and \(DQ\) intersect at point \(R\). Prove that \(S_{PQRS} \leq \frac{1}{4} S_{ABCD}\).

Answer: S_{PQRS} \leq \frac{1}{4} S_{ABCD}


There are $n$ participants at a conference, each knowing at most $k$ languages. For every group of three participants, there are at least two who speak the same language. Find the smallest value of $n$ such that for any distribution of languages satisfying these properties, one can find a language spoken by at least three delegates.

Answer: 2k + 3


Three compressor stations are not located on a single line but are connected by straight roads. The distance from the first station to the third via the second is four times longer than the direct path between them; the distance from the first station to the second via the third is \(a\) km longer than the direct path; the distance from the second station to the third via the first is 85 km. Determine all values of \(a\) for which such an arrangement of the compressor stations would be possible. Calculate the distances between the compressor stations for \(a = 5\).

Answer: 60\,


A kite is a quadrilateral whose diagonals are perpendicular. Let kite \(ABCD\) be such that \(\angle B = \angle D = 90^{\circ}\). Let \(M\) and \(N\) be the points of tangency of the incircle of \(ABCD\) to \(AB\) and \(BC\) respectively. Let \(\omega\) be the circle centered at \(C\) and tangent to \(AB\) and \(AD\). Construct another kite \(AB'C'D'\) that is similar to \(ABCD\) and whose incircle is \(\omega\). Let \(N'\) be the point of tangency of \(B'C'\) to \(\omega\). If \(MN' \| AC\), then what is the ratio of \(AB : BC\)?

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


In the city "N," there are 12 horizontal and 16 vertical streets, forming a rectangular border for the city, while the remaining streets divide the city into blocks shaped as squares with a side length of 100 meters. Each block has an address consisting of two integers \((i; j)\), where \(i=1,2, \ldots, 11\) and \(j=1,2, \ldots, 15\), representing the numbers of the streets that border it from below and from the left. A taxi transports passengers from one block to another, adhering to the following rules:

1. Passengers can board and alight at any point on the boundary of the block at their discretion.
2. It is forbidden to drive inside the block.
3. Transport occurs via the shortest route.
4. A fare of 1 coin is charged for every 100 meters traveled (with the distance rounded up to the nearest multiple of 100 meters in favor of the driver).

How many blocks are in the city? What is the maximum and minimum fare that the driver can charge for a ride from block \((7, 2)\) to block \((2, 1)\) without breaking the rules?

Answer: 8


Leonid has a white checkered rectangle. First, he colored every other column gray, starting with the very left column, and then he colored every other row gray, starting with the very top row. All cells adjacent to the boundary of the rectangle ended up being colored.

How many colored cells could there be in the rectangle if there are 74 white cells left? Indicate all possible options.

An example of coloring a \(5 \times 7\) rectangle is shown below:

![](https://cdn.mathpix.com/cropped/2024_05_06_5b323dccc142e96a9907g-32.jpg?height=226&width=311&top_left_y=448&top_left_x=575)

Answer: 373


Johnny the grad student is typing all the integers from 1 to infinity, in order. His computer's 2 key is broken, so he skips any number with a 2. What is the 2008th number he types?

Answer: 3781


The area of the square $ABCD$ in the figure below is $1$. $M$ is the midpoint of side $AD$. Find the area of the shaded region in the figure.

Answer: \frac{1}{3}


Given the ellipse \( C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where \( a > b > 0 \), let \( F \) be the right focal point of the ellipse \( C \). A line \( l \) is drawn through the intersection of the right directrix \( x = 2a \) and the \( x \)-axis, and intersects the ellipse \( C \) at points \( A \) and \( B \). The arithmetic mean of \( \frac{1}{|AF|} \) and \( \frac{1}{|BF|} \) is \( \frac{1}{24} \). Find the maximum area of triangle \( \triangle ABF \).

Answer: 192\sqrt{3}


On a ruler, three marks are indicated: 0, 2, and 5. How can you use this ruler to measure a segment equal to 6?

Answer: 6


There are two cylinders with a volume ratio of 5:8. Their lateral surfaces can be unfolded into identical rectangles. If the length and width of this rectangle are both increased by 6, the area increases by 114. What is the area of this rectangle?

Answer: 40


The height of a cone is 20, the radius of the base is 25. Find the area of the cross-section, passing through the vertex, if its distance from the center of the base of the cone is 12.

Answer: 500


Let $ k$ be a positive constant number. Denote $ \alpha ,\ \beta \ (0<\beta <\alpha)$ the $ x$ coordinates of the curve $ C: y\equal{}kx^2\ (x\geq 0)$ and two lines $ l: y\equal{}kx\plus{}\frac{1}{k},\ m: y\equal{}\minus{}kx\plus{}\frac{1}{k}$.　Find the minimum area of the part bounded by the curve $ C$ and two lines $ l,\ m$.

Answer: \dfrac{\sqrt{6}}{3}


Answer: 20


10:22 ** Let the side length of the equilateral $\triangle A B C$ be 1, and there are $n$ equal division points on side $B C$. Along the direction from point $B$ to point $C$, they are sequentially $P_{1}, P_{2}, \cdots, P_{n-1}$. Find $S_{n}=\overrightarrow{A B} \cdot \overrightarrow{A P_{1}}+\overrightarrow{A P_{1}} \cdot \overrightarrow{A P_{2}}+\cdots+\overrightarrow{A P_{n-1}} \cdot \overrightarrow{A C}$ (expressed in terms of $n$).

Answer: \dfrac{5n^2 - 2}{6n}


Let \(ABCD\) be a rectangle with \(BC = 2 \cdot AB\). Let \(E\) be the midpoint of \(BC\) and \(P\) an arbitrary point on the side \(AD\). Let \(F\) and \(G\) be the feet of the perpendiculars drawn from \(A\) to \(BP\) and from \(D\) to \(CP\), respectively. It is known that \(\angle BPC = 85^{\circ}\).

a) Verify that the triangles \(BEF\) and \(BEP\) are similar.

b) Determine the value of the sum of the angles \(\angle BEF + \angle CEG\).

Answer: 85^\circ


Let positive integers $p,q$ with $\gcd(p,q)=1$ such as $p+q^2=(n^2+1)p^2+q$. If the parameter $n$ is a positive integer, find all possible couples $(p,q)$.

Answer: (n+1, n^2 + n + 1)


6.4. The frame for three square photographs has the same width everywhere (see figure). The perimeter of one opening is 60 cm, the perimeter of the entire frame is 180 cm. What is the width of the frame?

Answer: 5


B. Let the circumcenter and orthocenter of $\triangle A B C$ be $O$ and $H$ respectively. If $B, C, H, O$ are concyclic, find all possible measures of $\angle B A C$ for all $\triangle A B C$.

The text is translated while preserving the original formatting and line breaks.

Answer: 60^\circ


6. (36th Austrian Mathematical Olympiad) Let $f$ be a function defined on $\{0,1, \cdots, 2005\}$, taking values in the set of non-negative integers. For any variable $x$ in the domain, we have $f(2 x+1)=f(2 x)$, $f(3 x+1)=f(3 x)$, $f(5 x+1)=f(5 x)$. How many different function values can this function take at most?

Answer: 536


9. Use five different colors to color the six vertices of the triangular prism $A B C-D E F$, requiring each point to be colored with one color, and the two endpoints of each edge to be colored with different colors. Then the number of different coloring methods is $\qquad$.

Answer: 1920


7. Mothers' Voting (9-11). In the final of the play competition for March 8, two plays made it to the final. The first play featured $n$ students from 5th grade A, and the second play featured $n$ students from 5th grade B. At the play, there were $2 n$ mothers of all $2 n$ students. The best play is chosen by the mothers' voting. It is known that exactly half of the mothers vote honestly for the best play, while the other half always vote for the play in which their child participates.

a) (2 pts.) Find the probability that the best play will win by a majority of votes.

b) (1 pt.) The same question, if more than two plays made it to the final.

Answer: 1 - \dfrac{1}{\dbinom{2n}{n}}


2. Real numbers $x, y, z$ satisfy the relations:

$$
4 x^{2}-2 x-30 y z=25 y^{2}+5 y+12 x z=9 z^{2}-3 z-20 x y .
$$

Find the maximum of the sum $a+b+c$, where $a=2 x+5 y, b=3 z+5 y, c=3 z-2 x$.

Answer: 2


Let \( X \) and \( Y \) be non-negative random variables such that there exist \(\alpha > 1\) and \(\beta > 0\) for which the inequality

$$
\mathrm{P}(X > \alpha x, Y \leq x) \leq \beta \mathrm{P}(X > x)
$$

holds for all \( x > 0 \).

Let \( f \) be a non-negative increasing function with \( f(0) = 0 \) and

$$
L_{f}(\alpha) := \sup_{x > 0} \frac{f(\alpha x)}{f(x)} < \frac{1}{\beta}
$$

Prove that

$$
\mathrm{E} f(X) \leq \frac{L_{f}(\alpha)}{1 - \beta L_{f}(\alpha)} \cdot \mathrm{E} f(Y)
$$

under the condition that \(\mathrm{E} f(X)\) and \(\mathrm{E} f(Y)\) are finite.

Answer: \mathrm{E} f(X) \leq \frac{L_{f}(\alpha)}{1 - \beta L_{f}(\alpha)} \cdot \mathrm{E} f(Y)


Given a line \( d \) passing through \( A \). The isogonal conjugate of \( d \) is the reflection of \( d \) with respect to the bisector of \(\widehat{A}\). Show that three lines passing through \( A, B, \) and \( C \) respectively are concurrent if and only if their isogonal conjugates (with respect to \( A, B, C \) respectively) are concurrent. If the point of intersection of the isogonals has barycentric coordinates \((\alpha, \beta, \gamma)\), what are the barycentric coordinates of the point of intersection of the three original lines?

Answer: \left( \dfrac{a^2}{\alpha}, \dfrac{b^2}{\beta}, \dfrac{c^2}{\gamma} \right)


1 In an exam, there are 30 multiple-choice questions. Correct answers earn 5 points each, incorrect answers earn 0 points, and unanswered questions earn 1 point each. If person A scores more than 80 points, and tells B the score, B can deduce how many questions A answered correctly. If A's score is slightly lower but still greater than 80, B cannot deduce the number of correct answers. What score did A get in this exam? (2nd American Mathematical Invitational Competition Question)

Answer: 119


$p$ is a prime number such that the period of its decimal reciprocal is 200 . That is,

$$
\frac{1}{p}=0 . X X X X \ldots
$$

for some block of 200 digits $X$, but

$$
\frac{1}{p} \neq 0 . Y Y Y Y \ldots
$$

for all blocks $Y$ with less than 200 digits. Find the 101st digit, counting from the left, of $X$.

Answer: 9


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Bob`, `Peter`, `Alice`
 - Everyone has a unique favorite cigar: `dunhill`, `blue master`, `blends`, `pall mall`, `prince`
 - Each person has an occupation: `lawyer`, `artist`, `engineer`, `doctor`, `teacher`
 - People own unique car models: `ford f150`, `toyota camry`, `honda civic`, `bmw 3 series`, `tesla model 3`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `dragonfruit`, `desert`, `lime`
 - Everyone has something unique for lunch: `stew`, `spaghetti`, `grilled cheese`, `pizza`, `stir fry`

## Clues:
1. There are two houses between the person who owns a BMW 3 Series and the person who owns a Tesla Model 3.
2. The person who is an artist is Bob.
3. The person who owns a Ford F-150 is the person who smokes Blue Master.
4. The person who loves the spaghetti eater is in the second house.
5. The person who is an engineer is the Watermelon smoothie lover.
6. Alice is the person who drinks Lime smoothies.
7. The person who is an artist is the Dunhill smoker.
8. The person who likes Cherry smoothies is not in the fifth house.
9. Peter is not in the fourth house.
10. The Desert smoothie lover is the person who loves stir fry.
11. The person who likes Cherry smoothies is somewhere to the right of the person who smokes many unique blends.
12. The person who is a teacher is the person who loves eating grilled cheese.
13. Arnold is the person who is a lawyer.
14. The Dragonfruit smoothie lover is the person who owns a Honda Civic.
15. Peter is somewhere to the right of the Desert smoothie lover.
16. The Prince smoker is the Watermelon smoothie lover.
17. There is one house between the person who owns a Honda Civic and the person who loves the stew.
18. The person who owns a Tesla Model 3 is the person who likes Cherry smoothies.
19. The person who owns a Ford F-150 is directly left of the person who is a doctor.
20. The person who is an artist is the Desert smoothie lover.
 What is the value of attribute Food for the person whose attribute Smoothie is lime? Please reason step by step, and put your final answer within \boxed{}

Answer: grilled cheese


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`
 - People have unique favorite sports: `basketball`, `soccer`, `tennis`, `baseball`, `swimming`
 - The people keep unique animals: `dog`, `fish`, `horse`, `bird`, `cat`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `samsung galaxy s21`, `iphone 13`, `huawei p50`
 - People have unique hair colors: `black`, `red`, `brown`, `blonde`, `gray`
 - Each person lives in a unique style of house: `modern`, `craftsman`, `victorian`, `ranch`, `colonial`

## Clues:
1. The person who has gray hair is the person who loves tennis.
2. The person in a ranch-style home is in the fourth house.
3. The person living in a colonial-style house is the person who loves swimming.
4. The dog owner is somewhere to the right of the person who has gray hair.
5. The fish enthusiast is the person who has gray hair.
6. The person who has black hair is not in the fifth house.
7. The person who uses a Samsung Galaxy S21 is the bird keeper.
8. There is one house between the fish enthusiast and the person who uses a Samsung Galaxy S21.
9. The person residing in a Victorian house is somewhere to the left of the person in a Craftsman-style house.
10. The person who has red hair is not in the fifth house.
11. There is one house between the person who uses a Samsung Galaxy S21 and the person who uses a Huawei P50.
12. The person who uses an iPhone 13 is the person in a Craftsman-style house.
13. Eric is the person who has red hair.
14. The person who has red hair is not in the fourth house.
15. The person who uses a OnePlus 9 is in the first house.
16. The person who loves soccer is the dog owner.
17. The person who keeps horses is Bob.
18. Arnold is the fish enthusiast.
19. The person who loves baseball is Peter.
20. The person who loves soccer is in the fourth house.
21. The person who has blonde hair is directly left of the person who has black hair.
 What is the value of attribute FavoriteSport for the person whose attribute HairColor is brown? Please reason step by step, and put your final answer within \boxed{}

Answer: swimming


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

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

Answer: Peter


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Carol`, `Peter`, `Bob`
 - Each person lives in a unique style of house: `mediterranean`, `modern`, `craftsman`, `ranch`, `colonial`, `victorian`
 - People have unique favorite music genres: `country`, `hip hop`, `pop`, `jazz`, `classical`, `rock`
 - Each person has a unique hobby: `cooking`, `painting`, `photography`, `woodworking`, `gardening`, `knitting`

## Clues:
1. The person who loves rock music is in the fifth house.
2. The person who loves classical music and the woodworking hobbyist are next to each other.
3. The person in a Mediterranean-style villa is the person who loves hip-hop music.
4. There are two houses between Arnold and the person residing in a Victorian house.
5. The person who loves jazz music is directly left of Eric.
6. The person who loves hip-hop music is somewhere to the left of the person who enjoys knitting.
7. Carol is the person who loves hip-hop music.
8. The person in a Craftsman-style house is Arnold.
9. The person in a ranch-style home is Eric.
10. The woodworking hobbyist is the person residing in a Victorian house.
11. The person who loves country music is in the first house.
12. There is one house between the person who paints as a hobby and the person living in a colonial-style house.
13. Alice is the photography enthusiast.
14. The person who enjoys gardening is Eric.
15. Bob is in the third house.
 What is the value of attribute HouseStyle for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: modern


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

Answer: gray


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`, `Alice`, `Arnold`
 - Each person has a unique level of education: `bachelor`, `high school`, `associate`, `master`
 - People have unique favorite music genres: `jazz`, `rock`, `pop`, `classical`
 - Each person has a favorite color: `green`, `red`, `yellow`, `white`
 - They all have a unique favorite flower: `lilies`, `carnations`, `daffodils`, `roses`

## Clues:
1. The person with a bachelor's degree is the person who loves a bouquet of daffodils.
2. The person who loves a carnations arrangement is not in the first house.
3. The person with a master's degree is Alice.
4. The person with a master's degree is directly left of the person who loves classical music.
5. Eric is not in the second house.
6. Arnold is not in the third house.
7. The person who loves yellow is directly left of the person who loves the rose bouquet.
8. The person who loves pop music is in the second house.
9. The person with an associate's degree is not in the fourth house.
10. The person who loves a carnations arrangement is not in the fourth house.
11. The person whose favorite color is red is directly left of the person who loves white.
12. The person whose favorite color is red is the person who loves rock music.
13. Arnold is the person who loves yellow.
14. The person who loves a bouquet of daffodils is the person who loves yellow.
 What is the value of attribute Color for the person whose attribute MusicGenre is jazz? Please reason step by step, and put your final answer within \boxed{}

Answer: yellow


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Carol`, `Alice`, `Eric`, `Bob`
 - People have unique favorite music genres: `pop`, `hip hop`, `rock`, `jazz`, `classical`, `country`
 - The people keep unique animals: `fish`, `dog`, `cat`, `rabbit`, `horse`, `bird`
 - Each person lives in a unique style of house: `modern`, `colonial`, `mediterranean`, `ranch`, `victorian`, `craftsman`
 - Each person has a unique hobby: `gardening`, `woodworking`, `knitting`, `cooking`, `painting`, `photography`
 - The people are of nationalities: `brit`, `german`, `norwegian`, `swede`, `dane`, `chinese`

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

Answer: modern


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

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

Answer: 5


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Eric`, `Arnold`, `Peter`, `Alice`
 - Everyone has a unique favorite cigar: `blends`, `prince`, `pall mall`, `blue master`, `dunhill`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `stew`, `pizza`, `stir fry`
 - The people are of nationalities: `norwegian`, `swede`, `brit`, `dane`, `german`
 - Each person has an occupation: `teacher`, `lawyer`, `doctor`, `artist`, `engineer`
 - People have unique heights: `average`, `very tall`, `short`, `very short`, `tall`

## Clues:
1. The person who smokes Blue Master is the person who is a lawyer.
2. Alice is the person partial to Pall Mall.
3. The person partial to Pall Mall is somewhere to the left of the Dunhill smoker.
4. The German is the person who loves stir fry.
5. The person who is tall is the person who smokes many unique blends.
6. The person who is an artist is the person who loves the spaghetti eater.
7. Peter is the person who is short.
8. The Dunhill smoker is not in the fourth house.
9. The person who is very short is the person who smokes Blue Master.
10. There are two houses between the person who loves the spaghetti eater and the person who loves stir fry.
11. The person who has an average height is in the second house.
12. The Prince smoker is the person who is a doctor.
13. The Swedish person is the person who is an engineer.
14. Eric is the person who loves stir fry.
15. There is one house between the British person and the person who loves the stew.
16. The person who is very tall is the Norwegian.
17. The Dane is the person who is short.
18. The person who is very tall is Arnold.
19. There are two houses between the person who is a pizza lover and the person who is very tall.
20. The person who is a teacher is not in the third house.
 What is the value of attribute Height for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: short


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

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

Answer: swimming


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Eric`, `Arnold`, `Peter`, `Alice`
 - Everyone has a unique favorite cigar: `blends`, `prince`, `pall mall`, `blue master`, `dunhill`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `stew`, `pizza`, `stir fry`
 - The people are of nationalities: `norwegian`, `swede`, `brit`, `dane`, `german`
 - Each person has an occupation: `teacher`, `lawyer`, `doctor`, `artist`, `engineer`
 - People have unique heights: `average`, `very tall`, `short`, `very short`, `tall`

## Clues:
1. The person who smokes Blue Master is the person who is a lawyer.
2. Alice is the person partial to Pall Mall.
3. The person partial to Pall Mall is somewhere to the left of the Dunhill smoker.
4. The German is the person who loves stir fry.
5. The person who is tall is the person who smokes many unique blends.
6. The person who is an artist is the person who loves the spaghetti eater.
7. Peter is the person who is short.
8. The Dunhill smoker is not in the fourth house.
9. The person who is very short is the person who smokes Blue Master.
10. There are two houses between the person who loves the spaghetti eater and the person who loves stir fry.
11. The person who has an average height is in the second house.
12. The Prince smoker is the person who is a doctor.
13. The Swedish person is the person who is an engineer.
14. Eric is the person who loves stir fry.
15. There is one house between the British person and the person who loves the stew.
16. The person who is very tall is the Norwegian.
17. The Dane is the person who is short.
18. The person who is very tall is Arnold.
19. There are two houses between the person who is a pizza lover and the person who is very tall.
20. The person who is a teacher is not in the third house.
 What is the value of attribute Food for the person whose attribute Occupation is teacher? Please reason step by step, and put your final answer within \boxed{}

Answer: grilled cheese


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

## Clues:
1. Alice is directly left of the person with a high school diploma.
2. The person with a master's degree and the person with a high school diploma are next to each other.
3. Eric is not in the fourth house.
4. The person who loves pop music is somewhere to the left of the person with a bachelor's degree.
5. The person who loves rock music is in the second house.
6. Peter is the person with a high school diploma.
7. Alice is somewhere to the right of the person who loves jazz music.
 What is the value of attribute MusicGenre for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: jazz


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Alice`, `Arnold`, `Peter`
 - Each person has an occupation: `engineer`, `doctor`, `artist`, `lawyer`, `teacher`
 - Everyone has a favorite smoothie: `desert`, `dragonfruit`, `lime`, `cherry`, `watermelon`
 - Each mother is accompanied by their child: `Fred`, `Bella`, `Meredith`, `Samantha`, `Timothy`

## Clues:
1. Arnold is somewhere to the right of the person who is a teacher.
2. The person's child is named Bella is in the third house.
3. The person who is the mother of Timothy is the person who is an artist.
4. Alice is in the third house.
5. The person who is a doctor is directly left of the person who is an engineer.
6. The Desert smoothie lover is the person who is an engineer.
7. The person's child is named Fred is the person who is a teacher.
8. The person who is an engineer is not in the third house.
9. Bob is the Dragonfruit smoothie lover.
10. Peter is somewhere to the right of the person who is an engineer.
11. The person's child is named Meredith is the person who drinks Lime smoothies.
12. The person's child is named Meredith is in the first house.
13. The person who likes Cherry smoothies is somewhere to the right of the Desert smoothie lover.
 What is the value of attribute House for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 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`, `Alice`, `Arnold`
 - Each person has a unique level of education: `bachelor`, `high school`, `associate`, `master`
 - People have unique favorite music genres: `jazz`, `rock`, `pop`, `classical`
 - Each person has a favorite color: `green`, `red`, `yellow`, `white`
 - They all have a unique favorite flower: `lilies`, `carnations`, `daffodils`, `roses`

## Clues:
1. The person with a bachelor's degree is the person who loves a bouquet of daffodils.
2. The person who loves a carnations arrangement is not in the first house.
3. The person with a master's degree is Alice.
4. The person with a master's degree is directly left of the person who loves classical music.
5. Eric is not in the second house.
6. Arnold is not in the third house.
7. The person who loves yellow is directly left of the person who loves the rose bouquet.
8. The person who loves pop music is in the second house.
9. The person with an associate's degree is not in the fourth house.
10. The person who loves a carnations arrangement is not in the fourth house.
11. The person whose favorite color is red is directly left of the person who loves white.
12. The person whose favorite color is red is the person who loves rock music.
13. Arnold is the person who loves yellow.
14. The person who loves a bouquet of daffodils is the person who loves yellow.
 What is the value of attribute House for the person whose attribute Education is bachelor? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


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

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

Answer: Peter


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Eric`, `Arnold`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `dragonfruit`, `cherry`
 - Each person lives in a unique style of house: `craftsman`, `victorian`, `ranch`, `colonial`
 - The people are of nationalities: `dane`, `brit`, `norwegian`, `swede`
 - People use unique phone models: `google pixel 6`, `samsung galaxy s21`, `iphone 13`, `oneplus 9`

## Clues:
1. The person who uses a Samsung Galaxy S21 is Arnold.
2. There is one house between Alice and the Norwegian.
3. There are two houses between the person who uses a OnePlus 9 and the person living in a colonial-style house.
4. The person living in a colonial-style house is the British person.
5. The person who uses an iPhone 13 is somewhere to the right of Alice.
6. The Watermelon smoothie lover is the British person.
7. The person residing in a Victorian house is somewhere to the left of the person who uses a Samsung Galaxy S21.
8. The Dane is the person who uses a OnePlus 9.
9. The Desert smoothie lover is not in the second house.
10. The person who uses an iPhone 13 is the Dragonfruit smoothie lover.
11. Eric is the Norwegian.
12. The person who uses a Samsung Galaxy S21 is the person in a Craftsman-style house.
 What is the value of attribute PhoneModel for the person whose attribute Nationality is norwegian? Please reason step by step, and put your final answer within \boxed{}

Answer: iphone 13


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Arnold`, `Alice`, `Eric`
 - Everyone has a unique favorite cigar: `blue master`, `dunhill`, `prince`, `blends`, `pall mall`
 - People have unique favorite music genres: `rock`, `classical`, `hip hop`, `pop`, `jazz`
 - Each person has a unique birthday month: `feb`, `april`, `sept`, `jan`, `mar`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`, `biography`, `fantasy`

## Clues:
1. The person who loves pop music is the person who loves romance books.
2. The person who loves fantasy books is in the second house.
3. The person who smokes Blue Master is the person who loves science fiction books.
4. The person who loves hip-hop music is the person whose birthday is in September.
5. The person who loves biography books is Arnold.
6. Alice is the person who loves science fiction books.
7. The person who loves jazz music is somewhere to the left of Eric.
8. The person who smokes Blue Master is the person whose birthday is in April.
9. The person who smokes many unique blends is not in the third house.
10. The Dunhill smoker is the person whose birthday is in February.
11. The person who loves science fiction books is in the fifth house.
12. The person who loves classical music is Bob.
13. The person who loves pop music is Eric.
14. The person who loves mystery books is the person partial to Pall Mall.
15. Peter is the person whose birthday is in February.
16. The person whose birthday is in March is the person who smokes many unique blends.
17. The person who loves classical music is not in the first house.
 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: 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: `Peter`, `Eric`, `Alice`, `Bob`, `Arnold`, `Carol`
 - Everyone has a unique favorite cigar: `yellow monster`, `blends`, `prince`, `dunhill`, `pall mall`, `blue master`
 - People use unique phone models: `xiaomi mi 11`, `iphone 13`, `google pixel 6`, `huawei p50`, `samsung galaxy s21`, `oneplus 9`
 - People own unique car models: `bmw 3 series`, `honda civic`, `tesla model 3`, `ford f150`, `chevrolet silverado`, `toyota camry`
 - The people are of nationalities: `dane`, `brit`, `chinese`, `swede`, `norwegian`, `german`

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

Answer: Peter


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Alice`, `Bob`, `Arnold`
 - The people are of nationalities: `norwegian`, `brit`, `swede`, `dane`, `german`
 - Each person prefers a unique type of vacation: `cruise`, `mountain`, `camping`, `beach`, `city`
 - Each person has a unique level of education: `bachelor`, `master`, `associate`, `doctorate`, `high school`
 - Each person has an occupation: `artist`, `doctor`, `engineer`, `teacher`, `lawyer`

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

Answer: teacher


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`, `Carol`, `Peter`, `Bob`
 - Each person lives in a unique style of house: `mediterranean`, `modern`, `craftsman`, `ranch`, `colonial`, `victorian`
 - People have unique favorite music genres: `country`, `hip hop`, `pop`, `jazz`, `classical`, `rock`
 - Each person has a unique hobby: `cooking`, `painting`, `photography`, `woodworking`, `gardening`, `knitting`

## Clues:
1. The person who loves rock music is in the fifth house.
2. The person who loves classical music and the woodworking hobbyist are next to each other.
3. The person in a Mediterranean-style villa is the person who loves hip-hop music.
4. There are two houses between Arnold and the person residing in a Victorian house.
5. The person who loves jazz music is directly left of Eric.
6. The person who loves hip-hop music is somewhere to the left of the person who enjoys knitting.
7. Carol is the person who loves hip-hop music.
8. The person in a Craftsman-style house is Arnold.
9. The person in a ranch-style home is Eric.
10. The woodworking hobbyist is the person residing in a Victorian house.
11. The person who loves country music is in the first house.
12. There is one house between the person who paints as a hobby and the person living in a colonial-style house.
13. Alice is the photography enthusiast.
14. The person who enjoys gardening is Eric.
15. Bob is in the third house.
 What is the value of attribute MusicGenre for the person whose attribute Hobby is photography? Please reason step by step, and put your final answer within \boxed{}

Answer: pop


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Arnold`, `Peter`
 - Everyone has a unique favorite cigar: `prince`, `dunhill`, `blue master`, `pall mall`
 - Each person lives in a unique style of house: `victorian`, `colonial`, `ranch`, `craftsman`
 - Each person has a unique birthday month: `april`, `sept`, `feb`, `jan`
 - Each person has an occupation: `teacher`, `artist`, `doctor`, `engineer`
 - Each mother is accompanied by their child: `Samantha`, `Bella`, `Meredith`, `Fred`

## Clues:
1. The person who is a doctor is the person's child is named Fred.
2. The person's child is named Samantha and the person whose birthday is in January are next to each other.
3. Arnold is the person whose birthday is in September.
4. The person in a Craftsman-style house is the person who is an engineer.
5. The Dunhill smoker is the person's child is named Samantha.
6. Alice is directly left of Arnold.
7. The person whose birthday is in April and the Prince smoker are next to each other.
8. Arnold is the person who is a teacher.
9. The person living in a colonial-style house is not in the first house.
10. The person who smokes Blue Master is somewhere to the left of Eric.
11. The person who smokes Blue Master is somewhere to the right of the person's child is named Fred.
12. The person whose birthday is in April is somewhere to the left of the person who is an engineer.
13. Eric is not in the fourth house.
14. The person's child is named Bella is somewhere to the left of the person whose birthday is in April.
15. The person in a ranch-style home is Arnold.
 What is the value of attribute Children for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: Fred


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Eric`, `Arnold`
 - People own unique car models: `honda civic`, `toyota camry`, `ford f150`, `tesla model 3`
 - Each person has a unique birthday month: `feb`, `april`, `sept`, `jan`
 - Everyone has something unique for lunch: `pizza`, `stew`, `spaghetti`, `grilled cheese`
 - People have unique heights: `average`, `tall`, `very short`, `short`
 - The mothers' names in different houses are unique: `Janelle`, `Kailyn`, `Holly`, `Aniya`

## Clues:
1. The person who is very short is the person who loves the stew.
2. The person who is a pizza lover is Alice.
3. The person who loves the spaghetti eater is the person who owns a Honda Civic.
4. The person who owns a Toyota Camry is somewhere to the left of the person who is a pizza lover.
5. The person whose birthday is in February is not in the fourth house.
6. Eric is the person who owns a Tesla Model 3.
7. The person who owns a Toyota Camry is somewhere to the right of The person whose mother's name is Janelle.
8. The person who is tall is in the fourth house.
9. The person whose birthday is in September is Arnold.
10. The person who has an average height is somewhere to the right of Arnold.
11. The person whose mother's name is Aniya and Alice are next to each other.
12. The person whose birthday is in April and The person whose mother's name is Holly are next to each other.
13. The person who is short is in the second house.
14. The person who loves the stew and The person whose mother's name is Janelle are next to each other.
 What is the value of attribute House for the person whose attribute CarModel is toyota camry? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Arnold`, `Alice`, `Peter`
 - People own unique car models: `bmw 3 series`, `toyota camry`, `honda civic`, `tesla model 3`, `ford f150`
 - People use unique phone models: `google pixel 6`, `huawei p50`, `oneplus 9`, `iphone 13`, `samsung galaxy s21`
 - Each person has a unique hobby: `painting`, `gardening`, `knitting`, `photography`, `cooking`
 - Each person has an occupation: `lawyer`, `artist`, `teacher`, `engineer`, `doctor`
 - Each person has a favorite color: `white`, `blue`, `green`, `red`, `yellow`

## Clues:
1. The person who owns a Ford F-150 is directly left of the person who is a teacher.
2. The photography enthusiast is Arnold.
3. The person who uses a Huawei P50 is Eric.
4. The person who enjoys gardening is somewhere to the right of the person who loves blue.
5. The person who enjoys knitting is the person who loves white.
6. The person who owns a Ford F-150 is the person whose favorite color is red.
7. There is one house between the person who owns a BMW 3 Series and the person who loves yellow.
8. The person who is an artist is the person who uses a OnePlus 9.
9. The person who is a doctor is Arnold.
10. The person who enjoys gardening is the person who uses a Huawei P50.
11. There is one house between the person who loves blue and the person who loves yellow.
12. The person who owns a BMW 3 Series and the person who uses a Google Pixel 6 are next to each other.
13. The person who uses a Google Pixel 6 is directly left of the person who owns a Honda Civic.
14. Peter is the person who is an engineer.
15. The person who uses a Samsung Galaxy S21 is the person who owns a Toyota Camry.
16. The person who owns a Honda Civic is Alice.
17. The person who owns a Tesla Model 3 is the person who paints as a hobby.
 What is the value of attribute Color for the person whose attribute Occupation is engineer? Please reason step by step, and put your final answer within \boxed{}

Answer: green


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Bob`, `Eric`, `Carol`, `Alice`
 - People own unique car models: `bmw 3 series`, `ford f150`, `tesla model 3`, `toyota camry`, `honda civic`, `chevrolet silverado`
 - Each person has a unique hobby: `woodworking`, `painting`, `knitting`, `photography`, `gardening`, `cooking`

## Clues:
1. The person who loves cooking is not in the first house.
2. The person who owns a Tesla Model 3 is in the fifth house.
3. The person who owns a Honda Civic is in the third house.
4. Alice and the person who owns a Honda Civic are next to each other.
5. There is one house between the person who owns a Chevrolet Silverado and Carol.
6. The photography enthusiast is directly left of the person who paints as a hobby.
7. Peter is somewhere to the right of the person who owns a Tesla Model 3.
8. The person who owns a Honda Civic is directly left of Bob.
9. The person who enjoys knitting is somewhere to the right of the person who owns a Toyota Camry.
10. The person who enjoys knitting is somewhere to the left of Eric.
11. The woodworking hobbyist is directly left of the person who owns a BMW 3 Series.
12. The person who paints as a hobby is in the third house.
 What is the value of attribute CarModel for the person whose attribute Hobby is knitting? Please reason step by step, and put your final answer within \boxed{}

Answer: ford f150


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`, `Alice`
 - People have unique heights: `very short`, `short`, `tall`, `average`
 - Each person prefers a unique type of vacation: `cruise`, `beach`, `city`, `mountain`
 - Each person has a favorite color: `green`, `red`, `yellow`, `white`
 - Each mother is accompanied by their child: `Fred`, `Bella`, `Meredith`, `Samantha`

## Clues:
1. The person who loves beach vacations is the person who is very short.
2. The person who is very short is the person who loves white.
3. The person who has an average height is the person's child is named Bella.
4. The person who has an average height is the person whose favorite color is red.
5. Peter is the person's child is named Samantha.
6. The person whose favorite color is green is directly left of the person who loves white.
7. Alice is in the third house.
8. The person who likes going on cruises is the person's child is named Bella.
9. The person who is short is the person who loves yellow.
10. The person who is very short is the person's child is named Fred.
11. Eric is the person who loves yellow.
12. The person who prefers city breaks is Eric.
13. The person who is short is somewhere to the left of the person who is tall.
 What is the value of attribute Height for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: very short